Lefschetz-thimble path integral and its physical applicationtanizaki/talk/YITP_seminar_2015.pdf ·...
Transcript of Lefschetz-thimble path integral and its physical applicationtanizaki/talk/YITP_seminar_2015.pdf ·...
Lefschetz-thimble path integraland its physical application
Yuya Tanizaki
Department of Physics, The University of Tokyo
Theoretical Research Division, Nishina Center, RIKEN
April 3, 2015 @ YITP, KyotoCollaborators: Takuya Kanazawa (RIKEN)
Kouji Kashiwa (YITP, Kyoto), Hiromichi Nishimura (Bielefeld)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 1 / 39
Motivation
Introduction and Motivation
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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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 3 / 39
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
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 4 / 39
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]
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 5 / 39
Lefschetz-thimble integrations
Introduction to Lefschetz-thimble integrations
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 6 / 39
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!
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 7 / 39
Lefschetz-thimble integrations
Steepest descent contoursWhat is the most appropriate contour for our purpose?
Re[S(x, a)] should be made as small as possible.⇔ The contour should be perpendicular to Re[S(x, a)] = cosnt.
Steepest descent ones J must satisfy Im[S(x, a)] = const.because of the holomorphy.
Figure : Contour plots for Re[S(x, a)] with a = exp±i0.1.Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 8 / 39
Lefschetz-thimble integrations
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))
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 9 / 39
Lefschetz-thimble integrations
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 10 / 39
Lefschetz-thimble integrations
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 .
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 11 / 39
Lefschetz-thimble integrations
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
].
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 12 / 39
Lefschetz-thimble integrations
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]
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 13 / 39
Lefschetz-thimble integrations
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 14 / 39
Gross–Neveu-like model
Phase transition associated with symmetry
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Gross–Neveu-like model
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}.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 16 / 39
Gross–Neveu-like model
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 .
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 17 / 39
Gross–Neveu-like model
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 18 / 39
Gross–Neveu-like model
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
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Gross–Neveu-like model
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)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 20 / 39
Gross–Neveu-like model
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]
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 21 / 39
Gross–Neveu-like model
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.
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Sign problem in MFA Formal discussion
Sign problem in MFA and its solution: Formal discussion
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Sign problem in MFA 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?
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 24 / 39
Sign problem in MFA Formal discussion
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!
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 25 / 39
Sign problem in MFA Formal discussion
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))
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 26 / 39
Sign problem in MFA Formal discussion
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 27 / 39
Sign problem in MFA Formal discussion
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 .
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 28 / 39
Sign problem in MFA Formal discussion
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 .
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 29 / 39
Sign problem in MFA Formal discussion
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−.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 30 / 39
Sign problem in MFA Formal discussion
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)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 31 / 39
Sign problem in MFA Application to QCD
Sign problem in MFA and its solution:Application to QCD-like theories
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 32 / 39
Sign problem in MFA Application to QCD
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 33 / 39
Sign problem in MFA Application to QCD
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)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 34 / 39
Sign problem in MFA Application to QCD
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 35 / 39
Sign problem in MFA Application to QCD
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)
)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 36 / 39
Sign problem in MFA Application to QCD
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)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 37 / 39
Sign problem in MFA Application to QCD
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)
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 38 / 39
Sign problem in MFA Application to QCD
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.
Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Apr. 3, 2015 @ YITP, Kyoto 39 / 39