2020/08/25 熱場の量子論とその応用

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経路最適化法を用いたU(1)ゲージ理論における符号問題への取り組み
Yuto Mori (Kyoto Univ.)
Kouji Kashiwa (Fukuoka Inst. Tech.)
Akira Ohnishi (YITP)
2020/08/25 熱場の量子論とその応用

2020/08/25 熱場の量子論とその応用

Content
1. Sign problem
2. Path Optimization
3. low dimensional QCD

4. U(1) one plaquette action
5. Summary1

2020/08/25 熱場の量子論とその応用

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Sign problem
 When, serious cancellation occurs in integration at large
volume. And, integrals cannot be obtained precisely.

 Seriousness of the sign problem… average phase factor (APF) e.g. finite density QCD

ex.)
J. Nishimura and S. Shimasaki,
PRD 92 (2015) 011501

2020/08/25 熱場の量子論とその応用

Path Optimization

3Y.M., K. Kashiwa, A. Ohnishi,
Phys. Rev. D96 (2017) no.11, 111501

Optimize the integral path in the complexified variable space
to weaken the sign problem.
(Integral of holomorphic(analytic) function is independent ofintegral path.)
We can regard the sign problem
as an optimization problem.
(ex.) One variable case Trial function (integral path) Cost function (function to minimize)oscillation of the integrand becomes weaker.

2020/08/25 熱場の量子論とその応用

Path Optimization

4Y.M., K. Kashiwa, A. Ohnishi,
Phys. Rev. D96 (2017) no.11, 111501

Optimize the integral path in the complexified variable space
to weaken the sign problem.
(Integral of holomorphic(analytic) function is independent ofintegral path.)
We can regard the sign problem
as an optimization problem.
(ex.) One variable case Trial function (integral path) Cost function (function to minimize)oscillation of the integrand becomes weaker.

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2020/08/25 熱場の量子論とその応用

Neural Network for field theories
Neural Network(NN) is powerful to represent any functions of many inputs.
 Combination of linear and non-linear transformation※:Activation fn. (ex. tanh): parameters Any fn. can be reproducedat (# of units of hidden layer) → ∞(Universal approximation theorem)

InputsG. Cybenko, MCSS 2, 303 (1989)
K. Hornik, Neural networks 4, 251 (1991)

 We input the real part of variables,
and obtain the imaginary part from outputs.

Hiddenlayer

Outputs

2020/08/25 熱場の量子論とその応用

Flowchart of POM: Original manifold: Modified manifold

Kashiwa and YM, arXiv:2007.04167.

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2020/08/25 熱場の量子論とその応用

0+1 dim. QCD
Application to gauge theories

1-spiecies of Staggered fermion

One link variable, No plaquette

7YM, Kashiwa, Ohnishi,
arXiv:1904.11140

cf. CLM: Aarts, et al.(2010)
LTM: Schmidt, et al.(2016),
Di Renzo, et al.(2017)

2020/08/25 熱場の量子論とその応用

8YM, Kashiwa, Ohnishi,
arXiv:1904.11140

1 dim. QCD one link Complexification the link variable：trial function
We generate the configurations by HMC andoptimize NN parameters by stochastic gradient descent(SGD, Adadelta).
 1 – average phase factor(APF) ()After some steps ofoptimization,
1-APF becomes
3~100 times
smaller than before.

2020/08/25 熱場の量子論とその応用

1+1D QCD

9YM, Kashiwa, Ohnishi,
in progress• Action with staggered fermion (strong coupling)• Polyakov gauge (without diagonal gauge fixing)※There still remains the gauge degreeof freedom depending on only x• Complexification

2020/08/25 熱場の量子論とその応用

1+1D QCD

10YM, Kashiwa, Ohnishi,
in progress

2×2 Lattice,
Average phase factor
Average phase factor can be slightly
enhanced by the deformation of manifold.

…Is there upper bound of APF?
Is the optimization failed?
(and Why?)→We study the U(1) gauge theory toinvestigate this reason

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2020/08/25 熱場の量子論とその応用

U(1) one plaquette action

Kashiwa and YM,
arXiv:2007.04167.

ActionPartition function

Modified Bessel fn.

Gauge fixing
1~3 link variables among 4 can be fixed as a unit element

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2020/08/25 熱場の量子論とその応用

U(1)｜integral path

Kashiwa and YM,
arXiv:2007.04167.

 In the following,is pure imaginary andwe use the gauge fixing: We complexify the variable

and optimize
 Scatter plot of 𝒫 and 𝒫 × 𝑒 𝑖𝜃 (𝜃:phase of 𝐽𝑒 −𝑆 , probability:|𝐽𝑒 −𝑆 |)Original path

Modified path13

2020/08/25 熱場の量子論とその応用

U(1)｜phase histogramNormalized histogram ofKashiwa and YM,
arXiv:2007.04167.
On original path

phase 𝜃 (𝜃:phase of 𝐽𝑒 −𝑆 ) 𝜃 distribution becomeswell localized by optimization.

 There remains

global sign problem→exchange Monte Carlois needed.
So we use exchange HMC
in this work,

On Modified path2020/08/25 熱場の量子論とその応用U(1)｜expectation valueAverage phase factor
 APF can be enhanced.
(solid line :APF on original path) The upper bound is caused
by global sign problem
 Expectation value of Plaquette Analytic results (solid line)

are reproduced except
the region near zero point of…exchange HMC works well.

.

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Kashiwa and YM,
arXiv:2007.04167.

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2020/08/25 熱場の量子論とその応用

U(1)｜effect of gauge fixing

Kashiwa and YM,
arXiv:2007.04167.

 Average phase factor as a function of optimization step

with/without gauge fixing With gauge fixing, APF canapproach to the maximal value.

 Without gauge fixing,optimization fails.

 The symmetry may bedifficult to learn

2020/08/25 熱場の量子論とその応用

16Summary In the path optimization method, we can regard the sign
problem as an optimization problem.
Neural Network and optimization methods (SGD) developed inmachine learning are helpful.

 We apply this method to U(1) gauge theory with one
plaquette actionWith gauge fixing, average phase factor becomes large and exact results
are reproduced in observable calculations.

Without gauge fixing, the optimization fails.

 We will study this reason.