Exact bosonization for interacting fermions in arbitrary dimensions.

(New route to numerical and analytical calculations)

K.B. Efetov, C. Pepin , H. Meier

Bosonization: mapping of electron models onto a model describing collective excitations (charge, spin excitations, diffusion modes, etc).

RUB Bochum, Germany, CEA Saclay, France

1 12

1 2

21.04.23 2

Origin of the word: 1D electron systems.

The main idea: writing the electronic operators as

dxii exp)exp(

dxNKH ][ 22

A simple Hamiltonian H

i ,

-Density fluctuations operator

K-compressibility, N-average density

The most general form conjectured by K.E. & A. Larkin (1975)

Bosonization: for Tomonaga-Luttinger model (long range interaction) (Luttinger, Tomonaga (196?))

Microscopic theory Haldane (1982)

…………………Importance of long wave length excitations!

21.04.23 3

Formal replacement of electron Green functions by propagators for collective excitations!

Equivalent representation

Spin excitations

Transformation from electrons to collective excitations: Bosonization

Very often direct expansions with electronic Green functions are not efficient (infrared divergences, high energy cutoffs respecting symmetries).

21.04.23 4

Difficulties in Monte Carlo simulations for fermionic systems: negative sign problem exponential growth of the computation time with the size of the system.

(From a lecture by M. Troyer)

NP-nondeterministic polynomial

21.04.23 5

Can one bosonize in higher dimensions?

Earlier attempts:

A. Luther 1979: Special form of Fermi surface (square, cube, etc.). Almost 1D.

F.D.M. Haldane 1992: Patching of the Fermi surface, no around corner scattering

Further development of the patching idea:

A. Houghton & Marston 1993; A.H. Castro Neto & E. Fradkin 1994; P. Kopietz & Schonhammer 1996; Khveshchenko, R. Hlubina, T.M. Rice 1994 et al; C. Castellani, Di Castro, W. Metzner 1994……

Main assumption of all these works: long range interaction.

21.04.23 6

I.L. Aleiner & K. B. Efetov 2006, Method of quasiclassical Green functions supplemented by integration over supervectors

Logarithmic contributions to specific heat and susceptibility are found. Good agreement with known results in 1D. No restriction on the range of interaction but not a full account of effects of the Fermi surface curvature in d>1.

In all the approaches only low energy excitations are considered: no chance for using in numerics.

Present work: Exact mapping fermion models onto bosonic ones. New possibilities for both analytical and numerical computations.

21.04.23 7

Starting Hamiltonian H

int0ˆˆˆ HHH

;', ;',

'',0ˆ

rr rrrrrrrr cccctH

rrrrrr

rr ccccVH ''''',;',

',int 2

1ˆ

The interaction, tunneling and dimensionality are arbitrary!

Small simplification of the formulas

0, 00',', VVV rrrr

r

rrrr ccccV

H2

,,,,0)0(

int 2ˆ

2/' V

)/ˆexp( THTrZ

21.04.23 8

Hubbard-Stratonovich Transformation with a real field )(

T/1),()(

DdrV

ZZr

0

2

0

,2

1exp

dTrZ rrr

0

, ')(ˆ/lnexp '

'',ˆr

rrrrr ftf

21.04.23 9

Another representation for Z

0, 'ˆexp1det dTZ rrr

Basis of auxiliary field Monte Carlo simulations (Blankenbecler, Scalapino, Sugar (1981))

Procedure: subdividing the interval into time slices and calculation for each slice recursively.

0

However, can in such a procedure be both positive and negative: sign problem! Not convenient for analytical calculations either.

Z

21.04.23 10

Good news: can exactly be obtained from a bosonic model!

][Z

Derivation of the model (main steps):

,

0

1

0 ;,0 0,expr

rrr dudGZZ

'','ˆ ',;',

rrrrrr Gru

',;', rrG -fermionic Green function in the external field.

Boundary conditions

,,', GGG

21.04.23 11

Looking for the solution: (in spirits of a Schwinger Ansatz)

''', 1','''

0''',''

''','''',', rrrr

rrrrrr TGTG ',

1',''

'''', rrrr

rrr TT

'0', rrG is the Green function of the ideal Fermi gas

Equations for and T 1T

TTBosonic periodic boundary conditions:

0ˆˆ ',','',

zTuzTzT rrrrrrrrr

0ˆˆ 1','

1','

1',

zTuzTzT rrrrrrrrr

},,{ uz

Solving these equations is not convenient. Solution is not unambiguous.

21.04.23 12

New function A:

zTnzTnzTnzTzArr

rrrrrrrrrr1

''',''',

1','''''','''',', ˆ,)(

',rrn -Fourier transform of the Fermi distribution 1

1'

pepn

or

00','', rrrr Gn

,

1

0 0

0 expr

rrr dudzAZZ

Then, the “partition function” is Z

What is good in that? How to find A without solving equation for T?

21.04.23 13

“Free lunch”:

0ˆˆ ',','',

zTuzTzT rrrrrrrrr

0ˆˆ 1','

1','

1',

zTuzTzT rrrrrrrrr

a)

b)

nffn rr ˆ,ˆ..., an operator

1. Act with on a) and multiply by from the right. n̂..., 1T

2. Multiply b) by from the left. nT ˆ,

3. Subtract the obtained equations from each other.

21.04.23 14

Final equation for A

'',',''ˆˆ rrrrrrrrrr unzAu

,

1

0 0

0 expr

rrr dudzAZZ

Partition function

Boundary conditions

',',, rrrrrr AA

DrV

ZZ

0

2

0

,2

1exp

Purely bosonic problem! Linear (almost separable) real equation for A

},,{ uz

New possibilities for both numerical and analytical studies!

21.04.23 15

How can one calculate numerically ? (Non-expert’s view)

Z

Introduce the Green function g for the linear equation and represent it in the form:

',',

1

1',1',1',1',;', 1111,ˆ0,ˆ1,ˆ, rrrrrrrrrrrrrr PPPg

where the operator P is

',,'exp,ˆ

1

',1',

duhTP rrrr

0'

''', ˆˆˆrrrrrr uh

',,,

1

0 0 0

'1',;,0

11

1111''',exp

rrrrrrrrrr duddgZZ

Possibility to do recursion in time (as usual)! Compensation of the Bose-denominator no singularities in g. In the absence of singularities, the exponent remains real for any subdivision of the interval into slices is positive: no sign problem!

Z ,0

21.04.23 16

Perturbation theory: expanding in and subsequent integration over this field.

r

Main order:

d

d

d

d pd

kpkpi

kpnkpnV

kdTZZ

22/2/

2/2/1ln

22exp 00

RPA-like formula (first order in the expansion in collective excitations).

21.04.23 17

Analytical calculations: BRST (Becchi, Rouet, Stora, Tyutin)

-possibility of integration over the auxiliary field before doing approximations

How to calculate if is the solution of the equation ?

0AB 0A 0AF

A well known trick: daa

FaFaBAB

det0

Next step: dfaifFCaF exp

ddaFa

F/expdet

-Grassmann variables ,

21.04.23 18

Description with a supersymmetric action and superfields *,Introducing new Grassmann variables and superfields

*',',

*',',', zzzfzaR rrrr

Trrrrrr

is anticommuting (!)

The “partition function” as the functional integral over Z

*,,,, uR

sbss SSZZ exp0

',

',',,'ˆ

2 rrrrrrrrss dRh

iS

r rr

rrrrrrrrrsb udRnidRS',

','',*

',

Gaussian averaging over can immediately be performed!

''', ˆˆˆrrrrrr uh

21.04.23 19

DSZZ ][exp0

-partition function of the ideal Fermi gas0Z

IB SSSS 0

',

',','0 ˆˆ2 rr

rrrrrr dRi

S

Final superfield theory (still exact).

is the bare action (fully supersymmetric)

0S

111*11,

*,1

0 22

dRdRRiRRV

S rrrrrB

111'10

2dRdRRR

VS

rrrI

'

',',',,'r

rrrrrrrrr nRnRuR

21.04.23 20

Logarithmic contributions exist in any dimensions and they can be studied by RG. Reduction of the exact model to a low energy one is convenient. Variety of phenomena for, e.g., cuprates and other strongly correlated systems can be attacked in this way.

The terms and are invariant under the transformation of the fields :

0S IS

',**

',*

', ,, rrrrrr n

*, -anticommuting variables

(Almost) supersymmetry transformation.

What to calculate?

21.04.23 21

Conclusions.

The model of interacting fermions can be bosonized in any dimension for any reasonable interaction: free lunch but…..

What about a free dinner? Can the bosonization be the key to the ultimate solution of the sign problem? Can one solve non-trivial models (e.g., models for high temperature cuprates or for quantum phase transitions) using the supersymetric model for collective excitations?

To be answered