Physics of a two-dimensional electron gas with cold atoms in non-Abelian gauge potentials

Indubala I. Satija,1,2 Daniel C. Dakin,3,1 J. Y. Vaishnav,2 and Charles W. Clark2

1

Department of Physics, George Mason University, Fairfax, Virginia 22030, USA 2

Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 3

Optical Air Data Systems, 10781 James Payne Court, Manassas, Virginia 20110, USA

Received 15 November 2007; published 15 April 2008

Motivated by the possibility of creating non-Abelian fields using cold atoms in optical lattices, we explore

the richness and complexity of noninteracting two-dimensional electron gases 2DEGsin a lattice, subjected

to such fields. In the continuum limit, a non-Abelian system characterized by a two-component “magnetic flux”

describes a harmonic oscillator existing in two different charge states mimicking a particle-hole pairwhere

the coupling between the states is determined by the non-Abelian parameter, namely, the difference between

the two components of the “magnetic flux.” A key feature of the non-Abelian system is a splitting of the

Landau energy levels, which broaden into bands, as the spectrum depends explicitly on the transverse momentum. These Landau bands result in a coarse-grained “moth,” a continuum version of the generalized Hofstadter

butterfly. Furthermore, the bands overlap, leading to effective relativistic effects. Importantly, similar features

also characterize the corresponding two-dimensional lattice problem when at least one of the components of

the magnetic flux is an irrational number. The lattice system with two competing “magnetic fluxes” penetrating

the unit cell provides a rich environment in which to study localization phenomena. Some unique aspects of the

transport properties of the non-Abelian system are the possibility of inducing localization by varying the

quasimomentum, and the absence of localization of certain zero-energy states exhibiting a linear energymomentum relation. Furthermore, non-Abelian systems provide an interesting localization scenario where the

localization transition is accompanied by a transition from relativistic to nonrelativistic theory.

DOI: 10.1103/PhysRevA.77.043410 PACS numbers: 37.10.Vz, 32.80.Qk, 71.30.h, 03.75.Lm

I. INTRODUCTION

Methods for creating fields that couple to neutral atoms in

the same way that electromagnetic fields couple to charged

particles have created the exciting possibility of studying the

effects of a generalized magnetism using cold atoms 1–3.

Using laser induced hopping, a controlled phase can be imposed upon particles moving along a closed loop in an optical lattice. The associated synthetic fields can be sufficiently

strong to enter the regime of exotic magnetic phenomena that

have been difficult to explore in condensed matter experiments, such as the fragmented fractal spectrum of a twodimensional electron gas 2DEGin a magnetic field, the

famous “Hofstadter butterfly” 4. Such fields need not obey

Maxwell’s equations, thus providing the possibility of discovering fundamentally new physics 5. For example, we

discuss here the generation of non-Abelian fields, by using

cold atoms that occupy two Zeeman states in the hyperfine

ground level 3; these two states may be thought of as “colors” of the gauge fields, and such a system may be used to

simulate lattice gauge theories in 2+1dimensions. Other

potential applications of non-Abelian fields are the creation

of counterparts of magnetic monopoles 1 and integer and

fractional quantum Hall effects 6.

In this paper, we adopt the 2DEG as a motif for the study

of cold atom systems. The subject of 2DEGs in a magnetic

field is a textbook topic 7, as the problem maps to a onedimensional harmonic oscillator. The discrete energy levels

of the oscillator are the Landau levels that describe free particle energies in terms of the quantized units c, where

c=eB/mc is the cyclotron frequency of the corresponding

classical motion. Each level is highly degenerate, reflecting

the fact that a classical electron spirals about a line parallel to

the magnetic field, with an arbitrary center in the transverse

plane. The degree of degeneracy is equal to L2 /22 where

=c/eB is the magnetic length and L2 is the area of the

system.

Beginning with the celebrated work of Onsager 8,

Harper 9, and then Hofstadter 4, the subject of 2DEGs in

a crystalline lattice in a magnetic field has fascinated physicists as well as mathematicians. In the presence of a lattice,

each Landau energy level splits into Q bands, where the

rational number P/Q is the magnetic flux through the unit

cell in units of the magnetic flux quantum fluxoid. The

heart of the problem is the two competing periodicities related to the ratio of the reciprocal of the cyclotron frequency

and the period of the motion of the electron in the periodic

lattice. Two key aspects that have been explored extensively

are the exotic multifractal spectrum Hofstadter butterfly,

and the metal-insulator transition obtained by tuning the ratio

of the tunneling along the two directions of the lattice 9. In

contrast to Mott transition, which has its origin in particle

interactions, the metal-insulator under consideration here has

its origin in the irrational flux quanta penetrating the unit

cell. Furthermore, unlike the Mott insulating phase which is

associated with the appearance of a gap and hence is accompanied by a transition from compressible to impressible state,

localization transition induced by competing periodicities is

characterized by exponential decay of the single particle

wave function.

This paper revisits the metal-insulator transition when the

2DEG is subjected to a non-Abelian gauge field which is a

natural generalization of the uniform magnetic field. Such

fields yield a much richer spectral and transport landscape

PHYSICAL REVIEW A 77, 043410 2008

1050-2947/2008/774/04341011043410-1 ©2008 The American Physical Society
than is encountered in the Abelian case. Recent studies have

shown that this transition can be demonstrated using ultracold atoms in an artificial magnetic field 2,10.

The generic experimental setup for producing nonAbelian U2gauge fields that we consider here, consists of

a two-dimensional optical lattice populated with cold atoms

that occupy two hyperfine states 2,3. Such systems exhibit

three competing length scales, associated with two distinct

“magnetic fluxes” denoted by 1 and 2that penetrate the

unit cell. Our aim is to describe some of the generic properties of such systems. Although our main focus is on optical

lattices, we first discuss the corresponding continuum problem, where the infinite degeneracy of the Landau levels is

lifted by non-Abelian interactions. The continuum problem

mirrors some of the features subsequently encountered in the

lattice system.

In the discussion of the metal-insulator transition in the

lattice, we focus on the ground state as well as the states at

the band center. These two cases are relevant for experimental systems involving Bose condensates and fermionic gases

near half-filling, respectively. Some of these results were described in an earlier paper 11. In addition to a detailed

analysis, here we describe new results, such as the simulation

of relativistic phenomena using cold atoms in non-Abelian

fields. By tuning lattice anisotropy, we can implement relativistic as well as nonrelativistic dynamics, with a particular

focus on the effects of disorder. Simulation and detection of

Dirac fermions using single-component cold atoms in a hexagonal lattice was recently proposed 12. The systems we

propose here provide the possibility of observing relativistic

particles and also of studying their localization properties.

We show that the non-Abelian systems provide an experimental realization of the defiance of localization by disordered relativistic fermions, a topic that has been the subject

of extensive study 13.

In Sec. II, we introduce non-Abelian gauge fields and the

corresponding effective “magnetic fields.” Section III examines the continuum limit of a single particle in a non-Abelian

gauge field. In Sec. IV, we discuss lattice systems subjected

to these fields, and describe methods of calculation. In Sec.

V, we study various spectral characteristics of the nonAbelian lattice systems. There, following long established

practice for studying metal-insulator transition in Abelian

systems, we fix 1=5−1/2, the golden mean, which we

denote as . The irrationality of 1 ensures the existence of a

localization transition 9,14. For 2, we consider a selected

set of both rational and irrational values.

Sections VI and VII discuss localization properties of the

states at the band center E= 0and at the band edge. The

localization of the E= 0 states brings out some of the most

important features of the non-Abelian cases, including the

dependence of the transition upon a conserved momentum.

Furthermore, a unique aspect of the non-Abelian system,

namely, the defiance of localization of the E= 0 states,

emerges when the energy-momentum relation mimics the behavior of relativistic particles. Section VIII describes the experimental realization of the metal-insulator transition in

cold atom lattices.

II. NON-ABELIAN GAUGE FIELDS

Effective non-Abelian vector potentials arise naturally in

systems where the atoms have N degenerate internal states.

The most general vector potential couples the states, and thus

gives rise to a UNgauge symmetry. We here consider the

case where N= 2. In our treatment of the non-Abelian case,

we follow the convention of an earlier study 3, adopting its

form of vector potential

A

 = c

ea 

2 

−1 1

1 −1 , 2

x

a



1 0

0 2

,0	. 1

The i determine the “magnetic fluxes” of the lattice with

lattice constant a. Equation 1is in the Landau gauge

A

x, y=(Ax ,Ayx, 0), where Ax is a constant and Ay depends

only on x.

We rewrite the vector potential in terms of Pauli matrices

i, separating the Abelian and the non-Abelian parts of the

gauge field

A

 = c

ea 
− 2

I − xxˆ + 2

x

a

Iyˆ + 	

x

a

zyˆ, 2

where we have defined quantities =1+2/2 and 	

=1−2. Here a, a=x, y ,z denotes Pauli matrices. The

parameter 	 characterizes the non-Abelian feature of the system.

For non-Abelian fields, the effective “magnetic field” is

given by

Bˆ =  
 Aˆ − ie

c

Aˆ 
 Aˆ . 3

The origin of the extra term Aˆ 
Aˆ can be traced to the

commutator for the generalized velocity operator p

− e

cA/M,

vn,vm = e

M2c

nAm − mAn − ie

c

An,Am = ie

M2c

mnrBr.

For the vector potential in Eq. 1, this gives

B

ˆ

z = B

ˆ

0 + 	 c

ea2 ˆ z − 

x

a

ˆ y, 4

where B0= 2 c

ea2 . Thus, =B0a2 /2c/edescribes the

Abelian flux quanta penetrating per unit cell of the lattice.

The non-Abelian gauge potential generates a nonuniform

magnetic field, as Bˆ depends explicitly on the spatial coordinate x when 	0.

III. CONTINUUM LIMIT OF THE NON-ABELIAN

SYSTEM

We now consider the continuum limit of the non-Abelian

problem. Although the Aˆ of Eq. 2is ill defined in the continuum limit a→0, the study is useful in illustrating some

key aspects of the non-Abelian systems. In general, continuum problems can also be experimentally realized, as in

Ref. 1.

SATIJA et al. PHYSICAL REVIEW A 77, 043410 2008

043410-2
It can be shown, after some algebra, that the twocomponent continuum Hamiltonian Hˆ c=pˆ − e

cAˆ 

2 /2Mresulting from the vector potential in Eq. 2is gauge equivalent to

Hˆ = 1

2M 

pˆ x + 

2 + VxC	x − x0

2

C	x − x0

2 pˆ x − 

2 + Vx

 5

up to a ky-dependent term. The transverse momentum ky is a

conserved quantity as the Hamiltonian Hˆ c with A

 given by

Eq. 2is cyclic in y. Here, = 

2a , C= 82

/a

2

, Vx

=M2

2x2 /2 with = 2

ma 1

2+22

/2 and x0= 2a

ky

M22a

12

 .

This particular form of the Hamiltonian provides an illuminating picture of the non-Abelian problem; the particle behaves as a two-component harmonic oscillator existing in a

positive as well as a negative charge state. The physics of

this system is that of a particle-hole pair, with the nonAbelian parameter 	, governing the coupling between states.

The spectrum of Hˆ c is obtained by numerical diagonalization in a basis of harmonic oscillator wave functions with

frequency . Figure 1 shows the six lowest energy levels.

For fixed ky, at 	= 0, each Landau level is twofold degenerate. For 	0 the degeneracy is lifted and the eigenstates

become superposition states of a particle-hole pair.

As Fig. 1 shows, the energy levels are equally spaced only

for the Abelian case 1=2. We can explicitly understand the

splitting of each Landau level via degenerate perturbation

theory, with 	 as a small parameter and using degenerate

eigenstates

f1

n = eikyy



e−ix
nx

0 , f2

n = eikyy

 0

eix


nx

,

where 
nxare the eigenstates of the corresponding harmonic oscillator. Perturbative correction, although linear in 	

exhibits rather complicated dependence on ky and other parameters. Figure 2 displays the perturbative splitting of the

lowest Landau level and shows it to be in good agreement

with the numerical results for small 	.

Figure 3 shows the variation of the energies with ky, obtained numerically. In highly non-Abelian cases, the energies

bear no relation to their Abelian values. Close to the Abelian

limit bottomthe energy levels are simply split around the

Abelian energies. The energies oscillate with ky, resulting in

actual and avoided crossings i.e., the Landau bands overlap. In the vicinity of the crossings, the bands exhibit a

linear dispersion relation. As we shall discuss, these features

reappear in the corresponding problem of the non-Abelian

gauge field on an optical lattice.

Figure 4 summarizes the effects of the non-Abelian gauge

potential on the lowest Landau level of the corresponding

Abelian problem. The figure describes the continuum limit of

the Hofstadter “moth” 3, which is the generalization of the

Hofstadter “butterfly” as the underlying gauge field becomes

non-Abelian. This coarse-grained “moth” illustrates the symmetry breaking feature of the non-Abelian system as it lifts

the degeneracy of the corresponding Abelian problem.

−0.16 0 0.16 0.31 0.47

−0.5

0

0.5

1

1.5

2

2.5

Δ

E

FIG. 1. Six lowest energy levels of the continuum non-Abelian

Hamiltonian in Eq. 5, =/2, M =1, ky= 1, and 21= 1. The

levels are equally spaced in and only inthe Abelian case 	=0,

where the system reduces to two decoupled harmonic oscillators.

−4.8 −3.2 −1.6 0 1.6 3.2 4.8

x 10−3

−0.737

−0.736

−0.735

−0.734

−0.733

−0.732

−0.731

E



FIG. 2. Color onlineThe ground-state splitting obtained numerically see Fig. 1solid black, compared with that obtained by

perturbation theory dashed redabout the Abelian point 	=0.

−2 0 2 0

2

4

6

8

k

y

E

−2 0 2 0

5

10

15

20

k

y

E

FIG. 3. Color onlineThe three lowest energy levels of the

non-Abelian system with 1= 1.32 and 2= 0.253 top, 2= 1.11

bottom. The dashed line shows the Abelian levels corresponding

to  1

2 1

2+22

.

PHYSICS OF A TWO-DIMENSIONAL ELECTRON GAS ... PHYSICAL REVIEW A 77, 043410 2008

043410-3
IV. TWO-DIMENSIONAL LATTICE IN NON-ABELIAN

GAUGE FIELDS

Our starting point is a tight binding model TBMof a

particle moving on a two-dimensional rectangular lattice

x, y, with lattice constants a,band nearest-neighbor hopping characterized by the tunneling amplitudes J,J.

When a weak external vector potential A

x, y=Ax ,Ay , 0is

applied to the system, the Hamiltonian

Hˆ = − Jcos
px − e

c

Ax a

 +  cos
py − ec

Ay b


 ,

where p is the momentum operator. Alternatively, the Hamiltonian of a 2DEG on a lattice in the presence of a magnetic

field can be written as

H = −

ij

Jijcj

†

eiijci + H.c., 6

where ci is the usual fermion operator at site i. The Jij is the

nearest-neighbor anisotropic hopping with values J and J

along the x and y directions. Throughout this paper, we will

use the vector potential defined in Eq. 1.

The phase factor ij=−ji defined on a link i, j is identified as 2e/chA·dl, where A is the vector potential and

1

2 

unit cell

ij = e/ch A · dl = 1

0

 B · dS 7

is the magnetic flux penetrating the unit cell in units of magnetic flux quantum 0=ch/e. We denote the eigenfunction

projected onto the x, y basis, corresponding to the eigenvalue equation H=E
 as x, y. With the transverse

wave number of the plane wave as k

 ̃

y=ky /a, the wave function can be written as ma,na=ei2kyn

gm with x=ma and

y=na.

Substituting the vector potential defined in Eq. 2, the

two-component vector gm =

m

m can be shown to result in the

equations



m+1

m+1

 + 

m−1

m−1

 −  0 E − Vm

E − Um 0 m

m

 = 0,

where

Um = 2 cos21m − 2ky,

Vm = 2 cos22m − 2ky.

For 1=2 mod 1, we recover the Abelian limit described

by the Harper equation 9

gm+1 + gm−1 + 2 cos2m − 2kygm = Egm. 8

For irrational values of the flux , the system exhibits a

metal-insulator transition at = 1, where the model exhibits

self-duality 9. For 1, all states are extended and Bloch

type while for 1, all states are exponentially localized.

The approach to the irrational values of i is studied by

considering a sequence of periodic systems obtained by rational approximants i=pi /qi

. This corresponds to truncating

the continued fractional expansion of 1 and 2. The resulting periodic system will have period Q, the least common

multiple of q1 and q2.

The 2Q-dimensional system can be cast in the form of

two Q-dimensional eigenvalue problems



U1 1 0 0 . .. e−ikA

1 V2 1 0 0 .. 0

0 1 U3 1 0 .. 0

0 01 V4 1 0. 0

0 001 U5 1. 0

. . . . . .. .

eikA . . 0 0 01 VQ



1

2

3

4

.

.

.

Q

 = EA

1

2

3

4

.

.

.

Q



and



U2 1 0 0 . .. e−ikB

1 V3 1 0 0 .. 0

0 1 U4 1 0 .. 0

0 01 V5 1 0. 0

0 001 U6 1. 0

. . . . . .. .

eikB . . 0 0 01 V1



2

3

4

5

.

.

.

1

 = EB

2

3

4

5

.

.

.

1



.

Here EA and EB denote the two sets of eigenvalues of the two

uncoupled systems. The allowed eigenenergies of the full

system are the union of these two sets.

In the above two eigenvalue equations, we have used the

Bloch condition

0.1

0.15

0.2

0.1

0.15

0.2

−0.8

−0.6

−0.4

−0.2

0

0.2

α1 α2

E

0.5

1

1.5

2

2.5

3

3.5

4

FIG. 4. Color onlineThe continuum version of the Hofstadter

“moth” 3. This plot shows the energies as a function of 1 and 2

for a range of ky; the color scale indicates the range of ky. Along the

line 1=2, the Abelian “backbone” of the moth, there is no ky

dependence.

SATIJA et al. PHYSICAL REVIEW A 77, 043410 2008

043410-4


2m−1+Q

2m+Q

 = eikAQ

2m−1

2m

,  2m+Q

2m+1+Q

 = eikBQ 2m

2m+1

.

An important consequence of this decoupling of the

2Q-dimensional problem into two Q-dimensional problems

is that the eigenstates of the system are of an “antiferrimagnetic” type, as shown schematically in Fig. 5. We will refer

to them as of the A and B type. The corresponding states

denoted as A and B are, in general, nondegenerate.

It is instructive to compare the eigenvalue formulation to

the transfer matrix approach discussed in earlier studies 3.

The TBM equation can be written as a transfer matrix equation



m+1

m+1

m

m

 = Tm

m

m

m−1

m−1  ,

where

Tm=

 0 E − Vm−1 0

E − Um0 0 −1

1 0 00

0 1 00  .

The allowed energies are those for which the product of Q

successive matrices Bmhas an eigenvalue on the unit circle

ei1 and ei2. Alternatively, the four-dimensional transfer

matrix equation can be reduced to two independent twodimensional transfer matrices as



m+2

m+1

 = TAm m

m−1

, 

m+1

m

 = TBm

m−1

m−1

, 9

where the 2
2 matrices TA and TB are given by

TAm= 

E − Vm+1E − Um−1−1 − E − Vm+1

E − Um− 1 

and

TBm= 

E − VmE − Um−2−1 − E − Um

E − Vm−1− 1 .

This decoupling of the four-dimensional transfer matrix

problem into two two-dimensional transfer matrices is

equivalent to the decoupling discussed earlier for the eigenvalue problem, which in turn implies the possibility of

antiferrimagnetic-type states as shown in Fig. 5. An important consequence of this type of state is that out of four,

only two of the eigenvalues of the four-dimensional transfer

matrix have to be on the unit circle. In other words, in contrast to the statement made in an earlier paper 3, the allowed energies include states where two of the four eigenvalues of the transfer matrix are not on the unit circle 11.

The existence of “antiferrimagnetic” states and the relationship between the direct diagonalization method and the

transfer matrix approach can be illustrated by considering a

simple non-Abelian system, namely, the one with 1= 1/2

and 2= 0 which can be treated analytically. Diagonalization

of two independent 2
2 matrices Eq. 9gives

EAkA,ky= 2 cos2ky  coskA,

EBkB,ky=  22 cos2

2ky + cos2kB

and the corresponding eigenvectors

A =



e−ikA

0

1

0

, B =

 0

e−ikB

E/2   cos2ky

coskB

0

1

 .

In general, the A and the B states are nondegenerate. As is

explicit in this example, the magnitude of the two components of the vectors are in general unequal, and hence the

solutions correspond to antiferrimagnetic states. However, at

ky=/2, the EA=EB and the two degenerate states are antiferrimagnetic.

The spectrum can also be obtained by iterating the transfer matrix problem where the energies can be written in

terms of the eigenvalues of the transfer matrix eiA ,eiB.

Comparison of the spectrum obtained by these two methods

shows that A=kA/2 and B=kB/2.

The results of this paper were obtained using the direct

diagonalization method. We shall henceforth set 1=, the

golden mean, and explore various complexities of the problem for different values of 2. Here we confine ourselves to

simple fractions such as 1/2, 1/3, 1/4, etc., while for the

irrational case, we choose 2=3 ,4

. In contrast to the continuum problem, in the lattice problem, the spectral and

transport properties are sensitive to number theoretical properties of i and hence the effects of non-Abelian feature cannot be studied by small variation in the non-Abelian

parameter	.

V. NON-ABELIAN SPECTRUM FOR IRRATIONAL 1

The energy spectrum of the system is the union over ky of

the individual energy spectra of the A and the B types of the

tight binding equations. In the Abelian case 1=2 and when

1 is rational, equal to p/q, the spectrum consists of q bands

which are usually separated by gaps. As ky varies, the bands

shift and their width may change, but they do not overlap,

except at the band edges. For irrational , the spectrum is

independent of ky.

( ) θ1

0

( ) 0

η1

1

( ) 0

η2

( ) θ2

0

2

( ) θ3

0

( ) 0

η3

3

( ) 0

η4

( ) θ4

0

4

A

B

FIG. 5. Color onlineThe two possible antiferrimagnetic states

designated as A and B.

PHYSICS OF A TWO-DIMENSIONAL ELECTRON GAS ... PHYSICAL REVIEW A 77, 043410 2008

043410-5
A striking aspect of the non-Abelian problem is the overlapping of the bands, as illustrated in Figs. 6 and 7. These

figures depict two different classes of typical non-Abelian

spectra with rational 2: In Fig. 6, type-A and -B solutions

result in a nondegenerate spectrum; and Fig. 7 shows the

case where A and B solutions are degenerate. Below we discuss various spectral characteristics of the system.

For rational 2=p2 /q2, the spectrum is found to be a periodic function of ky. We list below some of the characteristic

properties of the spectrum that are observed in numerical

studies: 1EA,B2ky=−EA,B2ky+ 22, 2EA2ky

=−EB2ky+ 22, 3EA,B2ky=EA,B2ky+ 42, 4

E2ky=E2ky+ 42.

For certain values of 2, A- and B-type of states are degenerate. This happens when the two sets V2mkyand

V2m+1−kycoincide. This degeneracy occurs when a2 is

an irrational number and b2=p/q with q odd as

Vm+qnky=Vq−m+qn−kywhere n is an integer and mq.

For even q, the A and B states are, in general, nondegenerate.

This distinction between the odd and even cases leads to

significant differences between these two cases.

VI. THE LOCALIZATION TRANSITION

The metal-insulator transition 14 in 2DEGs in the presence of a magnetic Abelianfield is a paradigm for the

Anderson localization transition. We now discuss the corresponding localization transition that exists in the nonAbelian systems. In the metallic phase, the wave function is

extended and its Fourier transform is characterized by sharp

Bragg fringes at the reciprocal lattice vectors of the lattice. In

the insulating phase the wave function is exponentially localized resulting in the broadening of the fringes.

In contrast to the Abelian case described by Harper equation, where all states localize at the same value of the tunneling anisotropy, localization in the non-Abelian case varies

throughout the spectrum. In this section, we will discuss the

localization properties of the E= 0 state a study is relevant

for fermionic atoms near half-filling. As shown below, for

2= 1

2 as well as for 2= 14 , the onset to a localization transition can be inferred from the well-known localization characteristics of the Harper equation 9.

A. Localization boundary for 2= 1Õ2

For 2= 1

2 , the coupled TBM equations Eq. 10 for E

= 0 reduce to

m+2 + m−2 + 2− 1

m cos21m − 2kym = m,

m+1 + m−1 + 2 cos21m − 2kym = 0,

where =−2 and = 22 cos2ky. For E= 0, the uncoupled

 equation maps to an E=−2 Harper-like equation 10,

where the on-site quasiperiodic potential is a sinusoidal function of ky. The eigenstates of this system localize at =1,

providing an explicit threshold for localization of the E= 0

state of the non-Abelian system provided =−2 is the eigenvalue of Eq. 10.

As shown in Fig. 8, =−2 is an eigenvalue of the system

provided =10.48 or =21.83. These critical values

determine the boundary curves for the localization of the E

= 0 state: in the Harper equation, all states are extended for

values of 1. These two localization boundaries are exhibited in Fig. 9

B. Localization boundary for 2= 1Õ4

For 2= 1

4 , the uncoupled  equations for the A and the B

sectors of the TBM for E= 0 reduce to



 ̄

m+2

A + 

 ̄

m−2

A + 2iA cos21m − 2ky

 ̄

m

A = 0,



 ̄

m+2

B + 

 ̄

m−2

B + 2iB sin21m − 2ky

 ̄

m

B = 0,

where

FIG. 6. Color onlineEnergy spectrum viewed as a function of

ky for 1= 89/144 and 2= 1

2 with = 1 for a range of kx values.

Family of bands with minima maximaat ky= 0 correspond to

EAEB. The set of bands that show no variation with ky describe the

corresponding Abelian case with 1=2=.

FIG. 7. Color onlineEnergy spectrum viewed as a function of

ky for 1= 89/144 and 2= 1

3 with = 0.5. The set of nonintersecting bands show a corresponding Abelian case with 1=2= 1

3 .

SATIJA et al. PHYSICAL REVIEW A 77, 043410 2008

043410-6
A = 22 cos2ky, B = 22 sin2ky.

The above two equations correspond to A- and B-type states

with E= 0, respectively. Here 

 ̄

m

A,B=i

mm

A,B. In a manner

analogous to the corresponding Hermitian problem, the system exhibits self-duality at A,B= 1 and this self-dual point

describes the onset of localization 15. For ky= 1

8 mod14 ,

both types of solutions localize simultaneously. However, at

other values of the transverse momentum, only one of the

states is localized. This is an example of two degenerate

states with different transport properties: depending upon

,ky, type-A states may be extended localizedwhile

type-B states will be localized extended. This localization

boundary in -ky space is shown for types A and B in Fig.

10.

The existence of conducting states for all values of  is

one of the most intriguing characteristics of the non-Abelian

system. Below we show that these states defying localization

describe relativistic particles.

C. Relativistic dispersion and defiance of localization

Figure 11 shows the energy-momentum relation for 2

= 1/2 near E=0, ky= 1/4. Although the level structure is

complicated, near ky= 1/4, the energy bands exhibit the linear dispersion characteristic of the one-dimensional relativistic particles. Thus, the non-Abelian system with A- and

B-type states, provides an interesting manifestation of the

positive and the negative energy states of a one-dimensional

relativistic particle.

An important characteristic of the states that reside at the

crossings is that they defy localization. It should be noted

FIG. 8. Color onlineThe eigenvalues of the effective TBM

Eq. 10 describing E= 0 states as  varies. The line red online

shows the values of  where E= 0 is an eigenvalue of the TBM Eq.

10.

FIG. 9. Color onlineWith 2= 1/2 shaded regime that begins

at = 0 shows the extended phase while shaded lobes that begin at

= 1 and extend to all values of 1 show the localized phase of

E= 0 state in the -ky plane.

FIG. 10. Color onlineFor 2= 1/4, shaded regimes show extended phase for two degenerate E= 0 states with lobes centered at

ky= 0.25 and ky= 0.75 corresponding to A type while the rest of the

regime correspond to B type.

FIG. 11. Color onlineBlowup of the E= 0 and ky= 1/4 neighborhood for 1= 89/144 and 2= 1

2 and = 10. Bands going left to

right right to leftcorrespond to Ea Eb.

PHYSICS OF A TWO-DIMENSIONAL ELECTRON GAS ... PHYSICAL REVIEW A 77, 043410 2008

043410-7
that a crossing at E= 0 exists irrespective of the value of .

In other words, we have a relativistic theory for all values of

 as shown in Figs. 6 and 11. Such states remain extended

irrespective of the quasiperiodic disorder in the system as the

linear dispersion exists for the full range of  values.

For 2= 1

4 , we obtain an effective relativistic theory for

zero-energy states near ky= 0 for type-A and near ky=/2 for

type-B states. These states remain conducting for all values

of .

We would like to note that in the Abelian system, a linear

energy-momentum relation resulting in a Dirac cone occurs

for rational values of  in the neighborhood of some special

values of kx ,ky near E= 0. However, for irrational , the

spectrum is independent of ky and the Dirac cone disappears.

Therefore, in the Abelian case, effective relativistic theory

bears no relationship to the transport properties as the states

are always extended for rational .

D. Localization transition and loss of relativistic dispersion

Our detailed investigation for various values of 2 shows

that the presence of conducting states for all values of  is

not a generic property of the system. In particular, for cases

where the type-A and type-B states are always degenerate, all

states are found to localize. Interestingly, the transition to

localization is accompanied by a loss of the relativistic character of the energy momentum relation.

For example, for 2=p/q where q is odd, as well as for

irrational 2, the crossings characterizing certain E= 0 states

disappear beyond a certain critical value of . Interestingly,

this threshold for the disappearance of the crossing is always

found to coincide with the onset to localization of that state.

Figures 12 and 13 illustrate this for irrational 2 as the disappearance of band crossings is accompanied by the broadening and flattening of the Bragg peaks.

VII. LOCALIZATION TRANSITION

OF BOSE-EINSTEIN CONDENSATES

The natural locus for BEC in ultracold atoms in optical

lattices is the band edge. We now explore the spectral and

transport properties of the states at band edges, namely, the

minimum energy states as ky varies. For very low filling

factor, this is also relevant for the electron problem. In contrast to the preceding analytical treatment of the band centers, we have investigated localization properties of the states

at the band edge with numerical methods.

As seen from Figs. 6 and 14, the energy spectrum for

2= 1

2 shows the existence of a linear dispersion relation near

the band crossings. As the lattice anisotropy  varies, we see

a transition from relativistic to nonrelativistic behavior near

2.5; this transition is accompanied by the loss of the

wave function’s spinor character, causing an effective spin

polarization.

The robustness of the linear dispersion in non-Abelian

systems is shown for various values of 2 in Figs. 15 and 16.

It appears that it is only in the even-q cases that the nature of

the dispersion changes as  varies. Similarly, for 2=4 an

odd harmonic of , as 4=2−3, linear dispersion at ky

FIG. 12. Color onlineThe top and bottom panels show the

spectrum for 2=3 with = 0.75, 1.25, respectively, which correspond to extended and localized E= 0 states, respectively.

FIG. 13. Color onlineFourier transform of the wave function

for E= 0 state with 2=3 with = 0.75 sharp fringesand 

= 1.25 smeared out fringes. The label k on the x axis corresponds

to the momentum 2k/L, where L is the size of the lattice.

FIG. 14. Color onlineMinimum energy as a function of ky for

2= 1

2 with = 0.5, 1.5, 2.5, 3.5 top to bottomillustrating the

change from linear to quadratic dispersion near ky= 1/4.

SATIJA et al. PHYSICAL REVIEW A 77, 043410 2008

043410-8
= 0 and at ky= 1/4 occurs for all values of , while for 2

=3 an even harmonic of , as 3= 2− 1, a relativistic

energy-momentum relation is seen for small and large values

of  as illustrated in Fig. 16. In other words, a “transition”

from relativistic to nonrelativistic behavior can be induced

by varying  for some values of 2. Another point to be

noted is that the ground state of the system may have nonzero momentum: for even q, the global energy minimum

occurs at ky= 0, while for odd q, it occurs at ky=2 /2.

Figures 17 illustrate the localization transition of the minimum energy states. Extended states in these figures are characterized by sharp Bragg peaks in the momentum distribution, and the localization transition is signaled by the

broadening of these peaks. As we increase the parameter ,

the ky= 1/4 states localize before the ky= 0 state. Our detailed investigation shows that ky= 0 is the last state to localize as  is varied for all values of 2. This is contrary to

familiar experience, in which localization begins at the band

edge. The localization for the minimum energy state is insensitive to the energy-momentum relation, in contrast to the

E= 0 states.

For irrational 2, we expect the localization threshold to

be lowered. Our numerical results show that the minimum

energy states begin to localize at a relatively small value of

0.15. As discussed earlier, E= 0 states resist localization

due to their linear dispersion but eventually localize. Our

numerical studies show that localization is complete at 

= 1, as in the Abelian case.

VIII. EXPERIMENTAL OBSERVATION

OF METAL-INSULATOR TRANSITION

An experimental setup for generating artificial Abelian

and non-Abelian fields consists of Refs. 2,3 a twodimensional optical lattice populated with cold atoms that

occupy two hyperfine states. The lattice laser polarization is

adjusted to confine these states to alternating columns. The

non-Abelian scheme requires atoms with two pairs of hyperfine levels: g1, e1, g2, e2 as shown in Fig. 18.

The typical kinetic energy tunneling along the y direction

is suppressed by accelerating the system or applying an inhomogeneous electric field in that direction such that the

lattice is tilted. Tunneling is instead accomplished with two

sets of laser-driven Raman transitions with space-dependent

Rabi couplings jeiqjy

, where j= 1, 2. The wave numbers qj

generate an effective magnetic flux where qj=2j/a,

where = 2a is the wavelength of the laser light. In an optical

lattice with a finite number of sites, the two components of

the “magnetic flux” 1 ,2can be adjusted, in a controlled

FIG. 15. Color onlineMinimum energy as a function of ky for

2= 1/4 cross, 2= 1/5 dotswith = 1.5, 2.5, 3.5, 4.5 top to

bottomillustrating the change from the linear to quadratic dispersion near ky= 1/4.

FIG. 16. Color onlineVariation of minimum energy with ky for

2=3 large crossesand 2=4 smaller crosseswith 

= 1.5, 2.5, 3.5, 4.5, 5.5, 6.5 top to bottom.

FIG. 17. Color onlineFourier transform of the wave function

for = 0.4, 0.5, 0.7 top to bottom. Each caption shows ky= 0.25

redand ky= 0 bluefor the minimum energy state for 2= 1/2.

The x axis index k corresponds to the Bloch vector 2k/L. There

are peaks corresponding to the irrational values of 1, occurring at

k values equal to half of the Fibonacci numbers as antiferromagnetic nature effectively doubles the size of the unit cell. Additional

satellite peaks characterize the non-Abelian feature of the system.

PHYSICS OF A TWO-DIMENSIONAL ELECTRON GAS ... PHYSICAL REVIEW A 77, 043410 2008

043410-9
manner, to a sequence of rational approximants to the golden

mean by tuning the qj. We direct readers to Refs. 2,3 regarding various details for generating these artificial gauge

fields.

We now describe the experimental feasibility of tuning 

to induce metal-insulator transitions by adjusting the lattice

beam intensity V0. For simplicity, we will initially discuss

the Abelian case. Let us first consider the laser assisted coupling Jy as a function of V0 /ER, where ER= 22

2 /M2 is the

photon recoil energy. The tunneling is defined as the matrix

element of the Rabi coupling between Wannier functions

w, evaluated at the two adjacent lattice sites

Jy =  wx − xi





2

 expiqxwx − xi−1d3

x, 10

where q=2/a. The Wannier functions for Vx

=V0 sin22x/have been computed 2; Jy decreases

monotonically with V0 /ER. This basic behavior can be demonstrated analytically by assuming a deep lattice approximated by a harmonic oscillator potential and taking the Wannier functions to have the corresponding Gaussian form. The

Gaussian approximation yields

Jy = 

2 exp
− 2

16V0/ERexp
− 2

V0/ER

. 11

The kinetic energy coupling in the x direction Jx also decreases monotonically with V0 /ER for sufficiently large values 16 as described by

Jx  1.397ER V0

ER



1.051

exp− 2.121V0/ER. 12

The ratio of =Jx /Jy=ER/fV0 /ER,is shown in Fig.

19 for a characteristic range of V0 /ER with the scale set by

the factor /ER. In order to generate a useful range of 

values e.g., 02, the parameter /ER must be set to

order unity.

We now argue that it is possible to achieve this with reasonable experimental settings. We consider the case of 87Rb,

where the g and e states are taken to be the hyperfine

levels of the 52

S1/2 level 17 and the Raman level is 52P3/2

 .

The parameter /ER can be fixed near unity if the Raman

laser beams have intensities on the order of 1 mW/cm2 and

are detuned from the 87Rb D2 line by about r100 GHz.

We require that the Rabi coupling of the Raman transition ,

the detuning 	, and the lattice trapping frequency x

=4ERV0 /have well separated magnitudes such that 

	x, to ensure that only the lowest band of the lattice is

occupied and no other excitations occur. Typical values of x

are on the order of tens of kilohertz. The Raman transition is

stimulated by two lasers with Rabi couplings g

1and e1

and intensities Ig and Ie with a large detuning r such that the

effective Rabi coupling magnitude is =g

1

e

1

/2r. The

Rabi coupling  can be written as a product of atomic factors and laser tuning parameters

 =  2

4Isat

  Ig

r

,

where  is the natural decay rate of the 52P3/2

° state and Isat is

the saturation intensity of the D2 line see Ref 17.. The

ratio =Ie /Ig must be less than 0.17 or greater than 5.8 to

satisfy the 	 condition. The separation of scales between

the Rabi coupling  of the Raman transition and lattice trapping frequency x necessary to generate the magnetic field in

the above scheme i.e., xis sufficient to generate a

reasonable range of  values.

In the non-Abelian case, there are generally two possible

values of  corresponding to 1 and 2, one for each

“color.” By adjusting 2 /1= fV0 /ER,1/ fV0 /ER,2,

we obtain a single  in correspondence with the theoretical

studies described here.

FIG. 18. Color onlineSchematic diagram illustrating the nonAbelian U2setup. The ground states are hollow and the excited

states are filled. Red and blue represent the two “colors” of the U2

group.

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

V0/ER

Λ (hΩ/2πER)

2.1

Λ

FIG. 19. Color online as the depth of the 2D optical lattice

is tuned with = 1

2 . The factor ER/4.2 with the following laser

parameters: Ig= 1 mW/cm2, = 11, r= 100 GHz, ER/h= 3.2 kHz.

SATIJA et al. PHYSICAL REVIEW A 77, 043410 2008

043410-10
IX. SUMMARY

This paper discusses spectral and transport properties of

the cold atom analog of a 2DEG in a lattice, subject to a

non-Abelian gauge field with U2symmetry. In the continuum limit of the lattice, the system maps onto two oppositely charged coupled harmonic oscillators, with a coupling

constant proportional to the strength of the non-Abelian field.

The Landau energy levels of the Abelian problem evolve into

entangled states of this particle-hole pair.

These features also characterize the energy spectrum of

the corresponding lattice problem. In fact, the transition from

Landau levels to Landau bands is the analog of the generalization from the butterfly to the moth spectrum as the Abelian

system becomes non-Abelian. The non-Abelian coupling

breaks the degeneracy of the Landau levels; the spectrum

depends explicitly on the transverse momentum.

The non-Abelian system exhibits antiferrimagnetic-type

ground states, whose components A and B, need not be degenerate, and in fact may have very different transport properties. A particularly interesting example of this is the zeroenergy state for 2= 1/4, where the degenerate A and B

components have different localization properties. Additionally, an intriguing relationship between the A and B components occurs for 2= 1

2 , as these two components correspond

to the positive and the negative energy states of the system.

Such novelties may open new avenues for exploring frontiers

of physics with cold atoms.

The use of ultracold atoms to simulate relativistic as well

as nonrelativistic theories and study the effect of disorder is

an exciting field of research. In a two-dimensional lattice

subject to a non-Abelian gauge field, one can induce not only

localization transitions, but also a transition from relativistic

to nonrelativistic theory by tuning the lattice anisotropy. A

well-known feature of the Dirac Hamiltonian is an extra term

in the conductivity attributed to Zitterbewegung ZB18

corresponding to interband transitions. It has been suggested

that such a term is responsible for the finite conductivity of

graphene described by a massless Dirac energy spectrum

18. In other words, it is ZB that makes it impossible to

localize relativistic particles, as it is connected with the uncertainty of the position of a relativistic quantum particle due

to the creation of particle-antiparticle pairs. Therefore, the

origin of delocalization characterizing the non-Abelian system that persists even for infinite disorder →!can be

attributed to ZB.

The detection of relativistic particle and a transition from

nonrelativistic to relativistic dispersion in cold atoms in optical lattices was recently discussed; it was shown that the

relativistic dispersion can be detected using atomic density

profiles as well as Bragg spectroscopy 12. Our detailed

study for various values of 2 captures some of the universal

features of non-Abelian systems. Exploration of the twodimensional space 1 ,2may reveal additional phenomena, and the richness of UNgauge systems with N2

remains to be explored. Moreover, the effects of interparticle

interactions remain to be investigated 19.

ACKNOWLEDGMENTS

We are grateful to Ashwin Rastogi for his efforts in initiating the study of the continuum limit of the problem and Jay

Hanssen for helpful discussions regarding experimental aspects of this work. We would also like to thank Ian Spielman

and Nathan Goldman for their comments and suggestions on

the paper.

1 J. Ruseckas, G. Juzeliunas, P. Ohberg, and M. Fleischhauer,

Phys. Rev. Lett. 95, 010404 2005.

2 D. Jaksch and P. Zoller, New J. Phys. 5, 56 2003.

3 K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein,

Phys. Rev. Lett. 95, 010403 2005.

4 D. R. Hofstadter, Phys. Rev. B 14, 2239 1976.

5 E. J. Mueller, Phys. Rev. A 70, 041603R2004.

6 N. Goldman and P. Gaspard, Europhys. Lett. 78, 60001

2007.

7 R. Prange, in The Quantum Hall Effect, edited by R. Prange

and S. Girvin Springer-Verlag, Berlin, 1990.

8 L. Onsanger, Philos. Mag. 43, 1006 1952.

9 P. G. Harper, Proc. Phys. Soc., London, Sect. A 68, 879

1955; S. Aubry and G. Andre, Ann. Isr. Phys. Soc. 3, 133

1980.

10 K. Drese and M. Holthaus, Phys. Rev. Lett. 78, 2932 1997.

11 I. I. Satija, D. C. Dakin, and C. W. Clark, Phys. Rev. Lett. 97,

216401 2006; 98, 269904E2007.

12 S.-L. Zhu, B. Wang, and L. M. Duan, Phys. Rev. Lett. 98,

260402 2007.

13 L. Balents and M. P. A. Fisher, Phys. Rev. B 56, 12970 1997.

14 For a review, see J. B. Sokoloff, Phys. Rep. 126, 189 1985.

15 A. Jazaeri and I. I. Satija, Phys. Rev. E 63, 036222 2001.

16 A. M. Rey, G. Pupillo, C. W. Clark, and C. J. Williams, Phys.

Rev. A 72, 033616 2005.

17 D. A. Steck data available at http://steck.us/alkalidata

18 M. I. Katsnelson, e-print arXiv:cond-mat/0512337.

19 N. Goldman, Europhys. Lett. 80, 20001 2007.

PHYSICS OF A TWO-DIMENSIONAL ELECTRON GAS ... PHYSICAL REVIEW A 77, 043410 2008

043410-11