Sign problem in Monte Carlo simulations and the …Sign problem in Monte Carlo simulations and the tempered Lefschetz thimble method Sep 2, 2019 熱場の量子論とその応用 @YITP

Sign problem in Monte Carlo simulations and the tempered Lefschetz thimble method

Sep 2, 2019熱場の量子論とその応用@YITP

Nobuyuki Matsumoto (Kyoto Univ) & Naoya Umeda (PwC)

Masafumi Fukuma (Dept Phys, Kyoto Univ)

Based on work with

-- MF and Umeda, “Parallel tempering algorithm for integration over Lefschetzthimbles” [arXiv:1703.00861, PTEP2017(2017)073B01]

-- MF, Matsumoto and Umeda, “Applying the tempered Lefschetz thimble methodto the Hubbard model away from half-filling”, [arXiv:1906.04243]

-- MF, Matsumoto and Umeda, [arXiv:1705.06097, JHEP1712(2017)001], [arXiv:1806.10915, JHEP1811(2018)060]

Also, for the geometrical optimization of tempering algorithms and an application to QG:Matsumoto’s poster (today)

1. Introduction

Summary

Typical examples:① Finite density QCD② Quantum Monte Carlo simulations of quantum statistical systems③ Real time QM/QFT

Today, I would like to

② Quantum Monte Carlo simulationsof strongly correlated electron systems,especially the Hubbard model away from half-filling

-- argue thata new algorithm “Tempered Lefschetz thimble method” (TLTM)is a promising method, by exemplifying its effectiveness for:

The numerical sign problem is one of the major obstacleswhen performing numerical calculations in various fields of physics

-- give a review on various methods towards solving the sign problem

Sign problemOur main concern is to calculate:

( )( ) ( )

: dynamical variable (real-val: action, : observab

ued)le

i Nx xS x x = ∈

( )

( )

( )( )

S x

S S x

dx e xx

dxe

−

−≡ ∫

∫

Markov chain Monte Carlo (MCMC) simulation:( ) ( )( /( ) )eqWhen as a PDF:, one can regard S x S xS x p x e dxe− −∈ ≡ ∫

0 ( ) 1, ( ) 1eq eq p x dx p x≤ ≤ =∫( )

1, ,{ } ( )conf eqGenerate a samp from le N

kkx p x= …

( )

1

1( ) ( )conf

conf

Nk

k

xN

x=

≈ ∑

Sign problem:( ) ( )( ) ( ) ( /)When , one cannot regard as a PDFS x S x

R IS x S x i S x e dxe− −= + ∈ ∫

( ) ( )/Reweighting method : treat as a PDFR RS x S xe d ex− −∫( )

( )

( )

( )

( ) (1 / )( )

(1 / )conf

conf

I

R

I

R

i S x

SS i S x

S

O N

O N

e x Ox

eN

Oee N

−

−

−

−

±≡

±≈

( )conf

O NN e

probability distribution function

conf

: DOF : sample size

NN

( )(1 / )confRequire O NNO e−<

Sign problemOur main concern is to calculate:

( )( ) ( )

: dynamical variable (real-val: action, : observab

ued)le

i Nx xS x x = ∈

Markov chain Monte Carlo (MCMC) simulation:( ) ( )( /( ) )eqWhen as a PDF:, one can regard S x S xS x p x e dxe− −∈ ≡ ∫

( )1, ,{ } ( )

conf eqGenerate a samp from le Nk

kx p x= …

( )

1

1( ) ( )conf

conf

Nk

k

xN

x=

≈ ∑

Sign problem:( ) ( )( ) ( ) ( /)When , one cannot regard as a PDFS x S x

R IS x S x i S x e dxe− −= + ∈ ∫

( )

( )

( )

( )

( ) (1 / )( )

(1 / )conf

conf

I

R

I

R

i S x

SS i S x

S

O N

O N

e x Ox

eN

Oee N

−

−

−

−

±≡

±≈

( )(

( )

(

( )

)

)

)

(

( )(

( ))

I

I

R

R

SS x

S

x

S x

x

xx

i S

i SS

edx e xx

xdx e

e ex edxd

−

−

−

−

−

−=≡ ∫∫

∫∫

( ) ( )/Reweighting method : treat as a PDFR RS x S xe d ex− −∫

probability distribution function

( )conf

O NN e

conf

: DOF : sample size

NN

( )(1 / )confRequire O NNO e−<

0 ( ) 1, ( ) 1eq eq p x dx p x≤ ≤ =∫

2

2

( ) ( ) ( ) ( )2

( )R IS x x i S x i S x

x x

β = − ≡ + =

2( ) ( 1)2

( )R

I

S x x

S x x

β

β

= − = −

( )

( )

( ) 2 1 / 22

/ 2( )

1 / 2

/ 2

(

1

1 )

)

1 /

(1 /conf

conf

I

R

I

R

i S x

SS i S x

S

e ex

ee N

x

eO

e O N

β

β

β

β

β

β

−

−

− − −

−

− −

−⟨ ⟩ =

− ±

=

≈±

NB :The num and the denomare estimated separately.

Let us consider

[Essence]

x

2( ) /2RS x xe e β− −∝( ) cosRe Ii S xe xβ− ∝

1β

1β

( )1 / 1 /In the limit ,the integration becomes highly oscillatory

β β β→∞ ∴

/ 2 )1 / ( ( )conf confN O e N O eβ β−< >⇔

Necessary sample size:

Example: Gaussian

1β

numerically

~large mimics large DOF ( )Nβ β

Approaches to the sign problemVarious approaches:

(1) Complex Langevin method (CLM)(2) (Generalized) Lefschetz thimble method ((G)LTM)(3) ... [to be commented later]

Advantages/disadvantages:(1) CLM

(2) LTM

( ):Cons: "wrong convergence problem"

O N∝Pros fast

[Parisi 1983][Cristoforetti et al. 2012, ...][Alexandru et al. 2015, ...]

Jacobian determinant + tempering

(2’) TLTM (Tempered Lefschetz thimble method) [MF-Umeda 1703.00861,MF-Matsumoto-Umeda 1906.04243]

“facilitate transitions among thimblesby tempering the system with the flow time”

~3 4( )thimbles are relevant

O N∝Pros: Works well even when multi Cons: Expensive

[Ambjørn-Yang 1985, Aarts et al. 2011,Nagata-Nishimura-Shimasaki 2016]

[Kashiwa-Mori-Ohnishi 2017, Alexandru et al. 2018][Di Renzo et al., Ulybyshev et al., ...]

( :DOF)N

3( )thimble is releva

Multimodal problem if more than one thimble are relev

nt

antO N∝

Pros: No wrong convergence problem only a single Cons: Expensive

iff

(wrong convergence de facto)

Jacobian determinant

Plan

1. Introduction (done)2. Complex Langevin method (CLM)3. (Generalized) LTM (GLTM)4. Tempered LTM (TLTM)5. Applying the TLTM to the Hubbard model

- 1D case- 2D case

6. Other approaches7. Conclusion and outlook

Complex Langevin method: basics (1/2)2 : Gaussian white noise with varianc e τν σ•

2

0 0( )2

Complex Langevin equation:

wi

th z S z xzτ τ τ τσν == −

•

′ =

1 2 1 2

21 22 1/ 2

2 2(1/2 )

2)

)((

dd eττ τ ν τ τ

τ

τ

σ τννν ν ν ν σ δ τ τπσ

− ⟨ ⟩ ≡

= −∫∏∫

0;( ) ( ) (0 ) for a given soln sz z x sτ τ ν ν ν τ= = ≤ ≤

( )( )

0 ( )

( )lim ; (( ) )

S x

S S x

dxex

dx

xz x

eτ ντν

−

−→∞

=

= ∫∫

x

iy

0x

( )0-independentx

0( ) ( ; ) in by and take the Replace average over :x x z xτ ν ν•

The limit gives the desired expec (under

tatisome

on value conditi n) o :

τ• → ∞

( ) ( )0 0( ) ( )( ) ; ;x xx z zτ τ νν ν→ →

0( );z xτ ν

ν

[Parisi PLB131(1983)393]

( )2

1 ( )2n n n nz z S z nσν τ+

′= + − =

Complex Langevin method: basics (2/2)2Introduce a PDF over :=

20 0) (( | ), ;( )z xx y x zτ τ νδ ν≡ ⟨ − ⟩

( )2

, ( , ) ( )2x y x x R y IS x y S zσ

′ ′ ≡ − ∂ ∂ + + ∂

“proof”

( )0 0; ) ( , |( ) ( ) x xdz d x y x x yy iτ τνν +=• ∫

( )z x iy= +

( , ) ( ), ( , ) Im ( )ReR IS x y S x iy S x y S x iy′ ′ ′ ′≡ + ≡ +

x

iy

0x

0| )( ,x y xτ

,0 0) ( )( , | ) ( x ye x yx y x xτ

τ δ δ−• = −

( ) ,0 0; ) (( ) ( ) ( )

Tx yz d x x yx x e x iy ydτ ν

τν δ δ −=∴ − +∫

( )

( )

2

,

2

2

( ) ( , ) ( ) ( , )( ) ( )2

( , ) ( ) ( , )( ) ( )2

( , ) ( , ) ( ) ( ) ( )2

Tx y x R x I y

R I

TR

z z

z I

z

zz

x iy S x y S x y x iy

S x y S x y i z

S x y iS x y z H z

σ

σ

σ

′ ′ + = − −∂ + −∂ + −∂ +

′ ′ = − −∂ + −∂ + − ∂

′ ′= − −∂ + + −∂ ≡

( )S z′=

Here,

( )0 0

0 0

; ) ( ) ( ) ( )

( ) ( ) ( ) ( )

(

x

Tz

Tx

H

H H

z d x x y e x iy

d x x e x d

x xdy

x xx e x xτ ν

τ

τ τ

ν δ δ

δ δ

−

− −

=

= =

∴ −

−

+

−∫∫ ∫

0( | )P x xτ≡2

0 0 0 0( | ) ( | ) ( | ) ( )] ( | )2

[ satisfies x x xP Px x P x x H x x S x x xPτ τ τ τσ ′+= − = ∂ ∂

( )0

1( | ) S xx x dxP eZτ

τ −→∞→ ∫

2

[ ( )]2z z zH S zσ ′= − ∂ ∂ +

( )0 0

0

lim ; ) lim ( , | ) ( )

lim ( | ) ( ) (

(

) S

z d x y x x iyx xdy

dx P x x x xτ τντ τ

ττ

ν→∞ →∞

→∞

=

=

+

= ⟨ ⟩∫

∫

[Aarts-James-Seiler-Stamatescu 1101.3270]

P.I.

P.I.

Complex Langevin method: wrong convergence,

0 ( )

| |)( , |

not be spread out largely in In order for the partial integration a

direction should

not have a significant suppo

nd to be mean

rt around zero

ingful,

Otherwis of

se, the

Tx y

S z

ey

ex y xτ

τ

−

−

→

∞

limit gives a wrong result.

Criterion2 2( , )| ( ) | ( , )The histogram of

must decrease rapidly (at least exponentially) at large valuesR IS x iy S x y S x y+′ ′ ′+ =

2 /4 2( ) (: )Example xS x x eiα −= + [Nagata-Nishimura-Shimasaki PRD94 (2016) 114515]

3.8rapid fall-offα ≥

3.7powerlike fall-offα ≤

| ( ) | ( )( )

: normalized histogramS z z x iy

pu

u′≡ = +

3.8good agreementα ≥

3.7wrong convergenceα ≤

[Aarts-James-Seiler-Stamatescu 1101.3270]

[Nagata-Nishimura-Shimasaki 1606.07627, PRD94 (2016) 114515]

CLM: attempts to solve the wrong convergenceAim: reduce the effects from dangerous configurations

:gauge cooling repeatedly make "gauge transformations" (if possible) to send the variables near N

(1) configurations far from the original integration region N

[Seiler-Sexty-Stamatescu 1211.3709]

( )(2) configurations close to zeros of S ze−

( ; ) ( ; ) ( ; ) ( ; )

:reweighting Use a parameter with which CLM works (assuming an enough overlap): e S x S x S x S xe eα β α β− − − +→ ×

regarded as a part of observable

[Ito-Nishimura 1710.07929]

“excursion problem”

“singular drift problem”

:( ( ; ),

0)

deformation Add a parameter s.t. CLM wo then take a

rkst

:i

l mi

SS xx αα

→→

[Bloch 1701.00986 , Bloch et al. 1701.01298]

3. (Generalized) Lefschetz thimble method (GLTM)[Cristoforetti et al. 1205.3996, 1303.7204, 1308.0233][Fujii-Honda-Kato-Kikukawa-Komatsu-Sano 1309.4371][Alexandru et al. 1512.08764]

Lefschetz thimble method (1/2)

Cauchy’s theorem

( ) ( ) i N i i i Nx x z z x iy= ∈ = = + ∈⇒ Complexify the variable:( ) ( ) ( ), entire functions : over S z S z Ne e z− −

Assumption:

0

| | to

(with the boundary at infin kept fixeity ) d :

N N

xΣ = ⊂→∞

Σ

Integral does not change under continuous deformations of the integration region from

0

0

( ) ( )

( ) ( )

( ) ( )( )

S x S z

S S x S z

dx dz

dx dz

e x e zx

e e

− −Σ Σ

− −

Σ Σ

≡ =∫ ∫∫ ∫

( )sign problem will get much reducedif Im is almost constant on S z Σ

severe sign problem

Σ

0Σx

{ }N z=iy

Lefschetz thimble method (2/2)

0( ) with i i it i t tz z xzS == ∂ =

Prescription:antiholomorphicgradient flow

Property: [ ] 2) ( ) ( ) 0( it i t t i tS z zz S S z∂ ∂= ≥=

[ ][ ]

) 0

)

(

0( :

Re : real part always increases along the flowIm imaginary part is kept fixed

t

t

S z

S z

= ′′

≥

, approaches a uni Lefschetz thimbleson of :In tt →∞ Σ

xσx 0Σ

( )tz x σz•σ

tΣ

( )

( ) ( ( ))

( ( )) ( ( )/ )( ( ) / )

eff Re

Im argdetdet

t

i jt

t t

jit ti x

x S z x

i S z x i z x x

S z xe

e

x

e

eθ

− −

− + ∂ ∂

≡ ∂ ∂

≡

tσ

σΣ →

0 0

0 0

( ) ( ( ))( )

( ) ( ( ))( )

( )

( )

( ) ( ) ( ( ) / ) ( ( ))( )

( ( ) / )

( ( )) eff

eff

det

det

t

t

t

t

t

t

t

t

t

t

S z S z xS x i jt t t

S S z S z xS x i jt t

i x

Si x

t

t

S

dx dze x e z z x x e z xx

e e z x x e

dx

dx dz dx

e z x

e

θ

θ

− −−

Σ Σ Σ

− −−

Σ Σ Σ

∂ ∂≡ =

∂

=

=∂

∫ ∫ ∫∫ ∫ ∫

Expectation value:( )Im is constant(on each of which ) S z

( )( )efftt i xxS ee θ−≡

( )( ) 0 : critical point

i

zS z

σ

σ

∂ =

Gradient flow:

( )( )( )

(1 )t tt

ttt

x xedz xx e

dx

z i e

J

β β

β

−=

= =

+ −

2 2 4( ) ( )2

( )

1eff

t

t tt x xS e te

x x

ββ

β

β

θ

β −

=

−= −

( )2

2

( ) 2 ( / 2) 1

( ) ( / 2)

( ) 1eff

eff

tt

tt

t

t

i x et S

i x e

S

z x e

e

e

e

β

β

θ β

θ β

β−

−

− −

−

= −

=2 1(1) 1 if t tO e eβ ββ

β− −

= ⇔

1 s.t. e , we can nTaking a umerical la ly estimate :rge TT β

β−

( )2

2

( ) 2

2( )

( / 2) 11

( / 2)

( )

11

eff

eff

T

T

T

TT

T

i x

Si xS

S

e

e

Tz xx

e

e

e

e

β

β

θ

θ

β

β

ββ

−

−

− −−

−

=

−= = −

(no small numbers appears!)

1 1te β

ββ−

1β

x

x

i yzz iσ =

xxσ

( )tz x

•σ

Example: Gaussian

t

{ { 0

0( ) ( 1) 0 with t t

t t t tt t

x x x xz x y S z y yi yββ

=

=

=′ ⇔ = −== + − ==

exponential growthof coefficient

2( ) ( / 2)( )S z z iβ = −

xσ

Multimodal problem and Generalized LTM (1/2)Flow time t needs to be large enough to solve the sign problem

However, this introduces a new problem “multimodal problem”

transitions among thimbles become indefinitely difficultas increasest

Dilemma between the sign problem and the multimodal problem

large

x

(for small )t (for large )t

2 /6 2( ) )( iS ezz π = −

Multimodal problem and Generalized LTM (2/2)[Alexandru-Basar-Bedaque-Ridgway-Warrington 1512.08764]

flow time (= 𝑇𝑇) small medium largesign problem NG △ OK

multimodal problem OK △ NG

s.t. it is large enough for the sign problembut at the same time is not too large for the multimodal problem

TChoose a middle value of

is not obvious a prioriTHowever, the existence of such Even when it exists,a very fine tuning will be needed

Tempered LTM:

Implement a tempering method by using the flow time t as a dynamical variable

Proposal in Generalized LTM:

no fine tuning needed!

[MF-Umeda 1703.00861] (cf. [Alexandru-Basar-Bedaque-Warrington 1703.02414])

flow time (= 𝑇𝑇) small medium largesign problem NG OK OK

multimodal problem OK OK OK

4. Tempered Lefschetz thimble method (TLTM)[MF-Umeda 1703.00861][MF-Matsumoto-Umeda 1906.04243]

Idea of tempering( ; )

( ; ) ( ) 1 gives a multimodal distribution

for the value of in our main concern (e.g. Suppose that the ac

withtion

)S x

S x V xβ

β β β β=

( ) ( : large)V xβ β ( ) ( : small)V xβ β

transition is difficultdue to the high potl barrier

transition is easy

{ } {( , )}.Then, transitions between we extend

two modes the config space fro

become easyby passin

In the tempe

g through

m to

configs with smaller

ring m h

et od,x x β

β

large β

small β

[Marinari-Parisi Europhys.Lett.19(1992)451]

1It often happens that multimodality disappearsif we take a different val (e.g. forue )o f β β

x

β

Tempered LTM (1/3)

( )0 1 2{ } ( 0,1, , ) 0 ,a At a A t t t t T= = … = < < < =<(1) Introduce co

and cons

pies of config space labeled by a finite set of f

truct a Markov chain that drives the enlarged sys

low time

tem

s

to global

equilibrium

sign problem : OKmultimodality：NG

sign problem： NGmultimodality：OK

Algorithm of TLTM

t

0 0t =

1t

at

( )effT

AS xw e−

00

( )0

( )eff Re S xS x w ew e −− =

At T=( )eff

taa

S xw e−

( ): prob wt factor of replica aw a

[MF-Umeda 1703.00861]

Tempered LTM (1/3)

( )0 1 2{ } ( 0,1, , ) 0 ,a At a A t t t t T= = … = < < < =<(1) Introduce co

and cons

pies of config space labeled by a finite set of f

truct a Markov chain that drives the enlarged sys

low time

tem

s

to global

equilibrium

Algorithm of TLTM

t

At T=

0 0t =

1t

at( )eff

taa

S xw e−

( )effT

AS xw e−

00

( )0


( ): prob wt factor of replica aw a

[MF-Umeda 1703.00861]

Tempered LTM (2/3)Algorithm of TLTM

t

0 0t =

1t

at

00

( )0


At T=(2) After the enlarged system is relaxed to global equilibrium, evaluate the expectation value by using the subsample at

( )effT

AS xw e−

At T=( )eff

taa

S xw e−

( )a A=

-th replicaA( ): prob wt factor of replica aw a

[MF-Umeda 1703.00861]

Tempered LTM (2/3)

t

0 0t =

1t

at

00

( )0


( )effT

AS xw e−

At T=( )eff

taa

S xw e−

Algorithm of TLTM


( )a A=


{ }( , )ax t× = ・ simulated tempering : enlarged system

・ parallel tempering(replica exchange MCMC) : enlarged system

[Marinari-Parisi 1992]

NB: various tempering methods { }( ) : original config spacex≡

{ }0 1

1( ), , , A

Ax xx

+

× × × …=

( )○most of relevant steps can bedone in parallel processes

aw

tedious task △ to detemine

the weights

[Swendsen-Wang 1986, Geyer 1991,Nemoto-Hukushima 1996]

[MF-Umeda 1703.00861]

Tempered LTM (2/3)

t

0 0t =

1t

at

00

( )0


( )effT

AS xw e−

{ }( , )ax t× = ・ simulated tempering : enlarged system

・ parallel tempering(replica exchange MCMC) : enlarged system

[Marinari-Parisi 1992]

At T=( )eff

taa

S xw e−

NB: various tempering methods { }( ) : original config spacex≡

{ }0 1

1( ), , , A

Ax xx

+

× × × …=

( )○most of relevant steps can bedone in parallel processes

Algorithm of TLTM


aw

tedious task △ to detemine

the weights

( )a A=


[Swendsen-Wang 1986, Geyer 1991,Nemoto-Hukushima 1996]

[MF-Umeda 1703.00861]

Tempered LTM (3/3)Important points in TLTM:

NO "tiny overlap problem" in TLTM

We can expect significant overlap between adjacent replicas!

t

0 0t =

1t

at

( )effT

AS xw e−

00

( )0


At T=( )eff

taa

S xw e−

Distribution functions have peaks at the same positions for varying tempering parameter (which is in our case)

xt

σ

The growth of computational cost due to the temperingcan be compensated by the increase of parallel processes

(1)

(2)

[MF-Umeda 1703.00861, MF-Matsumoto-Umeda 1906.04243]

Example: (0+1)-dim Massive Thirring model (1/3)Lorentzian action (dim reduction of (1+1)D model):

20 0 2

0 ( )2MS gdt i mψγ ψ ψψ ψγ ψ= ∂ − −

∫ ( )0 2 0† 02( 1 ,)γ γ γ= =

bosonization + discretization

( ),Grand partition functi tr on :e H QZ β µ

β µ− −=

( ), ( )

PBCSdZ e φ

β µ φ −= ∫

( )

( )

( )2

11

( ) ( )1, 1, , ,1 ,1 , ,

1( ) , ( )exp 1 cos2 2

1( )2

detwith

Nn Nn

N NSn

nnn

i i i inn n n n n n N n n n N n n

dd e Dg

D e e e e m

φ

φ µ φ µ φ µ φ µ

φφ φ φπ

φ δ δ δ δ δ δ δ

−

+

==

− −′ ′ ′ ′ ′ ′+ −

+ + +

−= =

− − +

− = +

∑∏

[Pawlowski-Zielinski 1302.1622, 1402.6042,Fujii-Kamata-Kikukawa 1509.08176]

[ ]( ; ) ( ; )D Dφ µ φ µ∗ = −det det ( )D µ∉ ∈ thus, for detOne can show

NSign problem will arise when is very large

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8Re

Example: (0+1)-dim Massive Thirring model (2/3)χχChiral condensate

2 (w/o temp)T = 2 (w/ temp)T =

dominated by a singlethimble

contributed bymultiple thimbles

large errorsdue to the sign problem

deviate from exact valuesdue to multimodality(see below)

2 (w/o temp)T =

2 (w/ temp)T =

0T =

dominant thimbleConfirmation of the resolution of multimodality

good agreement

(i.e. config space is well explored)

( : exact values)

Re χχ⟨ ⟩

reweighting

/θ π /θ π

[MF-Umeda 1703.00861]

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0denominator

Example: (0+1)-dim Massive Thirring model (3/3)

Confirmation of the resolution of sign problem

2 (w/o tempering)T =

2 (w/ tempering)T =

0T =

2T =no sign problemat

0

sign problemsurely existsfor the originalaction ( )T =

sign average

( )eff

NB: sign avarage is smaller

for the right

T

T

i

Se θ φ

sampling

( ( ))( ) ~e ffff eT

T

I T

T

iS zi

S See φθ φ −

reweighting

[MF-Umeda 1703.00861]

(

)

)

(

( )( )

ef

f

f

e fT

ta

T

TS

Si

ie

e θ

θ

φ

φ φφ

⟨ ⟩ =

⟩⟩

⟨⟨

We actually can go further...

( )

( )

( )

1

1

( )( )

( )( )

( ( )( ( )) )conf

conf

eff

eff

aa ta

tata

tat

k

aa

k

t

N

tt

k

kaS N

k

S

i xi x

i xi x

S

e z xe z x

e e

θθ

θθ

=

=

⟨ ⟩⟨ ⟩ ≈ ≡

⟨=

⟩

∑

∑

Consider the estimates of at various flow times :S at⟨ ⟩

( 0,1 , ), Aa = …

t

0 0t =

1t

at

At T=

:atHere the estimation on the RHS is made by using the subsample at

-th replicaa

[MF-Matsumoto-Umeda 1906.04243]

We actually can go further...Consider the estimates of at various flow times :S at⟨ ⟩

The LHS must be independent o due to Cauchy's the mf orea

The RHS must be the same for all 's within the statistical error marginif the system is in global equilibrium and the sample size is large enough

a

This gives a method with a criterion for precise estimation in the TLTM!

a

a

S⟨ ≈⟩

0 1 2

2 fit with a const fucn of aχ

( 0,1 , ), Aa = …

a

| |taie θ⟨ ⟩

0 1 2 maxaA=

mina mina maxaA=

they make the stat error ofth

to be disc

e ratio

arded because

untrustablea3

2 confN

12 confN

①②

③

④

discarded


( )

( )

( )

1

1

( )( )

( )( )

( ( )( ( )) )conf

conf

eff

eff

aa ta

tata

tat

k

aa

k

t

N

tt

k

kaS N

k

S

i xi x

i xi x

S

e z xe z x

e e

θθ

θθ

=

=

⟨ ⟩⟨ ⟩ ≈ ≡

⟨=

⟩

∑

∑

5. Applying the TLTM to the Hubbard model[MF-Matsumoto-Umeda 1906.04243]

Hubbard model (1/2)[Hubbard 1963]Hubbard model

modeling electrons in a solid

( )†, ,

,, : creation/anihilation op of an electron

on site with spin c cσ σ

σ

•

=↑ ↓x x

x↑

↓↑

↓

↑

, , 1 / 2 s.t.n nσ σ −→x x

( )†, , , , , ,

,

21

1 12

12

HH

K UH n nc c n nσ σσ

κ µ ↑ ↓ ↑ ↓

= − − + − + −

−∑∑∑ ∑x y x x x xxyx xx y

( )

( )

†,, ,

0

0

: hopping parameter : chemical potential

: strength of on-site replusive potential

n c

U

cσ σ σ

κµ

≡ > >

x x x

( )†, , , , , ,

,

Hamiltonian c nc nn nUH σ σ

σκ µ ↑ ↓ ↑ ↓⟨ ⟩

•

= − − + +∑ ∑ ∑∑x y x x x xx xx y

(fermion bilinear) (four fermion)

( ) : # of sitesSN

↓

↑ ↑

,,

0 1 / 2 0half-filling n σσ

µ=↑ ↓

= ⇔ − =∑ x

Hubbard model (2/2)


( ) ( ) ( )1 2 1 2 1 2( ) (

Quantum Monte Carlo

NH NH H H H HHe ee ee Nτ τ

τββ β+− + − − −−

•

= = ≅ =

( )( ), ,21/2 1/2

toTransform u singa fe a boson rmion bilinearnH U n

e e φ↑ ↓− − −− =∏ x x

x

2, ,

,,[ ]

,1

(1/2)[ [ ] [] ]det det

NSZ e d e M Md

τ

β µφ φ

φ φ φ φ↓↑

−

=

− ∑≡= ∏∏∫∫ x x xx

x

( ),/ [ ]] 1 diag[ matrix

s

KN s s

i UM e Ne e Nβµ κ φφ ±↑ ↓

±+≡ ×∏ x

：

,contGrand partition function (continuous imaginary time) tr HZ e ββ µ

−=• ：

2

0

[ ] [ ] [ ] 0det: For half filling ( )

No sigde

n problemt detM M M

µ

φ φ φ↑↓↑

=

=

⇒

≥

NB

0states ( )µ ≠This gives complex actions for non half-filling

We apply the Tempered LTM to this system ,) ( )1, , ((

)

i N

si Nx x

NN Nτ

φ = ∈ = …

==

x

Results for 1D lattice (1/3)

w/ temp

w/o temp

reweighting

( ), ,

1 1number density x xxsN

n nn ↑ ↓= + −∑

deviate from exact valuesdue to multimodality(but very small errors)

agree with exact values(small errors)


22

1, 16, 0.4

( )

max flow time 5,000

s

NN

U T

τ

β κ β

= =

= = =

spatial lattice: 1D periodic lattice witimaginary time : 2 steps

sample size:

h

µ

n⟨ ⟩

w/ tempw/o temp

[MF-Matsumoto-Umeda 2019]

βµ


w/ temp

w/o temp

reweighting

( ), ,

1 1number density x xxsN

n nn ↑ ↓= + −∑

deviate from exact valuesdue to multimodality(but very small errors)

agree with exact values(small errors)


µ

n⟨ ⟩

w/ tempw/o temp

focus on this

22

1, 16, 0.4

( )

max flow time 5,000

s

NN

U T

τ

β κ β

= =

= = =

spatial lattice: 1D periodic lattice witimaginary time : 2 steps

sample size:

h

βµ



peaked at several anglesbecause of sufficient transitionsamong thimbles(errors become a bit largerdue to the small size of sampling)

peaked at a single angle ~0.8 πdue to the trap to a single thimble(errors become smallbecause the thimble is well sampled)

0.4(projected on a plane)

T =Distribution of flowed configs at flow time

w/o temp w/ temp

w/ tempw/o tempreweighting

distributing uniformlyfrom –π to +π

severe sign problem

Histogram of ImS(z)/π

/ImS π /ImS π /ImS π


Results for 1D lattice (3/3)sign average

When only a single (or very few) thimble(s) is sampled,the sign average can become larger than the correct samplingdue to the absence of phase mixtures among thimbles

( )

( ) ( ( ))( )

eff

eff

T

T

T

T

i

S

x

x

T Si

e xx

e

zθ

θ

⟨ ⟩ ⟨ ⟩ = ⟨ ⟩

It is generally dangerous to regard the sign averageas an index of the "resolution of the sign problem"

( )eff

T

T

i xS

e θ⟨ ⟩

µ

w/ temp (T>0)w/o temp (T>0)

βµ


Results for 2D lattice (1/5)5

2 23 13, 0.5

( )

max flow time 00~25,000 depending on

s

NN

U T

τ

β κ ββµ

= = ×

= = =

imaginary spatial lattice: 2D periodic lattice wit

time : 5 steps

sample size: 5,0

h

discarded

5, 1 )( 1min maxaa = =

5Example: βµ =

2 fitχ

n⟨ ⟩

①

②

③

④

(

( )

) ( ( )) eff

efft

a

ta

t

ta a

aai x

i

S

tx

Sne

e

z xnn θ

θ

⟨ ⟩ ⟨ ⟩ = ≈

⟨ ⟩

discarded


2

0.221 0.012/ 0.45DOF

nχ⟨ ⟩ ≈ ± =

Results for 2D lattice (2/5)5, 2 23, 13

sN NU

τ

β κ β= = ×

= = ( ), ,

1 1s

nN

n n↑ ↓⟨ ⟩ −= +∑ x xx


reweightinglarge errorsdue to the sign problem

deviate from exact valuesdue to multimodality(but very small errors) agree with exact values

(small errors)w/o temp

w/ temp

focus on this

( 0)T >

( 0)T >

( 0)T =

βµ

Results for 2D lattice (2/5)( ), ,

1 1s

nN

n n↑ ↓⟨ ⟩ −= +∑ x xx

[MF-Matsumoto-Umeda 1906.04243]5, 2 23, 13

sN NU

τ

β κ β= = ×

= =

reweightinglarge errorsdue to the sign problem

deviate from exact valuesdue to multimodality(but very small errors) agree with exact values

(small errors)w/o temp

w/ temp

focus on this

( 0)T >

( 0)T >

( 0)T =

βµ

Results for 2D lattice (3/5)[MF-Matsumoto-Umeda 1906.04243]

0.5 ( 5) (projected on a plane)

T βµ= =Distribution of flowed configs at flow time

w/ temp w/o temp

distributed widelyover many thimbles

distributed over onlya small number of thimbles

Results for 2D lattice (4/5)w/ temp

w/o temp


0a = 1a = 2a = 3a = 4a = 5a =

6a = 7a = 8a = 9a = 10a = 11a =

0a = 1a = 2a = 3a = 4a = 5a =

6a = 7a = 8a = 9a = 10a = 11a =

unimodal distribution

many peaks (may not be so obviousbecause there are so many peaksand the peaks are broadened by Jacobian)

[ , ]Histogram of at

θ π π∈ −

Results for 2D lattice (5/5)[MF-Matsumoto-Umeda 1906.04243]sign average

( )

( ) ( ( ))( )

eff

eff

T

T

T

T

i

S

x

x

T Si

e xx

e

zθ

θ

⟨ ⟩ ⟨ ⟩ = ⟨ ⟩

It is generally dangerous to regard the sign averageas an index of the "resolution of the sign problem"

When only a single (or very few) thimble(s) is sampled,the sign average can become larger than that in the correct samplingdue to the absence of phase mixtures among thimbles

( )eff

T

T

i xS

e θ⟨ ⟩

βµ

Comment on the Generalized LTM

( )

52 2

3, 13, 0 0.4 0 10

( )

00~25,000 depending on

s

NN

U T a

τ

β κ ββµ

= = ×

= = ≤ ≤ ⇔ ≤ ≤

imaginary time : 5 steps spatial lattice: 2D periodic lattice with

sample size: 5,0

(

( )

) ( ( )) eff

efft

a

ta

t

ta a

aai x

i

S

tx

Sne

e

z xnn θ

θ

⟨ ⟩ ⟨ ⟩ = ≈

⟨ ⟩


5Example: βµ = large stat errors(due to sign problem)

wrong value(due to multimodality)

It is a hard task to find an intermediate flow timethat solves both sign problem and multimodality

6. Other approaches

Path optimization (sign maximization) method

( )| |where Find a sign-opti

takes a maximal mized manif

valueold

i ze θ⟨ ⟩Σ

Care must be paid not to miss good surfaceswhen multi thimbles are relevant

This may also be used as a complementary method to TLTMfor improving the precision after one obtainsa rough shape of thimble and the corresponding sign average

x

iy

1 23

thimbles

Σ

initΣ

[Kashiwa-Mori-Ohnishi 1705.05605][Alexandru et al. 1804.00697]

NB( )| | may take larger values

when only a small number of thimbles are taken into account

i ze θ⟨ ⟩

Single-thimble dominance

Develop a machinary so thatthe problem can be reduced to caluculations over a single thimble

- Works for the Hubbard model in some parameter region- May not be a versatile method ...

[Di Renzo-Zambello, Ulybyshev et al. ,...]

- May be combined with TLTM to further improve the precision

There had been an expectation that only a single thimble dominates at criticality.

First counterexample: (0+1)-dim Thirring model[Fujii-Kamata-Kikukawa 1509.08176]

Multi thimbles are taken care of in Generalized LTM and Tempered LTM

Other approach: sticking to the single-thimble dominance

•

Change of dynamical variables•

[Cristoforetti et al. 1205.3996, 1303.7204, 1308.0233]

[Ulybyshev et al. 1906.07678]

[History]

7. Conclusion and outlook

Conclusion and outlook

[MF-Matsumoto, work in progress]

What we have done:- We proposed the tempered Lefschetz thimble method (TLTM)

as a versatile method to solve the numerical sign problem- We further developed it and found an algorithm to estimate expec. values

with a criterion ensuring global equilibrium and the sample size

- GLTM can easily give incorrect results or large ambiguities- TLTM works for the Hubbard model and gives correct results,

avoiding both the sign and multimodal problems simultaneouslyOutlook:

- Investigate the Hubbard model of larger temporal and spatial sizesto understand the phase structure

- More generally, apply the TLTM to the following three typical subjects:① Finite density QCD② Quantum Monte Carlo (incl. the Hubbard model)③ Real time QM/QFT

- Develop a more efficient algorithm with less computational cost

should not depend on replica due to Cauch(the key: y's th )em eora a

3~4( )][computational cost: O N

Thank you.

Documents

Sign problem in Monte Carlo simulations and the …Sign problem in Monte Carlo simulations and the tempered Lefschetz thimble method Sep 2, 2019 熱場の量子論とその応用 @YITP