NCTS One-Day Workshop

Emergence of an excitonic collective modeand “light-fermion” problem
in the dilute electron gas
Yasutami Takada
The Institute for Solid State Physics, The University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581
takada@issp.u-tokyo.ac.jp
Rm. 916, Condensed Matter and New Physics Building, NTU, Taipei
16:30-17:30, Friday, 16 December 2016

cf. YT, Phys. Rev. B 94, 245106 (2016): Editors’ Suggestion
Excitonic collective mode in the dilute electron gas (Takada)

1Outline
1. Introduction○ The electron-gas model: Overview of its ground-state phase diagramand developments of theoretical tools (EPX, GW, etc.)

2. Momentum distribution function: n(p)○ Its rigorous asymptotic form and three sum rules (total electronnumber, total kinetic energy, total kinetic-energy fluctuation）○ Gori-Giorgi & Ziesche（GZ）theory for n(p) [PRB66, 235116 (2002)]○ Improved on GZ to obtain n(p) down to rs~20
 Discovery of an interesting anomalous feature in n(p)3. Dynamical structure factor: S(q,)○ Twin-peak structure; one for plasmon, the other for excitonic collective Discovery of a new singular point in the dielectric function (q,)4. Discussion: Anomalous behavior and physical interpretation○ Decrease of the quasi-particle effective mass m* (“light-fermion”) andcompetition with the exciton-formation instability

5. Summary
Excitonic collective mode in the dilute electron gas (Takada)2Electron-Gas Model: Reduction from First Principles◎ First-principles Hamiltonian○ Plane-wave expansion for the electron field=V(q)◎ If we write a0 for the lattice constant, H is reduced to an atomic or molecularsystem for a0∞, while H is reduced to the electron gas for a00, because
only K=0 is relevant.
 Universal properties of the dilute metals can be investigatedby the study of the dilute electron gas.

○ In general, for a0 pF≪1, H is reduced to “an generalized electron-gas model” inwhich the band effects are treated in terms of the k・p perturbation and the long-range
Coulomb interactions are included in the diagonal part of the k・p matrix.
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Excitonic collective mode in the dilute electron gas (Takada)

Ground States in the Electron Gas： Overview
The interaction part is the same in both the first-principles H and the electron gas.
 All the many-body effects must manifest themselves in the electron gas.
 Need to know all possible phases in the electron gas with the change of n.

=(4/9)1/3~0.5211

Dielectric catastrophe：

Compressibility  diverges at rs=5.25. 
For rs>5.25,  < 0: by the compressibility
sum rule (at q  0, (q,0)  n2 ),(q,0) [=1+V(q)(q,)] < 0 at small q.
rs :density parameterc（rs）:correlation energyF : for a free-electron systemExcitonic collective mode in the dilute electron gas (Takada)

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EPX: Perturbation Method for Metallic-Density Region1) Fermi gas for rs<1 RPA+Fock term:2) Fermi liquid for 1≦rs≦5Proper combination of RPA+Ladder+exchange
Variational version of the Coupled-Cluster method
 EPX (Effective-Potential eXpansion method) YT, PRA28, 2417 (1983);
PRB35, 6923 (1987)

H=H0+V
Variationally determining～
the effective potential V
by an Euler-Lagrange eq.
(equivalent to the BetheSalpeter eq.)
 No convergent results
obtained for rs>5！
Dynamic responses are notavailable by this method！
Excitonic collective mode in the dilute electron gas (Takada)

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GW: Nonperturbative Approach for Metallic Region
1) Solve a set of exact relations among physicalquantities nonperturbatively, rather than the sumof terms in the perturbation expansion.
 GWmethod based on the Hedin eqs. [PR139,A796(1965)] YT, PRL87, 226402 (2001)
2) Key quantity: Vertex part [G] as a functional of
the one-electron Green’s function G.
A trial functional form was proposed by using
the compressibility sum rule and Ward identity.
[G+(q): local-field factor]

No self-consistent solutions are obtained for rs>5 by this functional!
Excitonic collective mode in the dilute electron gas (Takada)

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Improvement for the Dielectric-Catastrophe Region
1) Exclude the polarization function (q) from the iteration loop.
2) Improve on the functional [G] by including the momentum sum rule(back-flow effect).
Maebashi &YT, PRB84, 245134 (2011)1) No convergent solutions are obtained even in this functional for rs>10.
2) When the convergent results are given, the effective mass m* at the Fermi
level decreases with rs, seemingly approaching zero. Is this physically correct?
Excitonic collective mode in the dilute electron gas (Takada)

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Momentum Distribution Function n(p)◎Definition：◎Exact asymptotic behavior（Kimball condition）：◎Sum rules： cf. YT &Yasuhara, PRB44, 7879 (1991)

Excitonic collective mode in the dilute electron gas (Takada)

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Charge Correlation Function◎Charge correlation function：◎Polarization：◎Local-field factor：G+(q)  Effective interaction between bare electron and hole

◎Asymptotic behavior：cf. Moroni, Ceperley & Senatore, PRL75, 689 (1995)

Excitonic collective mode in the dilute electron gas (Takada)

9Another Local Field Factor Gs(q)◎WI(q,i)：◎Local field factor: Gs(q)  Effective interaction between dressed electron and holecf. Niklasson, PRB10, 3052 (1974)◎ On-top value of the pair correlation：

=(4/9)1/3~0.5211

Excitonic collective mode in the dilute electron gas (Takada)

10New Sum Rule for Total Kinetic-Energy Fluctuation◎Relation between the two types of polarization functions：◎Asymptotic expansion of the polarization function：◎Sum rules： By comparing each term order by order, we obtain
Total electron number 
Total kinetic energy New sum rule : Total fluctuations of the kinetic energy
Excitonic collective mode in the dilute electron gas (Takada)11

Calculated n(p) in GW
○ Accuracy of n(p) can be assessed by the three sum rules.
 GW provides very accurate n(p) up to rs=10, but not for RMC!
cf. Maebashi & YT, PRB84, 245134(2011)

Excitonic collective mode in the dilute electron gas (Takada)12

Interpolation Scheme of GZ◎ In order to obtain n(p) for rs>10, let us combine the results of n(p)for rs<10 with that in the Wigner-crystal state at rs ∞： Sophisticated interpolation scheme of Gori-Giorge & Ziesche (GS)

cf. GZ, PRB66, 235116 (2002)◎ Parameters n0 and n± will be determined by interpolation;◎ parameters a± and a∞ will be determined by three sum rules.
Excitonic collective mode in the dilute electron gas (Takada)

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Determination of n0 and n±◎Determine n0 and n± by
interpolating the GW datafor 1≦rs≦10 and n(p) in the
Wigner-crystal state at rs ∞：cf. In GZ, the EPX data for1≦rs≦5 were used instead.

Excitonic collective mode in the dilute electron gas (Takada)

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Example of Calculated n(p)◎ Determine the parameters a±
and a∞ by three sum rules.
cf. In GZ, only first two sum rules
were used instead,with assuming a+=a-.

◎ No appreciable difference isseen by the choice of different
formulas for g(0).
◎ As seen at rs=8, the obtainedn(p) in this interpolation
scheme reproduces theaccurate n(p) given in GW
for rs≦10.

Excitonic collective mode in the dilute electron gas (Takada)

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Interpolated Momentum Distribution Function○ Behavior of n(p) in the wide range of rs up to about 20.
For the case of g(0) in the Yasuhara’s formula:

Excitonic collective mode in the dilute electron gas (Takada)16

Anomalous Behavior of n(p) near the Fermi Level◎Behavior of n(p) near the Fermi level:
A±(rs): Fermi-edge coefficient (coefficient of
the logarithmic singularity of dn(x)/dx at x=1± 0±)

cf. If the electron-hole symmetry strictly holds in theelectronic excitations, we obtain A+(rs)=A-(rs),as was assumed in GZ.

◎ For rs>12, electron-hole symmetry isclearly broken.
cf. In GZ, no solution for n(p) was obtained for rs>12.

◎ A-(rs) diverges at rs~22.
This may indicate some new phase transition at rs～22!
Excitonic collective mode in the dilute electron gas (Takada)

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Dynamical Structure Factor S(q,)◎ At T=0, S(q,) is given by◎ Retarded polarization function WR(q,)Due to the inclusion of multiple electron-holeexcitations, Im WR(q,) > 0 for >0.
Excitonic collective mode in the dilute electron gas (Takada)

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Dynamical Local-Field Factor GsR(q,)◎ On imaginary -axis, Richardson-Ashcroft (RA) form Gs(q,i)
is known to be very accurate. cf. RA, Phys. Rev. B 50, 8170 (1994).

◎The RA form is improved byoptimizing free parameters,
RA, 1, 2, 1, 2, and 3.
 Analytic continuation
to give GsR(q,)Excitonic collective mode in the dilute electron gas (Takada)19

Calculated S(q,)◎ At metallic densities, thereappears the shoulder structure
due to the excitonic effect in asingle electron-hole excitation.

◎ Twin-peak structure for rs>12!

cf. YT, PRB94, 245106 (2016)

cf. YT and Yasuhara, PRL89,216402 (2002).

◎ The shoulder structure
evolves as rs increases.
Excitonic collective mode in the dilute electron gas (Takada)

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Singularity in Dielectric Function R(q,)◎ Singularity in the normal metal
 Single point at (q,=(0,0)◎ Singularity in the dielectric catastrophe regime
 Double points at (q,=(0,0) and (qex,0),
where qex(~2pF) is determined by the ghost
exciton mode at which (q,i) diverges.

Excitonic collective mode in the dilute electron gas (Takada)

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Excitonic Collective Mode
◎ Effect of singularity increases as rs increases.
 For rs>12, R(q,) ~ 0 not only at theplasmon but also at another mode or theexcitonic collective mode, leading to thetwin-peak structure in S(q,).

Excitonic collective mode in the dilute electron gas (Takada)22

Dispersion of the Excitonic Collective Mode
◎ While the plasmon occurs
by the coherent excitationsof mutually uncorrelated
electrons and holes, theexcitonic collective modeis associated with theexcitations of a large
collection of excitons (ortightly bound electron-holepairs).
◎ At rs~30, ex(q)~0 at q=2pF, leading to the spontaneous
excitation of macroscopic number of the excitonic
collective modes.  Spontaneous exciton formation!
Excitonic collective mode in the dilute electron gas (Takada)

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Light-Fermion Problem◎ In the dielectric catastrophe regime, the quasiparticle
effective mass m* rapidly decreases as rs increases.
 “Light-electron problem”◎ Note that m* becomes zero in the Hartree-Fock approximation
due to the long-range nature of the bare Coulomb interaction.
◎ If the excitonic collective mode dominates the plasmon, themetallic screening effect becomes much weaker, making theeffective interaction approach the bare Coulomb interaction.
Excitonic collective mode in the dilute electron gas (Takada)24

Spontaneous Exciton Formation vs Light Electron○The physics of the spontaneous exciton formation can be consideredas the same as that in the emergence of an excitonic insulator insemimetals and semiconductors; namely, the dominance of theelectron-hole attraction over the kinetic-energy loss of the electronhole pair creation.
cf. W. Kohn, PRL19, 439 (1967)○ If mapproaches zero, however, the kinetic energy, which competes
with the attractive electron-hole interaction, becomes very large,
making it difficult to obtain the spontaneous multiple electron-holeexcitations. Then, instead of the spontaneously excited excitonic
phase, an anomalous liquid state characterized by m*=0 may appear!
Excitonic collective mode in the dilute electron gas (Takada)

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Summary10 Derived a new sum rule for n(p), based on which an improvement on
the interpolation scheme of Gori-Giorgi & Ziesche is successfully
made to obtain n(p) for rs>10.
20 Discovered an anomalous behavior of n(p) at rs～20, the enhancementof the excitonic collective mode in S(q,), and a new singularity inthe dielectric function R(q,).
30 Competition is predicted between a CDW-like phase of spontaneously
formed many excitons and a singular Fermi-liquid-like phase with m*=0for rs around 20; if the latter is the case, we may revise the overview ofthe phase diagram of the electron gas as follows:

Excitonic collective mode in the dilute electron gas (Takada)26