From Cooper Pairs to Molecules: Effective field
theories for ultra-cold atomic gases near
Feshbach resonances
byJ. N. Milstein
B.A., Northern Illinois University, 1996

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2004

This thesis entitled:
From Cooper Pairs to Molecules: Effective field theories for ultra-cold atomic gases
near
Feshbach resonances
written by J. N. Milstein
has been approved for the Department of Physics

Murray Holland

Prof. Paul Beale

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Milstein, J. N. (Ph.D., Physics)
From Cooper Pairs to Molecules: Effective field theories for ultra-cold atomic gases near
Feshbach resonances
Thesis directed by Prof. Murray Holland

Feshbach resonances in dilute atomic gases are a powerful tool used to control thestrength of atom-atom interactions. In practice, the tuning is accomplished by varying
a magnetic field, affording experiments on dilute atomic gases a knob with which they
can arbitrarily adjust the interactions. This precision control makes atomic gases anideal place to study many-body phenomena. The resonance works by introducing aclosed channel containing a bound, molecular state within the open channel of continuum scattering states. The molecular state greatly modifies the scattering responsible
for the interactions, as it is tuned near resonance, introducing pair correlations throughout the sample. The size of these correlations may range from either very small, where
they appear as molecules, to very large, where they resemble Cooper pairs. This leads
to a “crossover” problem of connecting the Bardeen-Cooper-Schrieffer theory of Cooper
pairing, which describes conventional superconductors, to the process of Bose-Einsteincondensation of molecules. To answer this question, it is necessary to develop an appropriate field theory for both Bosons and Fermions that can account for the Feshbach
processes and, therefore, properly describe resonant, ultra-cold atomic gases.

Dedication

This work is dedicated to Jack Shear for getting me to think about the world and to
Terry Robinson for not letting me stop.

v

Acknowledgements

I should express my deepest gratitude to Murray Holland for all his support during
the formation of this dissertation work. The environment in which I spent my graduate
career could not have been more ideal. The independence which I was granted helped
to both nurture and stimulate a variety of interests within me as can be seen in the
broad range of topics present within this work.
In fact, an innumerable number of people were instrumental in the developmentof this dissertation such as the various members of my research group, many of theexperimentalists and staff of JILA, as well as all the collaborators I have had the chance
to work with on the projects discussed within. I would particulary like to acknowledge
the collaboration and friendship I have had with Servaas Kokkelmans and would liketo thank Christophe Salomon for hosting me at ENS in Paris where I was fortunate
enough to be during the final stages of this work.
If I look back at those who helped me achieve this doctorate I should thank both
George Crabtree and Art Fedro for all their encouragement. If it had not been for thekind reception I received as I entered the physical sciences with a degree in the social
sciences it is not certain that I would ever have written this work.
This dissertation is dedicated, in part, to my grandfather, Jack Shear, who was notable to see it completed. I’m sure he hardly suspected that all those broken appliances
he used to have me fetch from the dumpsters when I was a kid were worth anything,but they were.

vi

Contents

Chapter
1 Introduction1

1.1

Degenerate atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Tunable interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Molecular superfluids and beyond . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Outlook and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2 Feshbach resonance formalism

8

2.1

Feshbach resonance theory . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Single resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Double resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.4

Coupled square-well scattering . . . . . . . . . . . . . . . . . . . . . . .

14

2.5

Comparison with coupled channels calculation . . . . . . . . . . . . . . .

17

2.6

Pseudo-potential scattering and renormalization

20. . . . . . . . . . . . .

3 Collapsing condensates and atomic bursts

23

3.1

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2

The “Bosenova” problem in collapsing condensates . . . . . . . . . . . .

23

3.3

Development of an effective field theory for bosons . . . . . . . . . . . .

26

3.4

Application to a spherical trap geometry . . . . . . . . . . . . . . . . . .

30

3.5

Numerical results and analysis . . . . . . . . . . . . . . . . . . . . . . .

32

vii
3.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 BCS Superfluidity

37
39

4.1

Bardeen-Cooper-Schrieffer theory . . . . . . . . . . . . . . . . . . . . . .

39

4.2

Cooper pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.3

BCS linearization and canonical transformation . . . . . . . . . . . . . .

40

4.4

The appearance of an energy gap . . . . . . . . . . . . . . . . . . . . . .

43

4.5

Populating the quasi-particles . . . . . . . . . . . . . . . . . . . . . . . .

43

4.6

Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

5 Resonant superfluid Fermi gases

46

5.1

Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

5.2

Resonant Hamiltonian for a Fermi gas . . . . . . . . . . . . . . . . . . .

47

5.3

Construction of dynamical equations for fermions . . . . . . . . . . . . .

49

5.4

Application to a trapped system . . . . . . . . . . . . . . . . . . . . . .

50

5.5

Thermodynamic results and signatures of superfluidity . . . . . . . . . .

51

5.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

6 Feshbach Resonant Crossover Physics
6.1

Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Path integral approach to the crossover problem

55
55
59

7.1

Resonant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

7.2

Saddle-point approximation . . . . . . . . . . . . . . . . . . . . . . . . .

61

7.3

Beyond the saddle-point approximation . . . . . . . . . . . . . . . . . .

62

7.4

Bound states and asymptotes . . . . . . . . . . . . . . . . . . . . . . . .

64

7.5

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

8 Pseudogapped Crossover Theory

69

viii
8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

8.2

Dynamical Green’s function method . . . . . . . . . . . . . . . . . . . .

70

8.3

Truncation of the equations of motion . . . . . . . . . . . . . . . . . . .

72

8.4

Extension to resonant Hamiltonian . . . . . . . . . . . . . . . . . . . . .

75

8.5

Self-energy and definition of the t-matrix . . . . . . . . . . . . . . . . .

77

8.6

Foundations of a pseudogapped resonant crossover theory . . . . . . . .

78

8.7

Extension below Tc and the appearance of an order parameter . . . . . .

80

8.8

Self-consistent equations for T < Tc . . . . . . . . . . . . . . . . . . . . .

82

8.9

Numerical solution of the pseudogapped theory . . . . . . . . . . . . . .

84

8.10 Conclusions and experimental implications . . . . . . . . . . . . . . . . .

89

9 Fractional Quantum Hall Effect

91

9.1

The Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

9.2

The integer quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . .

91

9.3

Fractional quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . .

94

9.4

The Laughlin variational wavefunction . . . . . . . . . . . . . . . . . . .

95

9.5

Composite fermions and Chern-Simons theory . . . . . . . . . . . . . . .

98

10 Resonant Manipulation of the Fractional Quantum Hall Effect100

10.1 Resonant theory of a rapidly rotating Bose gas . . . . . . . . . . . . . . 100
10.2 Application of Chern-Simons theory . . . . . . . . . . . . . . . . . . . . 102
10.3 Calculation of the ground state wavefunction . . . . . . . . . . . . . . . 104
10.4 Discussion of resonant effects . . . . . . . . . . . . . . . . . . . . . . . . 107

ix
Bibliography

110

Appendix
A Renormalization of model potentials

116

A.1 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.3 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B Bogoliubov Transformation

118

B.1 Hermitian quadratic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 118
B.2 Unitary canonical transformation . . . . . . . . . . . . . . . . . . . . . . 119
B.3 Basis transformations for bosons and fermions . . . . . . . . . . . . . . . 119
C Functional Integrals and Grassmann Variables

121

C.1 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.2 Fermions and Grassmann variables . . . . . . . . . . . . . . . . . . . . . 122
D Matsubara frequency summations

124

D.1 Common summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D.2 Evaluation of Matsubara summations . . . . . . . . . . . . . . . . . . . 125
E Tc equations from the pseudogap theory

128

E.1 Gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
E.2 Number equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
E.3 Pseudogap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
E.4 Coefficients a0 ,B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
F Landau levels

131

x

FiguresFigure1.1

Born-Oppenheimer curves illustrating the mechanism of a Feshbach resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

5

Scattering length as a function of magnetic field for the (f, mf ) = (1/2, −1/2)
and (1/2, 1/2) mixed spin channel of 6 Li.

. . . . . . . . . . . . . . . . .

13

2.2

Illustration of the coupled square well system. . . . . . . . . . . . . . . .

15

2.3

Comparison of coupled square-well scattering to Feshbach scattering. . .

18

2.4

Real and Imaginary part of the T -matrix for the Feshbach model, thecontact square-well model, and a coupled channels calculation. . . . . .

19

3.1

Illustration of the spherically symmetric geometry used. . . . . . . . . .

31

3.2

A direct comparison of the collapse between the Gross-Pitaevskii and theresonance approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

34

Simulation of the collapse showing the time evolution of the condensed
and noncondensed fraction. . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.4

The velocity fields for the condensate and non-condensed components. .

36

3.5

Density distribution illustrating the formation of an atomic burst. . . .

38

4.1

Cooper pairing at the fermi surface. . . . . . . . . . . . . . . . . . . . .

40

4.2

Cooper pairing for q 6= 0 states. . . . . . . . . . . . . . . . . . . . . . . .

41

xi
5.1

Real and imaginary components of the T -matrix for collisions of the lowest two spin states of 40 K at a detuning of 20 EF . . . . . . . . . . . . .

46

5.2

Density profile revealing an accumulation of atoms at the trap center. .

52

5.3

Emergence of the coherent superfluid for ν = 20 EF . . . . . . . . . . . .

53

5.4

Inverse isothermal compressibility C −1 . . . . . . . . . . . . . . . . . . .

54

6.1

Comparison between Bardeen-Cooper-Schrieffer (BCS) superfluidity and
Bose-Einstein condensation (BEC) in a Fermi gas. . . . . . . . . . . . .

6.2

Diagrammatic contribution of pair fluctuations to the thermodynamic
potential included by NSR. . . . . . . . . . . . . . . . . . . . . . . . . .

7.1

58

Binding energies for 40 K resonance at positive background scattering
length abg = 176a0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

58

65

Binding energies for 40 K resonance at negative background scattering
length abg < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

7.3

Critical temperature T /TF as a function of detuning ν. . . . . . . . . . .

67

7.4

Chemical potential as a function of detuning ν. . . . . . . . . . . . . . .

68

7.5

Population fraction as a function of detuning ν. . . . . . . . . . . . . . .

68

8.1

An illustration of preformed pairing. . . . . . . . . . . . . . . . . . . . .

69

8.2

Illustration of the three-particle Green’s function, G3 , truncation. . . . .

73

8.3

Diagrammatic representation of Lαβ equation. . . . . . . . . . . . . . . .

75

8.4

Comparison between NSR approach and pseudogap theory for the criticaltemperature of 40 K as a function of detuning. . . . . . . . . . . . . . . .

84

8.5

Tc vs. detuning ν for the pseudogap model. . . . . . . . . . . . . . . . .

86

8.6

Fermionic momentum distribution function at T = 0 and T = Tc . . . . .

87

8.7

Fermionic density of states vs. energy. . . . . . . . . . . . . . . . . . . .

89

9.1

Hall effect setup diagram. . . . . . . . . . . . . . . . . . . . . . . . . . .

92

xii
9.2

Experimental signatures of the integer quantum Hall effect. . . . . . . .

93

9.3

Experimental signatures of the fractional quantum Hall effect. . . . . . .

96

9.4

Illustration of composite particles. . . . . . . . . . . . . . . . . . . . . .

98

9.5

Illustration of particles at ν = 1/3. . . . . . . . . . . . . . . . . . . . . .

99

9.6

Illustration of composite particles at ν = 1 . . . . . . . . . . . . . . . . .

99

10.1 The resonance pairing mechanism illustrated for composite fermions. . . 103
D.1 Typical contour for Matsubara summation integrals. . . . . . . . . . . . 127

Chapter 1
Introduction

1.1

Degenerate atomic gases
Dilute atomic gases provide a fascinating setting in which to study an array offundamental questions in many-body physics. The precise control with which these gases
can be manipulated, combined with a detailed understanding of the atomic interactions,allows for an unprecedented level of comparison between theory and experiment. Since
the first observations of Bose-Einstein condensation(BEC)[1, 2, 3], innumerable works
have been published on the nature of quantum degenerate gases. With the achievementof degeneracy within a Fermi gas [4], the next big push is toward superfluidity, a contest
analogous to the race toward condensation almost a decade ago.
The main ingredient to creating a degenerate atomic gas is to cool an ensembleof atoms until the de Broglie wavelength of the particles, defined as:
λdb = h̄/(2M kB T )1/2 ,(1.1)

becomes large compared to the average inter-particle spacing. For a system of bosons,cooling leads to a condensate of atoms where a macroscopic fraction occupies a single
quantum state. For fermions, cooling will cause the atoms to fill each energy level,
allowing for no more than one of each spin state at a given level. The result is adegenerate “Fermi sea” of atoms.
Unfortunately, the onset of quantum degeneracy will often be proceeded by a2liquid or solid phase transition. In order to reach the quantum degenerate regime, thetimescale for formation of molecules by 3-body collisions must be much longer than
the time needed to reach degeneracy. Since equilibrium is reached through binaryelastic collisions (a process which leads to cooling of the gas) at a rate proportional
to the density ∼ n, and 3-body inelastic collisions (which result in losses) occur at a
rate proportional to ∼ n2 , a window may open at extremely low densities in which
degeneracy may be achieved. However, such low densities also depress the temperature
requirement for quantum degeneracy into the nanokelvin range. As an illustration,
current BEC experiments reach quantum degeneracy at temperatures between 500 nK
and 2 µk at densities between 1014 and 1015 cm−3 [5].
The first observation of Bose-Einstein condensation(BEC) was made in a dilute
alkali gas of 87 Rb at JILA [1], followed almost immediately by similar reports in 23 Na
at MIT [2] and 7 Li at Rice [3]. This achievement was made possible by the rapid
advances that had occurred in the field of laser cooling and trapping over the previous
few decades. Not long after, quantum degeneracy was achieved at JILA within a twocomponent Fermi gas of 40 K atoms [4].
Although similar to the methods employed in achieving BEC, the techniques used
to reach the degenerate regime within fermions had to be adapted to account for Pauli
statistics. A major hurdle results from the fact that identical fermions tend to avoid
each other. In this case, s-wave interactions, which dominate the low energy scatteringof these gases, would vanish between a single species of fermions. To overcome thisdifficulty, the JILA group used a mixture of atoms within the two hyperfine levels:|F = 9/2, mF = 9/2i and |F = 9/2, mF = 7/2i, where F = 9/2 is the total atomic spin
and mF is the magnetic quantum number. Runaway evaporation, by which the collision
rates increase as the temperature decreases, was achieved by carefully removing equal
populations of both spin states as the gas cooled. Another method for increasing elastic
collisions is to sympathetically cool the fermions with an auxiliary, bosonic population.

3This technique was soon employed to create degenerate mixtures of 7 Li and 6 Li at ENS[6], Rice [7], and Duke [8], a mixture of 6 Li and 23 Na at MIT [9], and a 40 K and 87 Rb
mixture at Firenze [10].

1.2

Tunable interactions
One of the more unique qualities of cold atomic gases is the ability to finely

tune the interatomic interactions. This control is achieved by making use of low-energy
Feshbach resonances which can be precisely tuned by variation of a magnetic field. The
observation of a Feshbach resonance within a dilute atomic gas is a relatively recent
event, first attained within a Na BEC at MIT [11]. Quite soon after, observations ofFeshbach resonances where reported in 85 Rb at U.T. Austin[12] and JILA[13], as wellas in Cs at Stanford [14]. Feshbach resonances have also been observed in degenerate
Fermi systems such as 40 K at JILA [15], and 6 Li at MIT [16], ENS [17], and Innsbruck[18], to name a few.
In cold alkali collisions, the atom-atom interactions are predominantly determinedby the state of the valence electrons. If two atoms involved in a collision form a triplet,where “triplet” refers to the electronic spin configuration of the two valence electrons,
the electrons will tend to avoid each other in order to preserve the overall antisymmetryof the wavefunction. This behavior acts to reduce the Coulomb repulsion felt by the two
atoms resulting in a somewhat shallower potential well. For a singlet state, however,
the atoms may sit upon one another resulting in a much stronger potential. Because of
these considerations, singlet potentials are in general much deeper than corresponding
triplet potentials.
The singlet and triplet spin states may couple to one another through hyperfine
interactions with the spin of the nucleus. If two atoms are scattering within a triplet
potential, for instance, the hyperfine coupling may flip the electronic and nuclear spins of
one of the atoms bringing the collision into the singlet potential. Another spin flip may

4then bring the collision back to the triplet potential. For large internuclear separations
we may define a quantity ∆ which gives the energy shift between the scattering thresholdof the singlet and triplet potentials. This relative separation is determined by the
Zeeman shifts of the internal states of the two atoms and may, therefore, be adjusted
by variation of a magnetic field.
Since the threshold of the singlet potential generally appears above the thresholdof the triplet potential it becomes energetically unfavorable for atoms to scatter out of
the singlet potential. The singlet is referred to then as a closed channel potential andthe triplet as an open channel potential. Of course, for real alkali atoms, the atoms
appear as a linear combination of a singlet and triplet states so it is best to talk interms of open and closed channels rather than singlet and triplet channels.
If a closed channel Q supports a bound state, we may form a Feshbach resonanceby tuning this bound state near the threshold of the open channel P (see Fig. 1.1).
The energy between the bound state in channel Q and the threshold of channel P isreferred to as the detuning ν. Two atoms scattering within the P channel may collide
to form a quasi-bound molecular state within the Q channel with a different internal
spin arrangement. The molecular state is only considered to be quasi-bound since itremains coupled to the continuum states resulting in a finite lifetime for the molecularstate. Another spin flip breaks apart the molecule and the system is returned to theinitial P channel.
The detuning of the two channels has a dramatic impact on the scattering properties of the system and results in a scattering length which depends on the external
magnetic field as:µa = abg¶∆B.
1−
B − B0(1.2)

Here abg would be the background scattering length of the open channel if the closed
channel was not accessible, ∆B is a measure of the width of the resonance, B0 is the5value of the magnetic field B just on resonance, and B − B0 ∼ ν. We will discuss more
the details of Feshbach resonance theory in the next chapter.

Figure 1.1: Born-Oppenheimer curves illustrating the mechanism of a Feshbach resonance. A Feshbach resonance results when a closed channel potential possesses a boundstate in proximity to the scattering threshold of an open channel potential. The detuning of the bound state from the edge of the collision continuum is denoted by ν. ∆represents the energy shift between the two channels at large, relative separation.

1.3

Molecular superfluids and beyond
The introduction of a Feshbach resonance not only allows direct control over theatom-atom interactions, but inherently introduces a process of molecular formation anddisassociation. The presence of molecules greatly increases the richness and utility ofthese systems. Molecular BECs, for instance, have remained out of reach to experiment
due to the difficulty involved in cooling molecules as compared to alkali atoms. One wayof overcoming this difficulty would be to create an atomic condensate through traditionallaser trapping and cooling techniques and to then ramp across a Feshbach resonanceconverting the atomic condensate into a molecular condensate. This technique was usedin the JILA 85 Rb experiment by Donley et al. [19] to create a superposition of atoms andmolecules–although the question of whether a true molecular condensate was formed is6still being debated. A similar experiment was performed by Dürr et al. [20] where they
were able to spatially separate the molecular component from the atomic population by
performing a Stern-Gerlach experiment.
Surprisingly enough, degenerate Fermi gases proved more adept at forming molecules
than Bose gases. Due to Pauli blocking of the available decay channels, these molecules
showed extremely long lifetimes, some remaining for up to several seconds [21, 22, 23, 24].
With such long lived molecules available, reports of molecular condensates quickly appeared at JILA [25], then MIT [26] and Innsbruck [27].
The production of a molecular condensate from a Fermi gas of atoms was the firststep in experimentally studying the “crossover problem” of moving between BardeenCooper-Schrieffer (BCS) superfluidity and Bose-Einstein condensation (BEC). Soon after these initial observations, reports of condensate formation above the resonance, wellwithin the crossover regime between the extremes of BCS and BEC, where made first
at JILA [28] and then at MIT [29]. These reports may be the first observations of“resonance superfluidity” within Fermi gases.

1.4

Outlook and overview
The structure of this thesis will be as follows. Chapter 2 will present the formalismof Feshbach resonant scattering. In Chapter 3 we will use the ideas discussed in theprevious chapter to develop a field theory for bosons and apply it to the collapse of
a condensate. We will review some of the fundamental notions behind BCS theory in
Chapter 4 and go on to extend these ideas to form a theory of resonance superfluidity
in Chapter 5. This theory will be used to determine signatures of a superfluid phasetransition in a degenerate Fermi gas. Chapter 6 will discuss the crossover problem,
briefly mentioned in the previous section of this introduction, in greater detail and in
Chapter 7 we will treat the crossover by extending a lowest order theory to account forthe production of molecules. Chapter 8 will extend the theory of the previous chapter

7
to formulate a nonperturbative theory of the crossover. In Chapter 9 we introduce thefractional quantum Hall effect, properties of which are predicted to occur in a rapidly
rotating Bose gas, and, in Chapter 10, discuss how a resonance may alter these properties
focusing on the ground state of such a system.

Chapter 2
Feshbach resonance formalism

2.1

Feshbach resonance theory
We now briefly describe the Feshbach formalism and derive the elastic S- andT -matrices for two-body Feshbach resonant scattering. These matrices represent thetransition probabilities for scattering between states within the resonant system. A moredetailed and extended treatment of this formalism can be found in the literature [30, 31].
In Feshbach resonance theory, two projection operators P and Q are introduced
which project onto the subspaces P and Q and satisfy the relations:
P = P†Q = Q†

P2 = P

Q2 = Q(2.1)

P + Q = 1.
These subspaces form two orthogonal components which together span the full Hilbert
space of both scattering and bound wavefunctions |ψi = P |ψi + Q|ψi. The open andclosed channels are contained in P and Q, respectively. The operators P and Q split
the Schrödinger equation for the two-body problem (E − H)|ψi = 0 into two parts:(E − HP P )|ψP i = HP Q |ψQ i,(2.2)(E − HQQ )|ψQ i = HQP |ψP i,(2.3)

where HP P = P HP , HP Q = P HQ, etc., and ψ is the total scattering wavefunction.
The projections on the two sub-spaces are indicated by P |ψi = |ψP i and Q|ψi = |ψQ i.

9The Hamiltonian H = H 0 + V consists of the sum of single-particle interactions H 0 andthe two-body interaction V . We may formally solve Eq. (2.3) to find:|ψQ i =

1
E+ − HQQ

HQP |ψP i,(2.4)

where E + = E + iδ, with δ approaching zero from positive values. Substituting thisresult into Eq. (2.2), the open channel equation can be written as(E − Heff )|ψP i = 0,

(2.5)

where
Heff = HP P + HP Q

1
E + − HQQ

HQP .

(2.6)

If we write Eq. (2.6) in the following form:
Heff = HP0 P + Veff ,(2.7)

we may identify an effective potential
Veff = VP P + HP Q

1
E+ − H

QQ

HQP ,

(2.8)

resulting from the coupling to the Q subspace. We have thus reduced the scattering
problem to that of scattering off an effective potential composed of an additive combination of the open channel potential and a coupling to the closed channels. We will returnto this result in our later discussion of the many-body physics of Feshbach resonant
systems.
The trick to solving Eq. (2.5) is to expand the resolvent operator into the discreteand continuum eigenstates of HQQ :Heff = HP P +X HP Q |φi ihφi |HQPi

E − ²iZ+

HP Q |φ(²)ihφ(²)|HQP
d².
E+ − ²(2.9)

Here the ²i ’s and ²’s are the uncoupled bound-state and continuum eigenvalues, respectively. In practice, only a few bound states will significantly affect the properties of

10
the open-channel. We will now assume that a small number of bound states dominate
the problem and will neglect the continuum expansion in Eq. (2.9). Then the formal
solution for |ψP i is given by|ψP i = |ψP+ i +X HP Q |φi ihφi |HQP |ψP i

1E+ − HPP

E − ²ii,(2.10)

where the scattered wavefunction |ψP+ i is an eigenstate of the direct interaction HP P
(i.e., (E + − HP P )|ψP+ i = 0) and satisfies outgoing, spherical wave boundary conditions.
Likewise, we may define a scattered wavefunction |ψP+ i satisfying incoming, sphericalwave boundary conditions, (E − − HP P )|ψP− i = 0. The scattered wavefunctions |ψP± i
may be formally solved for with the result|ψP± i = |χP i +VP P|χP i,
E ± − HP P

(2.11)

where the unscattered wavefunction |χP i is defined as an eigenstate of HP0 P .
We now quantify the scattering behavior of a Feshbach resonance system by
calculating the transition matrix (T-matrix). To begin, let us define a nonresonant
T-matrix for scattering within the P subspace as:TP= hχP |VP P + VP P

1
E + − HP P

VP P |χP i= hχP |VP P |ψP+ i.

(2.12)(2.13)

In moving from Eq. (2.12) to Eq. (2.13) we have made use of Eq. (2.11).
To calculate the full scattering T-matrix T , which accounts for the coupling to
the Q subspace, we must identify the effective potential as defined by Eq. (2.6). By
multiplying Eq. (2.10) from the left with hχP |Veff , we derive the full, scattering T-matrix:T= hχP |Veff |ψP i= TP +(2.14)X hψ − |HP Q |φi ihφi |HQP |ψP i
Pi

E − ²i,(2.15)

where we have again made use of the relation between the unscattered state |χP i andthe scattering wave-function |ψP− i as given in Eq. (2.11).

11
From the T-matrix, we can easily go to the S-matrix defined as:
S = hψP− |ψP+ i.

(2.16)For s-wave scattering, there exists a simple relation between the S- and T-matrix [32]:S = 1 − 2πiT .

(2.17)

This allows us to rewrite Eq. (2.15) as
S = SP −X 2πihψ − |HP Q |φi ihφi |HQP |ψP i
P

E − ²ii,(2.18)

where the non-resonant factor SP describes the direct scattering process within thesubspace P.

2.2

Single resonanceFor the case of only one resonant bound state and only one open channel, Eq. (2.18)

gives rise to the following elastic S-matrix element:"

S = SP#

2πi|hψP+ |HP Q |φ1 i|2.
1−
1
HP Q |φ1 i
E − ²1 − hφ1 |HQP E + −H
PP

(2.19)

This relation is found by acting on both sides of Eq. (2.10)with hφi |HQP and substituting the result into Eq. (2.18). The direct, non-resonant S-matrix is related to thebackground scattering length abg , for k ¿ a−1
bg , by SP = exp[−2ikabg ]. The term in
the numerator gives rise to the energy-width of the resonance, Γ = 2π|hψP+ |HP Q |φ1 i|2 ,which is proportional to the incoming wavenumber k and coupling constant ḡ1 [33]. The
bracket in the denominator gives rise to a shift of the bound-state energy, and to an
additional width term iΓ/2. If we denote the energy-shift between the collision continuum and the bound state by ν̄1 , and represent the kinetic energy simply by h̄2 k 2 /m,
the S-matrix element can be rewritten as"−2ikabg

S(k) = e

1−

2ik|ḡ1 |2
2

2 2

h̄ k
2
− 4πh̄m (ν̄1 − m ) + ik|ḡ1 |#.

(2.20)

12
The resulting, total scattering length, coming from the relation
lim T (k) =k→0

4πh̄2 a,m

(2.21)

has a dispersive shape of the formÃa = abgm|ḡ1 |21−
4πh̄2 abg ν̄1!

,(2.22)

which may directly be related to Eq. (1.2). The form of Eq. (2.22) allows us to extract theparameters of the resonance model from a plot of the scattering length versus magnetic
field.

2.3

Double resonance
Often more than one resonance may need to be considered. For example, thescattering properties for the (f, mf ) = (1/2, −1/2) and (1/2, 1/2) channel of 6 Li aredominated by a combination of two resonances: a triplet potential resonance and a
Feshbach resonance. A mechanism of this sort may be inferred from Fig. 2.1 where theresidual scattering length, which would arise in the absence of the Feshbach resonancecoupling, would be very large and negative and vary with magnetic field. This should becompared with the value of the non-resonant background scattering length for the triplet
potential of 6 Li which is only 31 a0 , which is an accurate measure of the characteristic
range of this potential. An adequate scattering model for this system, therefore, requires
inclusion of both bound-state resonances. The double-resonance S-matrix, with again
only one open channel, follows then from Eq. (2.18) and includes a summation over twobound states. After solving for the ψP wave function, the S-matrix can be written as"

S(k) = e−2ikabg#

2ik(|ḡ1 |2 ∆2 + |ḡ2 |2 ∆1 ),1−
ik(|ḡ1 |2 ∆2 + |ḡ2 |2 ∆1 ) − ∆1 ∆2(2.23)

with ∆1 = (ν̄1 − h̄2 k 2 /m)4πh̄2 /m, where ν̄1 and ḡ1 are the detuning and couplingstrengths for state 1. Equivalent definitions are used for state 2. Later we will show that

13
5000

0

a (Units of a )

2500

0

−2500

−5000
0

200

400

600
B (G)

800

1000

1200

Figure 2.1: Scattering length as a function of magnetic field for the (f, mf ) =(1/2, −1/2) and (1/2, 1/2) mixed spin channel of 6 Li.

this simple analytic Feshbach scattering model mimics the coupled channels calculationof 6 Li.
Once again, the parameters of this model, which are related to the positions andwidths of the last bound states, can be directly found from a plot of the scattering length
versus magnetic field as given, for example, by Fig. 2.1. The scattering length behavior
should be reproduced by the analytic expression for the scattering length following from
Eq. (2.23):ma = abg −4πh̄2

Ã|ḡ1 |2 |ḡ2 |2
+
ν̄1
ν̄2

!

.

(2.24)

The advantage of a double-pole, over a single-pole, S-matrix parametrization is that we
can account for the interplay between a potential resonance and a Feshbach resonancewhich, in principle, can radically change the scattering properties. This interplay is not
only important for the description of 6 Li interactions, but also for other atomic systems
which have an almost resonant triplet potential, such as bosonic 133 Cs [34, 35] and 85 Rb[36].
In the many-body theories discussed in this thesis, the scattering properties are
represented by a T -matrix instead of an S-matrix. We have shown for s-wave scattering14that there exists a simple relation between the two, however, the definition for T in
the many-body theory will be slightly different in order to give it the conventional
dimensions of energy per unit density:
T (k) =

2.4

2πh̄2 i[S(k) − 1] .
mk

(2.25)

Coupled square-well scattering
We now describe the coupled-channels extension of a textbook single-channelsquare-well scattering problem [37]. Our motivation for studying this model is that we
may take the limit of the potential range as R → 0, resulting in an explicit representationof a set of coupled delta-function potentials. Such a set of potentials greatly simplifies thedescription in the many-body problem. The applicability of delta-function potentials,
or contact potentials, to the many-body system is motivated by the fact that the lengthscale associated with the range of the interatomic potential is, typically, much smaller
than the length scale associated with the de Broglie wavelength, or other thermodynamic
correlation scales. This means that we need not concern ourselves with the actual
shape of the potential so long as the chosen potential reproduces the correct asymptotic
scattering properties. We may, therefore, chose the simplest potential which satisfies
the constraints of our problem.
The scattering equations for a system of coupled square wells may be written as
Ekin ψ P (r) =Ekin ψ Q (r) =hhi

H0 (r) + V P (r) ψ P (r) + g(r)ψ Q (r),(2.26)i

H0 (r) + V Q (r) + ² ψ Q (r) + g ∗ (r)ψ P (r),(2.27)

with ² the energy-shift of the closed channel and Ekin = h̄2 k 2 /m the two-body kinetic
energy. The coupled square well model encapsulates the general properties of two-bodyalkali interactions. In scattering events between alkalis, we can divide the internuclear
separation into two regions: the inner region where the exchange interaction (the difference between the singlet and triplet potentials) is much larger than the hyperfine

15

Ekin
0

V1
V2

R
Figure 2.2: Illustration of the coupled square well system. The solid and dotted lines
correspond to the molecular potentials V1 (P ) and V2 (Q) inner r < R (outer r > R)region, respectively. The dashed line corresponds to the kinetic energy Ekin in the open
channel. The detuning ² can be chosen such that a bound state of the square-wellpotential V2 enters the collision continuum causing a Feshbach resonance in the open
channel.

16
splitting, and the outer region where the hyperfine interaction dominates. Here we
make a similar distinction for the coupled square wells. In analogy to the real singlet
and triplet potentials, we use, for the inner region, two artificial square-well potentials
labelled as V1 and V2 . We take the coupling g(r) to be constant over the range of thepotentials , r < R, and to be zero outside this range (see Fig. 2.2). For the outer region,r > R, the open and closed channel wavefunctions are given by uP (r) ∼ sin kP r anduQ (r) ∼ exp(−kQ r), respectively, where u(r) = rψ(r). For the inner region r < R the
wave functions are given by u1 (r) ∼ sin k1 r and u2 (r) ∼ sin(k2 r). The relevant wavevecpp√tors are defined as: kP = mEkin /h̄, kQ = m(² − Ekin )/h̄, k1 = m(Ekin + V1 )/h̄,and k2 =pm(Ekin + V2 − ²)/h̄.

This problem may now be solved by means of a basis rotation at the boundary R
which gives rise to simple analytic expressions. The efficacy of this transformation may
be understood by realizing that the effect of the potentials is to cause the wavefunction
to accumulate a phase φ1 = k1 R and φ2 = k2 R at the boundary R. For r > R, wetherefore consider one open channel and one closed channel with wavenumbers kP (Ekin )and kQ (Ekin , ²). Ekin is the relative kinetic energy of the two colliding particles in thecenter of mass frame and ² is the energy-difference of the two outer-range channels.
In analogy with a real physical system, we can refer to the inner range channels as amolecular basis and the channel wave functions are just linear combinations of the u1
and u2 wave functions. The coupling strength is effectively given by the basis-rotation
angle θ for the scattering wave functions:








 uP (R)   cos θ

=

 

uQ (R)

sin θ   u1 (R) − sin θ cos θ

u2 (R)

,

(2.28)

allowing for an analytic solution of the scattering model. This leads to the following
expression for the S-matrix:
S = e−2ikP R [1 − (−2ikP (k2 cot φ2 cos2 θ + kQ + k1 cot φ1 sin2 θ)/(2.29)

17(kP kQ + k1 cot φ1 (kP sin2 θ − kQ cos2 θ) + ik2 cot φ2 (k1 cot φ1 + kP cos2 θ + kQ sin2 θ)].
An extension to treat more than two coupled potentials, which would be required tomodel more than one resonance, is straightforward.
The parameters of the two wells have to be chosen such that the results of areal scattering calculation are reproduced. In fact, all the parameters are completely
determined from the field dependence of the scattering length, and all other scattering
properties can then be derived, such as the energy-dependence of the scattering phaseshift. One way of selecting these parameters is to first chose a range R, typically of theorder of an interatomic potential range (100 a0 ) or less. Now we have only to determine
the set of parameters V1 , V2 , and θ. The potential depth V1 is chosen such that the
scattering length is equal to the background scattering length abg while keeping θ = 0.
Also, V1 should be large enough that the wavenumber k1 depends only weakly on the
scattering energy. Then we set θ to be non-zero and change the detuning until a boundstate crosses the collision threshold giving rise to a Feshbach resonance. The value of
V2 is more or less arbitrary, but we typically chose it to be larger than V1 . Finally, wechange the value of θ to give the Feshbach resonance the desired width.
In Fig. 2.3 the coupled square-well system is compared with the Feshbach scattering theory for the scattering parameters of 40 K. Despite the fact that there is a strong
energy-dependence of the T -matrix, the two scattering representations agree very well.
We will later show that the resulting scattering properties converge for R → 0.

2.5

Comparison with coupled channels calculation
In the last section we argued that the double-well system is in good agreementwith Feshbach scattering theory. Now we will show how well both the Feshbach theory
and the R → 0 coupled square-well, or contact square-well, theory agree with the full
coupled channels calculation [38]. In Fig. 2.4 we show the real and imaginary parts of the

18

3

0

Re[T] (Units of 10 a )

0

−1

−2

−3
0

0.5

1
E (µK)

1.5

2

Figure 2.3: Comparison of coupled square-well scattering (solid line), with a potential range R = 1a0 , to Feshbach scattering (dashed line) for a detuning that yields ascattering length of about -2750 a0 . Similar agreement is found for all detunings.

T -matrix applied to the case of 6 Li and compare the contact square well and Feshbach
scattering representations to a full coupled channels calculation. The agreement issurprisingly good and holds for all magnetic fields (i.e., similar agreement is found at
all detunings).
In this section we have discovered the remarkable fact that even a complex system,
including internal structure and resonances, can be described with contact potentials
and a few coupling parameters. This was trivially known for off resonance scattering
where only a single parameter, the scattering length, is required to encapsulate thecollision physics. However, this has not be pointed out before for the resonance system
where an analogous parameter set is required to describe a system with a scattering
length that may pass through infinity!

19

3

0

Re[T] (Units of 10 a )

−3

−4

−5

0

Im[T] (Units of 10 a )

0

3

−1

−2

−3
0

0.2

0.4

0.6

0.8

1

E (µK)

Figure 2.4: Real (top) and imaginary (bottom) part of the T -matrix , as a function ofcollision energy, for the Feshbach model and the contact square-well model (overlapping
solid lines), and for a coupled channels calculation (dashed line). The atomic species
considered is 6 Li, for atoms colliding in the (f, mf ) = (1/2, −1/2) and (1/2, 1/2) channel
near 800G.

20

2.6

Pseudo-potential scattering and renormalization
In forming a field theory to describe Feshbach resonant interactions we would

prefer not to work with the complete interatomic potentials since they contain much
more information than is actually needed and severely complicate the problem. Rather,
we would like to work with a suitable set of pseudo-potentials that incorporate all the
relevant scattering physics. As we have shown in the previous sections, a pair of coupledsquare wells may be accurately used to describe a Feshbach resonance. However, we had
to adjust the various parameters, such as the depth and coupling of the wells, in orderto produce the desired resonance. A correct field theoretic description of a resonant
system must, similarly, adjust the pseudo-potentials. This adjustment procedure istermed “renormalization” and connects the pseudo-potential parameters in the field
theory to the physical parameters describing the resonance. What’s more, if contact
potentials are incorporated in the many-body theory, which is often done out of initialconvenience, the renormalization works to remove divergences that will naturally arise
due to the singularity of such a potential. To show how one properly renormalizes
a Feshbach resonant field theory, we must first take a closer look at the underlying
two-body physics.
We begin by solving the Lippmann-Schwinger scattering equation for a Feshbachresonance system. Our goal will be to derive explicit expressions which match the twobody scattering parameters describing the Feshbach resonance (Ū , ḡi , ν̄i ) to the model
potential parameters (U, g, ν) entered into our field theory. The first step is to solve
the scattering Eqs. (2.26) and (2.27) for these pseudo-potentials. As we have seen, we
can formally solve the bound-state equations and make use of Eq. (2.9) to expand theGreen’s function in bound-state solutions. In this case we may writeQψ (r) =X φQ (r)ii

R 3 0 Q∗ 0 ∗ 0 P 0
d r φi (r )g (r )ψ (r ),E + − ²i

(2.30)

with φQi (r) a bound state wavevector and ²i its eigenenergy. For the purposes of this21
section we will only consider a single resonance potential so will henceforth drop the isubscript. We now define an amplitude for the system to be in the bound state φ =hφQ |ψ Q i. This definition will later prove useful in the mean-field equations. Together
with the open channel equation and the definition g(r) = g(r)φQ (r) we get a new set of
scattering equations
Ãh̄2 k2
ψ(r) =m!

h̄2− ∇2 + V (r) ψ(r) + g(r)φmh̄2 k2φ = νφ +m(2.31)

Z

d3 r0 g(r0 )ψ(r0 ),(2.32)

where we have dropped the P superscripts. A formal solution for ψ(r) may be given by
the following relation:
ψ(r) = χ(r) −

m
4πh̄2

Z

0

d3 r 0ei|k||r−r | £
0
0
0 ¤
V
(r
)ψ(r
)
+
g(r
)φ ,
|r − r0 |(2.33)

where χ(r) represents the unscattered component. The scattering amplitude f (θ) is
defined in the limit r → ∞ of the wavefunction by the relation
ψ(r) = χ(r) + f (θ)

eikr
.
r

(2.34)

By taking the r → ∞ limit of Eq. (2.33) we have an asymptotic form for the fullscattered wavefunction
ψ(r) = χ(r) −m eikr4πh̄2 r

Z

0

£

¤

d3 r0 e−ik·r V (r0 )ψ(r0 ) + g(r0 )φ .

(2.35)

A comparison of Eq. (2.35) to Eq. (2.34) gives the following form for the scattering
amplitude:f (θ) = −

m
4πh̄2

Z

0

£

¤

d3 r0 e−ik·r V (r0 )ψ(r0 ) + g(r0 )φ .

(2.36)In Fourier space this becomesmf (k, k ) = −
4πh̄2
0

"Z

#

¢
d3 p ¡0V(k−p)ψ(p)+ g(k0 )φ .
(2π)3(2.37)

22
With the aid of the relationship T (k, p) = −(4πh̄2 /m)f (k, p), linking the T-matrix tothe scattering amplitude, we may write the Fourier representation of the open channel
wavefunction asψ(p) = (2π)3 δ(k − p) +T (k, p)h̄2 p2h̄2 k2m − m + Iδ.

(2.38)

Substituting this result into Eq. (2.37) yields
Z

T (k, k0 ) = V (k0 − k) +d3 p U (k0 − p)T (k, p)+ g(k0 )φ.
(2π)3 h̄2 k2 − h̄2 p2 + Iδm

(2.39)m

By rewriting the original closed channel scattering equation, Eq. (2.27), asφ=Z

1h̄2 k2m −ν

d3 pg(p)ψ(p),(2π)3(2.40)

we may eliminate φ from Eq. (2.39). This results in the following integral equation forthe off the energy-shell T-matrix"

T (k, k0 ) = U (k0 − k) +g(k0 )g(k)h̄2 k2
m

−ν

µ

#

Z

+

d3 p




(2π)3 

¶

0

)g(p)
U (k0 − p) + g(kh̄2 k2−νm

T (k, p) h̄2 p2h̄2 k2m − m + Iδ.


(2.41)
We now have a general integral equation for the T-matrix. However, to make use ofthis relationship we will need to make some simplifying assumptions. The first being
the separability of the potential U (k − k 0 ) = λ(k)λ(k 0 ). The second, that the T-matrix
is basically constant over the range of the integral allowing us to pull it outside of theintegral. This is another way of saying that the T-matrix only contributes on the energyshell. Third, we will assume s-wave scattering so the wavenumbers k = k0 . This resultsin the algebraic equation for the T-matrix:"

T (k) = λ(k)2 +g(k)2h̄2 k2m−ν#Z+ T (k)µd3 p 
(2π)3 g(k)g(p)λ(k)λ(p) + h̄2 k2mh̄2 k2m−h̄2 p2
m

−ν

+ Iδ

¶

.


(2.42)The renormalization is performed by matching the potentials U, g and detuning ν to thephysical potentials Ū , ḡ and the physical detuning ν̄ as given by the form of the Feshbachtheory (Eqs. (2.20) and (2.25)). Results for several types of model pseudo-potentials
are given in Appendix. A.

Chapter 3
Collapsing condensates and atomic bursts

3.1

Outlook

Having thoroughly discussed the two-body physics underlying Feshbach resonant interactions, we are now in a position to apply these ideas to a many-body system. We begin,in this chapter, by discussing the formalism for Bosons and apply it to a particularly
novel experiment. In the next chapter we will expand upon this formalism to accountfor Fermions.

3.2

The “Bosenova” problem in collapsing condensates
The ability to dynamically modify the nature of the microscopic interactions in a

Bose-Einstein condensate—an ability virtually unique to the field of dilute gases—opens
the way to the exploration of a range of fundamental phenomena. A striking example ofthis is the “Bosenova” experiment carried out in the Wieman group at JILA [39] which
explored the mechanical collapse instability arising from an attractive interaction. This
collapse resulted in an unanticipated burst of atoms, the nature of which is a subject of
current debate.
The Bosenova experiment, conducted by the JILA group, consisted of the following elements. A conventional, stable Bose-Einstein condensate was created in equilibrium. The group then used a Feshbach resonance to abruptly switch the interactions
to be attractive inducing an implosion. One might have predicted that the rapid in-24crease in density would simply lead to a rapid loss of atoms, primarily through inelastic
three-body collisions. In contrast, what was observed was the formation of an energetic
burst of atoms emerging from the center of the implosion. Although the energy of theseatoms was much larger than that of the condensate, the energy was insignificant whencompared to the molecular binding energy which characterizes the energy released ina three-body collision. In the end, what remained was a remnant condensate whichappeared distorted and was believed to be in a highly excited collective state.
One theoretical method which has been extensively explored to explain this behavior has been the inclusion of a decay term into the Gross-Pitaevskii equation as a way
to account for the atom loss [40, 41, 42, 43]. Aside from its physical application to the
Bosenova problem, the inclusion of three-body loss as a phenomenological mechanism
represents an important mathematical problem, since the nonlinear Gross-Pitaevskii
equation allows for a class of self-similar solutions in the unstable regime. The local
collapses predicted in this framework can generate an outflow even within this zerotemperature theory. However, there are a number of aspects which one should consider
when applying the extended Gross-Pitaevskii equation to account for the observations
made in the JILA experiment.
The first problematic issue is the potential breakdown of the principle of attenuation of correlations. This concept is essential in any quantum or classical kinetic theory
as it allows multi-particle correlations to be factorized. The assumption of attenuationof correlations is especially evident in the derivation of the Gross-Pitaevskii equation
where all explicit multi-particle correlations are dropped. Furthermore, there is considerable evidence for this instability toward pair formation in the mechanically unstable
quantum theory [44].
A second difficulty with motivating the Gross-Pitaevskii approach is that, by thismethod, one describes the energy-dependent interactions through a single parameter,
the scattering length, which is determined from the s-wave scattering phase shift at

25
zero scattering energy. Near a Feshbach resonance, as we have seen in Chapter 1, theproximity of a bound state in a closed potential to the zero of the scattering continuum
can lead to a strong energy dependence of the scattering. Exactly on resonance, the swave scattering length passes through infinity and, in this situation, the Gross-Pitaevskii
equation is undefined.
These two fundamental difficulties with the Gross-Pitaevskii approach led us toreconsider the Bosenova problem [45]. We were motivated by the fact that the same
experimental group at JILA recently performed a complementary experiment [19]– the,so called, molecular Ramsey fringe experiment. Their results provided significant insight into this problem. What was remarkable in these new experiments was that, evenwith a large positive scattering length, in which the interactions were repulsive, a burstof atoms and a remnant condensate were observed. Furthermore, in the large positive
scattering length case, simple effective field theories, which included an explicit description of the Feshbach resonance physics, were able to provide an accurate quantitative
comparison with the data [46, 47, 48]. The theory showed the burst to arise from thecomplex dynamics of the atom condensate which was coupled to a coherent field of
exotic molecular dimers of a remarkable physical size and near the threshold binding
energy.
In this Chapter, based on the work of [45], we draw connections between the
two JILA experiments. We pose and resolve the question as to whether the burst of
atoms in the Bosenova collapse could arise in a similar way as in the Ramsey fringe
experiment—from the formation of a coherent molecular superfluid. This hypothesis istested by applying an effective field theory for resonance superfluidity to the collapse.
For fermions, the case of resonance superfluidity in an inhomogeneous system has beentreated in the local density approximation using, essentially, the uniform solution ateach point in space [49]. For the collapse of a Bose-Einstein condensate, as we wish
to treat here, the local density approximation is not valid and the calculation must be26
performed on a truly inhomogeneous system.

3.3

Development of an effective field theory for bosons
In the Feshbach resonance illustrated in Figure 1.1, the properties of the collisionof two ground state atoms is controlled through their resonant coupling to a bound state
in a closed channel Born-Oppenheimer potential. By adjusting an external magnetic
field, the scattering length can be tuned to have any value. This field dependence ofthe scattering length is characterized by the detuning ν̄ and obeys a dispersive profile
given by a(ν̄) = abg (1 − κ/(2ν̄)), with κ the resonance width and abg the backgroundscattering length. In fact, as described in Chapter 2.1, all the scattering propertiesof a Feshbach resonance system are completely characterized by just three parameters√Ū = 4πh̄2 abg /m, ḡ = κŪ , and ν̄. Physically, Ū represents the energy shift per unit
density on the single particle eigenvalues due to the background scattering processes,
while ḡ, which has dimensions of energy per square-root density, represents the couplingof the Feshbach resonance between the open and closed channel potentials.
We now proceed to construct a low order many-body theory which includes thisresonance physics. The Hamiltonian for a dilute gas of scalar bosons with binary interactions is given in complete generality byZH=1d x ψa† (x)Ha (x)ψa (x) +32Zd3 xd3 x0 ψa† (x)ψa† (x0 )U (x, x0 )ψa (x0 )ψa (x),(3.1)

where Ha (x) is the single particle Hamiltonian, U (x, x0 ) is the binary interaction potential, and ψa (x) is a bosonic scalar field operator. In cold quantum gases, where theatoms collide at very low energy, we are only interested in the behavior of the scattering
about a small energy range above zero. There exist many potentials which replicate thelow energy scattering behavior of the true potential; therefore, it is convenient to carryout the calculation with the simplest one, the most convenient choice being to take theinteraction potential as a delta-function pseudo-potential when possible.

27
For a Feshbach resonance, this choice of pseudo-potential is generally not available
since the energy dependence of the scattering implies that a minimal treatment must atleast contain a spread of wave-numbers which is equivalent to the requirement of a nonlocal potential. Since the solution of a nonlocal field theory is inconvenient, we take analternative, but equivalent, approach. We include into the theory an auxiliary molecularfield operator ψm (x) which obeys Bose statistics and describes the collision between
atoms in terms of two elementary components: i.) the background collisions between
atoms in the absence of the resonance interactions and ii.) the conversion of atom pairs
into tightly bound, molecular states. This allows us to construct a local field theory withthe property that when this auxiliary field is integrated out, an effective Hamiltonianof the form given in Eq. (3.1) is recovered with a potential V (x, x0 ) = V (|x − x0 |) which
generates the form of the two-body T -matrix predicted by Feshbach resonance theory(Eq. (2.20) with (2.25)). The local Hamiltonian which generates this scattering behavior
is:
Z

H =

Z

d3 x ψa† (x)Ha (x)ψa (x) +†d3 x ψ m(x)Hm (x)ψm (x)

Z

+
+

1
d3 xd3 x0 ψa† (x)ψa† (x0 )U (x − x0 )ψa (x)ψa (x)
2
¶
µ
Z
0
1
† x+x
)g(x − x0 )ψa (x)ψa (x0 ) + h.c. .
d3 xd3 x0 ψm
(
2
2

(3.2)

Equation (3.2) has the intuitive structure of resonant atom-molecule coupling [50, 51]
and was motivated by bosonic models of superfluidity [52, 53]. Here we have defined the2h̄
free atomic dispersion Ha (x) = − 2m
∇2x + Va (x) − µa and the free molecular dispersion2h̄
Hm (x) = − 4m
∇2x + Vm (x) − µm . Va,m are the external potentials and µa,m are thechemical potentials (the subscripts a, m represent the atomic and molecular contributions, respectively). The Feshbach resonance is controlled by the magnetic field whichis incorporated into the theory by the detuning ν = µm − 2µa between the atomic andmolecular fields.
The field operators which we have introduced ψa† (x), ψa (x) create and destroy

28† (x), ψ (x) create and destroy molecules at point x. For theatoms at point x and ψmmbosonic gas, both sets of these operators obey Bose commutation relations:hiψa (x), ψa† (x0 )hi†ψm (x), ψm(x0 )= δ 3 (x − x0 )(3.3)

= δ 3 (x − x0 ),and we assume that the fields ψa and ψm commute. It is important to keep in mind thatthe parameters introduced in the field theory are distinct from the physical parametersŪ , ḡ, and ν̄ behind the resonance. In order for the local Hamiltonian given in Eq. (3.2) tobe applicable, one must introduce into the field theory a renormalized set of parameters.
Since we will be working with contact potentials, each will contain a momentum cutoff
associated with a maximum wavenumber Kc . This need not be physical in origin but
should exceed the momentum range of occupied quantum states. The relationships
between the renormalized and physical parameters for the contact potentials are given
in Appendix A.1 . Of course, all of our final results must be independent of this artificial
cutoff; a condition which has been checked for all the results we will present.
To begin our development of a resonant field theory, we define the atomic andmolecular condensates in terms of the mean-fields of their respective operators
φa (x) = hψa (x)i and φm (x) = hψm (x)i.

(3.4)

The remainder is associated with the fluctuations about these mean-fields and can likewise be defined:χa (x) = ψa (x) − φa (x) and χm (x) = ψm (x) − φm (x).

(3.5)

Assuming the occupation of φm (x) to be small (less than 2% in the simulations we
present), we drop higher-order terms arising from fluctuations about this mean-field
which do not give a significant correction to our results.
Starting from Heisenberg’s equation of motioni∂A(x, t) = [A(x, t), H(x0 , t0 )],
∂t

(3.6)

29
we may derive a hierarchy of time dependent equations for the various fields involved
within our problem, each coupling to higher-order correlations. To truncate this hierarchy, we work within Hartree-Fock-Bogoliubov (HFB) theory. The simplification which
this introduces into the problem is that it allows us to factorize terms to quadratic orderin the field operators. Another way to say this is that we only account for correlations
between pairs of fields, but neglect all higher-order correlations. For example, we use
Wick’s theorem to factorize a four-point interaction as:
hA(x1 )A(x2 )A(x3 )A(x4 )i = hA(x1 )A(x2 )ihA(x3 )A(x4 )i(3.7)+ hA(x1 )A(x3 )ihA(x2 )A(x4 )i + hA(x1 )A(x4 )ihA(x2 )A(x3 )i.
Under this set of approximations we derive four equations: two corresponding toa Schrödinger evolution of the mean fields
ih̄

dφa (x)
dt

³

=

−

´
h̄2 2
∇x + Va (x) − µa + U [|φa (x)|2 + 2GN (x, x)] φa (x) (3.8)
2m+ [U GA (x, x) + gφm (x)]φ∗a (x),ih̄dφm (x)
dt

³

=

−

´
h̄2 2
g
∇x + Vm (x) − µm φm (x) + [φ2a (x) + GA (x, x)],
4m
2

(3.9)

and two corresponding to the Louiville space evolution of the normal density GN (x, x0 ) =hχ†a (x0 )χa (x)i and of the anomalous density GA (x, x0 ) = hχa (x0 )χa (x)i. The time evolution of the densities may be expressed in the compact form [54]ih̄∂G= ΣG − GΣ† ,dt(3.10)

where the density matrix and self-energy matrix are defined respectively as0† hχa (x )χa (x)iG(x, x0 ) = andhχ†a (x0 )χ†a (x)ihχa (x0 )χa (x)i hχa (x0 )χ†a (x)iΣ(x, x0 ) = ,(3.11)

H(x, x0 )∆(x, x0 )−∆∗ (x, x0 ) −H ∗ (x, x0 ).


(3.12)

30
The convenience of choosing a microscopic model in which the potential couplings are ofcontact form is now evident since the elements of the self-energy matrix Σ are diagonalin x and x0 with non-zero elements
H(x, x) = −h̄2 2∇ + Va (x) − µa + 2U [|φa (x)|2 + GN (x, x)],2m x∆(x, x) = U [φ2a (x) + GA (x, x)] + gφm (x).

3.4

(3.13)

Application to a spherical trap geometry
Equations (3.8), (3.9), and (3.10) form a closed set of equations for the inhomo-

geneous system. However, since the normal density and anomalous pairing field areboth six-dimensional objects, it is very difficult to solve these equations in an arbitrarygeometry. For this reason, we consider the case of greatest symmetry consisting of aspherical trap. Here we can reduce the problem to one of only three dimensions according to the following procedure. To begin, it is convenient to write the elements of thesingle particle density matrix in center of mass and relative coordinates:
0

R = x+x2

and r = x − x0 .

(3.14)The normal density then takes on a familiar structure corresponding to the Wigner
distribution [55]Z

d3 r hχ†a (R − r/2)χa (R + r/2)ie−ik·r

GN (R, k) =Z

d3 r GN (R, r)e−ik·r ,=(3.15)

which, in the high-temperature limit, will map to the particle distribution functionf (R, k) for a classical gas. Correspondingly, the anomalous density can be written in
Fourier space asZ

d3 r hχa (R − r/2)χa (R + r/2)ie−ik·r

GA (R, k) =Z=

d3 r GA (R, r)e−ik·r .

(3.16)

31

θRkFigure 3.1: Illustration of the spherically symmetric geometry used as defined by the~ relative momentum vector ~k, and the angle θ between them.
center of mass vector R,

In this geometry, the angular dependence of the center of mass vector R is irrelevant,and the cylindrical symmetry about R allows the wavevector k to be represented by its
length and the one remaining angle as illustrated in Figure 3.1.
This simplification allows us to represent the density distributions in three-dimensionsas
G(R, k) = G(R, k, θ),(3.17)

where G corresponds to either the normal (GN ) or anomalous (GA ) density. It is now
straightforward to rewrite Eqs. (3.9) and (3.10) in this coordinate system. It is worth
pointing out the simple structure of the kinetic energy contributions to Eq. (3.10) which,for the GN and GA components, take the corresponding forms, respectively:³´∇2x − ∇2x0 GN (x, x0 ) = 2 (∇R · ∇r ) GN (R, r)³´∇2x + ∇2x0 GA (x, x0 ) =µ(3.18)¶

1 2∇ + 2∇2r GA (R, r).
2 R(3.19)

One may now take the Fourier transform with respect to r as indicated by Eqs. (3.15)
and (3.16), replacing ∇r → ik. The gradient operator ∇R can be expressed in any
representation, but it is most convenient to use spherical polar coordinates aligned with32
the k direction vector∇R = R̂

∂
1 ∂
1 ∂
+ θ̂
+ ϕ̂
,
∂R
R ∂θ
sin θ ∂ϕ(3.20)

where ϕ is the azimuthal angle about k (which will eventually drop out in our chosen
symmetry), and (R̂, θ̂, ϕ̂) are the spherical unit vectors in the R, θ, and ϕ directions.
Noting that R̂ · k = k cos θ, θ̂ · k = −k sin θ, and ϕ̂ · k = 0, we arrive at the following
expression for the differential operator in Eq. (3.18):µ∇R · k = k cos θ¶∂
sin θ ∂
−
.
∂R
R ∂θ

(3.21)

Furthermore, the spherical Laplacian for a system with no azimuthal dependence, as
required in Eq. (3.19), is given by∇2R =1 ∂ ³ 2 ∂ ´
∂ ³
1
∂ ´
R
+
sin
θ
.
R2 ∂R
∂R
R2 sin θ ∂θ∂θ

(3.22)

In practice, we expand the θ dependence of GN and GA in terms of the orthogonal
Legendre polynomials and the angular derivatives are then easily implemented via theusual recursion relations.

3.5

Numerical results and analysis
As an initial test, we expect the resonance theory to give a similar prediction to

the Gross-Pitaevskii equation in the initial phase of the collapse, when the quantum
depletion is small. Figure 3.2 shows a direct comparison between the Gross-Pitaevskii
approach and the resonance theory. The same initial conditions were used for all our
simulations: 1000 rubidium-85 atoms in the ground state of a 10 Hz harmonic trap.
For all the images we present, the results of the three-dimensional calculation, in ourspherical geometry, are illustrated as a two-dimensional slice through the trap center.
In the Gross-Pitaevskii solution we used a scattering length of −200 a0 , where a0 is theBohr radius. For comparison, the Feshbach resonance theory uses a positive background33
scattering length of 50 a0 and a resonance width and detuning , respectively, of 15 kHz
and 2.8 kHz. These parameters give the same effective scattering length as the one usedin the Gross-Pitaevskii evolution, but nowhere in the resonance theory does the effective
scattering length appear explicitly. As is evident, there is no noticeable discrepancy
between the two approaches over this short timescale. Eventually, we expect these
theories to diverge significantly as the density increases and the coupling between theatomic and molecular degrees of freedom become stronger. However, at this stage, theagreement is a demonstration that our renormalized theory correctly allows us to tune
the interactions in an inhomogeneous situation.
We now proceed to a more complex situation in which the timescales for the atommolecule coupling and the collapse dynamics are more compatible. From a numericalpoint of view, it becomes convenient to increase the resonance width to 1.5 MHz and thedetuning to 14 kHz so that the effect of the atom-molecule coupling will appear in thefirst stage of the collapse. This allows us to form a complete picture of the dynamics
involving the atomic collapse and the simultaneous coupling to a coherent molecularfield. The numerical calculation is shown in Fig. 3.3 for both the condensed and noncondensed components. We see the formation of a significant fraction of non-condensed
atoms–a feature not described within the Gross-Pitaevskii framework. During a time
evolution of 0.8 ms the condensate fraction falls to approximately 80% of its initial value
while the non-condensate fraction reaches a peak at around 20%. The amplitude of thescalar field φm remains below the 2% level at all times.
To better illustrate the behavior of the atoms during the collapse, we present theflow of the different distributions involved. The condensate velocity field is shown in
Fig. 3.4. It exhibits similar characteristics to those predicted by the Gross-Pitaevskii
theory, which, without loss, predicts that the condensed atoms will always accelerate
toward the trap center. In contrast, the velocity field of the atoms outside of thecondensate is radially outward.

34

Figure 3.2: A direct comparison of the collapse between the Gross-Pitaevskii (left) andthe resonance approach (right) within the regime of applicability of the Gross-Pitaevskii
equation. Each horizontal pair is at the same time step with time increasing from top tobottom. As expected, we observe no appreciable difference between the two methods.

35

Figure 3.3: The simulation of the collapse in the resonance theory showing the time
evolution of the condensed fraction φa (x) (left) and noncondensed fraction GN (x, x)(right). Each horizontal pair is taken at the same instant of time with time increasingfrom top to bottom. It is evident that noncondensate atoms are produced during thecollapse forming rings which propagate from the center of the cloud outward.

36

Figure 3.4: The velocity fields for the condensate component φa (x) (top) and the noncondensed component GN (x, x) (bottom) midway through the simulation (0.4 ms). The
color contours indicate the densities and the velocity fields are represented in direction
and strength by the arrows. This clearly shows that in the resonance theory, as thecondensate collapses inward, the non-condensate atoms that are generated flow outward.

37
An important quantity to calculate for these expanding non-condensed atoms isthe effective temperature, or energy per particle, since this quantity is observed experimentally. This is illustrated in Fig. 3.5 where we show, superimposed on an illustrationof the density, a colormap of the temperature. The hottest atoms generated in the center of the cloud are of comparable energy scale to those seen in the experiment, being
on the order of 100 nK.

3.6

Conclusion
These numerical simulations illustrate the feasibility of generating atomic bursts

purely through a coupling between an atomic and molecular component. However,there are a number of important distinctions with the experimental situation which
would have to be accounted for before making a direct comparison. These simulations
contain no inelastic three-body loss and particle number is absolutely conserved. In
reality, three-body loss may be important to the experiment, but we suggest with thiswork that three-body loss is not the only mechanism for producing a non-condensed
burst during the collapse.
It should be emphasized that, if our hypothesis for the burst generation is correct, the non-condensate atoms that are produced by this mechanism are not simply
generated in a thermal component, but are instead generated in a fundamentally intriguing quantum state. The process of dissociation of molecules into atom pairs produces
macroscopic correlations reminiscent of a squeezed vacuum state in quantum optics.
This means that every atom in the burst with momentum k would have an associatedpartner with momentum −k. In principle, the correlations could be directly observed inexperiments through coincidence measurements providing clear evidence as to whether
this is the dominant mechanism for the burst generation in the Bosenova.

38

Figure 3.5: Density distribution of the non-condensate atoms near the end of the simulation (0.8 ms) on which we have superimposed the energy per particle as a colormap.
The range of energies, of order 100 nK, is consistent with the characteristic scale of the
burst particle energies in the Bosenova experiment. Note that hot atoms are generated
in the center of the cloud during the atom-molecule oscillations since this is where theatom-molecule coupling is strongest (the coupling strength varies as the square root of
the density). As the hot particles radiate outward a ring can be observed.

Chapter 4
BCS Superfluidity

4.1

Bardeen-Cooper-Schrieffer theory
The classic theory of superfluidity/superconductivity was put forth by Bardeen,

Cooper, and Schrieffer in 1957 [56]. Entitled BCS theory, after its founders, the ideas
presented in this seminal work were able to successfully explain almost all observed
properties of conventional superconductors. We will extend BCS theory in Chapter 5 toincorporate Feshbach resonant interactions, but first it will prove instructive to review
some fundamentals.

4.2

Cooper pairing
The cardinal process behind superfluidity is the formation of Cooper pairs. Cooper,

who originally studied the problem [57] , showed that fermions, interacting above aclosed Fermi sea, will show an instability towards forming pairs regardless of the weakness of the interaction, so long as the interaction remains attractive. Due to the sharpness of the Fermi surface, the main contribution to this population of paired states willarise between atoms on opposite ends of the Fermi surface. For instance, if we consider
the s-wave pairing of a two spin state system (↑, ↓), pairs will prefer to form between
atoms of k, ↑ and −k, ↓ (see Fig. 4.1). As long as the Fermi surface remains relatively
sharp, it is unlikely that other combinations of pairs will form. This statement is moreclearly illustrated in combination with Fig. 4.2.

40

Figure 4.1: For s-wave collisions, pairs of atoms with opposite spin will form at the
Fermi surface. The interior region represents the filled Fermi sea whereas the dark line
around the edge shows the range of available states. In order to conserve energy thestate shown on the left may scatter into any available state (right) which lies along
its equipotential surface. For this example, the equipotential surface lies within theavailable states at the Fermi surface.

In a true many-body system, these atom pairs do not truly form bound states,
but should rather be thought of as resulting from strong correlations. The energy gap,
and most of the observed properties of superconductors, would be absent if it weren’t
for the strong correlations between these pairs. It should be noted, however, that these
correlations result more from the Pauli blocking of available states within the Fermi
surface than from the actual dynamical interactions. As a consequence, it is often a very
good approximation to treat the system as though the interactions only occur between
the Cooper pairs. This scheme may be called the BCS or pairing approximation.

4.3

BCS linearization and canonical transformation

Let us begin by discussing BCS theory as derived from the method of canonical transformation [58, 59]. We start with the following interacting Hamiltonian which accounts
for the effects of pair-wise interactions:
H=

X
kσ

²k a†kσ akσ +X
kk0

Vkk0 a†k↑ a†−k↓ a−k0 ↓ ak0 ↑ ,

(4.1)

41

Figure 4.2: If we attempt to form a Cooper pair between k, ↑ and k0 , ↓, where k 6= −k0 ,the number of states which the pair may scatter into is severely limited. This may
be seen by transforming to a frame in which the pairs form with equal and opposite
momentum, but with center of mass momentum q. The equipotential surface is givenby the dotted line and only rarely falls within the available region of states.

where a†kσ (akσ ) are fermion creation(annihilation) operators, ²k = h̄2 k 2 /2m, and Vkk0
define the matrix elements of the interaction potential. A fundamental idea behind
BCS theory is that the many-body wavefunction will be constructed of a phase coherent
superposition of paired states|φN i =

X
k1

X

···

kN/2

gk1 · · · gkN/2 ...a†k1 ↑ a†−k1 ↓ · · · a†kN/2 ↑ a†−kN/2 ↓ |ϕ0 i,(4.2)

where gkN/2 is the weight of each of the N/2 paired states and |ϕ0 i is the vacuum ket.
Because of this phase coherence, operators such as a−k↓ ak↑ may not average to zero as
they would in the normal state where the phases are at random. What’s more, for alarge number of particles one may assume that the fluctuations about these averages
should be small. With this in mind, we rewrite the paired operator asa−k↓ ak↑ = ha−k↓ ak↑ i + (a−k↓ ak↑ − ha−k↓ ak↑ i),(4.3)

which separates the average (first term) from the fluctuations about the average (secondterm). Substitution of Eq. (4.3) into Eq. (4.1) results in
H=

X
kσ

ξk a†kσ akσ +X

Vkk0 (a†k↑ a†−k↓ ha−k0 ↓ ak0 ↑ i + ha†k↑ a†−k↓ ia−k0 ↓ ak0 ↑ ),

(4.4)

kk0

where ξk = ²k − µ. We have introduced a chemical potential since the linearized Hamiltonian no longer conserves number. The chemical potential may be adjusted to gain

42
the desired number of atoms. We may now define the gap
X∆k = −

Vkk0 ha−k0 ↓ ak0 ↑ i.

(4.5)

k0

For the moment, the name may seem a bit arbitrary, but this function will later beshown as equivalent to the value of the gap in the energy spectrum. The Hamiltonian
may now be written as:H=Xξk a†kσ akσ −kσX(∆k a†k↑ a†−k↓ + ∆∗k a−k↓ ak↑ ).

(4.6)

k

This Hamiltonian may be diagonalized by an appropriate canonical transformation (seeAppendix B) of the general form ak↑  

=

 

a†−k↓





u∗k
−vk∗



vk   γk0 
uk




†
γk1

,


(4.7)

where the γ’s are a new set of Fermi operators and the weights uk and vk must satisfy
the relationship|uk |2 + |vk |2 = 1,

(4.8)

for the resulting transformation to remain unitary. Note that an inversion of Eq. (4.7)
shows that the new operators γ have the net effect of mixing the spins. By insertingthese substitutions into Eq. (4.6) we will recover a diagonal Hamiltonian, that is, onethat only contains density type pairs of operators γk† γk , if the following condition issatisfied:∆∗k vk2 − ∆k u2k + 2ξuk vk = 0.

(4.9)

Multiplying through by ∆∗k /u2k and solving the resulting quadratic equation in terms of∆∗k vk /uk we have the following condition for the coefficientsvk
Ek − ξk=,uk∆∗k(4.10)q

where we define Ek =

ξk2 + |∆k |2 . Equation (4.10), which is the positive root, is chosen

since it corresponds to the stable minimum (as opposed to the unstable maximum)

43
energy solution. By combining Eq. (4.8) and Eq. (4.10) we have the following result for
the value of the coefficients:
µ

¶

1
ξk
|vk | = 1 − |uk | =
1−
.
2
Ek
2

2

(4.11)It should be noted that, although the phases up until know are written arbitrarily, thephase of uk , vk , and ∆k must occur so that ∆∗k vk /uk is real, as implied by Eq. (4.10).

4.4

The appearance of an energy gap

The resulting diagonal Hamiltonian (Eq. (4.6) written in terms of the basis defined in
Eq. (4.7)) isH=

X(ξk − Ek ) +

X

k

†
†
Ek (γk0
γk0 + γk1
γk1 ).

(4.12)k

The first term gives a constant shift to the energy spectrum while the second term
describes the elementary quasi-particle excitations of the system. The energy of these
excitations is given byq

Ek =ξk2 + |∆k |2 ,(4.13)

so that, even at ξk = 0, there remains a finite energy ∆k which must be invested in order
to excite the system. This gap in the energy spectrum justifies our earlier labelling of
∆k .

4.5

Populating the quasi-particles

In this diagonal representation, we may write the gap as∆k = −

X
k0

Vkk0 ha−k0 ↓ ak↑ i = −X

Vkk0 u∗k0 vk0 h1 − γk† 0 0 γk0 0 + γk† 0 1 γk0 1 i.

(4.14)

k0

In thermal equilibrium, the quasi-particles behave as noninteracting fermions so they
distribute themselves according to Fermi statistics:
f (Ek ) =1eβEk + 1.

(4.15)

44
By writingh1 − γk† 0 0 γk0 0 + γk† 0 1 γk0 1 i = 1 − 2f (Ek0 ),(4.16)

we may convert Eq. (4.14) to the following form∆k = −

∆k0
βEk0
1X
Vkk0
tanh
.
2 k0
2Ek0
2

(4.17)In deriving Eq. (4.17) we have used the trigonometric identity tanh(x/2) = 1 − 2f (x).
Equation (4.17) is called the gap equation. It is often assumed that the potential isconstant Vkk0 = V which results in a constant gap ∆k0 = ∆. This considerably simplifies
the problem since a factor of ∆ may be removed from both sides of the equation resultingin
X tanh(βEk /2)
1
=−
.
V
2Ek
k

(4.18)

We may perform the same transformation on the expectation value of the number
operator to derive a second equationnσ =Xha†kσ akσ i =kX|uk |2 f (Ek ) + |vk |2 (1 − f (Ek ))

k

=

X1µ
k

2

1−

¶

ξk
tanh(βEk /2) .
Ek(4.19)

Equations (4.18) and (4.19) are a closed set of equations forming the foundations of
BCS theory. Unfortunately, due to the form of the potential Vkk0 that we have chosen,
these equations will diverge. This divergence, however, may be remedied by a proper
renormalization. In the case of superconductors, the divergence is often remedied by
cutting the integral at the Debye energy since only a very limited shell of states about
the Fermi surface are thought to contribute. However, for atomic systems, the renormalization may be performed by matching the potentials to the T-matrix describing thefull 2-body scattering processes. This was explained in detail in Chapter 2.6.

45

4.6

Critical temperature
For dilute interactions, we may assume that the potential is given by V =4πh̄2 asc /m, where m is the mass of each fermion and asc is the s-wave scattering length.
At the critical point the gap vanishes, ∆ = 0, and we may solve Eqs. (4.18) and (4.19)
for the critical temperature Tc . Under the assumption that kF asc ¿ 1, we find for thecritical temperature:T≈ .614e−π/(2kF |asc |) .
TF

(4.20)The exponential form of Eq. (4.20), although only approximate, shows that we mustenhance the scattering in order to raise the critical temperature. For kF asc ∼ 1 wehave a critical temperature Tc /TF ∼ 0.1–a temperature which is accessible to modern
experiments. One method to enhance the scattering, and obtain critical temperaturesof this magnitude, would be by tuning the particle-particle interactions with a Feshbachresonance. In the next section, we discuss how to incorporate these resonant scatteringinteractions into this model.

Chapter 5
Resonant superfluid Fermi gases

0
−1
−2

−4

−2

−1

∆B (Gauss)
0
1

2

10
−5
0

a (10 a )

−6

3

3

0

T−Matrix (10 a )

−3

−7
−8

5
0
−5

−10
−0.2

−9

−0.1

0
ν (mK)

0.1

0.2

0

−10
0

0.1

0.2

0.3
0.4
0.5 E 0.6
F
Scattering Energy (µK)

0.7

Figure 5.1: Real (solid line) and imaginary (dashed line) components of the T -matrix
for collisions of the lowest two spin states of 40 K at a detuning of 20 EF , shown in length
dimensions, i.e., Tk /(4πh̄2 /m). The inset shows the scattering length as a function ofdetuning, with 20 EF detuning indicated by the dashed-dot line.

5.1

Signatures
In this chapter we will develop a Feshbach resonance field theory for interacting

Fermi gases and show that the enhanced interactions do indeed increase the criticaltemperature. Once the critical point is reached it remains to distinguish a clear signatureof the phase transition. Various approaches have been proposed to detect the superfluid
state: measurements of the pair distribution [60, 61], experiments involving the breakup

47
of Cooper pairs [62], measurements of the moment of inertia [63], and probes of collective
excitations [64, 65, 66]. In this chapter we will show that a more direct signature of thetransition to superfluidity is provided by the density distribution in an inhomogeneous
system. We demonstrate that, in a harmonic trap, the superfluid state manifests in theappearance of a density bulge in the central atomic density.

5.2

Resonant Hamiltonian for a Fermi gas
We will now extend the concepts of BCS theory, detailed in the last chapter,and well accounted for in the literature of trapped atomic gases [67, 68, 69, 70, 71],
to incorporate resonantly interacting Fermi gases [49, 72] . To begin, we consider aresonance such as that of 40 K, illustrated in Fig. 5.1, for s-wave scattering of atoms in
the lowest two hyperfine spin states σ ∈ {↑, ↓}. The scattering length is the intercept
at zero scattering energy which, for this case, is approximately -10000 a0 , where a0 is
the Bohr radius. The large variation in the T -matrix over the relevant energy range
indicates that a quantum field theory developed from this microscopic basis will needto correctly incorporate this strong energy dependence. The inset shows the resonant
form of the scattering length which obeys the usual dispersive form a = abg (1 − κ/ν̄).
For 40 K, the dispersion parameters are abg = 176 a0 and κ = 0.657 mK[73].
In analogy to the Hamiltonian presented in Eq. (3.2), we write down the following
Hamiltonian for the Fermi system:
H =

XZ
Zσ

+

Z

d3 x ψσ† (x)Hσ (x)ψσ (x) +d3 xd3 x0 ψ↓† (x)ψ↑† (x0 )U (x − x0 )ψ↑ (x)ψ↓ (x)
µ

Z

+

†
d3 x ψ m(x)Hm (x)ψm (x)d3 xd3 x0

¶

†
ψm
(

x + x0)g(x − x0 )ψ↓ (x)ψ↑ (x0 ) + h.c. .
2

(5.1)

2

h̄
We define the free atomic dispersion for each spin state as Hσ (x) = − 2m
∇2x +Vσ (x)−µσ2h̄and the free molecular dispersion as Hm (x) = − 4m
∇2x + Vm (x) − µm . Vσ,m are theexternal potentials and µσ,m are the various chemical potentials for each species. The

48
Feshbach resonance is controlled by the magnetic field which is incorporated by the
detuning ν = µm − µ↑ − µ↓ between the chemical potential of each of the two spin states
and the molecular fields.
The field operators ψσ† (x), ψσ (x) create and destroy atoms of spin σ at point x† (x), ψ (x) create and destroy molecules at point x. The atomic field operatorsand ψmm

now commute like fermions whereas the molecular fields retain a bosonic nature:hi

ψσ1 (x), ψσ† 2 (x0 )
h

A

i

†
ψm (x), ψm(x0 )= δσ1 ,σ2 δ 3 (x − x0 ),(5.2)= δ 3 (x − x0 ),where the above subscript A refers to anti-commutation. The bosonic nature of themolecular field, although comprised of a pair of fermions, is justified with the understanding that the closed channel molecules are extremely well bound and have a relative
size much smaller than all other length scales in the problem. They, therefore, are wellcharacterized as bosons.
We now consider the general structure of the theory for the homogeneous case.
Equation (5.1) is written in position space, however, we would now like to change tomomentum space. This may be done by introducing the plane wave expansion of theoperators:ψσ (x) =ψm (x) =1 X√
a eik·x
V k kσ1 X√
b eik·x ,
V k k

(5.3)

in a volume V . With this expansion, Eq. (5.1) transforms to:
H =

X
kσ

+ U

(²kσ − µσ ) a†kσ akσ +

X †

qkk0

X

(²km + ν)b†k bk

kaq/2+k↑ a†q/2−k↓ aq/2−k0 ↓ aq/2+k0 ↑ + g

(5.4)
X³

´

b†q aq/2−k↓ aq/2+k↑ + h.c. ,kqwhere h.c. denotes the hermitian conjugate. The free dispersion relation for the fermionsand bosons respectively are ²kσ = h̄2 k 2 /2m and ²km = h̄2 k 2 /4m, and ν denotes the49
detuning of the boson resonance state from the zero edge of the collision continuum. In
deriving Eq. (5.4) we have introduced contact potentials so must again renormalize as
in Appendix A.1.

5.3

Construction of dynamical equations for fermions
From this Hamiltonian (Eq. (5.4)), we construct the dynamical Hartree-Fock-

Bogoliubov (HFB) equations for both the bosonic and fermionic mean-fields. Theseequations involve the mean fields corresponding to the spin density n =(taken to be identical for both spins), the pairing field p =PP†k hakσ akσ ik ha−k↓ ak↑ i,and thecondensed boson field φm = hbk=0 i. Note that since we only consider the condensate
portion of molecules, and are dealing with a homogeneous system, we may neglect the
kinetic contribution of the molecules. The single particle density matrix [54] is defined
as:





 ak↑ 




 a

 k↓ 

,
A=

 a†

 −k↑ 
Gi,j = hA†j Ai i ,

(5.5)

a†−k↓

and evolves according to the Bogoliubov self-energy Σih̄dG= [Σ, G] .
dt

(5.6)

The differences between Eq. (5.6) and Eq. (3.10) arise due to the particle statistics. The
self-energy for fermions now has a Hermitian structure


 Uk


 0

Σ=

 0




∆∗

0

0

∆

Uk

−∆

0

−∆∗ −Uk

0

0

0

−Uk







,






(5.7)

where the single particle energy is Uk = ²k − µ + U n, we define a gap ∆ = U p + g φm ,and µ is the chemical potential. Finally, the dynamical equations are closed by the

50evolution equation for the boson mode:
ih̄

5.4

dφm
= ν φm + g p .
dt

(5.8)

Application to a trapped system
In order to solve this set of equations, the self-energy Σ is diagonalized lo-

cally at each k by the Bogoliubov transformation (see Appendix B) generating quasiq

particles with an energy spectrum Ek =Uk2 + |∆|2 . In equilibrium, the quasi-particle

states are occupied according to the Fermi-Dirac distribution nk = [exp(βEk ) + 1]−1 .
The corresponding maximum entropy solution for the molecule amplitude is found byih̄dφm /dt = µm φm , where µm = 2µ, so that Eq. (5.8) implies
φm = g p/(µm − ν).

(5.9)

The mean fields can then be determined by integration of the equilibrium single particle
density matrix elements given by:
Z

n =
p =

K
1
dkk 2 [(2nk − 1) cos 2θk + 1] ,(2π)2 0
Z K
1
dkk 2 (2nk − 1) sin 2θk ,(2π)2 0(5.10)

where tan 2θk = |∆|/Uk is the Bogoliubov transformation angle. Since θk depends onn and p, these equations need to be solved self-consistently. As a technical point, it
should be noted that the p = 0 solution should be discarded when the superfluid stateis present since it then corresponds to an unstable point on the free energy surface.
Perhaps it is useful to rewrite Eqs. (5.10) in the following formÃg2
1 = − U−ν − 2µ
n =

1
2

Z

µ

!Z

d3 k tanh(βEk /2)
(2π)3
2Ek

(5.11)

¶

d3 k
Uk
1−
tanh(βEk /2) .
(2π)3
Ek

The form of Eqs. (5.11) clearly relate the resonance equations to the BCS gap andnumber equation (Eqs. (4.18) and (4.19)).

51
So far this theory has been presented for a homogeneous system, while we areinterested in a gas of N atoms confined to an external trapping potential V (r). However, a full quantum mechanical treatment of the trapping states is not required. For
instance, in our case, with a temperature on the order of T = 0.2 TF , the harmonic
oscillator level spacing is smaller than both the Fermi and thermal energies. Under
these conditions, we may incorporate the effect of the trap through a semi-classical,
local-density approximation [75, 76]. This involves replacing the chemical potential bya local oneµ(r) = µ − V (r),(5.12)and determining the thermodynamic solution at each point in space as for the homogeneous system.
In general, the validity of the semi-classical approximation requires a slow variation in the occupation of the discrete quantum levels as a function of energy. Remarkably, in both bosonic and fermionic gases, this condition can often be satisfied even atvery low temperatures because of strong correlations in a BEC due to repulsive interactions and because of exchange effects in a quantum Fermi gas. In both cases, the zerotemperature, semi-classical approximation for dilute gases is usually referred to as theThomas-Fermi approximation.

5.5

Thermodynamic results and signatures of superfluidity

We evaluate the thermodynamic quantities at given T and N in three steps:(1) For given µ, we determine the local chemical potential µ(r) = µ − V (r) and use
this value to find the self-consistent solution for the density n(r) and pairing
field p(r) at each point in space, according to the solution of Eqns. (5.10).
(2) We modify the global chemical potential µ until the density integral N =R 3
d r n(r) gives the desired atom number.

52(3) We use the resulting solution for µ to calculate observable quantities, such as
the density, gap, compressibility, and so forth.
The resulting solution for the density distribution is illustrated in Fig. 5.2. A striking
signature of the resonance superfluidity is evident in the predicted density profile which
has a notable bulge in the region of the trap center. This signature would appear to be
directly accessible experimentally. One approach to measure this density bulge would
be to fit the expected density profile for a quantum degenerate gas with no superfluid
phase to the wings of the distribution (outside the dotted lines shown in Fig. 5.2). The
excess density observed at the trap center could then be recorded. Fig. 5.3 illustrates
the emergence of the superfluid as the temperature is decreased. Qualitatively, thissituation is reminiscent of the central condensate peak observed for a Bose-Einsteincondensed gas in a harmonic potential.
13

4

x 10

3.5

−3

Density (atoms cm )

3
2.5
2
1.5
1
0.5
0
−30

−20

−10

0
10
Radial Position (µm)

2030Figure 5.2: Density profile at temperature T = 0.2 TF and detuning ν = 20 EF showing
accumulation of atoms at the trap center (solid line). We compare with the profile
resulting from the same µ but artificially setting the pairing field p to zero so that nosuperfluid is present (dashed line).

53
12

x 10

−3

Excess density (cm )

10
8
6
4
2
0
0
0.1
0.2

20

Temperature (T )

0

0.3

F

0.4

−20
Radial Position (µm)

Figure 5.3: Emergence of the coherent superfluid for ν = 20 EF . The superfluid
occupies an increasing volume as the temperature is reduced. Shown is the excess
density (difference between the dashed and solid lines in Fig. 5.2) at each temperature.

We explain the observed behavior by considering the compressibility of the normal
and superfluid gas. Thermodynamically, the isothermal compressibility C is defined as
C −1 = n(∂P/∂n)T ,(5.13)

where P is the pressure and is shown in Fig. 5.4. The compressibility is positive everywhere, indicating that, in spite of the large attractive interactions, the Fermi pressure
makes the configuration mechanically stable. A significant feature is the discontinuous behavior at the radius from the trap center at which the superfluid changes fromzero to a non-zero value. This discontinuity is a manifestation of a second-order phasetransition occurring in space. The discontinuity is a consequence of the local density
approximation and cannot occur in a finite system. However, a rapid change in thecompressibility is expected. In principle, this could be probed by studies of shock waves
generated by the abrupt jump in the speed of sound as a density fluctuation passes
through the discontinuous region.

54

5.6

Conclusions
This section has shown how we may derive a resonant theory for dilute Fermi

gases and has applied this theory to the case of a trapped ensemble. We demonstrated
that there exists a direct signature of superfluidity in trapped Fermi gases. The onsetof superfluidity leads to a density bulge in the center of the trap which can be detectedby absorption imaging. The critical conditions for superfluidity are satisfied initially inthe trap center and the region of non-zero pairing field spreads out from the center as
the temperature is lowered further. The increase in the density profile in the superfluid
region is caused by a jump in the compressibility. Direct measures of this behavior
are possible by the study of the propagation of sound waves. We have applied our
method here to 40 K, but a similar approach is easy to derive for other interesting
atoms, including, in particular, 6 Li which is another fermionic alkali currently being
investigated experimentally.

Inverse Compressibilty C−1F/ (2nE )

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−30

−20

−10
0
10
Radial Position (µm)

2030Figure 5.4: Inverse isothermal compressibility C −1 in units of the Fermi energy (solidline). Here ν = 20 EF and T = 0.2 TF (as can be seen from the limiting behavior atlarge radial position). A discontinuity appears at the radius at which the superfluid
emerges (dotted line). We compare this solution to that corresponding to zero pairing
field and no superfluid phase transition (dashed line).

Chapter 6
Feshbach Resonant Crossover Physics

6.1

Historical backgroundIn the early stages of developing a theory for superconductivity it was suggested

that superconductivity was a result of a Bose-condensation of pairs of electrons into localized bound states. This mechanism came to be known as “Schafroth condensation”
and was extended upon in the work of Schafroth, Blatt, and Butler in, what they hadcoined, “quasi-chemical equilibrium theory” [77, 78]. In this theory, they considered
the size of the bound pairs to be small compared to the inter-particle spacing. Superconductivity would result as a continuum of Bose-Einstein excitations above a groundstate without an energy gap. Due to the complexity of this theory, and the extreme
success of the soon to be unveiled theory of Bardeen, Cooper, and Schrieffer (BCS), thequasi-chemical equilibrium theory fell into obscurity. Later, however, it was shown by
Blatt and coworkers [79] that their Bose-condensation approach could be extended togive the same results as the BCS pairing theory.
At the time, the differences between these two approaches were highly emphasized
and it was thought that superfluidity should be understood in terms of momentumspace pairing and not of real-space pairing. However, the similarities between the two
approaches remain so the question may be asked: How does one reconcile the difference
between BCS superfluidity of highly-overlapping, long-range fermion pairs and that of
Bose-Einstein condensation of non-overlapping, localized pairs. These two cases are

56
the extremes and the problem of how to move from one to the other is known as the“crossover problem” (see Fig. 6.1).
One of the first attempts to understand this crossover was put forth by Eagles ina 1969 paper on pairing in superconducting semiconductors [80]. He proposed moving
between these two limits by doping samples, in this case, by decreasing the carrier
density in systems of SrT iO3 doped with Zr. In a 1980 paper by Leggett [81], motivatedby the early ideas of quasi-chemical equilibrium theory, he modelled the crossover atzero temperature by way of a variational wavefunction:
ψBCS =

Y(uk + vk a†k a†−k )|0i.

(6.1)

k

This wavefunction is simply the BCS wavefunction and assumes that at T = 0 all thefermions form into Cooper pairs. What Leggett was able to show was that he could
smoothly interpolate between conventional BCS theory and the process of BEC.
In 1985, Nozières and Schmitt-Rink (NSR) extended this theory to finite temperatures in order to calculate the critical temperature Tc [82]. NSR derived the conventional
BCS gap and number equation, but introduced into the number equation the self-energy
associated with the particle-particle ladder diagram (or scattering T-matrix) to lowest
order (see Fig. 6.2). This very influential paper would be built upon by many othergroups and was transformed into a functional form by Randeria et al. [83].
A compelling reason for understanding the crossover problem comes from thefact that many high-Tc superconductors seem to fall within the region intermediate
between BCS and BEC. In the copper oxides, for instance, the coherence length of the
Cooper pairs has been measured to be only a few times the lattice spacing. In contrast,
in conventional superconductors, the coherence lengths are usually much greater thanthe lattice spacings. An understanding of the crossover may be one of the keys tounderstanding and manipulating high-Tc materials. In Chapter 7 we will extend the
NSR model of Randeria to account for Feshbach resonant interactions and the formation

57
of molecules. In Chapter 8 we will attempt to go beyond this formalism by adapting a
dynamical Green’s function approach, originally designed to model high-Tc systems, tothe case of Feshbach resonant interactions.

58

Figure 6.1: An illustration of the differences between Bardeen-Cooper-Schreiffer (BCS)
superfluidity and Bose-Einstein condensation (BEC) of molecules in a Fermi gas. BCS
theory describes weakly-bound fermions which are paired along the Fermi surface in
k-space so may be highly overlapping in real space. The Cooper pairs break up at thecritical temperature Tc signifying the phase transition. BEC results from from tightlybound, real-space fermion pairs. Above the critical temperature Tc , these bosonic pairsloose their mutual phase coherence, but may remain bound into pairs.

Figure 6.2: Diagrammatic contribution of pair fluctuations to the thermodynamic potential included by NSR. The solid lines are free propagators and the wavy lines areinteractions.

Chapter 7
Path integral approach to the crossover problem

7.1

Resonant action
To begin our discussion of the resonant crossover problem we consider a Feshbachresonance for s-wave scattering of atoms in the lowest two hyperfine states σ ∈ {↑, ↓} of afermionic alkali atom [84]. For a homogeneous system we have the following generalized
Hamiltonian:Ĥ(t) =

XZ
Zσ

+

†
ψm
(x)(Ĥm − 2µ + ν)ψm (x)d3 x

ψ↑† (x)ψ↓† (x0 )U (x − x0 )ψ↓ (x0 )ψ↑ (x)d3 xd3 x0

Z µ

+

Z

ψσ† (x)(Ĥσ − µ)ψσ (x)d3 x +

¶
0
0
0
† x+x
)g(x − x )ψ↓ (x)ψ↑ (x ) + h.c. d3 xd3 x0 ,
ψm (

2

(7.1)

† (ψ )
where the operators ψσ† (ψσ ) create (annihilate) fermions at x = (x, t), and ψmmcreate (annihilate) composite bosons. The free dispersion Hamiltonian for fermions(bosons) is Ĥσ (Ĥm ) and ν is the detuning of the resonant molecular state from thecollision continuum. The collisional interactions are described by both backgroundfermion-fermion scattering, U , and an interconversion between composite bosons andfermions, g.
Functional methods prove to be especially convenient in describing the thermodynamics of the resonant system. For a finite-temperature field theory, the connection
with statistical mechanics is made by Wick rotating the time coordinate t → −iτ sothat one works in terms of the spatial coordinate x and temperature τ [85]. The action

60
is defined in the usual way:
S=

XZ β
0

l

Z

dτ

d xψl† (x, τ )∂τ ψl (x, τ ) −
3

Z β
0

Ĥ(τ )dτ,(7.2)

where the sum in l runs over both the Fermi and the Bose degrees of freedom. In thisfunctional formulation we treat the fermion fields ψσ as Grassmann variables and thecomposite Bose fields ψm as classical fields [86].
Let us consider a system comprised of fermions at some finite temperature τ inside
a box of volume V (for convenience, let us work in the set of units where h̄ = kb = 1).
By imposing periodic boundary conditions upon the fields ψσ and ψm , we form thefollowing Fourier series expansions:ψσ (x, τ ) = (βV )−1/2Xei(ωτ +p·x) aσ (p),k,ωψm (x, τ ) = (βV )

−1/2

X

ei(vτ +q·x) b(q),

(7.3)

q,v

with even thermal (Matsubara) frequencies for the bosons (v = 2πn/β, where n is
an integer) and odd frequencies for the fermions (ω = 2π(n + 1)/β) [87]. Here aσ (p)annihilates a fermion at p = (k, ω) and b(q) annihilates a molecule at q = (q, ω).
By making use of the above transformation, Eq. (7.3), we may write out theaction for the resonant system in terms of the Fourier coefficients aσ (p) and b(q). To
help clarify the following calculation, we split the resulting resonant action into twoparts, the first being the usual BCS action
SBCS =

X

(iω −

p,σ

1 X
p2
+ µ)a∗σ (p)aσ (p) −U a∗↑ (p1 )a∗↓ (p2 )a↓ (p3 )a↑ (p4 ), (7.4)
2m
βV
p1 +p2 =p3 +p4

and the second we will label the molecular action
SM

=

X

(iv −

q

q2
− ν + 2µ)b∗ (q)b(q)
4m

(7.5)

´
X ³
1
g b∗ (q)a↓ (p1 )a↑ (p2 ) + a∗↑ (p2 )a∗↓ (p1 )b(q) .
−√βVq=p1 +p2

In deriving Eqs. (7.4) and (7.5) we have inserted contact potentials for the couplings
U (x − x0 ) → U δ(x − x0 ) and g(x − x0 ) → gδ(x − x0 ). The full partition function for our

61resonant system, under the model Hamiltonian of Eq. (7.1), can now be written as
Z=

Z ÃY

!

Da∗σ Daσ³Y

Db∗ Db´

eSBCS +SM ,

(7.6)

σ

with the functional integral, Dc ≡

7.2

Q

i
i dc , ranging over all Fermi and Bose fields.

Saddle-point approximation
From the form of the action in Eq. (7.6), it should be apparent that all of theresonant contributions are contained within the molecular action. In practice this gives
rise to the integral of a displaced Gaussian that can be easily evaluated (see Appendix
C). After integrating out the molecular degrees of freedom, we are left with the following
partition function:Z

Z = ZB (q 2 /4m + ν − 2µ)Da∗σ Daσ eSBCS0 .

(7.7)

Here ZB (q 2 /4m + ν − 2µ) is a Bose partition function describing the formation of boundmolecules and SBCS 0 is the BCS action with a potential that is now dependent on
both thermal frequencies and momentum. The interaction potential in the BCS action,
therefore, is modified in the presence of a Feshbach resonance in the following way:
U →U−

g2
.
q 2 /4m + ν − 2µ − iv

(7.8)

With the above partition function, Eq. (7.7), we may go on to calculate all thermodynamic properties of interest. Here we are primarily interested in calculating the criticaltemperature of the superfluid phase transition. This can be done by solving for a gap
and number equation and then self-consistently solving these two equations for boththe chemical potential µ and the critical temperature τ . The procedure is straightforward since the full resonant calculation has been reduced to the usual BCS calculation(however, we must now deal with the interaction given by Eq. (7.8)). Following Popov’s
derivation [86], we perform a Hubbard-Stratonovich transformation about an auxiliary

62
Bose field c(q) (see Appendix C) to derive the gap equation at the critical point

1 = −U +Xj

X tanh(β(k 2 /2m − µ)/2)
gj2
.
νj − 2µ2(k 2 /2m − µ)
k

(7.9)

A self-consistent equation for the total particle number can be found by expanding
the action to lowest order about the auxiliary field c(q) = c∗ (q) = 0. This is commonly referred to as the saddle-point approximation. The thermodynamic identity
N = −∂ ln Z/∂µ yields:
N = Nb + Nf = 2

X

1

k

eβ(k2 /4m+ν−2µ) − 1

+2

X

1

k

eβ(k2 /2m−µ) + 1.

(7.10)

Thus, our number equation counts all free fermions, Nf , plus an additional boson population Nb . Equations (7.9) and (7.10) provide a set of equations for determining thecritical temperature Tc and chemical potential µ. In conventional BCS theory, this levelof approximation proves reasonable for calculating Tc for weak, attractive interactions,but diverges as the interaction strength grows. The reason for this is that the mechanism which signals the phase transition within the weak coupling BCS limit is theformation and disassociation of Cooper pairs. As the coupling increases, the particles
tend to pair at higher and higher temperatures which means that the critical transition
is no longer signaled by the formation of Cooper pairs, but rather by a coherence acrossthe sample caused by condensation of pre-formed Cooper pairs. Since we are interested
in describing the resonant system at all detunings, the equations that we have derived so
far are insufficient because they do not account for this process. We should, therefore,
next focus on how to more accurately incorporate atom pairing into our model.

7.3

Beyond the saddle-point approximation
To account for fluctuations in the fermion field, we follow the method of Nozières

and Schmitt-Rink [82] in its functional form as put forth by Randeria et al. [83, 88].
This procedure will introduce a next order (Gaussian) correction to the saddle-point

63
calculation of the previous section. By expanding the action to second order in c(q),and then calculating the number equation in the same way as was done when deriving
Eq. (7.10), we introduce an additional population into the equation. The action becomesS(c(q), c∗ (q))BCS 0 ≈ SBCS 0 (0, 0) +X

|c(q)|2 χ(q),(7.11)q

where we have defined the auxiliary function χ(q) asÃχ(q) =

U − q2!

g2

X 1 − f (² q +k ) − f (² q −k )24m + ν − 2µ − iωk2

² q +k + ² q −k − iω2.

(7.12)

2

Here f (²k ) is the Fermi distribution function and ²k = k 2 /2m − µ. χ(q) is often referred to as the susceptibility (or the pair propagator) and results from the Matsubara
summation of the single particle, free Green’s functions G0 (see Appendix D):
1X
G0 (p, ν)G0 (k, ω − ν).
β ω(7.13)

Once again we make use of the thermodynamic identity N = −∂ ln Z/∂µ resulting in a
modified number equation
N= Nb + Nf + Np
= 2

X
k

(7.14)1eβ(k2 /4m+ν−2µ)

−1

+2

X
k

1
e

β(k2 /2m−µ)

+1

−

1X ∂
log[1 − χ(q, iω)].
β q,ω ∂µThis inclusion of the first order fluctuations introduces a new population which wewill refer to as atom pairs Np to distinguish them from the other bosonic population
Nb already accounted for at the saddle-point level. We are now able to solve for thefluctuation corrected critical temperature from a self-consistent solution of Eqs. (7.9)
and (7.14).
Due to the contact form of the couplings that we have chosen, however, we are
immediately plagued with problems of divergences in our equations. These may beremedied by a proper renormalization as oulined in Chapter 2.6. We may also accountfor mean-field shifts due to all 2-body scattering processes by adjusting the chemical

64
potential µ = µ̄ − hU nk i where nk is the Fermi distribution and hi denotes an averaging. This shift, however, is sufficiently small to neglect; inclusion has demonstrated
corrections of the order of 1% or less.

7.4

Bound states and asymptotes
The renormalization of the resonance theory forces us to take a closer look at thebound state physics of the system. In Fig. 7.1 we show the bound state energies for asingle resonance system with a positive background scattering length. The figure resultsfrom a coupled square well calculation of the bound state energies, as in Chapter 2.4,
and shows an avoided crossing between two molecular states. The upper state behavesto a fairly good approximation as Eb = (ma2eff )−1 , which is the molecular binding
energy regularly associated with a contact interaction [89]. The lower state, however,is offset from the detuning by an energy ∼ κ and goes linear with the detuning. We
find this similar behavior in the first term of Eq. (7.14). Taking the cutoff to infinity,
which is justified since this term does not diverge, the renormalized detuning approachesν → ν̄ −ḡ 2 /Ū . This produces a constant shift of ḡ 2 /Ū = κ between the detuning and themolecular binding energy. Keeping this term in the number equation would incorrectly
cause a transfer of the entire population into the wrong molecular state. In order toavoid this unwanted behavior, we set this term to zero, i.e., Nb = 0. In the case ofa negative background scattering length, we would not have encountered this problem
and only one molecular state would have appeared (see Fig. 7.2). We will show in thenext section that the pairing term fully accounts for the correct population of moleculesin this system with abg > 0.
However, before we present the full crossover solution for the case of 40 K, let
us look at the analytical solutions to Eqs. (7.9) and (7.14) in the strong (BEC) andweak (BCS) coupling regimes. We will first turn our attention to the weak coupling(BCS) limit. In this limit we would expect only free fermions to contribute to the

65
0
−1

−3
−4
1000

−5

500

aeff (a0)Eb (mK)

−2

−6

0
−500

−7
−8
−70

−1000

−60

−50

−40

−30

ν (G)

−50

0

ν (G)
−20

−10

50

0

Figure 7.1: Binding energies for 40 K resonance at positive background scattering lengthabg = 176a0 . A single resonance with a positive background scattering length produces
an effective scattering length aeff = abg (1 − κ/ν), as seen in the figure inset where weplot the effective scattering length vs. detuning (the dotted line is at 176a0 ). A positive
abg , which is larger than the range of the potential, implies that another bound state
is not far below threshold (dash-dotted line). In combination with the Feshbach state(dashed detuning line) this results in an avoided crossing and the molecular state ofinterest asymptotes quickly to the dash-dotted line.

0−1

Eb (mK)

−2

−3

−4

−5

−6
−70

−60

−50

−40

−30

ν (G)

−20

−10

0

Figure 7.2: Same as Fig. 7.1, but now for an artifical situation with abg < 0. A resonancesystem with a negative background scattering length has only one bound state relatively
close to threshold, which is shifted positive of the detuning. The next bound state in
the potential is too far away to be of any significant influence. The line styles are thesame as for Fig. 7.1.

66
population, so from equation (7.14) we find that the chemical potential is at the Fermi
surface (µ = EF ). With this information, we solve the gap equation for the criticaltemperature. The result is the usual exponential dependence on the effective scattering
lengthTc /TF ≈

8 γ−2
−π
e
exp(
),
π
2kF |aeff |(7.15)

where γ ∼ 0.5772 is the Euler-Mascheroni constant, kF is the Fermi wave number,and aeff < 0 is the effective scattering length produced by the Feshbach resonanceaeff = abg (1 − κ/ν̄).
The other limit we may consider is the strong coupling (BEC) limit. Whenthe argument of the tanh function in the gap equation, Eq. (7.9), becomes sufficiently
negative, it is a good approximation to use its asymptotic value of unity. What this
implies, physically, is that the fermion statistics are unimportant in determining thevalue of the gap. This allows us to solve the gap equation for the chemical potential
as a function of detuning. In the limit of large negative detuning we find that µ →−Eb /2, where Eb ≈ 1/ma2eff . Within this limit the entire population has been converted
to molecules and we can solve the number equation to get the noninteracting BEC
condensation temperature of Tc /TF ∼ 0.218.

7.5

Numerical results
To study the transition between the BEC and BCS regimes, we numerically solve

Eqs. (7.9) and (7.14) for 40 K. The single resonance curve is produced using a backgroundscattering length of 176a0 and κ = 7.68G at a density of 1014 cm−3 .
Figure 7.3 shows the critical temperature as a function of magnetic field detuning.
The crossover calculation clearly merges with the BEC result for large-positive detunings
and smoothly connects between positive and negative detunings, limiting to the Bose
condensation temperature of Tc /TF ∼ 0.218 for large negative detuning. This approach67
gives a maximum near zero detuning (Tc /TF ∼ 0.26).
Figure 7.4 shows the chemical potential as a function of detuning, beginning at
the Fermi energy for positive detuning and approaching half the bound state energy atlarge negative detuning µ → −Eb /2. Figure 7.5 shows the change in population as afunction of detuning. For large positive detuning, the system is composed solely of freefermions. As the detuning is decreased (i.e., from positive to negative) the contributionof the fermions begins to decrease until all the population is transferred into the atom
pair component at ν ∼ −0.5G. The chemical potential is then equal to −Eb /2 andwe may identify the atom pairs from that point on as the molecules. The superfluid
behavior then comes from the condensation of these molecules.
0.4

Tc / TF

0.3

0.2

0.1

0
−0.5

0

0.5

1

1.5

ν (G)

Figure 7.3: Critical temperature T /TF as a function of detuning ν in Gauss. The dashedline corresponds to the usual BCS solution, which limits to the full crossover theory
at large positive detuning. At negative detuning, Tc drops to the BEC condensation
temperature of Tc /TF ∼ 0.218.

68

1µ / EF

0.5

0

−0.5
−0.5

0

0.5

1

1.5

ν (G)

Figure 7.4: Chemical potential as a function of detuning ν in Gauss. For large negative
detuning 2µ approaches the bound state energy of the molecular state. At increasing
positive detuning, the chemical potential slowly approaches the Fermi energy.

1
0.9

Nf

0.8
0.7

N/N

T

0.6
0.5
0.4
0.3
0.2

Np

0.1
0
−0.5

0

0.5

ν (G)

1

1.5

Figure 7.5: The fraction of the total population as a function of detuning ν in Gauss.
The dot-dashed line corresponds to the pairing fraction Np and the solid to the freefermion fraction Nf .

Chapter 8
Pseudogapped Crossover Theory

8.1

Introduction

Figure 8.1: For weak coupling (left)the normal phase consists of interacting fermions.
As the coupling is increases (middle) preformed pairs may appear. At lower temperatures these pairs may become coherent resulting in a superfluid transition. For strong
couplings (right), the system is composed only of fermion pairs which may condense at
low enough temperature.

The resonant crossover work discussed in the previous chapter introduced theidea of preformed pairs which form above the critical temperature for superfluidity. One
would expect that the presence of these pairs would alter the nature of the gas fromthat of a simple degenerate Fermi gas. For instance, just as the presence of coherent,
Cooper pairing modifies the energy spectrum below Tc , leading to the effects witnessedin superfluidity, we might expect that incoherent pairs might also modify the energy
spectrum leading to other novel phenomena (see Fig. (8.1)). These above Tc effects
are often referred to as “pseudogap” effects and the region between superfluidity and a

70normal Fermi gas (the temperature noting this second phase is often referred to as T ∗ )is known as the pseudogap region.
Unfortunately, although beginning to account for the presence of preformed pairs,
the work of the previous section has several shortcomings [90, 91], most notably, it does
not account for the effects of the pseudogapped region above Tc . In fact, the form of thepair propagator given in Eq. (7.12) treats the pairs as composed of two free fermions
uncoupled from the medium (i.e., there is no self-energy term in the propagators). In
this section we wish to account for pseudogap effects by adapting a theory originallyproposed by Kadanoff and Martin [92] and extended to account for high-Tc effects byseveral authors [93, 94, 95]. An alternative approach to including these pseudogap
effects, in a non-resonant system, may be found in reference [96], although we feel thisapproach may overemphasize the importance of these contributions.
Although, at first it may appear quite different, the mode coupling approach weare about to present [97] is quite similar to the dynamical approach followed in Chapter
5. Now, however, higher-order correlations are to be accounted for. These correlations
will result in the process of pair formation and will lead to the pseudogapped behavior
we wish to incorporate.

8.2

Dynamical Green’s function method

We begin with a discussion of dynamical Green’s functions [98]. One method to solve
for the single-particle Green’s function, which encompasses all the information of asystem, is to derive an infinite hierarchy of dynamical equations and then to truncate,or decouple, those equations at a given order. To see how this may be done we first
define the single-particle Green’s function:DE

Gα (x1 t1 , x10 t10 ) = −i T(ψα (x1 , t1 )ψα† (x10 , t10 ))
³

D

(8.1)
E

D

E´

= −i Θ(t1 − t01 ) ψα (x1 , t1 )ψα† (x10 , t10 ) − Θ(t01 − t1 ) ψα† (x10 , t10 )ψα (x1 , t1 ),

71
where ψα† (x, t) (ψα (x, t)) creates(destroys) a particle of spin index α at position x attime t and Θ(t1 − t01 ) is the step function defined as:(0Θ(t − t ) =0, t < t0
1,

.

(8.2)

t > t0The inclusion of the step function in the second line of Eq. (8.1) defines the time ordering
T in the first. Higher-order Green’s functions may be defined in a similar way
D2
Gαβψα (x1 , t1 )ψβ (x2 , t2 )ψβ† (x20 , t20 )ψα† (x10 , t10 )
2 (x1 t1 , x2 t2 ; x10 t10 , x20 t20 ) = (−i)

Gαβγ
3 (x1 t1 , x2 t2 , x3 t3 ; x10 t10 , x20 t20 , x30 t30 ) = ...etc.

E

(8.3)

We may also define the noninteracting Green’s function by the useful relation
G−1
0 (x1 , t1 )G0 (x1 t1 , x10 t10 ) = δ(x1 t1 , x10 t10 ),(8.4)

where we have suppressed the spin indexes in this last formula.
We would now like to generate a set of equations from which we may solve forthe single-particle Green’s function describing the interacting system. In what follows,
we will focus on a two component system of interacting fermions and begin by defining
the single-channel Hamiltonian (h̄ = 1):
HA =

XZ

d3 x ψα† (x)(−α∇2)ψ (x) +2m αZ

d3 xd3 x0 ψ↑† (x)ψ↓† (x0 )U (x, x0 )ψ↓ (x0 )ψ↑ (x),(8.5)

where α runs over all spin components. An entire hierarchy of dynamical equations canbe generated starting with the equation of motion for the field operatori∂∇2ψα (xt) = −ψ (xt) +
∂t
2m α

Z

d3 x̄ U (x, x̄)ψβ† (x̄t)ψβ (x̄t)ψα (xt).

(8.6)

For example, the equation of motion for the first order Green’s function is
Ã

!

∇2
∂
+ 1 Gα (x1 t1 , x10 t10 ) = δ 3 (x1 − x10 )δ(t1 − t10 )i∂t1 2m
Z

− i

(8.7)

+
d3 x̄ U (x1 , x̄)Gαβ
2 (x1 t1 , x̄t1 ; x10 t10 , x̄t1 ).

72
Here we have introduced the argument t+ to denote an infinitesimal positive time above
t so that we may cast the equation in the above form. Likewise, we may solve for theequation of motion of the second order Green’s function
Ã

!

∂
∇2
3
α
i
+ 1 Gαβ2 (x1 t1 , x2 t2 ; x10 t10 , x20 t20 ) = δ (x1 − x10 )δ(t1 − t10 )G (x2 t2 , x20 t20 )∂t1 2mZ−idx̄3 U (x1 , x̄)Gαββ(x1 t1 , x2 t2 , x̄t1 ; x10 t10 , x20 t20 , x̄t+
3
1 ).

(8.8)

This procedure, of course, may be continued ad nauseam, but, in order to make use of
these equations, the hierarchy will need to be truncated at a given order. The method
chosen for performing this truncation sets the level of approximation at which we arecalculating the single-particle Green’s function. The next section discusses various ways
in which this may be done.

8.3

Truncation of the equations of motion
For the work that follows in this chapter, we will chose to truncate this hierarchyof equations at the order of the two-particle Green’s function G2 . We must, therefore,
decide how to properly account for the coupling of Eq. (8.8) to the three-particle Green’s
function G3 . For the moment, we will not consider anomalous correlations, but instead
will focus on the region T > Tc where we assume only conserving correlations are
relevant. Under this condition, and the assumption that we have an equal mixture ofspin states, we may keep only correlations which conserve both spin and number. A
lowest order approximation of this type would be to factorize G3 as:
Gαββ(1, 2, 1̄; 10 20 , 1̄+ ) ≈ G(1, 10 )G(2, 20 )G(1̄, 1̄+ ) − G(1, 10 )G(2, 1̄+ )G(1̄, 20 ).
3

(8.9)

We have introduced the 4-vector notation (1) = (x, t) for convenience in writing thevarious correlation functions. The above factorization is just the usual Hartree-Fock(HF) approximation encountered throughout this thesis and neglects the effects of all
but pairwise correlations. This is the level of approximation used in Chapter 5, however,

73
we would like to go beyond this approximation to begin to account for higher ordercorrelations between the particles.
A next order factorization would involve writing G3 in terms of factors of GG2and GGG. This is known as a cumulant expansion and may be written [92]:
αβ
0 +
+
0 0
Gαββ
(1, 2, 1̄; 10 , 20 , 1̄+ ) ≈ G(2, 20 )Gαβ32 (1, 1̄; 1 , 1̄ ) + G(1̄, 1̄ )G2 (1, 2; 1 , 2 ) (8.10)
αβ
0 0
0
0 +
− G(2, 1̄+ )Gαβ2 (1, 1̄; 1 , 2 ) − G(1̄, 2 )G2 (1, 2; 1 , 1̄ )− G(1, 10 )G(2, 20 )G(1̄, 1̄+ ) + G(1, 10 )G(2, 1̄+ )G(1̄, 20 ).
0 +In the above equation we have dropped the correlated component of G(1, 10 )Gββ
2 (2, 1̄; 2 , 1̄ )since only particles of opposite spin may collide. Notice that if we were to Wick expand
the various G2 functions we would indeed get the HF result of Eq. (8.9). Physically,
the form for the three-particle Green’s function G3 , which we have written, describes
the interaction of three particles taking into account that two of the particles may becorrelated during the interaction process and neglects all higher order correlations (seeFig. 8.2).

Figure 8.2: The factorization performed in Eq. (8.10) approximates the collisions of
three particles by the scattering of a particle off a correlated pair. The approximation
should remain valid, even for strong pair-correlations, so long as the pair size ξ remains
smaller than the mean inter-particle spacing L.

Substituting the form of G3 given by Eq. (8.10) into Eq. (8.8), we may write Eq. (8.8)

74
as:
Ã

!

∂
∇2
0 0
i
+ 1 Lαβ2 (1, 2; 1 , 2 ) = −i∂t1 2mZ

0 0
d1̄ U (1, 1̄)[G(1̄, 1̄+ )Lαβ2 (1, 2; 1 , 2 ))(8.11)

αβ
0 0
0
0 +
− G(2, 1̄+ )Gαβ2 (1, 1̄; 1 , 2 ) − G(1̄, 2 )L2 (1, 2; 1 , 1̄ )],where I have made use of Eqs. (8.4) and (8.7), and have introduced the notation
αβ
0 0
0 0 0
0
0
Lαβ
2 (1, 2; 1 , 2 ) = G2 (1, 2; 1 , 2 ) − G(1, 1 )G(2, 2 ).

(8.12)

This new function L2 accounts for the correlated part of the Green’s function G2 . Now,
by absorbing the shift given by the first term on the right hand side of Eq. (8.11) into
the definition of the free-particle Green’s function, we may simplify things even further:
αβ
0 0
G̃−1
0 (1)L2 (1, 2; 1 , 2 ) = i

Z

i

h

αβ
0 +
0 0
0
d1̄ U (1, 1̄) G(2, 1̄+ )Gαβ2 (1, 1̄; 1 , 2 ) + G(1̄, 2 )L2 (1, 2; 1 , 1̄ ) ,(8.13)
where

Z
−1
G̃−1
0 (1) = G0 (1) + i

d1̄ U (1, 1̄)G(1̄, 1̄+ ),(8.14)

and we have identified the free particle Green’s functionÃG−1
0 (1) =

!

∂
∇2
i
+ 1 .
∂t1 2m(8.15)

The first term on the right hand side of Eq. (8.13) will produce a series of ladder
diagrams. The second term, which represents all the multiple scattering events where
the interaction lines cross, we will drop. It is hoped that the ladder summation willaccount for the dominant processes involved. These statements might be made clearer
by representing Eq. (8.13) diagrammatically, as shown in Fig. 8.3.
The above approximation, as noted in the caption of Fig. 8.2, should remain valid
in the regime where the correlation length between pairs of atoms remains smaller thanthe mean inter-particle spacing. This is just the sort of term that we are interested insince it introduces the process of an atom scattering off a correlated pair, in analogy to75

Figure 8.3: (a) Diagrammatic representation of Eq. 8.13. The thin lines denote thefree Green’s functions G̃0 , thick lines are the full Green’s functions G, and dotted lines
represent the interaction U. (b) Diagrammatic representation of the resulting laddersummation.

the scattering of an atom and a molecule. The final result is an integral equation for
G2 which may be written as:
0 0
0
0
Gαβ
2 (1, 2; 1 , 2 ) = G(1, 1 )G(2, 2 ) + i

Z

0 0d1̄ d2̄ G̃0 (1, 1̄)U (1̄, 2̄)G(2, 2̄)Gαβ2 (1̄, 2̄; 1 , 2 )(8.16).
We have written the above equation in this form to illustrate the extension from theHartree-Fock approach (notice the lowest order term is just the HF result).

8.4

Extension to resonant Hamiltonian

Equation (8.16) was derived for the single-channel Hamiltonian of Eq. (8.5). We would
like to extend these results to account for a molecular channel so, as we have done
earlier, we include the resonant molecular processes in the hamiltonian:ÃZ
3d xϕ†m (x)

HB =
Z

+

·

!

∇2
−
+ ν ϕm (x)4md3 xd3 x0 ϕ†m ((8.17)¸x + x0)g(x − x0 )ψ↓ (x)ψ↑ (x) + h.c. .
2In analogy to the previous section, we first define the molecular Green’s function
D(x1 t1 , x10 t10 ) = −ihT (ϕm (x1 , t1 )ϕ†m (x10 , t10 ))i.

(8.18)

76
We could likewise define higher order molecular Green’s functions in much the same
way as we defined the higher order Fermi Green’s functions. For the moment, let us
only introduce this definition. What’s more, we must now write down what we may
refer to simply as “mixed” Green’s function
M (x1 t1 , x2 t2 ; x10 t10 ) = (−i)2 hT (ψ↑ (x1 , t1 )ψ↓ (x2 , t2 )ϕ†m (x10 , t10 ))i,

(8.19)

which accounts for correlations between atoms and molecules. We will quickly see thatthese two definitions are all we will need.
Using the full resonant Hamiltonian we may derive the equation of motion for themolecular field operator as:i∂−∇2ϕm (xt) =ϕ (xt) +
∂t
4m m

Z

d3 x̄g(x, x̄)ψ↓ (x̄)ψ↑ (x)(8.20)

From this equation, naturally, follows the equation for the lowest order molecular
Green’s function
!

Ã

∇2
∂
+ 1 D(x1 t1 , x10 t10 ) = δ 3 (x1 − x10 )δ(t1 − t10 )i∂t1 4m

(8.21)

Z− i

d3 x̄ g(x1 , x̄)M (x̄t1 , x̄t1 ; x10 t10 ).

From Eq. (8.21) we see that the coupling to higher order terms is through the mixed
function M . If we now look at the equation of motion for the mixed Green’s function,
once again making use of Eq. (8.20), we find:
Ã

!

∂
∇2
i
+ 1 M (x1 t1 , x1 t1 ; x10 t10 ) = −∂t1 4mZ+
d3 x̄g(x1 , x̄)Gαβ
2 (x1 t1 , x̄t1 ; x10 t10 , x̄t1 ).

(8.22)
Notice that this equation can be solved for M and plugged into our earlier Eq. (8.21)
to derive a relation for the full molecular Green’s function in terms of G2 . From thisresult we find that the modification to Eq. (8.7) is equivalent to the substitution
Vef f (x, x0 ) = U (x, x0 ) − D0 (x, x0 )g(x, x0 )2 ,(8.23)

77
where we have defined the free molecular Green’s function
D0−1 = i∂∇2+ 1.
∂t1 4m(8.24)

Therefore, as far as the fermionic part of our problem is concerned, the only modificationthat the resonant part of the Hamiltonian incurs is that the fermions now interact
through the effective potential Vef f . This is an equivalent result as to what we foundin Chapter 7 (see Eq. (7.8)) in our adaptation of the NSR approach to the resonant
molecular system.

8.5

Self-energy and definition of the t-matrix

It is much easier to work in momentum space since the integral equations found in theprevious section convert into algebraic equations. From Eqs. (8.7) and (8.16) we may
immediately write an equation for the full Green’s function
G = G̃0 + iG̃0

χVef fG̃0 G,
1+ξ

(8.25)

where we define the pair susceptibility to beχ = −iVef f G̃0 G.

(8.26)

This should be compared to the resonant NSR susceptibility of Eq. (7.12) which iscomposed of two free propagators. The inclusion of the fully dressed Green’s function
G in the susceptibility is one of the major extensions of this approach since the presenceof a self-energy couples the pairs to the medium.
In deriving Eq. (8.25) we have made the minor simplification of replacing one GG
pair, appearing in the numerator, with a G̃0 G pair. This is equivalent to replacing theupper, left-most, full Green’s function arm, in the ladder diagram of Fig. 8.3, with abare Green’s function G̃0 . This results in the irreducible self-energy:−1=iΣ = G̃−1
0 −G

χVef fG̃0 .
1+ξ

(8.27)

78
To further simplify the equations, we make an approximation in how we incorporate
the Hartree shift of the free Green’s function (as given in Eq. 8.14), that is, we put itback into the self-energy at lowest order and replace G̃0 with G0 . This results in thefollowing equation for the self-energy.
−1Σ = G−1=
0 −G−iVef f G0
≡ tG0 .
1+χ(8.28)

This last equation gives a general definition for the t-matrix t. The t-matrix will prove
to be a useful quantity since it may be used to define the critical point of superfluidity.
This statement, known as the Thouless criterion, states that the onset of superfluidity
occurs as an instability in the t-matrix or:
t−1 = 0.

(8.29)

We will often refer back to this condition in the following section.

8.6

Foundations of a pseudogapped resonant crossover theory
Before moving to a discussion on the application of sections 8.3-8.5 to the super-

fluid regime, T < Tc , let us first summarize the major results which we have found.
To this end, we begin with the Hamiltonian describing the atomic resonance system
written in momentum space:H − µN =+k,σXq,k,k0+X³

X²k a†k,σ ak,σ +

X

†
(²m
q + ν)bq bqq

U (k, k0 )a†q/2+k,↑ a†q/2−k,↓ aq/2−k0 ,↓ aq/2+k0 ,↑´g(k)b†q aq/2−k,↓ aq/2+k,↑ + h.c. .

(8.30)

q,k

The above Hamiltonian, which should be familiar to the reader by now, contains both
fermions ak,σ and (spinless) bosonic molecules bq (the sum in σ runs over both spinstates {↑, ↓}). The free dispersion relations for fermions and bosons are given by:
2²k = k 2 /2m − µ and ²m
q = q /2M − 2µ,(8.31)

79
respectively, where we assume an equal proportion of spins µ↑ = µ↓ and the mass of theboson is just twice that of the a single fermion M = 2m. Once again, ν is the detuningof the resonance state.
Until this point we have solely made use of contact potentials within all our manybody theories. However, for the work that follows, we will use a set of Gaussian, separable potentials. For instance, we set the direct s-wave interaction U (k, k0 ) = U ϕk ϕk0 and
the Feshbach coupling g(k) = gϕk , where the function ϕ2k = exp{−(k/Kc )2 } provides
a momentum cutoff Kc (note, we have set h̄ = kB = 1). This choice of potentials,
although at the same level of accuracy as the contact potential model, proves moreamenable to our later numerical calculations and may be renormalized by the relations
given in Appendix A.3.
We have recognized the relevant propagators for the resonant system. These
are the propagator for the fermion pairs t(Q), for the molecular bosons D(Q), andfor the individual fermions G(Q). As we have seen, the interaction effects induced by
the molecular field are equivalent to an effective pairing interaction. For the separable
Gaussian potentials we have introduced, this may be written asg̃ef f (Q, K, K 0 ) = gef f (Q)ϕk ϕk0 ,(8.32)

gef f (Q) ≡ U + g 2 D0 (Q),(8.33)

withwhere D0 (Q) ≡ 1/[iΩn − q 2 /2M − ν + 2µ] is the non-interacting molecular boson propagator (see Eq. (8.23)).
We have also found a particular form for the fermion pair-susceptibility, or thepair propagator χ(Q):χ(Q) =X
K

G(K)G0 (Q − K)ϕ2k−q/2 ,(8.34)

80
provided that the fermion self-energy appearing in the dressed propagator G isΣ(K) =

X

t(Q)G0 (Q − K)ϕ2k−q/2 .

(8.35)Q

Here and throughout, we take the convention

P

K ≡T

P
ωn

P

k , where K is a 4-vectorof wavenumber k and Matsubara frequency ωn . The form of Eq. (8.34) will allow us toaccount for non-condensed atom pairs and, in contrast to that found by NSR, accounts
for a dressed fermion pair.

8.7

Extension below Tc and the appearance of an order parameter
So far we have only considered the region T > Tc and would now like to extend our

results to temperatures below the critical temperature, T < Tc . To properly incorporate
the onset of the phase transition one could imagine going back through the work ofsections 8.2-8.5 to incorporate anomalous contributions into the various propagator
expansions. In fact, this is just what Kadanoff and Martin did [92] at the level of the
Hartree-Fock approximation (for the nonresonant case), resulting in the conventional
equations of BCS theory. The resulting t-matrix they found was the following:t(Q) = −

∆2
δ(Q).
T

(8.36)The singular nature of such a function leads us to think that it is this contribution to thet-matrix that dominates the onset of long-range order and that, perhaps, higher orderanomalous contributions are negligible. We propose that the full scattering t-matrix
should be composed of the BCS pairing interaction and an additional interaction whichleads to non-condensed Cooper pairs:t = tsc + tpg ,(8.37)

where tsc is the superfluid condensate component and tpg is the pseudogapped contribution. For the resonant system these may be defined as
tsc (Q) = −

˜2
∆
sc
δ(Q),
T

(8.38)

81
tpg (Q) =gef f (Q),1 + gef f (Q)χ(Q)

Q 6= 0.

(8.39)

Equation (8.38) is just a restatement of Eq. (8.36) for the resonant system and Eq. (8.39)
comes from our T > Tc result of the pseudogap process, Eq. (8.28). The effective,
resonant interaction gef f (Q) is defined in Eq. (8.33) and the susceptibility χ(Q) is given
by Eq. (8.34).
We identify the order parameter previously encountered in Chapter 5:
˜ sc = ∆sc − gφm ,∆where φm = hbq=0 i is the condensed molecular component and ∆sc = −U

(8.40)
Pk ha−k↓ ak↑ iϕkis the standard BCS gap. The order parameter is a linear combination of both a pairing
field and the closed-channel condensed molecules. It is, perhaps, enlightening to contrast this choice of an order parameter with the classic work of Fano on resonances [99].
For a potential resonance dominated by a bound state Fano chose the following ansatz
for the wavefunction of eigenvalue E:Z

ΨE = aϕ +dE 0 bE 0 ψE 0 .

(8.41)

The above form is composed of a linear combination of a bound state ϕ and a range
0 (a and b being constants). Fano found that this choice ofof continuum states ψEwavefunction could be used to exactly solve the resonant problem. Although, strictly
speaking, Eq. (8.41) is derived from a 2-body picture it shows a striking resemblance tothe order parameter of Eq. (8.40).
It is worthwhile to write the order parameter, Eq. (8.40), in a form more closely
resembling that of BCS theory. In Chapter 5.4 we found that the two components of
the order parameter, φ and ∆sc , are not independent, but are connected by the relation
φm =

g∆sc
.
(ν − 2µ)U(8.42)

This implies that˜ sc = −gef f (0)∆X

ha−k↓ ak↑ iϕk ,k(8.43)

82
as one might expect.

8.8

Self-consistent equations for T < Tc
The form of the order parameter, Eq. (8.40), implies that the molecular bosons

condense at the same temperature as a pairing field develops. Another way to state thiswould be:D−1 (0) = t−1 (0) = 0,

T ≤ Tc ,(8.44)

where D is the full bosonic Green’s function. This statement simply relates the Thoulesscriterion to the criterion for Bose-Einstein condensation. Equation (8.44) leads to theimportant result:−1
t−1 (0) = gef
f (0) + χ(0) = 0,

T ≤ Tc .

(8.45)

Equations (8.44) and (8.45) demand the following form for the bosonic self-energy:ΣB (Q) ≡ −g 2 χ(Q)/[1 + U χ(Q)],(8.46)

which remains valid for T ≤ Tc . Therefore, the full bosonic propagator is given by
D(Q) ≡

1.
iΩn − Eq0 − ν + 2µ − ΣB (Q)(8.47)

Equation (8.45) implies that tpg is extremely sharp for all temperatures T ≤ Tc .
We may make use of the sharpness of the t-matrix to approximate Eq. (8.35) to yield a
BCS-like self-energy
Σ(K) ≈ −G0 (−K)∆2 ϕ2k ,(8.48)

with the following definition for the full gap:
˜ 2sc + ∆2pg ,∆2 ≡ ∆(8.49)

and where we have defined the pseudogap parameter ∆pg∆2pg ≡ −X
Q6=0

tpg (Q).

(8.50)

83
For a more detailed discussion of the validity of this approximation see Maly et al. [100].
With these results, Eq. (8.45) can now be written as−1
gef
f (0) +

X 1 − 2f (Ek )2Ekk

ϕ2k = 0,T ≤ Tc ,(8.51)q

where Ek =²2k + ∆2 ϕ2k . Equation (8.51) is identical in form to the conventional BCS

˜ sc ,
equation, but the full excitation gap ∆, as distinguished from the order parameter ∆now appears in the dispersion Ek .
To close these equations, we must now calculate the total particle number fromthe propagators involved. The number of molecular bosons is given directly by nb =−P

Q D(Q). The number of fermions is

nf =

X·

¸

²k
²k
+ 2 f (Ek ) ,
Ek
Ek

1−

k

as follows from the condition nf = 2

P

K G(K).

(8.52)The total number N of particles is,therefore, given by
nf + 2nb + 2n0b = N ,(8.53)

where n0b = φ2m is the number of molecular bosons in the condensate. Equations (8.50),(8.51), and (8.53) form a closed set of equations for our resonance system. It should
be noted that at T = 0 the pseudogap parameter, Eq. (8.50), vanishes (∆pg → 0)
˜ sc (0) = ∆(0) and we reproduce BCS theory. In fact, one of the defining
so that ∆characteristics of this theory is that, at zero temperature, we reproduce the Leggett
groundstate suggested in [81]. This would say that all the pseudogap behavior resultsfrom finite-temperature effects and at T = 0 the system is accurately described byconventional BCS theory.
An essential distinction between this and previous crossover studies based on the
NSR approach [84, 101], below Tc , is that the modification to the energy spectrum is
˜ sc . This distinction, also
not fully accounted for by the superconducting gap, ∆ 6= ∆absent in strict BCS theory, is to be associated with non-condensed bosonic excitations

84
of the superfluid. The spectrum of these excitations are modified due to the presence ofother excitations. That is, the particles acquire a self-energy from other pairs present
within the medium, which is fed back into the propagator for the pairs.

8.9

Numerical solution of the pseudogapped theory
We now numerically solve Eqs. (8.50), (8.51), and (8.53) for a homogeneous sys-tem. Although most atomic gas experiments are confined to a harmonic trapping potential, we don’t expect the inhomogeneity of the trap to qualitatively alter the results wepresent here. To aid us in this task, we make use of the divergence of the t-matrix, expressed by Eq. (8.45). This allows us to Taylor expand the quantity χ(Q) in Eqs. (8.53)
and (8.50) to first order in Ω and q 2 :χ(Q) ≈ χ(0) + a0 (iΩn − B0 q 2 ).

(8.54)

We may expand gef f (Q) and ΣB (Q) in a similar fashion. These expansions considerably
simplify the analysis (see Appendix E for a discussion of the expansion coefficients a0
and B0 and for a detailed discussion of implementing the pseudogap equations).
0.3

0.25

T/Tf

0.2

0.15

0.1

0.05

0
−1

−0.5

0

0.5

ν (Gauss)

1

1.5

Figure 8.4: Comparison between the NSR approach (dashed line) and the pseudogap
theory (solid line) for the critical temperature Tc /TF of 40 K as a function of detuning
ν. All parameters are chosen to be the same as those of Fig. 7.3.

85
We begin by comparing the results of the pseudogap theory to our earlier work of
Chapter 7 where we analyzed the crossover for the case of 40 K. Figure 8.4 shows a directcomparison between the critical temperature prediction of the resonant NSR approachand the pseudogap equations we have just presented. With the inclusion of higher ordercorrections we still retain a pronounced maximum in Tc near the resonance at about
0.26TF .
To aid in illustrating the behavior of the resonant pseudogap equations we will
leave the case of 40 K aside for a moment and instead explore a model atom more
conducive to our discussion. We make this switch due to the positive value of the
background scattering length in 40 K. This implies the presence of a bound state in the
open channel which leads to an avoided crossing when the resonant molecular state islowered well below threshold. Since we are not interested in this process, but would
rather discuss the asymptotic case of a fully populated molecular fraction, we study amodel atom with negative background scattering length. We choose ḡ/EF = −42 andŪ /EF = −3 as indicative of currently trapped atomic Fermi gases. These parameters
are, in fact, the same as those of 40 K, but we have flipped the sign of the backgroundscattering length.
First we calculate Tc as a function of ν̄ which is plotted in Fig. 8.5. For ν̄ → −∞,Tc approaches the ideal BEC limit. As ν̄ → +∞, the bosonic populations become
irrelevant and the asymptote of the curve is dictated by the behavior of the open channel
fermions in the presence of the background interaction Ū . It should be noted that inthis work we have chosen a small Ū deliberately so that ν̄ → ∞ limits to BCS theory.
The figure indicates that, while there are differences in the shape of the Tc vs. ν̄ curve,
the overall magnitudes are not so different from those found within our earlier NSRbased approach. More generally, we could contemplate a moderate value of Ū where
the asymptotic superfluid state will have a pseudogap. This might be relevant in thecase of an atom such as 6 Li where the open channel contains a bound state close to86
threshold leading to a large negative background scattering length, abg ∼ −2500a0 .

Figure 8.5: Tc vs. detuning ν with a pronounced maximum near threshold. The upper
right inset plots the pseudogap at Tc vs. ν̄ with a maximum at large negative detuning.
The lower left inset plots the molecular contribution to the condensate weight and thefermionic chemical potential.

Indicated as an inset in the figure is the behavior of ∆(Tc ) as a function of ν̄. Asν̄ approaches the resonance from +∞, Tc first increases following the BCS trend, whichreflects an increasingly attractive pairing interaction. Likewise, the pseudogap, whichstarts at zero, begins to grow. At the point where the pseudogap becomes appreciable,
the rise in Tc slows down. Eventually Tc decreases as fewer fermions are available topair. An interesting feature of the pseudogap plot is that a maximum arises at large
negative detuning. This behavior has not been witnessed in the non-resonant theory.
In the inset to Fig. 8.6, we plot the temperature dependence of the normalized
excitation gap ∆(T )/∆(0) for three values of the detuning ν̄/EF = −200, −5, +200.
The middle value roughly corresponds to the maximum in Tc , where pseudogap effects
should be the most apparent. The first and last are illustrative of the BEC and BCS
limits, respectively. We, thus, refer below to these three values as the BEC, P G, and˜ sc , is not plotted here, but for all three cases it
BCS cases. The order parameter, ∆is rather close to the solid line in the inset. That ∆(T ) is relatively constant with T

87∆(T)/∆(0)

1
0.8
0.6

1
BEC
PG
BCS

nk

0
0

0.4

T/Tc

1

BEC, T = 0, T = Tc
PG, T = 0
PG, T = Tc
BCS, T = 0
BCS, T = Tc

0.2
0

0

1

2

3

k/kF
Figure 8.6: Fermionic momentum distribution function at T = 0 and T = Tc for thethree regimes. P G corresponds to maximal Tc . Inset plots the T dependent excitation
gap below Tc .

through the superfluid transition is to be expected in the presence of preformed pairs,
as we see for ν̄/EF = −5 and −200.
It should be stressed that the critical temperature Tc for the phase transition tosuperfluidity is only apparent in ∆(T ) for the BCS case. To stress this point, in the
body of Fig. 8.6, we plot the fermionic momentum distribution function nk , which isthe summand in Eq. (8.52):
·

¸

²k
²k
+ 2 f (Ek ) ,nk = 1 −Ek
Ek(8.55)

at T = 0 and T = Tc . From the figure, it is clear that there is very little temperaturedependence between T = 0 and T = Tc . This is a clear sign that the momentum
distribution is not a good indicator of phase coherent pairing [102]. For the P G case,where the critical temperature is a maximum, this results in the near temperature
independence of ∆(T ). In the far BEC limit the excitation gap does not vary at all
through Tc . In the BCS regime, ∆(T ) is so small as to be barely perceptible in the figure.
In order to address the question of whether a clear signature can be seen in the particle
density distribution [49, 60] these results must be generalized to a trapped system. A
simple approach would be to treat the system under a local density approximation as

88
was done in Chapter 5.
This pseudogap based phenomenology is well documented in high-Tc superconductors. For these materials, penetration depth data provides a direct probe of thetransition to superconductivity. Interestingly enough, densities of state measurementsin the Cuprates also show some indications of the establishment of order throughout
the sample. To see how phase coherence enters in the context of atomic physics, we
relax the approximation of Eq. (8.48) by noting that incoherent, or finite momentum,
pairs (pg) are distinguishable from coherent, or zero momentum, pairs (sc) through lifetime effects. The incoherent nature of the non-condensed pairs is affected by scattering
events and therefore incorporates a finite broadening into the self-energy Σpg . With thisintuition, the self-energy can be replaced by the model self-energy [95]:Σ(ω, k) ≈

˜2
∆2pg
∆
sc
+
,
ω + ²k + iγ ω + ²k

(8.56)

where γ is the finite broadening. We will not directly calculate γ, but will instead treat
it as a phenomenological parameter which is independent of T . To calculate the densityof states we remember thatG−1 = G−1
0 − Σ,(8.57)

relates the self-energy Σ (in this case we use the form of Eq. (8.56)) to the full propagator
G. The density of states, defined by [98]:N (ω) = −2

X

Im G(ω + i0, k),(8.58)kcan be directly evaluated given Eq. (8.56).
Figures 8.7(a)-8.7(c) show the results of evaluating Eq. (8.58), and correspondto the BEC, P G, and BCS cases displayed in the previous figures. The temperatures
shown are taken just above Tc , T = 0.75Tc , and T = 0.5Tc . The BCS case is indicatedby an abrupt transition at the critical temperature Tc . The density of states in the
BEC regime shows very little temperature dependence throughout, since the fermions

89
are tightly bound into molecules at all temperatures. The P G case, which is taken nearresonance at the point where Tc is a maximum, is the most interesting. A signatureof superfluid order, seen by the presence of sharp coherence features, is not clearly
perceptible until the temperature is well below the critical temperature. For the present
case these features do not appear until a temperature of about T = 0.5Tc is reached.
However, the appearance of a pseudogap is clearly signalled by a suppression in theavailability of low energy excitations displayed within the density of states plot. Similar
observations to these are made in the high-Tc cuprates.
0.01

2

(a)
DOS

BEC

BCS

(c)

0

0.05

1

0
-20
0.3

-10

(b)

0

10

20

DOS

PG

0

0

-0.05

ω/EF

T =Tc
T =0.75 Tc
T =0.5Tc
-2

-1

0

ω/EF

1

2

Figure 8.7: Fermionic density of states vs. energy for the three regimes at three indicatedtemperatures. Note the difference in the scales.

8.10

Conclusions and experimental implications
These plots have important implications for the interpretation of several predictedsignatures of superfluidity. One such proposal is based on laser probing of “atomic
Cooper pairs” where it has been argued that there is a conceptual analogy between
normal metal-superconductor tunnelling (which measures N (ω)), and the resonant scat-

90
tering of laser light [62]. The present work introduces a caution in interpreting future
trapped atomic gas experiments. Because of the presence of a pseudogap, the signaturesof superfluid onset are not as simple as in BCS or the related Bogoliubov-de Gennes
theory. In general, one has to distinguish between the superfluid order parameter andthe fermionic excitation gap. Nevertheless, superfluid coherence appears to be visible
as fine structure effects in the fermionic density of states. While several experimentally
accessible signatures, such as those displayed in the quantities N (ω) and nk , do notprovide a clear indication of superfluidity, they do establish the nature of the pairing
regime as that of a pseudogapped superfluid.

Chapter 9
Fractional Quantum Hall Effect

9.1

The Hall effect
The Hall effect, discovered in 1879, occurs when a current carrying conducting

plate, placed in the x-y plane, is subjected to a magnetic field Bz perpendicular to theplane. The magnetic field causes the electrons, of current density jx moving at a velocity
vx , to experience a Lorentz force vx Bz which results in a voltage drop across the sample(see Fig. 9.1). The Hall conductance, which is a measure of the conductance across the
sample is
σxy =

nevx
ne
jx
=
=
,
Ey
vx B
B

(9.1)

where n is the number density. Therefore, classically, we would expect the Hall conductance to vary linearly with ne/B.

9.2

The integer quantum Hall effect
In the early 1970’s, due to progress in the material sciences, new semiconductors

were being produced which could be layered to form very flat, thin interfaces. By coolingthese materials and applying an electric field perpendicular to the interface surface theelectrons would be forced to situate themselves deep within quantum wells. The result
was to quantize the motion of the electrons perpendicular to the interface. This severely
limited the motion of the electrons constraining them to move, essentially, within only
two-dimensions.

92

Figure 9.1: The Hall effect occurs when an electrical current is subjected to a transverse
magnetic field. The Hall voltage (perpendicular to the current) drops due to the Lorentz
force felt by the electrons.

In 1980, Klaus von Klitzing [103] found that at temperatures of only a few Kelvinand high magnetic field (3-10 Tesla), the Hall resistance did not vary linearly withthe field. Instead, he found that it varied in a stepwise fashion (see Fig. 9.2). It wasalso found that where the Hall resistance was flat, the longitudinal resistance disappeared. This dissipation free transport looked very similar to superconductivity. The
field at which the plateaus appeared, or where the longitudinal resistance vanished,quite surprisingly, was independent of the material, temperature, or other variables ofthe experiment, but only depended on a combination of fundamental constants -h̄/e2 .
The quantization of resistivity seen in these early experiments came as a grand surprise
and would lead to a new international standard of resistivity, the Klitzing, defined as
the Hall resistance measured at the fourth step.
The Integer Quantum Hall Effect (IQHE), as it later came to be known, may simply be explained in the context of non-interacting quantum mechanics. The Hamiltonian
for a particle subjected to a magnetic field perpendicular to its direction of motion canbe written down as:
H=NX(pi − eA(xi ))2
i

2m

,

(9.2)

93

Figure 9.2: The top figure shows the stepwise behavior of the transverse resistivity
and the bottom the longitudinal resistance, both as a function of magnetic field. Note,the plateaus(top) coincide with the dissipationless behavior(bottom) at each value of ν.
(reprinted from reference [104])94
with the vector potential A(x) chosen in the following gauge
1
A(xi ) = B(yi , −xi ),2

B = B ẑ.

(9.3)

The above Hamiltonian results in energy eigenvalues En,ky = (n+1/2)h̄ω for each “Landau level” (LL) which depend on the cyclotron frequency ω = eB/mc (see Appendix F).
Since the LLs do not depend on the quantum number ky , they are highly degenerate.
This degeneracy, defined as the number of states per unit area, may be quantified by
the relation ρB = eB/hc. This relation leads to a useful quantity for describing the
IQHE, known as the filling factor:ν = ρ/ρB ,(9.4)

which is the number of electrons per Landau level and acts as a measure of the appliedmagnetic field.
In Fig. 9.2, the plateaus occur at each integer value of the filling factor ν. Whatoccurs is that, as each of the degenerate states of a LL is filled, the conductivity decreases, or the resistivity increases, because fewer and fewer states remain unoccupied
within that energy level. Once the LL level is completely full, a gap exists requiring afinite jump in energy to reach the next set of degenerate energy states (i.e., the next LL).
However, due to impurities in a sample, localized states exist which may be filled, but
do not contribute to the conductivity. This mechanism explains the stepwise behaviorof the resistivity.

9.3

Fractional quantum Hall effectBy 1982, semiconductor technology had greatly advanced and it became possible

to produce interfaces of much higher quality than where available only a few years
before. That same year, Horst Stormer and Dan Tsui [105] repeated Klitzing’s earlier
experiments with much cleaner samples and higher magnetic fields. What they found95
was the same stepwise behavior as seen previously, but to everyone’s surprise, steps also
appeared at fractional filling factors ν = 1/3, 1/5, 2/5 . . . (see Fig. 9.3).
Since these fractional values occurred in the middle of the various highly degenerate Landau levels, where no gap was apparent, these observations could not be explainedby the non-interacting quantum mechanical theory.
It would quickly be realized that these observations were a result of the manybody effects of interacting electrons. We should once again consider the Hamiltonian
for a population of electrons in the presence of a transverse magnetic field, but now
include the effects of interactions:H=NX(pi − eA(xi ))2

2m

i

+

N
X

e2
|xi − xj |
i<j

(9.5)

For the IQHE the potential energy e2 /r̄, where r̄ is the average electron spacing,was small compared to the cyclotron energy ω = eB/mc and could be neglected. Now,
however, the kinetic term may be neglected so that e2 /r̄ is no longer a small term andwe are left with a problem which is intrinsically one of strong correlations.

9.4

The Laughlin variational wavefunction
Strongly correlated systems are notoriously difficult to understand, but in 1983,

Robert Laughlin [106] proposed his now celebrated ansatz for a variational wavefunction
which contained no free parameters:
ψL =Y(zi − zj )2p+1 e−

P |zi |2
i 4l2

,

(9.6)

i<j

where the position of the ith particle is given by the complex coordinate zi = xi +iyi andl2 = hc/eB is the magnetic length . The Laughlin wavefunction, as it would come to be
known, gave an accurate description of all filling fractions ν = 1/(2p + 1) and was shown
to almost exactly match numerical ground state wavefunctions found for small FQHE
systems. There are several key points that should be mentioned about this choice of
wavefunction:

96

Figure 9.3: The figure shows the stepwise behavior of the transverse resistivity, superimposed with the longitudinal resistance, as a function of magnetic field. The same
behavior as in figure 9.2 is seen except now at fractions of ν. (reprinted from reference[104])

97• The wavefunction is extremely efficient at minimizing the Coulomb energy.
Since the wavefunction has multiple zeros, occurring whenever two particles
approach, the wavefunction results in a very uniform distribution. This distribution is much more efficient at lowering the Coulomb energy than, for example,
a random distribution.
• The Laughlin wavefunction is the lowest energy state and is no longer degenerate, as one might naively expect. The state, therefore, is incompressible andwill generate a finite energy gap for all excitations.
• The quasi-particles which result above the ground state carry with them afractional charge.
One of the main reasons the Laughlin wavefunction is so celebrated is because it
is a true, highly-correlated wavefunction. In contrast, the majority of many-body wavefunctions are simply Slater-determinants of single-particle wavefunctions. For example,
the wavefunction for one filled Landau level can be written as proportional to:
¯
¯
¯ 1
¯
¯
¯
¯ z
¯
1
¯
¯
ψ(z1 , z2 , . . . zN ) ∼ ¯¯ z12
¯
¯
..
¯
.
¯
¯
¯
¯ z N −1
¯ 1

1

···

z2

···

z22
..
.

···
..
.

z2N −2 · · ·

¯
¯
1 ¯¯
¯
¯
zN ¯¯
¯
¯
2
zN ¯¯ ,
¯
.. ¯¯
. ¯
¯
¯
N −1 ¯zN¯(9.7)

where we have not bothered to write out the normalization or the usual decaying exponential factors. The Laughlin wavefunction, however, cannot be written in terms ofa Slater determinant and is, therefore, something much more complicated than is often
encountered. In order to explain the remaining filling fractions, such as 2/5, 3/7... it
was necessary to build off the work of Laughlin to create what has come to be known
as the hierarchy picture.

98

9.5

Composite fermions and Chern-Simons theory
In 1989, Jainendra Jain [107] identified a suitable set of quasi-particles for the

FQHE system, calling them “composite fermions”. In terms of these quasi-particles, the
FQHE behaves exactly as does the IQHE. When each of the composite fermion Landaulevels is filled there exists an energy gap separating it from the next composite Landaulevel. This scheme was able to explain both the Laughlin fractions 1/3, 1/5 . . . as wellas many of the remaining fractions 2/5, 3/7 . . .. The idea of composite particles may beclarified by the illustrations of Figs. 9.4, 9.5, and 9.6.
A system of composite fermions may seem strange at first thought since the quasiparticles are composed of both particles (the electrons) and fields (the flux quanta). A
field theory of this sort is known as a gauge field theory since it will require a gaugetransformation to move from particles to quasi-particles. In the context of the FQHE,
this transformation is known as the Chern-Simons theory [108]. We will discuss the
Chern-Simons theory more in the next chapter where we apply it to describe a resonantgas of Bosons.

Figure 9.4: The IQHE may be depicted by the above illustration. For each particle(balls) there exists one associated flux quanta (arrows) and the particles are weaklyinteracting. This results in a filling fraction of ν = 1.

99

Figure 9.5: At a filling fraction of ν = 1/3, for example, there exist three flux quanta
for each particle, but the particles are now strongly interacting.

Figure 9.6: By associating two flux quanta with each particle, the system at ν = 1/3
reduces to a weakly interacting system of composite particles (one particle and two flux
quanta) at ν = 1. We are left with the IQHE of figure 9.4, but in terms of composite
fermions.

Chapter 10
Resonant Manipulation of the Fractional Quantum Hall Effect10.1

Resonant theory of a rapidly rotating Bose gas
There is a considerable amount of interest in the behavior of rotating Bose gases

confined to an effective two-dimensional space. Beginning with the generation of a single
vortex [109], experiments have progressively created increasingly dense vortex arrays[110, 111]. By driving the stirring frequency ever closer to the critical trap frequency,
experiments are now creating scenarios where there exist a similar number of vortices
as particles. One of the major goals is to push these systems well into the regime where
the system is thought to display strong correlation effects.
Recently, Wilkin et al. [112] have shown that there exists a direct mapping between the two-dimensional Hamiltonian of a rotating Bose gas and of electrons subjected
to a transverse magnetic field. The analog of the magnetic field is realized by the angular rotation and the Coulomb interaction is replaced by the two-particle scattering. The
group later went on to show, through exact diagonalization studies of a small number ofparticles, that one could generate a Laughlin state (as well as many other novel FQHE
type states) when there exists on the order of one vortex per particle [113]. This was
also confirmed by Paredes et al. [114]. Unfortunately, if a state such as the Laughlin
state were to be created in the laboratory, it would remain a major challenge to resolve
it from the next excited state due to the extreme fineness of the gap. The size of theenergy gap, however, is directly related to the strength of interatomic interactions.

101
A natural strategy for increasing the gap, and thus making the FQHE states
more accessible to experiment, would be to exploit a mechanism, such as a Feshbachresonance, to enhance the interaction strength. To account for the full effects of theresonance, however, we cannot simply scale the mean-field energy, but must incorporate
the entire resonant structure into the model. We must, therefore, include the process ofmolecular formation to the description of the rotating system.
By introducing a bound state, we not only modify the relative interaction strength,
we also introduce a physical mechanism for generating pair correlations between particles (see Fig. 10.1). In the context of two-dimensional condensed matter systems, sucha mechanism, which may arise from the long range nature of the Coulomb interaction[115], can have a dramatic effect on the properties of the ground state. In a resonant
atomic gas, although the range of the potential is quite small, the effective scattering
length may well exceed the inter-particle spacing so that even a small overlap between
particles may lead to significant interaction effects. The closed channel bound state responsible for this enhancement, in effect, matches pairs of atoms within the gas resulting
in a large increase in correlations.
A natural starting point for studying resonant effects in a rotating Bose gas is
to look at the effect of the resonant interactions on the ground state wavefunction. By
focusing on the most fundamental FQHE state, the Laughlin state, we will show thatas the detuning approaches the Feshbach resonance, the Laughlin state transforms intoa unique, strongly correlated state [116].
We begin by writing down an effective Hamiltonian, in second quantized form, fora resonant gas of Bosons of mass m rotating in two dimensions with stirring frequencyΩ approaching the trapping frequency ω (i.e., ω − Ω → 0+ ):·Z
2d x ψ̂a† (x)

Ĥ =
+

1
2

Z

Z

¸

−1
(∇ − iA(x))2 ψ̂a (x)2md2 x0 d2 x ψ̂a† (x)ψ̂a† (x0 )U (x, x0 )ψ̂a (x0 )ψ̂a (x)(10.1)

102
·

Z

¸

−1
(∇ − 2iA(x))2 + ν ψ̂m (x)
4m
·
¸
Z
Z
0
1
† x+x
+
d2 x0 d2 x ψ̂m()g(x, x0 )ψ̂a (x)ψ̂a (x0 ) + H.c. .
2
2

+

†
d2 x ψ̂m(x)† (x), ψ̂
Here ψ̂a,ma,m (x) are the creation and destruction operators for atoms and molecules

which satisfy the commutation relations [ψ̂1 (x), ψ̂2† (x0 )] = δ (3) (x, x0 )δ1,2 , where 1, 2 ∈{a, m}. We define the two-dimensional vector potential A(x) = (mωy, −mωx), U (x, x0 )is the two-particle background scattering potential, g(x, x0 ) is the resonant coupling between the open and closed channel potentials, and ν is the detuning of the open channel
continuum from the level of the bound state in the closed channel.
From the work of Wilkin et al. [112], if we were to neglect resonant effects, wewould expect the many-body ground state to be described by the Laughlin wavefunction [106]ΨL (x1 , x2 , ...xN ) =Yi<j(zi − zj )2

Y

³

´

exp −|zk |2 /2 ,(10.2)kwhere the products run over the indices i, j, k = (1, 2, ...N ) at position z = x + iy for N
particles (as noted earlier, we are focusing on the Laughlin wavefunction arising from a
1/2 filling fraction). We will see that the inclusion of the resonant terms in Eq. (10.1) cansignificantly modify the form of Eq. (10.2) due to the growth of two-particle correlations.

10.2

Application of Chern-Simons theory
We approach this problem by an extension of Chern-Simons theory [108] whichallows us to develop a mean-field theory for the rotating system that has removed thecomplications of the associated rotation. This is done by constructing a composite
particle composed of the original particle and an artificially attached number of flux
quanta. The composite particle is designed so that the attached flux quanta cancel
the total rotation of the original system leaving a system of non-rotating, interacting
composite particles. For the resonant system, the composite particles can formally be

103

Figure 10.1: Pairs of composite atoms composed of single atoms and an associatednumber of angular momenta quanta (represented by the arrows) approach within an
open channel potential of background value U (x, x0 ). They may form a composite
molecule due to the presence of a closed channel bound state, at a detuning ν from thescattering continuum, which is coupled to the open channel with strength g(x, x0 ).

obtained by the following transformation:·

ϕ̂a,m (x) = exp − 2i¸Z

d2 x0 θ(x − x0 )(ρa (x0 ) + 2ρm (x0 )) ψ̂a,m (x),(10.3)

where ρa (x0 ) and ρm (x0 ) are the atomic and molecular spatial densities, respectively,and θ(x − x0 ) is the topological phase [104].
In the composite picture, we replace the ψ̂ operators in Eq. (10.1) with the corresponding composite operators ϕ̂ and introduce the statistical Chern-Simons field a(x)through the gauge transformation
A(x) → A(x) + a(x).

(10.4)

These modifications generate the Hamiltonian formulation of our composite atom/moleculetheory. The composite picture can be shown to be equivalent to the single particle picture of Eq. (10.1).
We will now shift to a functional representation of the composite atom/molecule
system to clarify the resonant modifications of the Chern-Simons theory and then returnto the Hamiltonian formulation to derive the ground state wavefunction. Defining the

104
action within one temporal and two spatial dimensions
Z

S=

3

d x

X

Z

ϕ∗σ (x)i∂0 ϕσ (x) −dx0 Ĥ,(10.5)

σ=a,m

we generate a Chern-Simons term which couples to the statistical vector field aµ (x)Z

SCS = − d3 x

1 µνλ
² aµ (x)∂ν aλ (x),8π(10.6)

where the indices of µ, ν, and λ run over the three dimensions (0, 1, 2) and a summation
convention over repeated indices is invoked. We have also introduced the antisymmetrictensor ²012 = 1. To simplify the following calculations we assume contact interactions of
the form U (x, x0 ) = U δ(x, x0 ) and g(x, x0 ) = gδ(x, x0 ). Any complications arising fromthis replacement of the true potentials with contact potentials should be remedied as
explained in Appendix A.
We next perform the lowest order variation of the action. Varying with respect
to the zeroth component of the gauge field
∂S
= 0,
∂a0

(10.7)

reproduces the Chern-Simons condition³´∇ × a(x)|z = −4π |ϕa (x)|2 + 2|ϕm (x)|2 .

(10.8)

Equation (10.8) is a statement of Gauss’s law for the statistical gauge field associating
an even number of rotational flux quanta with each particle. This relation is simply arestatement of our choice of quasi-particle.

10.3

Calculation of the ground state wavefunction
Since we will be interested in the ground state properties of the atom/molecule

system, let us assume that the fields ϕa (x) and ϕm (x) are uniform. By minimizing theaction with respect to the atomic and molecular fields∂S= 0 and
∂ϕa

∂S
= 0,
∂ϕm

(10.9)

105
we generate the following constraint equation for the molecules:
ϕm =

gϕ2a
.
2[ν + |A + a|2 /m](10.10)Equation (10.10) allows us to eliminate the molecular field from the theory and arrive
at a self-consistent relationship for the gauge fieldÃ2|A + a| =!

g2
U+2m|ϕa |2 .
4(ν + |A + a|2 /m)(10.11)Equation (10.11) is the usual result relating the gauge field to the background density
only now it is dependent upon the detuning from the resonance.
We next switch back to the Hamiltonian form of our theory to derive the groundstate wavefunction. After Fourier transforming the composite form of Eq. (10.1), by
substitution of the field operators ϕ̂a (x) =Pk âk eik·x and ϕ̂

m (x) =Pk b̂k eik·x , wefollow the usual Hartree-Fock-Bogoliubov (HFB) approach to construct a quadratic
Hamiltonian which accounts for the lowest order pairing. As before, we assume contact
interactions and now make the additional assumption that we may neglect the excitedmodes of the molecular field keeping only the lowest condensed mode b̂k = b0 . Thisshould remain a valid assumption so long as the molecular mode is not too greatly
populated. The resulting Hamiltonian for the composite system can be written as:
H = H0 +X

A†k Mk Ak .

(10.12)

k6=0

H 0 is composed of all terms of less than quadratic order in the operator âk , we define acolumn vector Ak = (âk , â†−k ), and Mk is the self-energy matrix. For our purposes weneed only concern ourselves with the structure of the second term in Eq. (10.12). Herethe self-energy matrix is expressed as



 Ek

Mk = 


∆∗

∆ 
E−k




(10.13)

106
with the additional definitions for the diagonal and off-diagonal terms
Ek = Ek0 + U (|ϕa |2 + n),(10.14)∆ = U (ϕ2a + p) + gϕm .

(10.15)Equations (10.14) and (10.15) are expressed in terms of the pairing-field p =the normal-field n =P

k0 hâk0 â−k0 i,

P†0
k0 hâk0 âk0 i, and Ek is the effective kinetic term which contains

the contribution from the gauge field A(x).
By rewriting Eq. (10.12) in terms of the quasi-particles
´
³
1
†
(E
+
ω
)â
+
∆â
k
k k
−k ,
(Ek + ωk )2 + |∆|2
³
´
1
†
†
α̂−k
= p
(E
+
ω
)â
+
∆
â
,
k
k
−k
k
(Ek + ωk )2 + |∆|2α̂k = p(10.16)(10.17)the result is a diagonal Hamiltonian
H = H0 +Xωk α̂k† α̂k ,(10.18)k6=0

where H0 contains the ground state contribution to the energy and the excitations areq

given by the spectrum of frequencies ωk =Ek2 − |∆|2 .

Since there are no quasi-particles present in the ground state |gsi , which is what
one would expect from an interacting bosonic system at T = 0, the ground state mustsatisfy the conditionα̂k |gsi = 0.

(10.19)Substitution of Eqs. (10.16) and (10.17) for the quasi-particle operators result in therelation(Ek + ωk )âk |gsi = −∆â†−k |gsi.

(10.20)Because âk and â†k are canonically conjugate variables, there is no loss of generality inmaking the replacement âk → ∂/∂â†k [117]. This converts Eq. (10.20) into a simple
differential equation for the ground state with the solution:"|gsi = exp

X

#

−∆
â†−k â†k |0i.
E
+
ω
k
k
k

(10.21)

107
To derive the many-body wavefunction, we must now move from second to first quantization. We follow the method of reference [118] in what follows. The relationship whichlinks the second quantized ground state |gsi with the first quantized wavefunction ΨCB
can be written for an even number of noncondensed particles 2N asΨCB (x1 , x2 , ...x2N ) = h0|ϕ̂a (x2N )...ϕ̂a (x2 )ϕ̂a (x1 )|gsi,(10.22)where it should be noted that ΨCB is the full-many body wavefunction for the composite
Bose particles. If we are able to assume that Ek + ωk À ∆ , an assumption which
will remain valid as long as we are not too close to resonance, we may truncate thepower expansion of the exponent in Eq. (10.21). This results in the composite boson
wavefunctionΨCB (x1 , x2 , ...x2N ) = S(ψ12 ψ34 ...ψ(2N −1)2N ),(10.23)comprised of a symmetrized product S of paired wavefunctions
ψij =

X

−∆
eik·(xi −xj ) .
E
+
ω
k
k
k

(10.24)

If we were dealing with a system of fermions, Eq. (10.23) would be antisymmetrized andwould result in a Pfaffian wavefunction [119]. Here, because of the statistics of the particles, we generate a bosonic analogue to this result. The many-body wavefunction forthe bare particles can now be extracted from the composite wavefunction (see reference[118]) resulting in:ΨM B = ΨCB (x1 , x2 ..., x2N ) × ΨL (x1 , x2 ..., x2N ),(10.25)which is a product of the composite particle wavefunction of Eq. (10.23) and the Laughlin wavefunction of Eq. (10.2).

10.4

Discussion of resonant effects
Equation (10.25) is the final result for the ground state wavefunction of the reso-

nant rotating Bose system. This result has important consequences for the generation

108of a Laughlin state within a resonant atomic gas. It would imply that since a Feshbachresonance necessarily increases two-particle correlations between composite particles,
resonantly increasing the interactions will eventually result in a modification to theground state wavefunction. As is clear from the form of Eq. (10.23), for large detuningfrom the resonance, corresponding to small pairing and molecular field, the many-body
wavefunction reduces to the Laughlin wavefunction. As one moves nearer to the resonance, however, the off-diagonal part of the self-energy matrix, ∆, grows. This results in
an increasing modification of the many-body wavefunction from the Laughlin wavefunction. The ability to tune a Feshbach resonance, therefore, allows for the direct study ofthis crossover from a Laughlin wavefunction to a paired wavefunction.
A similar, yet distinct, Moore-Read type wavefunction, as in Eq. (10.25), wasfound for electronic FQHE systems [120]. This has been used to explain the previously
unresolved, even denominator filling fractions which result from a pairing instability,
such as the observed incompressibility of the 5/2 filling. In this case, a straightforward
generalization of the Laughlin wavefunction would result in a symmetric wavefunction,
violating the asymmetry of the fermions. However, the generation of an antisymmetric Pfaffian wavefunction which multiplies the generalized Laughlin state allows theoverall ground state to be correctly antisymmetrized. For the bosonic system we havetreated, the overall wavefunction must remain symmetric, so the corresponding paired
wavefunction is symmetric in comparison to the antisymmetric Pfaffian wavefunction.
In conclusion, the introduction of Feshbach resonance interactions to the problem
of a rotating Bose gas forces one to account for the effects of molecular formation. The
growth of pair correlations among the composite particles leads to a modification of
the nonresonant ground state for a rapidly rotating Bose gas. The new ground state
wavefunction, which is generated by the Feshbach resonance, exists as a strongly correlated state unique to trapped Bose gases. Currently, the major challenge in realizing
the simplest of FQHE states, the Laughlin state, is the need to reach extremely low

109
temperatures to resolve the ground state from the excited states. Since the size of thisenergy gap is directly related to the strength of interatomic interactions, by increasing
the interactions, it would seem natural that one could increase the energy gap, making
the system more accessible to experiment. The results of this paper would imply thatany attempt to increase the gap with a Feshbach resonance would result in modifying
the ground state from the desired Laughlin state. Unfortunately, at the level of thepresent calculation, we are unable to comment on the resonant behavior of the gap sodefer this discussion for now. These results, nonetheless, reveal the exciting possibilityof directly studying the crossover transition between a bosonic Moore-Read type state
and the Laughlin state.
These conclusions have several other important implications for the resonant production of FQHE states within atomic gases. For instance, many of the observable
properties of the gas may be modified, such as the density profile for both atoms andmolecules and the nature of collective excitations. It should also be noted that thecrossover transition we have discussed is only a part of a much more general crossovertheory made accessible by the tunability of a Feshbach resonance. Although the methods presented here are invalid close to the resonance, one could imagine extending theseideas to describe the resonant system as one passes from a gas of interacting rotating
atoms, through the resonance, to a system of tightly bound, rotating molecules. This
would be the analog for the rotating system of the crossover problem we have discussed
at such length within this thesis.

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Appendix A
Renormalization of model potentials

Throughout this thesis we have made use of contact and separable pseudo-potentials
to simplify the evaluation of the resonant field theories. This has introduced a need torenormalize the resulting equations. Equation (2.41), which defines our renormalization
scheme, will now be solved for three different pseudo-potentials: contact, Lorentzian,and Gaussian.

A.1

Contact

The contact potential has the Fourier representation: λ(k) =√U and g(k) = g. To

prevent Eq. (2.41) from diverging we must impose a cutoff K bounding the integral.
After performing the integrals, Eq. (2.41) may be solved for T (k). The Feshbach theory,
as given by Eqs. (2.20) and (2.25), may be compared to this result by series expanding
to order O(k). Matching terms, order by order, we get a series of algebraic equations
which may be solved to find the following relationships between U, g, ν and the physicalquantities Ū , ḡ, ν̄:U= Ū Γ

g = ḡΓν = ν̄ + αΓḡ 2 ,

(A.1)

117
where we introduce the following definitions:
Γ = (1 − αŪ )−1 and α = mK/(2π 2 h̄2 ).

A.2

(A.2)

Lorentzian

In Fourier space the Lorentzian has the form: λ(k) =√U (1 + (k/K)2 )−1 and g(k) =g(1 + (k/K)2 )−1 . The form of the pseudo-potentials regularizes the integrals so we do
not need to impose a cutoff and the integrals may extend to infinity. The relations given
in Eqs. (A.1) again hold and we find
Γ = (1 − αŪ )−1 and α = mK/(4πh̄2 ).

A.3

(A.3)

Gaussian

In Fourier space we have: λ(k) =√ −(k/K)2
2
Ueand g(k) = ge−(k/K) . Again the poten-

tials regularize the integrals. Equations (A.1) still hold and we find
Γ = (1 − αŪ )−1 and α = mK/(4π 3/2 h̄2 ).

(A.4)

Appendix B
Bogoliubov Transformation

Very few many-body problems allow for an exact solution. Among the few that do, one of
the more commonly encountered forms are Hamiltonians which are quadratic in creation
and annihilation operators. In this appendix we will sketch the general method of solving
these quadratic Hamiltonians for the case of both bosons and fermions. A much morethorough treatment can be found in various books such as that of reference [54].

B.1

Hermitian quadratic Hamiltonian

A general, quadratic Hamiltonian of this sort may be written in the form
H=X

Aij a†i aj +

ij

1X
∗
(Bij a†i a†j + Bij
aj ai ),
2 ij

(B.1)

where the following conditions are satisfied
A = A† ,B > = −εB.

(B.2)

The variable ε is defined such that ε = 1 for fermions and ε = −1 for bosons. Equation(B.1) may be written in matrix form as1ε
H = α† M α + trA,
2
2

(B.3)

by introducing the row and column vectors




 a 
,


α=


a†

α† = (a† a),

(B.4)

119
and defining the Matrix





A

B

−εB ∗

−εA∗



M =


B.2


.


(B.5)

Unitary canonical transformation

We define the matrix T which moves us from the basis α to a diagonal basis ββ = T α.

(B.6)

We would like this transformation to be canonical, meaning that the operators in the
new basis satisfy the same commutation relations as the original operators. This canbe expressed by the matrix η defined as:



η=








 1
=
 

0 

[a, a† ]ε[a, a]ε [a† , a† ]ε[a† , a]ε

0 ε

.


(B.7)

In order for a transformation T to remain canonical, it must satisfy the following criteria:
T ηT † η = 1

T ∗ = γT γ,(B.8)

where we have introduced the matrix




 0

1 

γ=


1 0

.


(B.9)

It can clearly be seen that a canonical transformation of Eq. (B.3) results in the diagonal
Hamiltonian1²
H = β † ηT ηM T −1 β + trA,
2
2

(B.10)

where T ηM T −1 is a diagonal matrix.

B.3

Basis transformations for bosons and fermions

A canonical basis transformation, known as the Bogoliubov transformation, that brings
Eq. (B.3) into diagonal form is given by

120






 bν  

=

 

b†ν

 

cosh ωνeiθν sinh ωνe−iθν sinh ωνcosh ων  aν 
 
,
 


(B.11)a†ν

for a single component system of bosons and by b↑ν  
 

 

 b   ↓ν  
=

 

 b†   ↑ν  
 

 


b†↓ν







cos ων

00−ieiθν sin ων0cos ων

ieiθν sin ων00ie−iθν sin ων

cos ων

0−ie−iθν sin ων00cos ων

  a↑ν  a   ↓ν ,  a†   ↑ν 





a†↓ν

(B.12)
for a two-spin system of fermions. These transformations, of course, can be generalized
to systems of higher degrees of freedom.

Appendix C
Functional Integrals and Grassmann Variables

Several useful integrals are often encountered when working with path integrals, themost ubiquitous being Gaussian integrals. We begin by discussing the evaluation ofthese integrals for the case of Bosons, where we may work with classical fields, and thenproceed to discuss fermions and Grassmann variables [86, 125].

C.1

BosonsFor complex fields it is often useful to first decompose the fields into real and imaginaryparts. For the bosonic fields x, x∗ we may expand them as:x=x1 + ix2
√
,
2

x∗ =

x1 − ix2
√
.
2

(C.1)

Before solving for the basic Gaussian functional integral, it is useful to first look at theevaluation of a Gaussian integral of the complex variables x, x∗ :Z∗dx∗ dxe−x ax =

Z

1

1

2

µZ

2

dx1 dx2 e− 2 ax1 e− 2 ax2 =1dxe− 2 ax

2

¶2

=

2π
.
a

(C.2)

We have made use of Eq. (C.1) in the second step of Eq. (C.2). Equation (C.2) allows
us to evaluate the Gaussian functional integral of the complex fields x, x∗ acting on thematrix A with eigenvalues ai :Zdx∗ dx exp(−x∗ Ax) =



X
YZ

dx∗ dxj  exp(−
xi Ai,j xj )
i

i,j

i,j

(C.3)

122
=

Ã Z
Y

!

dx∗i dxi exp(−i=x∗i ai xi )

i

Y 2π

(2π)n
.
=
ai
det A

i

C.2

X

Fermions and Grassmann variables

Fermions require the introduction of Grassmann variables in order to satisfy the correct
particle statistics. Grassmann variables are defined such that the exchange of any twovariables is antisymmetric:xη = −ηx.

(C.4)

What’s more, the product of more than one Grassmann variable yields zero:x2 = 0.

(C.5)

This definition may yield an ambiguity in sign when integrating these variables so wemust choose a convention. Here we impose the convention

Rdx∗ dx(xx∗ ) = 1. Let’s firstlook at how a simple Gaussian integral over Grassmann variables is modified:ZZdx∗ dx exp(−x∗ ax) =Zdx∗ dx(1 − x∗ ax) =dx∗ dx(1 + xx∗ a) = a.

(C.6)

This should be contrasted to the result of Eq. (C.2) for a standard, classical variable.
With this information we may evaluate the Gaussian functional integral for fermions:Zdxdx∗ exp(−xAx) = 
Ã

=
=

YZ



dx∗i dxj  exp(−

i,j

YZ

Y

i

xi Ai,j xj )

(C.7)

i,j

!

dx∗i dxi

X

exp(−

X

x∗i ai xi )iai = det A.

i

Note that for bosons the result was (2π)n / det A. Another useful integral that is often
encountered is:Rdx∗ dx exp(−x∗ Ax + η ∗ x + x∗ η)R= exp(η ∗ A−1 η),dx∗ dx exp(−x∗ Ax)

(C.8)

123
where
η∗x =

P

∗
i ηi xi ,x∗ η =

P∗i xi ηi .

(C.9)

The variables ηi , ηi∗ anticommute with each other and xi , x∗i . This can be proved by ashift transformationx → x + η̃,x∗ → x∗ + η̃ ∗ ,(C.10)

which will cancel the linear terms in the exponent of Eq. (C.8). When applied in practice,
this shift transformation is often known as a Hubbard-Stratonovich transformation.

Appendix D
Matsubara frequency summations

In finite-temperature field theories we often encounter Matsubara frequency summations
over products of free Green’s functions. In this appendix we will list some of the morecommon summations and detail how they are evaluated. A discussion of evaluating
more complex summations than are presented here can be found in reference [87].

D.1

Common summations

In this thesis we have made use of the following useful relations:
1X
G0 (p, iωpn ) = nf (ξp ),
β iωn

(D.1)

1X
D0 (p, iνpn ) = nb (ξp ),β iν n(D.2)

1 − nf (ξp ) − nf (ξq )1X
G0 (p, iωpn )G0 (q, iωqn − iωpn ) =.
β iωniωqn − ξp − ξq

(D.3)

p

−

p

−

p

Equations (D.1) and (D.2) result in Fermi and Bose distributions defined, respectively,as:nf (ξp ) =1eβξp − 1and nb (ξp ) =1eβξp + 1,

whereas Eq. (D.3) was encountered in our discussion of the pair susceptibility χ.

(D.4)

125

D.2

Evaluation of Matsubara summations
We begin our discussion of evaluating Matsubara series with the simple case of

Eq. (D.1). If we explicitly write the free Fermion Green’s function, the sum we wish toevaluate is:
∞
1 X
1
S=
,
n
β n=−∞ iωp − ξp

(D.5)

where the summation runs over integer values for the odd Matsubara frequencies ωpn =(2n + 1)π/β. Let us write this in the representative formS=−

1X
f (iωpn ),
β n

(D.6)

where f (iωpn ) is defined through Eq. (D.5). To evaluate the sum, we transform thesummation into an equivalent contour integration. Let us look at the following integral:Z

I = limR→∞dzf (z)nf (z).
2πi

(D.7)

We now chose the contour to be a circle of radius R → ∞ and impose that the function
nf (z) generates poles at all points zn = (2n + 1)iπ/β. An appropriate function wouldbe:nf (z) =1
eβz + 1,

(D.8)

which has a residue of −1/β at each pole zn . The contour and these periodic poles areillustrated in Fig. D.1. The function f (z) is defined as:
f (z) =

1,z − ξp(D.9)

which contributes an extra pole at z = ξp . To clarify these results, we list all the poles
and residues:11
β (2n + 1)iπ/β − ξpzn = (2n + 1)iπ/β,Rn = −

z1 = ξp ,R1 = nf (ξ).

(D.10)(D.11)

126
According to Cauchy’s theorem [126], Eq. (D.7) may be written as a sum of residues
I=−

1X
1
+ nf (ξp ).
β n (2n + 1)iπ/β − ξp

(D.12)In the limit that we take the contour R → ∞, assuming Jordan’s lemma is satisfied[126], the integral I → 0. We, therefore, have the result:
1X
1= nf (ξp ),β n (2n + 1)iπ/β − ξp

(D.13)∞1 X
1= nf (ξp ).
nβ n=−∞ iωp − ξp

(D.14)

which may be written as

This final result proves Eq. (D.1).
For bosons, the method is similar except the contour integral should be chosen
to have poles at the even values zn = 2niπ/β. It is therefore appropriate to replace
Eq. (D.7) with:Z

I = limR→∞dzf (z)nb (z),2πi(D.15)

where the function nb (z) is given by
nb (z) =1eβz − 1

which now has a residue of 1/β at each pole zn .

,(D.16)

127

Figure D.1: A typical contour for Matsubara summation integrals. The x’s represent
evenly spaced poles along the vertical axis at points zn and the contour radius is takenin the limit R → ∞.

Appendix E
Tc equations from the pseudogap theory

The pseudogap equations derived in Chapter 8 are presented here in a more clear andexplicit form. The Matsubara summations are performed and the expansion coefficientsof Eq. (8.54) are clearly listed. A detailed discussion of their derivation is given in the
text. They are valid for T ≤ Tc and may be thought of as an extrapolation for a small
region above Tc .

E.1

Gap equation

The Gap equation is given by
1 = −gef f (0)X 1 − 2f (Ek )

2Ek

k

ϕ2k ,

(E.1)

q

where Ek =ε2k + ∆2 ϕ2k is the gapped energy spectrum, εk = k 2 /2m − µ, and the gap

is defined in terms of the superconducting gap ∆sc and the pseudogap ∆pg :∆2 = ∆2sc + ∆2pg .

(E.2)

The Fermi distribution function is defined as f (x) = 1/(exp(βx)+1). The resonant inter2gaction contribution is given as gef f (0) = U − ν−2µ
and the function ϕ2k = exp[−(k/Kc)2 ]is introduced to impose a cutoff Kc upon the integral. For this regularization to work
we must define the renormalized parameters as in Appendix A.3.

129

E.2

Number equation

The equation for the total number of particles in the system isN= Nf + 2Nb + 2Nb0
= 2

X·
k

¸

Ã

1
+ BC
1
εk
2 X
f (Ek ) + (1 −)(1 − 2f (Ek )) +n q 2 2M
2
Ek
1+C q
1+C

!

+ 2Nb0(E.3)

The Bose distribution function is introduced as n(x) = 1/(exp(βx) − 1). Here we havealso introduced the following constants a0 , B, C. The first two, a0 and B, will bedefined in a following section; the third:C=E.3

g 2 a0.
−1
2(1 − U gef
f (0))(E.4)

Pseudogap equation

The equation for the pseudogap is given byÃ∆2pg =

E.4

1
X
+ BC
1
n q 2 2M
a0 (1 + 1/C) q
1+C

!

.

(E.5)

Coefficients a0 ,BWe now define the expansion coefficients coming from Eq. (8.54) and appearing in theabove equations. To begin
·

¸

1 X
εk
(1 − 2f (εk )) −(1 − 2f (Ek )) .
2
2∆ k
Ek

a0 =

(E.6)

For the Gaussian potential
2~ 2 ϕ2 = 2ϕ2k (2( k )2 − 3) and (∇~ k εk ) · (∇~ k ϕ2 ) = − 2 ( k )2 ϕ2 ,
∇
k k
k
k
Kc
m Kc
K
c

(E.7)

130
so
B = −
+
−
−

1 X nh 4(²k + µ)f 0 (²k )6a0 ∆2 kmi
2 Ek (εk + µ) h
ε2k
εk
(1
+
)(1
−
2f
(E
))
−
2
(1
−
2f
(ε
))
k
k
m ∆2 ϕ2k
Ek
Ek2
ih
k i
1 h
εk
(1 − 2f (Ek )) 3 − 2( )2(1 − 2f (εk )) −
2m
Ek
Kc
2
h
io
∆ 1 − 2f (Ek ) 2
kϕk 3 − 2( )2 ,
2
2Kc
2Ek
Kc

where we have introduced the shorthand notation f 0 (x) = −f (x)(1 − f (x))β.

(E.8)

Appendix F
Landau levels

The Hamiltonian for a charged particle moving in a transverse magnetic field may be
written as
H=

1
1 2
(p − eA(r))2 =
π ,
2m
2m

(F.1)

where we have introduced the operator π which satisfies the commutation relation[πx , πy ] = −ih̄2 /l2 , with l =ph̄/|e|B the magnetic length or Larmor radius. Equa-tion (F.1) may be rewritten in terms of the ladder operators
a =
a† =

l
√ (πx − iπy ),2h̄l√ (πx + iπy ),
2h̄

(F.2)

which satisfy the commutation relation [a, a† ] = 1. This substiturion results in thefollowing Hamiltonian:µH = h̄ωc

¶

1
a a+
,
2
†

(F.3)

where ωc = |eB|/m is the cyclotron frequency. Equation (F.3) has eigenvalues En =³´h̄ωc n + 12 , like those of the harmonic oscillator, each associated with a highly degenerate set of eigenvectors, known as Landau levels [104]. Often it is convenient to workwithin the lowest of these degenerate energy levels, referred to as the lowest Landaulevel (LLL), whose wavefunction may be written as1
2
ψ0,m (r) = √z m e−|z| /4 ,m
2π2 m!l

(F.4)

132
where the quantum number m = 0, ±1, ±2 . . . and the complex position z = x + iy is
introduced.