使用路径优化解决符号问题的简单方法资源-CSDN文库

Open

Access

119 浏览量 2020-04-09 02:54:12 上传评论收藏 630KB PDF 举报

资源详情

资源评论

JHEP12(2018)054

Published for SISSA by Springer

Received: May 18, 2018

Revised: November 7, 2018

Accepted: December 1, 2018

Published: December 10, 2018

A simple approach towards the sign problem using

path optimisation

Francis Bursa

and Michael Kroyter

School of Physics and Astronomy, University of Glasgow,

Kelvin Building, University Avenue, Glasgow, G12 8QQ, United Kingdom

Department of Sciences, Holon Institute of Technology (HIT),

52 Golomb St., Holon 5810201, Israel

E-mail: francis@semichrome.net, michaelkro@hit.ac.il

Abstract: We suggest an approach for simulating theories with a sign problem that

relies on optimisation of complex integration contours that are not restricted to lie along

Lefschetz thimbles. To that end we consider the toy model of a one-dimensional Bose gas

with chemical potential. We identify the main contribution to the sign problem in this

case as coming from a nearest neighbour interaction and approximately cancel it by an

explicit deformation of the integration contour. We extend the obtained expressions to

more general ones, depending on a small set of parameters. We ﬁnd the optimal values

of these parameters on a small lattice and study their range of validity. We also identify

precursors for the onset of the sign problem. A fast method of evaluating the Jacobian

related to the contour deformation is proposed and its numerical stability is examined. For

a particular choice of lattice parameters, we ﬁnd that our approach increases the lattice

size at which the sign problem becomes serious from L ≈ 32 to L ≈ 700. The eﬃcient

evaluation of the Jacobian (O(L) for a sweep) results in running times that are of the order

of a few minutes on a standard laptop.

Keywords: Lattice ﬁeld theory simulation

ArXiv ePrint: 1805.04941

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP12(2018)054

JHEP12(2018)054

Contents

1 Introduction 1

2 Simple integration contours 7

3 A general ansatz for the integration contour 10

4 The Jacobian 11

5 Results 15

5.1 Fitting the parameters 16

5.2 The observables and the mean phase factor for the various contours 17

5.3 Resolving the numerical instability 21

6 Discussion 23

1 Introduction

The sign problem poses a challenge for a lattice simulation of many physical theories,

ranging from widely-studied systems such as QCD and other theories with a ﬁnite chem-

ical potential [1, 2], through the simulation [3] of PT-symmetric theories [4, 5], to the

simulations [6] of more exotic systems such as string ﬁeld theory [7]. It also appears in the

simulations of various condensed matter theories as well as in other systems. In general,

one faces a challenge whenever e

−S

is not a positive quantity. In such a case e

−S

cannot

be identiﬁed as a (non-normalized) probability density and the Metropolis algorithm [8]

cannot be applied without a modiﬁcation. A simple modiﬁcation is reweighting. How-

ever, for fast oscillations of the phase of the action the computational cost of evaluation

using reweighting grows exponentially with the lattice volume. The sign problem is this

exponential behaviour.

Several methods have been introduced in order to circumvent the sign problem: one

could attempt to expand the problematic part of the integrand in a Taylor expansion such

that all terms in the expansion involve only integrals with a constant phase [9]. While

this method works for some systems, it introduces an error in the estimate of observables

on top of the standard statistical error. Also, one could not expect it to be useful for

large values of the parameters used in the expansion, e.g. the chemical potential. Another

approach is to analytically continue the problematic parameters to values that lead to no

sign problem and then continue back [10, 11]. While this method could also be used in

some cases, numerical analytical continuation can be quite challenging and the approach

suﬀers from problems similar to those of the Taylor expansion approach. A third approach

is the complex Langevin method [12]. Here, stochastic dynamics is used for the calculation

– 1 –

JHEP12(2018)054

of observables and there is no reference to e

−S

as a probability density. Hence, the sign

problem is avoided. Nonetheless, this method also has its limitations [13–15].

An important ingredient of the complex Langevin method is the complexiﬁcation of the

degrees of freedom. Complexiﬁcation can be used in other ways as well: one could attempt

to avoid the phase oscillation causing the sign problem by using (the multi-dimensional

form of) Cauchy’s integral theorem for deforming the original integration contour to one

without (or at least with less) phase oscillations. When using Cauchy’s theorem, one must

take care not to pass any singularities of the integrand and to deform the asymptotic

integration range only when it vanishes fast enough, e.g. using Jordan’s lemma. For most

physical theories the analytical continuation of the action is regular. Hence, the ﬁrst issue

is of no concern for these theories. The asymptotic behaviour of the integrand, on the other

hand, can become singular for speciﬁc contours. For potentials that behave asymptotically

as V ' φ

with N variables (N equals the product of ﬁeld components by the number

of lattice points) there are generically (n − 1)

diﬀerent homology classes of integration

contours [16, 17]. One should therefore be careful not to deform the contour away from

the original homology class.

A possible prescription for the contour deformation is to use Lefschetz thimbles [17],

which also give the steepest descent of the (real part of the) measure. The implementation

of Lefschetz thimbles in lattice simulations was introduced in [18]. Despite the success of

this approach, it is also not without faults:

1. While it is known that Lefschetz thimbles are manifolds, explicit expressions deﬁning

these manifolds are absent. This leads to complicated and expensive algorithms for

verifying that the integration contour does not leave the thimble.

2. While in some cases only a single thimble contribution is relevant in the continuum

limit [18], in other cases one needs many thimbles. Since the mean phase factor, i.e.



i Im(S)



P Q

, the mean value of the phase in the phase quenched ensemble, can diﬀer

among diﬀerent thimbles, this could lead to reemergence of the sign problem as a

“global sign problem”, especially in light of the fact that the number of homology

classes (the number of independent thimbles) goes to inﬁnity in the continuum limit.

3. Lefschetz thimbles are constructed in a way that keeps the imaginary part of the clas-

sical action constant. However, when working with thimbles the integration measure

changes. This leads to the “residual sign problem”.

4. For any given lattice there is a diﬀerent set of thimbles. One should therefore identify

the thimbles as well as their contribution to the desired cohomology class indepen-

dently for every lattice size, lattice spacing, mass, etc. This becomes more and more

complicated as the lattice size is increased.

The above issues can be summarized by noting that Lefschetz thimbles improve, but do

not completely resolve the sign problem, while reducing the running cost from exponential

to O(V

). In some cases the computational cost of a thimble simulation can be further

decreased [19]. Nonetheless, it is still advisable to look for improvements and alternatives

– 2 –

JHEP12(2018)054

to the Lefschetz thimble method. Indeed, generalisations of the thimble method have

been proposed, in which the above mentioned issues are addressed by choosing a single

integration contour as an approximation to the sum of all the relevant thimbles [20–22].

In this work, we propose a method for a diﬀerent deformation of the integration con-

tour. We do not attempt to approximate the thimbles. Thimbles are deﬁned by a gradient

ﬂow, which deﬁnes a real N-dimensional submanifold of the complex N -dimensional space.

On the other hand, the requirement of a vanishing imaginary part puts only one real con-

straint and hence its solution is a (2N −1)-dimensional real manifold. We intend to choose

an N-dimensional subspace of this manifold, or an approximation to such a subspace, in

order to obtain a proper integration cycle. This might appear to be too arbitrary as com-

pared to a cycle deﬁned by a gradient ﬂow. However, there is much arbitrariness also in

the deﬁnition of a gradient ﬂow: in order to deﬁne the gradient operator, a metric should

be introduced in the complex N-dimensional space. It is possible to choose this metric

to be the ﬂat metric, but this choice is not canonical [17]. The only relevant attribute

of this metric is its compatibility with the complex structure, which guarantees that the

imaginary part is constant along the ﬂow, i.e. the metric must be K¨ahler [17]. While not

all the manifolds with constant imaginary part and appropriate boundary conditions can

be obtained by varying the metric (also, some metric variations lead to a reparametrisation

rather than a change of the obtained manifold), some can be obtained this way. This illus-

trates that even “the thimble” is not a uniquely deﬁned object. Moreover, an integration

cycle which is not the thimble can be chosen to be a single contour that already takes into

account the change in the integration measure. Hence, with a proper choice of integration

contour, both the residual sign problem and the global sign problem could be avoided.

For illustrating our method we consider the speciﬁc case of a Bose gas at a ﬁnite

chemical potential. This model is often used for the purpose of examining new methods

for dealing with the sign problem, e.g. [18, 23]. On the lattice (in lattice units) the action

of the model is,

S =



d +



+ v

) + λ

+ v

)

(1.1)

−

d−1

ν=1



k+ˆν

+ v

k+ˆν



− cosh(µ)(u

+ v

) + i sinh(µ)(u

− v

)

Here, u

, v

are respectively the real and imaginary parts of a single complex ﬁeld,

= u

+ iv

, (1.2)

the index k runs over all lattice sites and d is the dimensionality of space-time. While the

signature is Euclidean, the “time” direction is special, since the interactions related to it

diﬀer from those of the other coordinates due to the presence of the chemical potential

µ. Moreover, the terms related to the chemical potential are the ones responsible for the

imaginary part of the action. The action enjoys a U(1) symmetry, even for a non-zero

chemical potential.

– 3 –

JHEP12(2018)054

Rescaling the ﬁelds by a factor of

2d+m

the action becomes,

S =

λα

)

(1.3)

−α

d−1

ν=1

k+ˆν

)+cosh(µ)(u

)+i sinh(µ)(v

−u

)

where we deﬁned

α ≡

2d + m

. (1.4)

For ﬁxed d, this constant ranges from α = 0, which corresponds to the inﬁnite mass limit

to α =

, which corresponds to the zero mass limit. For small values of the chemical

potential the vacuum of the theory is in an unbroken phase of the U(1) symmetry, while a

spontaneously broken phase is obtained for

cosh(µ) > 1 +

. (1.5)

In the continuum limit the above equation takes the simpler form µ > m.

For concreteness we concentrate in the rest of the paper on d = 1. In this case the ﬁrst

term in the second line of (1.3) is absent. Assuming periodic boundary conditions and L

lattice points one can write

S =

λα

k=1

+ v

)

(1.6)

− α



cosh(µ)(u

k+1

+ v

k+1

) + i sinh(µ)(v

k+1

− u

k+1

)



As long as µ is small enough, taking the integration contour to be the product of the real

and v

axes would not lead to a signiﬁcant phase factor. As µ is increased the phase

variations along this integration contour become larger and larger and the sign problem

appears. To illustrate this (well known) situation we attempted to simulate the system with

the standard integration contour on lattices of size 16, 24, 32, . . . , 72. For simplicity we set,

as we do for all the simulations in this paper, λ = m = µ = 1 and estimated the number of

conﬁgurations that should be sampled in order to obtain a standard error of hSi that would

be about 2% of hSi. Since this is only for an illustrative purpose we were content with

identifying a number of conﬁgurations that leads to a relative error in the range of 1.6% −

2.4%. We ﬁnd that while for 16 and 24 lattice points we need roughly the same number

of conﬁgurations, for larger lattices the required number of conﬁgurations grows fast. We

performed a least squares ﬁt for the logarithm of the number of conﬁgurations as a function

of lattice size for the range L = 24 . . . 72 and found that the number of conﬁgurations needed

grows as e

0.21L

. We plot the number of conﬁgurations needed together with the ﬁt found in

ﬁgure 1. Assuming this exponential behaviour continues, we can estimate that the running

time (on a standard laptop) with no contour modiﬁcation needed for the stated precision

– 4 –

剩余27页未读，继续阅读

评论收藏

内容反馈

使用路径优化解决符号问题的简单方法

评论0

最新资源

使用路径优化解决符号问题的简单方法

评论0

最新资源

相关推荐

运筹学优化软件

Java开发实战1200例(第1卷).(清华出版.李钟尉.陈丹丹).part3

配置好的傻瓜化小狼毫全拼+多种双拼+五笔86-98+笔画输入法v200312 - 17种方案

《你必须知道的495个C语言问题》

grub4dos-V0.4.6a-2017-02-04更新

面向对象与UML资料

康成制作的某汽车租赁公司网站源码 v89

亮剑.NET深入体验与实战精要2

亮剑.NET深入体验与实战精要3

你必须知道的495个C语言问题

Oracle SQL高级编程(资深Oracle专家力作，OakTable团队推荐)--随书源代码

C#编程经验技巧宝典

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

NPPJSONViewer.zip

GD32替换STM32注意事项.pdf

XCP协议的规范文档

VS2015安装证书，JavaScript_ProjectSystem.msi，JavaScript_LanguageService.msi

CANoe通过CAPL脚本实现自动测试

蓝牙BLE协议中文版.pdf

BaiduOCR.zip

Qt 5实现串口调试助手（源工程文件、0积分下载）