量子モンテカルロ法における負符号問題とその克...
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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