lefschetz-thimble path integral and its physical...

Lefschetz-thimble path integraland its physical application

Yuya Tanizaki

Department of Physics, The University of Tokyo

Theoretical Research Division, Nishina Center, RIKEN

April 8, 2015 @ Komaba, TokyoCollaborators: Takuya Kanazawa (RIKEN)

Kouji Kashiwa (YITP, Kyoto), Hiromichi Nishimura (Bielefeld)

Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 8, 2015 @ Komaba 1 / 38

Motivation

Introduction and Motivation


Motivation

Motivation

Path integral with complex weights appear in many importantphysics:

Finite-density lattice QCD,spin-imbalanced nonrelativistic fermions

Gauge theories with topological θ terms

Real-time quantum mechanics

Oscillatory nature hides many important properties of partitionfunctions.


Motivation

Example: Airy integralLet’s consider a one-dimensional oscillatory integration:

Ai(a) =

∫R

dx

2πexp i

(x3

3+ ax

).

RHS is well defined only if Ima = 0, though Ai(z) is holomorphic.

-60

-40

-20

0

20

40

60

-8 -6 -4 -2 0 2 4 6 8

’10*’cos(x3/3+x)exp(-0.5x)*cos(x3/3)

Figure: Real parts of integrands for a = 1 (×10) & a = 0.5i


Motivation

Contents

How can we circumvent such oscillatory integrations?⇒ Lefschetz-thimble integrations[Witten, arXiv:1001.2933, 1009.6032]

[YT, Koike, Ann. Physics 351 (2014) 250]

Applications of this new technique for path integralsI Study on phase transitions of matrix models.I Lefschetz-thimble method elucidates Lee-Yang zeros.

[Kanazawa, YT, JHEP 1503 (2015) 044]

I General theorem ensuring that Z ∈ R.Sign problem in MFA can be solved.

I Application of the theorem to the SU(3) matrix model[YT, Nishimura, Kashiwa, in preparation]


Lefschetz-thimble integrations

Introduction to Lefschetz-thimble integrations



Again Airy integral

Airy integral:

Ai(a) =

∫R

dx

2πexp i

(x3

3+ ax

).

The integrand is holomorphic w.r.t x⇒ The contour can be deformed continuously without changing thevalue of the integration!



Steepest descent contours for the Airy integral

There exists two Lefschetz thimbles Jσ (σ = 1, 2) for the Airyintegral:

Ai(a) =∑σ

nσ

∫Jσ

dz

2πexp i

(z3

3+ az

).

nσ: intersection number of the steepest ascent contour Kσ and R.

Figure: Lefschetz thimbles J and duals K (a = exp(0.1i), exp(πi))



Tips: Airy integral & Airy equation

Let us consider the “equation of motion”:∫dz

2π

d

dzei(z3/3+az) = 0.

This is nothing but the Airy equation:(d2

da2− a)

Ai(a) = 0.

Two possible integration contours= Two linearly independent solutions of eom.



Generalization to multiple integrals

Model integral:

Z =

∫Rn

dx1 · · · dxn expS(xi).

What properties are required for Lefschetz thimbles J ?

1 J should be an n-dimensional object in Cn.

2 Im[S] should be constant on J .



Short note on technical aspects

Complexified variables (a = 1, . . . , n): za = xa + ipa.

Regard xa as coordinates and pa as momenta,so that Poisson bracket is given by

{f, g} =∑a=1,2

[∂f

∂xa

∂g

∂pa− ∂g

∂xa

∂f

∂pa

].



Short note on technical aspects

Hamilton equation with the Hamiltonian H = Im[S(za)]:

df(x, p)

dt= {H, f}

(⇔ dza

dt= −

(∂S

∂za

))

This is Morse’s flow equation (= gradient flow):

d

dtRe[S] = −

∣∣∣∣∣(∂S

∂z

)2∣∣∣∣∣ ≤ 0

⇒ We can find n good directions for J around saddle points![Witten, 2010]



Multiple integral: Lefschetz-thimble method

Oscillatory integrals with many variables can be evaluated using the“’steepest descent” cycles Jσ:∫

Rndnx eS(x) =

∑σ

nσ

∫Jσ

dnz eS(z).

Jσ are called Lefschetz thimbles, and Im[S] is constant on it.

nσ: intersection numbers of duals Kσ and Rn.


Gross–Neveu-like model

Phase transition associated with symmetry



0-dim. Gross–Neveu-like model

The partition function of our model study is the following:

ZN(G,m) =

∫dψ̄dψ exp

{N∑a=1

ψ̄a(i/p+m)ψa +G

4N

( N∑a=1

ψ̄aψa

)2}.



Hubbard–Stratonovich transformation

Bosonization:

ZN(G,m) =

√N

πG

∫R

dσ e−NS(σ),

with

S(σ) ≡ σ2

G− log[p2 + (σ +m)2].

For simplicity, we putm = 0 in the following.

S has three saddle points:

0 =∂S(z)

∂z=

2z

G− 2z

p2 + z2=⇒ z = 0, ±

√G− p2 .



Behaviors of the flow

Figures for G = 0.7e−0.1i, 1.1e−0.1i at p2 = 1[Kanazawa, YT, arXiv:1412.2802]:

z = 0 is the unique critical point contributing to Z if G < p2.

All three critical points contribute to Z if G > p2.



Stokes phenomenon

The difference of the way of contribution can be described byStokes phenomenon. [Witten, arXiv:1001.2933, 1009.6032]

Figures of G-plane for ImS(0) = ImS(z±) [Kanazawa, YT, arXiv:1412.2802]:

=G

<G



Dominance of contribution

The Stokes phenomenon tells us the number of Lefschetz thimblescontributing to ZN .However, it does not tell which thimbles give main contribution.

ZN ∼ # exp(−NS(0)) + # exp(−NS(z±))

In order to obtain 〈σ〉 6= 0 in the large-N limit, z± should dominatez = 0.

⇒ ReS(z±) ≤ ReS(0)



Connection with Lee–Yang zero

=G

<G

Blue line: ImS(z±) = ImS(0).Green line: ReS(z±) = ReS(0).Red points: Lee-Yang zeros at N = 40. [Kanazawa, YT, arXiv:1412.2802]



Conclusions for studies with GN-like models

1 Decomposition of the integration path in terms of Lefschetzthimbles is useful to visualize different phases.

2 The possible link between Lefschetz-thimble decomposition andLee–Yang zeros is indicated.


Sign problem in MFA Formal discussion

Sign problem in MFA and its solution: Formal discussion



Airy integral with real a

If a ∈ R, the following partition function is real:

ZAiry(a) =1

2π

∫R

dx exp i

(x3

3+ ax

).

However, the integrand is complex-valued.

Why does this happen?What is the mean-field approximation of this partition function?



Airy integral: Antilinear symmetry

The system have an antilinear symmetry:

S(x) = −i

(x3

3+ ax

)= S(−x).

By summing up in pairs ±x, the result becomes manifestly real!



Airy integral: Lefschetz thimble

Extend the linear map x 7→ −x to the antilinear map z 7→ −z.

The Lefschetz thimbles form invariants under the antiliner map:

Figure: Lefschetz thimbles J and duals K (a = exp(0.1i), exp(πi))



Generalization: Set up

Consider the oscillatory multiple integration,

Z =

∫Rn

dnx exp−S(x).

To ensure Z ∈ R, suppose the existence of a linear map L, satisfying

S(x) = S(L · x).

L2 = 1.

Let’s construct a systematic computational scheme of Z withZ ∈ R.



Generalization: Flow eq. and Complex conjugation

Morse’s downward flow:

dzidt

=

(∂S(z)

∂zi

).

Complex conjugation of the flow:

dzidt

=

(∂S(L · z)

∂zi

),

and thus z′ := L · z satisfies Morse’s flow equation .



Generalization: Immediate conclusion

Denote the antilinear map as K(z) := L · z.

1 When zσ is a saddle point, so is K(zσ).

2 Let Jσ is the Lefschetz thimble around zσ.⇒ J K

σ := {K(z) | z ∈ Jσ} coincides with the Lefschetz thimblearound K(zσ) (up to orientation).

3 Intersection numbers are the same: 〈Kσ,Rn〉 = 〈KKσ ,Rn〉 underappropriate choice of the orientation.

4 The contributing thimbles always form an invariant pairJσ ∪ J K

σ .



General Theorem

Let us decompose the set of saddle points into three parts,Σ = Σ0 ∪ Σ+ ∪ Σ−, where

Σ0 = {σ | zσ = L · zσ},Σ+ = {σ | ImS(zσ) > 0},Σ− = {σ | ImS(zσ) < 0}.

The antilinear map gives Σ+ ' Σ−.



General Theorem

The partition function:

Z =∑σ∈Σ0

nσ

∫Jσ

dnz exp−S(z)

+∑τ∈Σ+

nτ

∫Jτ+JKτ

dnz exp−S(z).

Each term on the r.h.s. is real.(YT, Nishimura, Kashiwa, in preparation)


Sign problem in MFA Application to QCD

Sign problem in MFA and its solution:Application to QCD-like theories



The QCD partition function at finite density

The QCD partition function:

ZQCD =

∫DA detM(µqk, A) exp−SYM[A],

w./ the Yang-Mills action SYM = 12tr∫ β

0dx4

∫d3x|Fµν |2 (> 0), and

detM(µ,A) = det[γν(∂ν + igAν) + γ4µqk +mqk

].

is the quark determinant.



Charge conjugation

If µqk 6= 0, the quark determinant takes complex values⇒ Sign problem of QCD.

However, ZQCD ∈ R is ensured thanks to the charge conjugationA 7→ −At:

detM(µqk, A) = detM(−µqk, A†)

= detM(µqk,−A).

(First equality: γ5-transformation,Second equality: charge conjugation)



Polyakov-loop effective model

The Polyakov line L:

L =1

3diag

[ei(θ1+θ2), ei(−θ1+θ2), e−2iθ2

].

Let us consider the SU(3) matrix model:

ZQCD =

∫dθ1dθ2H(θ1, θ2) exp [−Veff(θ1, θ2)] ,

where H = sin2 θ1 sin2 (θ1 + 3θ2)/2 sin2 (θ1 − 3θ2)/2.



Charge conjugation in the Polyakov loop model

Charge conjugation acts as L↔ L†:

Veff(z1, z2) = Veff(z1,−z2).

Simple model for dense quarks (` := trL):

Veff = −h(32 − 1)

2

(eµ`3(θ1, θ2) + e−µ`3(θ1, θ2)

)



Behaviors of the flow

0 1 2 3

0

-0.2

-0.1

-0.3

Figure: Flow at h = 0.1 and µ = 2 in the Re(z1)-Im(z2) plane.

The black blob: a saddle point.The red solid line: Lefschetz thimble J .The green dashed line: its dual K. (YT, Nishimura, Kashiwa, in preparation)



Saddle point approximation at finite density

The saddle point approximation can now be performed.

Polyakov-loop phases (z1, z2) takes complex values, so that

〈`〉, 〈`〉 ∈ R.

Since Im(z2) < 0 and

` ' 1

3(2eiz2 cos θ1 + e−2iz2),

we can confirm that` > `.

(YT, Nishimura, Kashiwa, in preparation)



Summary for the sign problem in MFA

Lefschetz-thimble integral is a useful tool to treat multipleintegrals.

Saddle point approximation can be applied without violatingZ ∈ R.

Sign problem of effective models of QCD is (partly) explored.


lefschetz-thimble path integral and its physical...

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