F.F. Assaad.

MPI-Stuttgart. Universität-Stuttgart.

21.10.2002

Numerical approaches to the correlated electron problem:

Quantum Monte Carlo.

The Monte Carlo method. Basic.

Spin Systems. World-lines, loops and stochastic series expansions.

The auxiliary field method I

The auxiliary filed method II

Ground state

Finite temperature

Hirsch-Fye.

Special topics (Kondo / Metal-Insulator transition) and outlooks.

Ground state method:CPU V3

TT

TT

eeOeH

HH

O||

||lim

22

0

00 T

t

)( ,S Hubbard 6X6

e

OeO H

H

Tr

Tr

Finite temperature: CPU V3

Ground state.

Finite temperature.

Hubbard.

0 , , ,i, j, i, i,j i

2

i, ,

,( 1/ 2)( 1/ 2)

j

i i i

id

it U n n nH c c ce c

A l

),0,0( B AB

The choice of the trial wave function for the Projector method.

Magnetic fields and size effects.

cctH jiji

,

t010.

LFK vT 2

ccetH jid

j

i

i

ji

lA0

2

,

),0,0( B AB

Scaling:

LB 2

0

Electronic system:X-Y plane.

L=16: More than an order of magnitude gain in temperature before results get dominated by size effects.

Cv/T

T T

0BL = 4,6,...16

02LB

L = 4,6,...16

Thermodynamic quantities.

T T

s

0B 02LB

L = 4,6,. ..16

L = 4,6,...16

FFA PRB 02

I. Basic formalism for the case of the Hubbard model.

),0,0( B AB

H t HU

Magnetic field in z-direction:

cccce nnnUtH iiidlA

ij

i

,i,i,i

,,,j,i,ji,

2

,)2/1)(2/1(0

)(ΨΨΨΨ τΔ)( 2HtτΔHτΔ U OZ T

L

TTH

T eee

)(TrTr τΔ)( 2HtτΔHτΔ U OZLH eee

τΔL

Trotter.

Ground state:

Finite temperature:

Hubbard.

1

)1('

1

)(τ~~s

si

s

sH eeennnnU

e

eU

U

2/

2/

)'cos(

)cosh(

0,)2/1)(2/1( UnnUHU

Breaks SU(2) spin symmetry.Symmetry is restored after summation over HS. Fields.

Complex but conservs SU(2) spin symmetry.

The choice of Hubbard Stratonovich transformation. (Decouples many body propagator into sum of single particle propagator interacting with extermal field.)

e φdφπ

A Aφe 22/

2 2

2

1 )τ()η()(γ 4τ

2,1

2τ

Oe AllA

le

Generic.

s

HH see tU )(B

ee cTccAcss

)(

)(Βwith:

γ γ( 1) 1 6 / 3 ( 2) 1 6 / 3

( 1) = 2(3- 6) ( 2) = 2(3+ 6)

01

,,

NAcAc

p

yyxxT

yyxx Pece

(1) Propagation of a Slater determinant with single body operator remains a Slater determinant.

Properties of Slater Determinants.

)'*det(Ψ'Ψ PPTT(2) Overlap:

01 1

,

N p

y

N

xyxxT Pc

, ,Tr det 1A Bc c c cA Bx y x yx y x ye e e e

(3) Trace over the Fock space:

01 1

,

N p

y

N

xyxxT Pc

Trial wave functionis slater determinant:

P is N x Np matrix.

ee TAsΒs)(

)(

Ground state.

ss

PsBsBPssss LL

LTLTZ...

1*

1...

11

)(....)(det)(....)( ΨΨ BB

Finite temperature.

ss

sBsBL

LZ...

1

1

)(....)(1det

ee cTccAcss

)(

)(Β

1

*1

...

1 ....det ( ) ( ) ( )L

LB B P O sP s sZ s s

O

1 1

1

.. ..( ) ( ) ( ) ( )( , )

...( ) ( )

LT T

LT T

Os s s sO s

s s

B B B B

B B

1

( ) 1G U U U U

For a given HS configuration Wick‘s theorem holds. Thus is suffices to compute Green functions.

,( ) ( )

i j i jG c c

1 1.... ...., LB B P B Bs s s sPU U

Observables ground state.

Observables finite temperature.

1 1

1

.. ..Tr ( ) ( ) ( ) ( )( , )

..Tr ( ) ( )L

L

Os s s sO s

s s

B B B B

B B

1( ) 1 ( ,0) ( , )G B B

1 1.... ....( ,0) , ( , ) LB B B B B Bs s s s

Wick´s Theorm

Upgrading, single spin flip.

)(1)()( ' sss lll BBB so that G

UU

UU

W

W

Old

New

11det

det

1det

Thus the Green function is the central quantity. It allows calculation of observables and determines the Monte Carlo dynamics. Same is valid in the finite temperature approach.

This form holds for both the finite temperature and ground state algorithm!

If the spin flip is accepted, we will have to upgrade the equal time Green function

1 11 1

1,

where1 i ji j

u vA A

v uAA u v u v u vA

Upgrading of the Green function is based on the Sherman Morrison formula.

Outer product.

II. Comments.

:....1 PsBsBU l:P

Gram Schmidt. > >d vU U :U

UUUU 11

UUUUGreen functions remains invariant. .

(A) Numerical stabilzation T=0.

Since the algorithm depends only on the equal time Green function everything remains

invariant!

< <d vU USimilarly:

The Gram Schmidt orthogonalization.

Numerical stabilization finite T.

Use:

To calculate the B matrices without mixing scales use.

Diagonal elements are equal time Green functions.

As we will see later, the off-diagonal elements correspond to time displaced Green functions.

You cannot throw away scales.

The inversion.

Measuring time displaced Green functions.

(a) Finite temperature.

1 2 Note:

,

1 2,s x yG

Thus we have:

But we have already calculated the time displace Green functions. See Eq 81.

Time displaced Green functions for ground state (projector) algorithm.

Consider the free electron case: so that 0 00( ) and assume that 0,1k k k k

k

k c c c cH

L=8, t-U-W model: W/t =0.35, U/t=2, <n>=1, T=0

Ψ)0,(),(Ψ3

1ln 00 QSQS

Ψ)0,(),(Ψln 00 QQ SS zz

Ψ)0,(),(Ψ3

1ln 00 QSQS

t

SU(2) invariant code.

SU(2) non-invariant code.

SU(2) non-invariant code.

(C) Imaginary time displaced correlation functions.

1

)1('τΔ~

s

siH eennU

Note: Same CPU time for both simulations.

1

)(τΔ~s

sH eennU

. Gaps. Dynamics (MaxEnt).

(B) Sign problem.

5 βt

6 βt

10 βt

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<n>

<si

gn>

U/t = 4, 6 X 6

a) Repulsive Hubbard.

1

)(τΔ~s

sH eennU

1

)1('τΔ~

s

siH eennU

)(....)(1detWeight 12

sBsB L

Particle-hole symmetry:

is real even in the presence of a magnetic field.

)(....)(1det 1sBsB L

b) Attractive Hubbard, U<0.

1

)1(''τΔ~

s

sH eennU

)(....)(1detWeight 12

sBsB L )(....)(1det 1sBsB L is real for all band fillings(no magntic field.)

General: Models with attractive interactions which couples independently to an internal symmetry with an even number of states leads to no sign problem.

Away from half-filling.

cc i

i

i 1

)(....)(1det)(....)(1det 11 sBsBsBsB LL

Half-filling.

Impurity models such as Anderson or Kondo model (Hirsch-Fye).

( , ),, ,

i j

f

I

c

i jt c c JH

S S No charge fluctuations on f-sites.

)(||1

)( ||2

,EEmn

ZS mn

f

I

En

mnf Se

)(S f

t/

T <TK

T >TK

Dynamical f-spin structure factor J/t=2

00

SSf

I

f

IId

Numerical (Hirsch-Fye impurity algorithm) CPU:V0(Nimp3

T/TK

J/t = 1.2J/t = 1.6J/t = 2.0

T

TK/t 0.21

TK/t 0.06TK/t 0.12

is the only low energy scaleeTJt

K

/

1 120, 0, 2

( ),

)( )(, ,f f

ff f nc c Vt c cH nU

i, j

i j

The Hirsch Fye Impurity Algorithm.

H

H U

fff nnUcffcVcctH ))((,, 2

121

0

,0,0),(

ji,

ji

Finite temperature:

ss

ssZL

LH

1

)()(1deteTr 1- BB We only need Green functions on f-sites

(1) Upgrading (equal time).(2) Observables.

)(1)(/)'(1)(11

)'( sGsDsDsGsG

G(s): f-Green function for HS s. (L x L matrix)

Start with D(s) = 1. G(s) is the f-Green function of H0. Exact solution in thermodynamic limit.

From G(s) compute G(s´) at the expense of LxL matrix inversion

CPU time ~ L3 (i.e. 3).

1

)(τΔ~s

sH eennU

With

e,ediag)( ,1 sssD L in say up spin sector.

T/TK

J/t = 1.2J/t = 1.6J/t = 2.0

T

Single impurity (Hirsch-Fye algorithm)

TK/t 0.21

TK/t 0.06TK/t 0.12

SSjiji,

f

IIcJcctH

,,),(

00

SSf

I

f

IId

(IV) Related algorithm: Hirsch Fye Impurity Algorithm.

(III) Approximate strategies to circumvent sign problem

sT

L

EL

L

TL

EL

LT

se

e

Ψlim

ΨlimΨΨΨ

)(1

Δ

Δ

00

0

0

BBB

BB

0ΨΨ,)( 0ΔΔ

Ts

HH see tU BB

Recall:

Assume that we know and for s´

then we can omit all paths evolving from this point since:

0,0

~)'(

10)(Δ

GStoOrthogonal1

Δ

10

0

lim

Ψlim

EE

s

e

e

EEL

L

TL

EL

L

BBB

But: We do not know ! Approximate it to impose constraint and method becomes approximative.

CPQMC (Zhang, Gubernatis). Approximate by a single Slater determinant to impose constraint.

Ψ0

0Ψ)'(Ψ0 TsB

Ψ0

Ψ0

4x4 Hubbard: U/t = 8, <n>=0.625

Ene

rgy/

t

First orderTrotter.

Second orderTrotter.

Exact: -17.510t, Extrapolated : -17.520(2)t