単調性を用いたレプリカ交換モンテカルロ法の拡張と行列幾何学
渡辺展正(KEK → 9月よりYITP)

共同研究者 : 花田政範 (Univ. Surrey), 松浦壮 (慶應大), 菅野聡 (筑波大)(Work in progress)
2022/09/22
基研研究会 「熱場の量子論とその応用」

Contents• Motivation from the gauge/gravity duality• How to identify the location of D-branes from QFT side
• Notion of wave packet in operator/path-integral formalism• How to solve the optimization problem• Replica-Exchange Monte Carlo method and its extension• Numerical demonstration• Summary & Prospectives

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基研研究会「熱場の量子論とその応用」2022

Contents• Motivation from the gauge/gravity duality(c.f. lectures by Hanada)• How to identify the location of D-branes from QFT side
• Notion of wave packet in operator/path-integral formalism• How to solve the optimization problem• Replica-Exchange Monte Carlo method and its extension• Numerical demonstration• Summary & Prospectives

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基研研究会「熱場の量子論とその応用」2022

Introduction:gauge/gravity duality
conjecture from 2 descriptions of Dp-branes in string theory;
D-branes & string

E ective theory ofopen strings on branes

Theory of closed stringsin D-brane geometry

QFT
GR in curved spacetime(Super Yang-Mills theory)
Expected to be obtained by the nonperturbative aspects of string theory

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基研研究会「熱場の量子論とその応用」2022

Position of D-branes & open stringsxμ

E ective action
211g
dp+1x trFμν2 + (Dμ XI )2 + [XI, XJ ]2 + (fermion terms)∫24(4)xνμ, ν = 0,⋯, pI, J = p + 1,⋯,9XI =(XI (x) : N × N hermitian matricesxI⋱)

Low-energy or classical states→ X : simultaneously diagonal
diagonal : position of D-branes
o -diagonal : open strings among D-branes[Witten, (1995)]

XI =()and strong coupling (:low energy),
where the dual gravity has been known.
基研研究会「熱場の量子論とその応用」2022

fi

ff

in the ʼt Hooft limit (N → ∞, λ = g 2 N ∼ N 0 : xed, E ∼ N 2)

[Polchinski, (1998/1999) / Susskind, (1999) / Hanada, (2021)]

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The above interpretation is also feasible

Review: Matrix Quantum Mechanics
Quantum Mechanics with dN 2 degrees of freedom(I, J = p + 1,⋯, d + p + 1)

→ each matrix element is operator

X̂I,ij =N2∑
a=1

X̂I,aτija,

̂ =PI,ijN2∑
a=1

̂ τaPI,aijτ a : generator of G = U(N )tr(τaτb) = δab
1 ik jlij kl(τ τ ) = δ δ∑ a aNa

Uncertainty relation

̂ ] = iδIJ δab[X̂I,a, PJ,b

• Hilbert space;̂ | P⟩ = PI,a | P⟩
ℋ = Span { | X⟩; X̂I,a | X⟩ = XI,a | X⟩} = Span { | P⟩; PI,a}“coordinate basis”“momentum basis”• Partition function
1̂̂Z(T ) =dg Trℋ (ĝ e−H/T) = Trℋinv (e−H/T)
VolG ∫G

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基研研究会「熱場の量子論とその応用」2022

Notion of wave packet[Hanada (2021)]

How to identify the position of D-branes?

❌

X̂I,a | X⟩ = XI,a | X⟩

: coordinate eigenvalue

Inconsistent in terms of the uncertainty relation and far from the classical picture…

In order to identify the location, consider the wave packet in dN 2-dim space✔

| Φ⟩ =∫ℝdN 2and the center of Φ(X),7 / 23dX | X⟩⟨X | Φ⟩ =YI,ij∫ℝdN 2dX Φ(X) | X⟩

(c.f. coherent state)[Hanada (2021)]

基研研究会「熱場の量子論とその応用」2022

Wave packet in color space[Hanada (2021)]

the wave packet in dN 2-dim space and the center of it| Φ⟩ =∫ℝdN 2dX | X⟩⟨X | Φ⟩ =∫ℝdN 2

YI =dX Φ(X) | X⟩(⋱)

The center YI determines the location of D-branes!
Gauge transformation
X̂I,ij → (U X̂IU −1) =ijN∑k,l=1(U′)
YI,ij

Uik X̂I,klUlj−1 =: X̂ (U)I,ij

provides the gauge orbit of YI,a• the position of the wave packet moves(U)
YI,ij

dN 2

ℝYI,ijO(1)tr YI2“diagonalizability” of Y• the shape and distance from originof the wave packet are invariant￼

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基研研究会「熱場の量子論とその応用」2022

Determination of wave packet
How to identify the low-energy wave function for interacting theory?
Proposal 1 (Hamiltonian formalism)min⟨Φ | Ĥ | Φ⟩Φ[Hanada (2021)]⟨Φ | X̂I | Φ⟩ = YI,

with given⟨Φ | PÎ | Φ⟩ = QI,⋯If we set YI = 0, QI = 0, | Φ⟩ = | YI = 0, QI = 0⟩ is the ground state.
We can prepare such a state by the technique of quantum computation.

e.g.) Adiabatic state preparation
221m(s)g(s) ̂ ̂ 22̂2̂
̂
HA(s) = tr
PI +XI −[XI, XJ ]24(2)Ĥ A(s = 0) = Ĥ 0
Ĥ A(s = sf ) = Ĥ target

ground state of Ĥ target is constructed from known & unique ground state of Ĥ 0sf

| ground state⟩ = lim T exp −i ds Ĥ A(s) | ground state⟩0
sf→∞
( ∫0)

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基研研究会「熱場の量子論とその応用」2022

Determination of wave packet
How to identify the low-energy wave function for interacting theory?
Proposal 1 (Hamiltonian formalism)min⟨Φ | Ĥ | Φ⟩[Hanada (2021)]

with givenΦProposal 2 (Path-integral formalism)⟨Φ | X̂I | Φ⟩ = YI,⟨Φ | PÎ | Φ⟩ = QI,⋯[Hanada, Kanno, Matsuura, HW, in progress](trial)X
Y
Prepare,andwithgivennd a unitary matrix U minimizing R∞{}II•

R∞(U, X, Y(trial)) := maxI,a(U)(trial)X−Y( I)
I(U′)
YI,aa2YI,a: L∞-distance or Chebyshev distance•(U )YI,aℝdN

tr YI2

Vary YI(trial) searching min R∞(U, X, Y )Y→ ⟨R∞(Umin, X, Ymin)⟩ is gauge invariantnd the location and width of wave packet with guessing the center of it

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基研研究会「熱場の量子論とその応用」2022

Contents• Motivation from the gauge/gravity duality• How to identify the location of D-branes from QFT side
• Notion of wave packet in operator/path-integral formalism• How to solve the optimization problem• Replica-Exchange Monte Carlo method and its extension• Numerical demonstration• Summary & Prospectives

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基研研究会「熱場の量子論とその応用」2022✔︎

Search of minimum
A common problem we face; F(X) : some function to be minimizedF(X )Xitrial

Ximin

F(X )
Xi

❌ hardXi

easyc.f. loss surface of DNN
from [Ulmer, (arXiv:2101.00674)]

The idea of importance sampling (Monte Carlo method)
is applicable and powerful to this problem.
However, itʼs often troublesome by being
trapped in the local minima.
→ we sometimes give up to obtain the true minimum.

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基研研究会「熱場の量子論とその応用」2022

Replica-Exchange Monte Carlo (REMC)[Swendsen, Wang, (1986) / Geyer, (1991)]

known also as the parallel tempering, is good at searching global minimum con g.;• Prepare M replica (copy of system)

F(X ) : same function among replicas

with di erent “temperature”

β1F(X )

β1 < β2 < ⋯ < βM

β2F(X )

which changes the scaling of pot. barrier⋮• Perform MCMC on each replica
independently and generate

βM F(X )X1, X2, ⋯, XM• Exchange con gurations Xm & Xm+1(m = 1,⋯, M − 1) with weight(:Metropolis test)

ΔS := βmF(Xm+1) + βm+1F(Xm)−βmF(Xm) − βm+1F(Xm+1)→sampling e ciency has been improved!
c.f. [textbook by Hanada, Matsuura, (2021/2022)]fifi

基研研究会「熱場の量子論とその応用」2022
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ff

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Escapable from minima via high-T replicaAn extension of REMC
Still severe to minimize the L∞-distance due to the huge #local minima
new problem

original problem

R∞(X (U)) = max | XI(U) − YI |aRp(X (U)) =I,a: L∞-distance(∑a1/p): Lp-distance| XI(U) − YI |a pextend by introducing “evaluation function” on each replica

β1F(X )

β1R2(X )β2F(X )

β2 R3(X )⋮⋮βM F(X )14 / 23βM RM+1(X )基研研究会「熱場の量子論とその応用」2022

Algorithm of extended REMC
• Prepare M replica (copy of system)with di erent “temperature”

β1 < β2 < ⋯ < βM
which changes the scaling of pot. barrier• Perform MCMC on each replica
independently and generate

X1, X2, ⋯, XM

Rp(X (U)) =(∑a1/p| XI(U) − YI |a p)

β1R2(X )β2 R3(X )⋮βM RM+1(X )• Exchange con gurations Xm & Xm+1
(m = 1,⋯, M − 1) with weight by p

ΔSp := βm Rp(Xm+1) + βm+1Rp+1(Xm)−βm Rp(Xm) − βm+1Rp+1(Xm+1)

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基研研究会「熱場の量子論とその応用」2022
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15 / 23• Introducing some ʻcuto ʼ M• optimizing Rp on each replicaProperties of extended REMC
• MCMC algorithm in each replica → guaranteed it could work• Di erent pot. structure among replicas → many minimizing path) for an X

R2(X) ≥ R3(X) ≥ ⋯ ≥ R∞(X) ≥ 0: monotonic series of X

but not satis ed

β2 R2(X) ≥ β3R3(X) ≥ ⋯ ≥ βM RM(X),• Less local minima for smaller p
F2(X (U)) =)tr(XI(U) − YI )2 is gauge inv.

• Rp(X) ≈ Rp+1(X) for su ciently large p

β1R2(X )β2 R3(X )⋮βM RM+1(X )ffi

基研研究会「熱場の量子論とその応用」2022
fi

ff

16 / 23β2 < β3 < ⋯ < βM

Contents• Motivation from the gauge/gravity duality• How to identify the location of D-branes from QFT side
• Notion of wave packet in operator/path-integral formalism• How to solve the optimization problem• Replica-Exchange Monte Carlo method and its extension• Numerical demonstration• Summary & Prospectives

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基研研究会「熱場の量子論とその応用」2022

Mock-data setup
Consider the simple setup (I = 1 case)

Xij =N 2−1∑
a=1

Xaτija,Xa = 1tr τ aτ b = δ ab

τ a : SU(N ) generator

and prepare the mock data Z by randomly generating the unitary matrices VZ = VXV −1
R∞(U, Z) := max Z (U)aa(C = O)
R∞(U, X) = 1• Demonstration that the algorithm works well and optimize the distance• Comparison with the standard REMC
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基研研究会「熱場の量子論とその応用」2022

Minimization : 4 × 4 matrix
Demonstration: 4 × 4 matrix(pmin, pmax) = (2, 500), βp = p

Za (exact value : Xa = 1)

preliminary

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standard

extended

0.974580
0.982464
0.985314
0.989367
0.993501
0.994553
1.000397
1.002344
1.003228
1.009786
1.010196
1.012892
1.013082
1.013474
1.013684

0.990662
0.991926
0.994632
0.997921
1.000019
1.000333
1.000921
1.001330
1.001429
1.001482
1.001692
1.002493
1.003807
1.005307
1.005908

基研研究会「熱場の量子論とその応用」2022

MC time : 4 × 4 matrix(#replica = 100, (threshold) = 1.025)
MC time (iteration)

preliminary

standard REMC
extended REMC

Z constructed by 10 di erent unitary matrix V

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基研研究会「熱場の量子論とその応用」2022

Minimization : 6 × 6 matrix
Demonstration: 6 × 6 matrix(pmin, pmax) = (2, 1000), βp = p

Za (exact value : Xa = 1)

preliminary

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standard

extended

0.552661
0.683123
0.710418
0.824777
0.862196
0.939672
0.959044…
1.068943
1.072758
1.072943
1.0753201.075803
1.077198
1.077872

0.576161
0.736379
0.824971
0.882625
0.910872
0.918226
1.003542…
1.043846
1.045241
1.045690
1.045949
1.046511
1.047045
1.047336

基研研究会「熱場の量子論とその応用」2022

MC time : 6 × 6 matrix(#replica = 500, (threshold) = 1.025)
MC time (iteration)

preliminary

standard REMC
extended REMC

Z constructed by 10 di erent unitary matrix V22 / 23ff

基研研究会「熱場の量子論とその応用」2022

Summary & Prospectives• Locations of D-branes can be determined by identifying thecenter of the wave packet in the color space.

• In the path-integral formalism, it corresponds to ndmin max (U †XIU − YI)aU∈SU(N) I,afor the eld XI.

• We employ the Replica-exchange Monte Carlo method and its
extended version to solve the optimization problem.

• Application to more physically-meaningful models.
• (0+0)-dim, (0+1)-dim matrix model (e.g. w/ 3 matrices)• Application to more generic setup (including outside the physics)
基研研究会「熱場の量子論とその応用」2022
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Backup

A puzzle[Polchinski, (1998/1999) / Susskind, (1999)]e.g.) p = 3 case; D3-brane e ective theory⟨tr XI2⟩ ∼ N 2: ʼt Hooft counting☆ eigenvalues of XI2 is of order N → eigenvalues of XI is of order

N

Note also that XI are not diagonalized simultaneously.

AdS5/CFT4 correspondence[Maldacena, (1997) / Gubser, Klebanov, Polyakov, (1998) / Witten, (1998)]

QFT sidegravity (string theory) side

rtypical ≃ N 1/2r ∼ (factor) ⋅ RAdSRAdS ∼ N 1/2
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基研研究会「熱場の量子論とその応用」2022✔︎✔︎

Reason for the discrepancy[Hanada (2021)]⟨tr XI2⟩ ∼ N 2❌Wrong!

: ʼt Hooft counting☆ eigenvalues of XI2 is of order N → eigenvalues of XI is of order

NAssumption;
The operator X̂I s.t. X̂I,a | X⟩ = XI,a | X⟩ provides the Hermitian matrix XI.
X̂I,ij =N2∑
a=1

X̂I,aτija,

̂ =PI,ijN2∑
a=1

̂ τaPI,aijτ a : U(N ) generators

However, uncertainty relation

̂I,a, PJ,b̂
X[] = iδIJ δab,ΔXI,aΔPJ,b > N 0δIJ δab∼the coordinate eigenstate | X⟩ contains various momentum modes
and therefore it is inappropriate for the low-energy state!

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基研研究会「熱場の量子論とその応用」2022