量子モンテカルロ法における負符号問題とその克服： 経路積分繰り込み群法の開発とその発展

渡辺真仁　　　東大工

基研研究会 「熱場の量子論とその応用」 2008年9月4日

今田正俊 東大工

水崎高浩　　

専修大

共同研究者

Outline of this talk

2. Basic idea of Path-Integral Renormalization Group

3. Quantum-number projection algorithm

4. Grand-canonical algorithm

5. Summary

1. Approach to material science

Approach to material science

Approach to material science

Quantum mechanicsCoulomb repulsionKinetic energy vs

Macroscopic nature of materials

2310 atoms & electrons

Microscopic principles for electrons & atoms

Solid-liquid transition, Ferro magnetism, Superconductivity, …, etc

High-temperature superconductivity, Heavy-electron system, …

Unexpected phenomena

Model and number of states

Hubbard model

tU

( ) ∑∑=

↓+↓↑

+↑

σσ

+σσ

+σ ++=

N

iiiii

jiijjiij ccccUcccctH

1,

N = number of lattice

0

0

0

0

+↓

+↑

+↓

+↑

ii

i

i

cc

c

c

4 states per site

operators Fermion: , σ+σ ii cc

Dimension of Hilbelt space of N-sites systemN4=

12101.1 20 ex. ×→=N

Diagonalization of matrix of NN 44 ×

Model and established methods

Hubbard model

tU

( ) ∑∑=

↓+↓↑

+↑

σσ

+σσ

+σ ++=

N

iiiii

jiijjiij ccccUcccctH

1,

Exact diagonalizationDensity Matrix Renor-malization Group (DMRG)

Quantum Monte Carlo(QMC)

exact Small N ( ~20 )

N = number of lattice

highly accurate Large N, but 1D

Large N, 1D, 2D, 3D(except fornegative-sign problem)accurate

Established method accuracy system size

Negative-sign problem in QMC

In QMC, expectation value is calculated as

Negative sign problem

samples ofnumber :sN

Basic idea of Path Integral Renormalization Group (PIRG)

∑=

ϕ=ΦL

aaaw

1

Variational wavefunction

Increase LOptimize at fixed LΦ

Extrapolation

ΦΦΦΦ

=H

EΦΦΦΦ

=A

A variational

: Slater determinant cf. DMRG

ba A ϕϕ ba ϕϕ

0 ≠ϕϕ ba non-orthogonal basis orthogonal basisaϕ

M. Imada & T. Kashima, JPSJ69(2000)2723T. Kashima & M. Imada, JPSJ70(2001)2287

Projection to ground state by path-integral operation

[ ] 0 exp ΦτΔ−=Φ H

[ ] [ ] [ ] )( exp exp exp 2τΔ+τΔ−τΔ−=τΔ− OHHH Ut

[ ] 0' exp ΦτΔ−=Φ tH

Ut HHH +=

[ ] ∑ Φ=ΦτΔ−=Φ ) ( )( exp 0' iHU

In Slater determinant representation

Stratonovich-Hubbard transformation

; branchingss = 1

0g ] exp[-lim Φτ=Ψ∞→τ

H ( ) 00g ] exp[-limlim ΦτΔ=Ψ∞→→τΔ

n

nH

↓↑ Φ⊗Φ=Φ0

τΔ=τ n

σ+σσ = iii ccn

Renormalization process

projection

truncation

Solve generalized eigenvalue problemafter each projection

bb

L

babb

L

ba wEwH

11ϕϕ=ϕϕ ∑∑

==

∑=

ϕ=ΦL

aaaw

1

Lowest eigenstate: approximate ground statein a truncated Hilbert space

determine { }aw

Projection to the ground state[ ] 0 exp ΦτΔ− H

Select the subspace which lowers the energy

or

Good agreement to QMC results (ex. half-filled caserelative error less than 0.3%)

Extrapolation to ∞→ L

Extrapolation with energy variance

( ) 222 / HHHE −=ΔEnergy variance

EEH Δ∝− exact

HE =

EΔ

∑=

=ΦL

aaaw

1 ϕ

for large L )5,5(),( ,26 ,4 ,1 =×=== ↓↑ NNNUt

E

1010×=N

Whole procedure of PIRG

∑=

ϕ=ΦL

aaaw

1 cf. L=1: Hartree-Fock solution(1) Initial state

(2) Projection by kinetic term [ ] ata H ϕτ−=ϕ exp'

[ ] aUaa iH ϕτ−=ϕ+ϕ −+ )( exp(3) Projection by interaction term

(4) Generate new states (L increases) and repeat (2)-(4)

(5) Variance extrapolation of physical quantities

Generalized eigenvalue problem

Generalized eigenvalue problem

Laaa ϕϕϕϕϕ +− ..., , , , , ... , 111

Laaa ϕϕϕϕϕ +− ..., , ,' , , ... , 111

Laaa ϕϕϕϕϕ +− ..., , , , , ... , 111

Laaa ϕϕϕϕϕ ++

− ..., , , , , ... , 111

Laaa ϕϕϕϕϕ +−

− ..., , , , , ... , 111

Select one set

Select one set

Ut HHH +=

022 →− HH

Numerical accuracy and efficiency

Ground-state energy per siteN ),( ↓↑ NN PIRG QMC

66× (18,18)

88× (31,31)

1010× (50,50)

8589.0−004.0902.0 ±−9031.0−

8646.0−

002.0860.0 ±−

003.0867.0 ±−

4 ,0' ,1 === Utt

Relativeerror

0028.00012.00013.0

Relative error is less than 0.3%

Hubbard model

Spin correlation

NH /

),( yx qq PIRGExactdiagonalization

Relativeerror

0.0036 0.279 0.278 )1,3( 0.0070 0.286 0.284 )1,2(

0.0071 0.281 0.283 )1,1(0.014 0.277 0.281 )1,0(0.013 0.0457 0.0451 )0,3(0.0044 0.0458 0.0454 )0,2(0.012 0.0488 0.0482 )0,1(

26×=N

)5,5(),(

=↓↑ NN

Remarks on PIRG

No negative sign problem

No restriction of lattice structure, boundary condition

1D, 2D, 3D, frustrated systems

Explicit variational wavefunction is given

Systematic scheme of improvement from “mean-field” approximation such asHartree-Fock or variational Monte Carlo method

Physical quantities for ground stateare obtained

Controlled way to reach exact ground statebased on variational principle ∑

=

ϕ=ΦL

aaaw

1

Application of PIRG

Electron gas (continuous space, long-range Coulomb interaction)

Geometrically frustrated Hubbard models

Shell model for atomic nucleus

square lattice triangular lattice

Wigner crystal – Charge order – Mott insulator

Magnetic transition, Metal-insulator transition

∑∑ −+

∂∂

=ji jii i

em

H,

2

2

2

21

rrr Phys. Rev. Lett. 89 (2002) 176803

J. Phys. Soc. Jpn. 71 (2002) 2109J. Phys. Soc. Jpn. 70 (2001) 3052

Phys. Rev. C 65 (2002) 064319

cuprates κ-(ET) X2

?

ex. diamond

Metal & Insulator

Mott insulator

Band insulator

odd number of electrons per unit cell

even number of electronsper unit cell

Metal

2310 atoms & electrons

ex. Cu

“Strong Coulomb repulsion”

Genuine Mott insulator except in 1D system?

Ground-state phase diagram of 2D Hubbard model

half filling:

Non-Magnetic Insulator phaseappears

T. Kashima & M. Imada, JPSJ 70 (2001) 3052 H. Morita, S. Watanabe & M. Imada, JPSJ 71 (2002) 2109

Hatree Fock: W. Hofstetter & D. Vollhardt(1998)

∑σ=

σ+σ ==

N

iii cc

Nn

,111

Non-Magnetic Insulator

In NMI phase, no indication ofmagnetic order, charge order, dimer order, plaquette order, and density-wave order (s-wave & d-wave flux states)

No connection to band insulator

No simple translational symmetry breaking

“Genuine Mott insulator” in 2D system

S. Watanabe, JPSJ72 (2003) 2042

Further development of PIRG

Quantum number projection (QP-PIRG)

Improvement of accuracyExcitation spectra

Canonical

Grand canonical

Input :

Input :

chemical potentialOutput : μeN

μeNOutput :

Extension to grand-canonical framework (GPIRG)

'ee

'ee )()(

NNNENE

−−

=μ

(PIRG is canonicalframework)

number of electrons

charge excitationμ-U phase diagram

spin excitationdispersion

ex. quantum number: total spin, momentum, parity, etc.

{{

Spin & momentum projection algorithm

Spin rotation operator

Translation operator

)(ϕR)(rT

[ ] Φ⋅ϕϕϕ=Φ ∑∫ )r(rkexp)(),()k,(r

TiRSWdS R

Wavefunction specified by total spin and momentum

Quantum number projection

∑=

ϕ=ΦL

aaaw

1

Superposition of wavefunction by spin-rotation & translation operators

M. Imada, T. Mizusaki & S. Watanabe, cond-mat/0307022T. Mizusaki & M. Imada, PRB69(2004)125110

Spin projection operator

∫ ΩΩΩπ+

≡ )()(8

12 *2 RDdSL S

MKSMK ),,( γβα=Ω

zyz SSSR γβα=Ω iii eee)(

)(ee)()( ii β=Ω=Ω γα SMK

KMSMK dSKRSMD

''''

'' KMSSSMK

SKM

SMK LLL δδ=

2/)(0 ↓↑ −= NNN

SSSNN

SNN PddSL y ≡βββ

+= ∫

π β

0

ie )( sin2

120000

Spin projection operator is given by

Euler angle:

Rotation operator:

Wigner’s D function:

By using the relation

spin projection operator is rewritten as

where

SKSMd ySSMK

β=β ie)(

P. Ring & P. Shuck, Nuclear Many Body Problem (Springer-Verlag, New York 1980)

Spin-projection algorithm

Ground-state energy projected onto state is represented as0SS =

⎟⎟⎠

⎞⎜⎜⎝

⎛βββ−β

=β

cossinsincos

e i- yS

Expectation value is calculated as T. Otsuka, et al Progr. Part. Nucl. Phys. 47(2001)319

Examination of spin & momentum projection

44,4,0',1 ×==== NUtt )8,8(),( =↓↑ NN

Gutzwiller+α

VMC -13.47

PIRG + spin projectionL=256, -13.60Extrapolation -13.62

QP-PIRG + spin projectionL=20 -13.600L=128 -13.615Extrapolation -13.622

Exact -13.62185

22 HH −

H

Spin excitation in the NMI phase

(8,8)),( ,44 7.5 ,5.0' ,1 =×==−== ↓↓ NNNUtt

Size scaling of S=1 excitations

In NMI phase, spin gap is closed

∞→→Δ NE for 0

“Genuine Mott insulator”

)0()1( =−=≡Δ SESEESpin gap:

U=4.0, t’=-0.2U=5.5, t’=-0.5U=5.7, t’=-0.5U=2.7, t’=-0.2

U=4.0, t’= 0.0

NMI phase

in two dimensional system

}

half filling

Melting of magnetic transitionby frustration & charge fluctuation

T. Mizusaki & M. Imada, PRB74(2006)014421

-possible realization of the NMI state32(CN)CuET- 2)(κ

Y. Shimizu, K. Miyagawa, K. Kanoda, M. Masato and G. Saito, PRL91(2003)107001

No signature of magnetic transition down to 32mK

11 ofbehavior

lowpower −T

gaplessspin excitations

anisotropic triangular latticewith single band at half filling

2.8~/family) ET among(largest , 1.1~/'

tUtt

strong frustration & correlation

Gapless “Spin liquid” phaseK. Ishida, et al, PRL79(1997)3451R. Masutomi, et al, PRL92(2004)025301plate graphite on He3

↑kc kc

↓−kc +kd

Grand-canonical Path Integral Renormalization Group (GPIRG)

Particle-hole Transformation

NUHH Ut ⎟⎠⎞

⎜⎝⎛ μ+−+=

4

Transformed Hamiltonian:

( ) ii

iijjiij

ij ccUcccct ∑∑ +++ ⎟⎠⎞

⎜⎝⎛ μ−++−

2

( ) ii

iijjiij

ij ddUddddt ∑∑ +++ ⎟⎠⎞

⎜⎝⎛ μ++++

2

∑ ++−=i

iiiiU ddccUH

=tH

( ) ⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −++−= ↓

+↓

=↑

+↑

σσ

+σσ

+σ ∑∑ 2

121

1,ii

N

iii

jiijjiij ccccUcccctH

{

S. Watanabe & M. Imada, JPSJ 73 (2004) 1251

Extended basis

+↓−

+↑ kk cc=+

kk dc

cf. superconductivity

∑∑=

++

σσ

+σ −+==

N

iiiii

iii ddccNccN

1e

Total electron number

0~][1

2

1∏ ∑= =

+⎟⎟⎠

⎞⎜⎜⎝

⎛φ=φ

N

k

N

iiikaa c

:][ ikaφ

+ic~ =

+ic+−Nid

forfor

Ni ,...,1=NNi 2,...,1+=

H. Yokoyama & H. Shiba (1988)

{canonical

grandcanonical

∑=

φ=ΦL

aaaw

1

N = number of lattice

N

2Nc

d

↑N

↓N

N

N

cf. diagonal in PIRG case

Projection to ground state

Interaction-term projection

( ) ⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧ −+τ− ++++

iiiiiiii ddccddccU21exp

[ ] ∑±=

φ=φτ−1

)()( exps

aaU siH

Kinetic-term projection[ ] atH φμτ− )'( exp

c-d hybridization

μ’: pseudo chemical potentialSelect the state which gives the lowest energy

Select the state which gives the lowest energySeveral sets of μ’ ( )μ≠

[ ] 0 exp Φτ−=Φ

[ ] [ ] [ ] )( exp exp exp 2τ+τ−τ−=τ− OHH Ut ∑=

φ=ΦL

aaaw

1

Extrapolation to ∞→ L

EΔexactE

L=1

( )[ ]∑±=

++ +α=1exp

siiii cddcsi

0

Numerical accuracy and efficiency

Ground-state energyN ),( ↓↑ NN GPIRG

Exact diagonalization or QMC

26× (5,5)

66× (13,13)

66× (18,18)

0229.07008.25 ±−072.032.58 ±−0902.04456.58 ±−

0472.09394.66 ±−

6952.25−

07.096.66 ±−

4 ,0' ,1 === Utt

Relativeerror

0003.00022.00002.0

GPIRG is useful to calculate chemical potential dependence of physical quantities

*

**

Correct electron number is obtained after variance extrapolation

Ground-state phase diagram of 2D Hubbard model

μ

2.0' ,1 −== tt

V-shaped structure of Mott insulator phaseappears

(ban

dwid

th c

ontr

ol)

(filling control)

(lattice structure control)

Bandwidth control :1st order transition

μ

Filling control:continuoustransition

U

S. Watanabe & M. Imada, JPSJ 73 (2004) 1251

Relation between shape of phase diagram and order of transition

1st order :

2nd order :

Metal-insulator transition

Bandwidth control :1st order transition

μ

Filling control:continuoustransition

U

MI

MI

DDnnU

−−

=δμδ

μ

⎟⎠⎞

⎜⎝⎛∂∂χ

−=δμδ

Un

UM

c

μ μ μ

Binding energy between holon and doublon

Binding energybetween holons (doublons)

>

( )

V-shaped structure Sharp contrast of characters of MI transitions

∑=

↓↑≡N

iii nn

ND

1

1

μ∂∂

≡χ Mc

n

S. Watanabe & M. Imada, JPSJ 73 (2004) 1251

M M M

I I I

SummaryPath-Integral Renormalization Group (PIRG)

Quantum-number projection algorithm (QP-PIRG)

Grand-canonical algorithm (GPIRG)A unified framework to study correlated-electron systems whose control parameters are bandwidth, filling and lattice structure

spin, momentum, parity, etc.excitation spectraimprovement of accuracy

no negative sign problemno restriction of lattice structure, boundary conditionany type of Hamiltonian for Fermions

Non-magnetic insulator

Overall understanding of Mott transitions in 2D

“Genuine Mott insulator” in 2Dno simple translational symmetry breaking, closed spin gap

first-order BCMT and continuous FCMT, V-shaped structure

ReferencesM. Imada and T. Kashima, J. Phys. Soc. Jpn. 69 (2000) 2723T. Kashima and M. Imada, J. Phys. Soc. Jpn. 70 (2001) 2287T. Kashima and M. Imada, J. Phys. Soc. Jpn. 70 (2001) 3052H. Morita, S. Watanabe and M. Imada, J. Phys. Soc. Jpn. 71 (2002) 2109S. Watanabe, J. Phys. Soc. Jpn. 72 (2003) 2042M. Imada, T. Mizusaki and S. Watanabe, cond-mat/0307022T. Mizusaki and M. Imada, Phys. Rev. B 69 (2004) 125110S. Watanabe and M. Imada, J. Phys. Soc. Jpn. 73 (2004) 1251

PIRG

Non-magnetic-insulator phase

Spin-projectionPIRGGrand-canonicalPIRG

{

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渡辺真仁、水崎高浩、今田正俊:固体物理, 39 (2004) 9月号 p565-576