Introduction to Lefschetz-thimble path integral and …tanizaki/talk/OIQP_2015.pdfIntroduction to...

Introduction to Lefschetz-thimble path integraland its various applications

Yuya Tanizaki

Department of Physics, The University of Tokyo

Theoretical Research Division, Nishina Center, RIKEN

January 9, 2015 @ Okayama Institute for Quantum Physics

Collaborators: Takayuki Koike (Tokyo), Takuya Kanazawa (RIKEN)

Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Jan 9, 2015 @ OIQP 1 / 42

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]

Applications of this new technique for path integralsI Quantum tunneling in the real-time path integral

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

I Phase transition of the matrix model[YT, arXiv:1412.1891; Kanazawa, YT, arXiv:1412.2802]


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 contoursWhat is the most appropriate contour for our purpose?

Re[iS(x, a)] should be made as small as possible.⇔ The contour should be perpendicular to Re[iS(x, a)] = cosnt.

Steepest descent ones J must satisfy Im[iS(x, a)] = const.because of the holomorphy.

Figure : Contour plots for Re[iS(x, a)] with a = exp±i0.1.



Steepest descent contoursWhat is the most appropriate contour for our purpose?

Re[iS(x, a)] should be made as small as possible.⇔ The contour should be perpendicular to Re[iS(x, a)] = cosnt.

Steepest descent ones J must satisfy Im[iS(x, a)] = const.because of the holomorphy.

Figure : Contour plots for Re[iS(x, a)] with a = exp±i0.1.Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Jan 9, 2015 @ OIQP 8 / 42


Rewrite 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))



Generalization to multiple integrals

Model integral:

Z =

∫Rn

dx1 · · · dxn exp I(xi).

What properties are required for Lefschetz thimbles J ?

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

2 Im[I] should be constant on J .



Short note on technical aspectsComplexified variables (a = 1, . . . , n): za = xa + ipa.Regard xa as coordinates and pa as momenta, so that Poissonbracket is given by

{f, g} =∑a=1,2

[∂f

∂xa

∂g

∂pa− ∂g

∂xa

∂f

∂pa

].

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

df(x, p)

dt= {H, f}

(⇔ dza

dt= −

(∂I∂za

))

This is Morse’s flow equation: ddt

Re[I] ≤ 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 eiS(x) =

∑σ

nσ

∫Jσ

dnz eiS(z).

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

nσ: intersection numbers of duals Kσ and Rn.


Lefschetz-thimble integrations Harmonic oscillator

Example of QM: Harmonic oscillator

The action functional

I[z] = i

∫ T

0

dt

[1

2

(dz

dt

)2

− 1

2z2

]

Saddle-point condition (=Euler–Lagrange eq.):

d2zcl(t)

dt2= −zcl(t).

The classical solution:

zcl(t) =xf − xi cosT

sinTsin t+ xi cos t.



Example of QM: Flow equation

Flow equation

∂

∂u∆z(t;u) = −i

(∂2

∂t2+ 1

)∆z(t;u),

Boundary conditions: ∆z(t;−∞) = 0, and ∆z(0;u) = ∆z(T, u) = 0.

The set of solutions ∆z are spanned by

∆zn(t;u) =

eiπ/4 exp[((

πnT

)2 − 1)u]

sin nπtT, (nπ > T ),

e−iπ/4 exp[(

1−(πnT

)2)u]

sin nπtT, (nπ < T ).



Example of QM: Lefschetz thimble

Lefschetz thimble J (' R∞):{zcl(t) + e−iπ/4

ν∑`=1

a` sinπ`t

T+ eiπ/4

∞∑`=ν+1

a` sinπ`t

T

∣∣∣∣∣ a` ∈ R

}

ν: the maximal non-negative integer smaller than T/π.Integration measure on J becomes∫JDz = N

∫ ν∏n=1

e−iπ4 dan

∞∏m=ν+1

eiπ4 dam = e−iπν/2N

∫ ∞∏`=1

√ida`.



Example of QM: Path integral

Feynman kernel for the harmonic oscillator:

Kh.o.(xf , xi, T )

= exp

(I[zcl]

~− i

πν

2

)N∏`

∫ √ida` exp

(−

∣∣∣∣∣(π`

T

)2

− 1

∣∣∣∣∣ a2`

)

= exp

(I[zcl]

~− i

πν

2

)√1

2πi~T

∞∏`=1

√1

|1− (T/π`)2|

=

√1

2πi~| sinT |exp

(I[zcl]

~− i

πν

2

).

Appearance of the Maslov–Morse index ν is quite manifest.[YT, Koike, Ann. Physics 351 (2014) 250]



Proposed Applications of Lefschetz-thimble methods

Analytic properties of partition function for CS theories[Witten]

Real-time description of tunneling[YT, Koike; Cherman, Unsal]

Sign problem of the lattice Monte Carlo simulation[M. Cristoforetti, et al., Kikukawa, et al.]

Geometrization of resurgence transseries[Basar, Dunne, Unsal; Cherman, Dorigoni, Unsal]


Quantum tunneling

Quantum tunneling and real-time path integral


Quantum tunneling

Double-well potential

Complex-energy conservation:

(dz

dt

)2

+ (z2 − 1)2 = p2

Two different origins of oscillations⇒ Solutions show double-periodicity!

x

x

real-time oscillation imaginary-time oscillation

The list of parameters p can be obtained as (n,m ∈ Z)

nK(√

(p+ 1)/2p)√2p

+miK(√

(p− 1)/2p)√2p

=T

2+ (bdry term).


Quantum tunneling

Complex classical solutionsAnalytic complex solution with z(0) = xi, z(T ) = xf :

z(t) =

√p2 − 1

2psd

(√2pt+ sd−1

(√2p

p2 − 1xi,

√1 + p

2p

),

√1 + p

2p

).

List of possible parameters p: (YT, Koike, Ann. Physics 351 (2014) 250)

-50 50ReHpL

-150

-100

-50

50

100

150

ImHpL

Let’s describe quantum tunneling semi-classically using these“complex” solutions.


Quantum tunneling

Imaginary-time instantons “⇒” SphaleronsInstantons in imaginary times (p ' 0):

2 4 6 8 10-ä t

-1.0

-0.5

0.5

1.0

Hn,mL=H0,0L

Im@zD

Re@zD

2 4 6 8 10-ä t

-1.0

-0.5

0.5

1.0

Hn,mL=H1,0L

Im@zD

Re@zD

⇓ Wick rotation

Sphalerons in real times (p ' 1):

2 4 6 8 10t

-1.0

-0.5

0.5

1.0

Hn,mL=H0,0L

Im@zD

Re@zD

2 4 6 8 10t

-1.0

-0.5

0.5

1.0

1.5

Hn,mL=H1,0L

Im@zD

Re@zD


Quantum tunneling

Important!Any complex solutions behave as sphalerons in the limit(tf − ti)→ +∞.

Energy p2 of the solution is determined by

nK(√

(p+ 1)/2p)√2p

+miK(√

(p− 1)/2p)√2p

=T

2+ (bdry term).

Labels n,m roughly determines the number of oscillations during T .⇒ If n,m� T , the solution must lie on the top. Thus, p ' 1.

2 4 6 8 10t

1.0

0.5

0.5

1.0

1.5

Imz

Rez


Quantum tunneling

Complex solutions for quantum tunneling

In order for the one-instanton action iS[z] ' −Sinst. = −4/3,complex solutions must be (highly-)oscillatory in the complexifiedspace, i.e., n,m ' O(tf − ti)(YT, Koike, arXiv:1406.2386; Cherman, Unsal, arXiv:1408.0012):

20 40 60 80 100t

10

5

5

10

15

Imz

Rez

50 100 150t

20

10

10

20

Imz

Rez


Summary & Perspectives

Summary & Perspectives of this section

Summary

Real-time dynamics becomes calculable in a nonperturbative waywith Lefschetz-thimble path integrals.

Exact semi-classical description of quantum tunneling isconsidered;

1 Imaginary-time instantons are connected to sphalerons byanalytic conti.

2 Real-time tunneling must be described by highly-oscillatorycomplex solutions.

Perspectives

Application to real-time dynamics of QCD: Sphaleron rate Γ.


Gross–Neveu-like model

Phase transition associated with symmetry



MotivationWhat is SSB in this context?

Let’s consider two questions using zero-dimensional fermionic modelswith N species:

What happens when discrete symmetry is broken at large N?

Can we also treat continuous symmetry?



0-dim. Gross–Neveu-like model

The partition function of our model study is the following:

ZN(G,m) =

∫dψdψ exp

{N∑a=1

ψa(i/p+m)ψa +G

4N

( N∑a=1

ψaψa

)2}.

The Hubbard–Stratonovich transformation gives

ZN(G,m) =

√N

πG

∫R

dσ e−NS(σ),

with

S(σ) ≡ σ2

G− log[p2 + (σ +m)2].

For simplicity, we put m = 0 in the following.



Properties of S(σ)S has three saddle points:

0 =∂S(z)

∂z=

2z

G− 2z

p2 + z2=⇒ z = 0, ±

√G− p2 .

Figures for G = 0.7e∓0.1i and p2 = 1 [Kanazawa, YT, arXiv:1412.2802]:



Properties of S(σ)

Figures for G = 1.1e∓0.1i and p2 = 1 [Kanazawa, YT, arXiv:1412.2802]:

From these figures, we learn that, for real G,

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

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



Stokes phenomenonThe difference of the way of contribution can be described byStokes phenomenon.⇐ At some special values of coupling, several critical points areconnected by the flow. [Witten, arXiv:1001.2933, 1009.6032]

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

=G

<GYuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Jan 9, 2015 @ OIQP 30 / 42


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.

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 thimble decomposition and Lee–Yangzeros is indicated.


Lefschetz thimbles with continuous symmetry

Lefschetz thimbles with continuous symmetry


Lefschetz thimbles with continuous symmetry Simple bosonic model

Lefschetz thimble with approximate O(2) symmetry

Let’s consider approximately O(2)-symmetric sigma model:

S(σ) =1

4(σ2 − 1)2 + εσ1.

“Angular momentum” gives an extra conserved quantity at ε = 0!

pθ = η2σ1 − η1σ2.

At ε = 0, the flow is singular, and naive construction of J failsdown.Under this circumstance, how does the flow look like?



Structure of the flow equation

Hamilton equation (= flow equation):

dr

dt= r

(r2 − p2

θ

r2− 3p2

r − 1

)+ ε cos θ,

dprdt

= pr

(1− 3r2 + pr −

p2θ

r2

)− εpθ

r2sin θ,

dθ

dt= −2prpθ

r− εsin θ

r,

dpθdt

= ε(pr sin θ +

pθr

cos θ).

In the limit ε→ 0, flows along symmetric direction disappears.⇒ Naive construction of Lefschetz thimbles breaks down at ε = 0.



Flow with approximate O(2) symmetry

Set of saddle points at ε = 0: σ2 − η2 = 1 & σ · η = 0.

Flows in the σ1σ2-plane & in the σ1η2-plane for small ε:

0 1 2-2 -1-2

-1

0

1

2 2

-2

-1

0

1

0 1 2 3 4

Due to the approximate O(2) symmetry, flow walks around thepseudo saddle points. [YT, arXiv:1412.1891]



Flow with approximate O(2) symmetrySchematic pictures for Lefschetz thimbles of the system

S = ei0+(

1

4(σ2 − 1)2 + εσ1

).

⇒ In order to get an O(2) symmetric result, we must sum up twothimbles J(1+ε/2,π) and J(1−ε/2,0). [YT, arXiv:1412.1891]



Singularity of symmetry-restoration limit

What is the fate of J(1+ε/2,π) and J(1−ε/2,0) in the limit ε→ 0?

Integration on J(1−ε/2,0):∫J(1−ε/2,0)

d2z exp [−S/~]

∼ −i

∫ ∞0

λdλ exp

(−(λ2 − 1)2

4~

)∫ ∞−∞

dφ exp

(−ελ

~coshφ

).

Logarithmic divergence:∫

dφ exp(− ελ

~ coshφ)∼ 2 ln ~

ε.

⇒ Thimbles related by the symmetry need to be summed up to givefinite results!


Lefschetz thimbles with continuous symmetry Nambu–Jona-Lasinio-like model

Nambu–Jona-Lasinio-like modelLet’s apply this observation to the following NJL-like model:

ZU(1) =

∫dψdψ exp

(N∑a=1

ψa(i/p+m)ψa

+G

4N

(

N∑a=1

ψaψa

)2

+

(N∑a=1

ψaiγ5ψa

)2 .

Bosonization:

ZU(1) =N

πG

∫R2

dσdπ exp (−NS(σ, π)) ,

with

S(σ, π) ≡ − log(p2 + (σ +m)2 + π2

)+σ2 + π2

G.



Lefschetz thimbles for NJL-like modelIn this case, m( 6= 0) breaks the O(2) symmetry:

m = 0 m = 0.4

Flow for NJL-like model at m 6= 0, p = 0 [Kanazawa, YT, arXiv:1412.2802]:

< z<w

=w



Conclusions for the continuous symmetry

Lefschetz thimbles with approximate continuous symmetries arecomputed.

The remnant of the symmetry implies existence of slow flowson the set of pseudo saddle points.

Such slow flow relates different thimbles via the approximatesymmetry.


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