Equation-of-Motion Series Expansion of

Double-Time Greens Functions

Department of Physcis

Renmin University of China

October 23, 2015 Institute of Physics, CAS

Ning-Hua Tong

Outline

1. Introduction to GF series expansion

2. Introduction to GF EOM

3. EOM expansion & resummation

4. Application to Anderson impurity model

5. Discussion and Summary

Greens function G

an important tool for studying quantum many-body systems

1. Introduction to GF series expansion

Methods of Quantum Field Theory in Statistical Physics

A.A.Abrikosov, L.P.Gorkov, and I.E.Dzyaloshinski 1975.

GFThermodynamicsE, F, U, S, C, M,

Dynamics

H: the electron + ion Hamiltonian for the material

a quantum many-body problem

(1) weak-coupling expansion

Small parameter expansion of GF

Exactly solvableH0 perburbationH1

H0 non-interacting part H1 interaction.

Interacting picture

Taylor expansion

A Guide to Feynman Diagrams in the Many-body Problem

R. D. Mattuck 1992.

Wicks theorem

Diagram representation

(2) strong-coupling expansion

(a) Shvaika Hubbard operator + Wick-like theorem

A. M. Shvaika, Phys. Rev. B 62, 2358 (2000).

(b) W. Metzner Cumulant expansion

Cumulant

W. Metzner, Phys. Rev. B 43, 8549 (1991).

1-particle GF

(c) Sarker dual fermion approach

S. K. Sarker, J. Phys. C: Solid State Phys. 21, L667 (1988).

Grassmann Hubbard-Stratonovich transfromation

Cluster Pertrubation Theory Senechal et al. 2000

2-D Hubbard model Pairault et al. 1998

Dual fermion DMFT Rubtsov et al. 2008

transform strong-coupling problem of original fermions

into weak-coupling problem of dual fermions

many applications

(3) Quantum Monte Carlo expansion of GF

(a) Determinant QMC

(b) Diagrammatic QMC

E. Gull, A. J.Millis, A. I. Lichtenstein, A. N. Rubtsov, M.Troyer,

and P. Werner, Rev. Mod. Phys. 83, 349 (2011).

K. Van Houcke, F. Werner, N. V. Prokofev, and B. V. Svistunov, arXiv: 1305.3901;

Y. Deng et al., arXiv:1408.2088

Features of these methods:

(1) One needs H0- dependent rules to construct

the expansion.

(2) Many time/momentum integration involved.

2. Introduction to GF EOM

2-time retarded GF

EOM

N. N. Bogolyubov and S. V. Tyablikov, Sov. Phys. Dokl. 4, 589 (1959).

P. W. Anderson, Phys. Rev. 124, 41 (1961).

J. Hubbard, Proc. Roy. Soc. Lond. A 276, 238 (1963).

not close in finite order truncation required.

Problem of EOM method

empirical, only for specific problem, causality no guarantee !

Some well known truncation schemes

Heisenberg model H. B. Callen, PR 130, 890 (1963).

C. Lacroix, JPC 11, 2389 (1981).

H. G. Luo et al. PRB 59, 9710 (1999).Anderson impurity

Hubbard model J. Hubbard, Proc. Roy. Soc. Lond. 281, 401 (1964).

J. X. Zhu et al, arXiv: 0409215.

causality

violated

analytical structure of GF no guarantee

example

3. EOM expansion & resummation

H0 is exactly solvable H1 is regarded as perturbation.

Suppose H0 is EOM-solvable. That is,

closes in finite order for any operator A

The exact definition of EOM-solvability is still an open problem.

Some H with special symmetry of Liouvillian is EOM-solvable.

If eigen-energies and -states of H0 is obtained,

the EOM of GF can be solved exactly.

suppose

standard basis operator (SBO)

Formally,

substitue them into EOM of GF, comparing

coefficients of on two sides of equation

Derivation of expansion formula

EOM of Gi(A|B)

EOM for GF residue

RecursionGn(A|B) Gn-1(A|B) G0(A|B)

expansion of average

residue of average

Analytical structure of GF

real simple

poles

1/T only on

exponent

Taylor expansion

Resummation

real

simple

poles,

right !

real 2nd

order poles

problem !

1/T factor,

problem !

Lehmann representation of order i = 0, 1

S. Pairault et al, Phys. Rev. Lett. 80, 5389 (1998).Example-1

Example-2

A. Sherman et al,

Phys. Rev. B. 73, 155105 (2006).

Example-3 G. Stefanucci et al, Phys. Rev. B. 90, 115134 (2014).

Truncated series expansion of GF

(1) has wrong analytical structure

(2) diverges at T = 0

Do resummation to remove these problems.

(1) self-energy resummation

suppose we obtain GF expansion

self-energy

it cannot solve either problem ! See data.

Put it into Dyson equation, one obtains

(2) continued fraction resummation

S. Pairault et al, Phys. Rev. Lett. 80, 5389 (1998).

If bk is real and real ak >= 0, GCF has only real simple poles !

solve problem (1)

but not (2)

(3) CF resummation with Self-Consistent

EOM expansion

definerenormalized 0th order GF

full average

definerenormalized 1st order GF

EOM of 0th order residue

EOM of 1st order residue

Iterate this process, one produces

EOM of residue

Self-consistent calculating of averages

self-consistent EOM expansion scheme

GF expansion using sc-EOM formula

continued fraction resummation

self-consistent average calculation

restore analytical

structure

overcome 1/T

problem

4. Application to Anderson impurity model

hybridyzation function

(a) weak-coupling expansion

agree with Yamadas result

K. Yamada, Prog. Theor. Phys. 53, 970 (1975);

A. Georges and G. Kotlair, PRB 45, 6479 (1992).

Choose

1/T problem, disappear in p-h

symmetric and paramagnetic case.

(b) strong-coupling expansion

Standard Basis Operator (SBO)

H0 is written as

0-th order GF

EOM of the i-th order GF (i >= 1)

For p-h symmetric case

SE resummation

CF resummation

causality not fulfilled ! X. Dai et al., Phys. Rev. B 72, 045111 (2005).

when extended to magnetic case, both results break down

due to zero-temperature divergence problem.

causality fulfilled !

not real

simple pole !In paramagnetic bath

Self-consistent EOM expansion

self-consistent calculation of averages

Using the self-consistent averages, calculate the LDOS

With CF resummation

Calculation Results

Anderson impurity model with Lorentzian hybridization function

This system has particle-hole symmetry.

U=

0.0, 0.2, 0.5,

1.0, 2.0, 3.0,

4.0, 5.0,

from top to

bottom at

small omega.

FDM-NRG,

self-energy

trick, Nz=8

paramagnetic

In PM phase,

CF = SC ~ NRG

SE is poor !

paramagnetic

U=

0.0, 0.2, 0.5,

1.0, 2.0, 3.0,

4.0, 5.0,

from top to

bottom at

small omega.

magnetic bath

Qualitative

agreement

at large U

and high T

In Magnetic

phase,

CF neq SC,

~ NRG

SE is poor !

magnetic

T=

0.005 (black)

0.01 (red)

0.03 (green)

0.1 (purple)

magnetic

SE and CF:

poles shifts

with T !

SC: no shift !

magnetic bathValidity range of present approach

5. Discussion and Summary

(1) Features of this method

EOM for GF expansion residue available

universalany solvable H0 and A, B operators

1-particle, 2-particle GF

(2) Possible extensions

more effective resummation methods

(3) Possible applications

spin excitations in large-U Hubbard model

perturbations based on exact diagonalization

DMFT impurity solver for multi-band models

multiple-variable expansion

simultaneous weak-and strong-coupling expansion

eg, both and are correct.

Summary

1. We developed an expansion method for double-time GF,

which is universal and practical.

2. CF resummation method is used to restore causality

of GFself-consistent EOM expansion used to remedy

T=0 divergence problem.

4. Future extensions and applications are possible.

3. We apply this method to Anderson impurity model and

compare the results with NRG agree well in validity range.

N. H. Tong, arXiv: 1507.06407.

Thank you