Real-time dynamics of quantum tunneling
motivated by Lefschetz-thimble path integral
Yuya Tanizaki
The University of Tokyo & RIKEN
March 24, 2015 @ JPS meeting, Waseda U.
Collaborator: Takayuki Koike (Tokyo)
Reference: Ann. Physics 351 (2014) 250
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Introduction
Motivation
Direct semiclassical description of quantum tunneling from the
viewpoint of real-time path integral,
Z
Dx(t) exp(iS[x(t)]/~).
Problem: Classical eom δS = 0 does not have tunneling solutions!
Naively, semiclassical description seems to be impossible.
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Introduction
Simple example: Airy integral
Let’s consider a one-dimensional oscillatory integration:
3
Z
dx
x
Ai(a) =
exp i
+ ax .
3
R 2π
What is the MFA of this system?
60
3
’10*’cos(x /3+x)
exp(-0.5x)*cos(x3/3)
40
20
0
-20
-40
-60
-8
-6
-4
-2
0
2
4
6
8
Figure: Real parts of integrands for a = 1 (×10) & a = 0.5i
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Introduction
New technology on path integral
Complex paths open a new way to compute path integrals.
It’s better to circumvent the sign problem in order for a semiclassical
method.
⇒ Lefschetz-thimble path integral (E.Witten, arXiv:1001.2933, arXiv:1009.6032)
Z
X Z
Dx exp iS[x]/~ =
nσ
Dz exp iS[z]/~.
σ
Yuya Tanizaki (U. of Tokyo, RIKEN)
Jσ
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Introduction
Find complex classical solutions
Idea:
1
2
3
Find complex classical solutions zσ of eom δS = 0.
Consider appropriate integration contours Jσ around zσ .
Path integral on Jσ can be computed without the sign problem.
Figure: Lefschetz thimbles for the Airy integral a = exp ±i0.1.
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Quantum tunneling
Double-well potential
Complex-energy conservation:
dz
dt
2
+ (z 2 − 1)2 = p2
Two different origins of oscillations
⇒ Solutions show double-periodicity!
real-time oscillation
imaginary-time oscillation
x
x
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Quantum tunneling
Double-well potential
The list of parameters p can be obtained as (n, m ∈ Z)
p
p
iK( (p − 1)/2p)
K( (p + 1)/2p)
tf − ti
√
√
+m
=
n
+(bdry. terms).
2
2p
2p
Short-time asymptotic behaviors of the classical action:
√
2K(1/ 2)4 (n + im)4
.
I[z(n,m) ] ' i
3
(tf − ti )3
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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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 complexified
space, i.e., n, m ' O(tf − ti )
(Me & Koike, arXiv:1406.2386):
15
20
10
10
5
20
40
60
80
100
t
50
100
10
5
Rez
10
Yuya Tanizaki (U. of Tokyo, RIKEN)
t
150
Imz
Rez
20
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
Imz
8 / 11
Quantum tunneling
Connection to the instanton calculus
This solution has a close relationship with one-instanton
¨
(Cherman & Unsal,
arXiv:1408.0012):
xα (τ ) = tanh[(τ − τ0 )e−i(α−π/2) ]
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Quantum tunneling
Connection to the instanton calculus
The action of this solution is that of the one-instanton solution at
¨
any α (Cherman & Unsal,
arXiv:1408.0012):
Z
4
e−iα
dτ [e2iα (∂τ xα )2 − (x2α − 1)2 ] =
Sα =
2~
3
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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Summary & Perspectives
Summary & Perspectives
Summary
Real-time dynamics will become calculable in a nonperturbative
way with Lefschetz-thimble path integrals.
Exact semi-classical description of quantum tunneling is
considered.
Perspectives
Application to real-time dynamics of topological objects in field
theories.
Sign problem, Resurgence trans-series, ....
Yuya Tanizaki (U. of Tokyo, RIKEN)
Lefschetz-thimble path integral & Quantum tunneling
Mar. 22, 2015 @ Waseda
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