Physics Letters B 793 (2019) 169–174
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Chaos in the fishnet
Robert de Mello Koch
a,b
, W. LiMing
a,∗
, Hendrik J.R. Van Zyl
b
, João P. Rodrigues
b
a
School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
b
National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, Wits, 2050,
South Africa
a r t i c l e i n f o a b s t r a c t
Article history:
Received
1 March 2019
Received
in revised form 16 April 2019
Accepted
17 April 2019
Available
online 23 April 2019
Editor:
M. Cveti
ˇ
c
We consider the computation of out-of-time-ordered correlators (OTOCs) in the fishnet theories, with a
mass term added. These fields theories are not unitary. We compute the growth exponent, in the planar
limit, at any value of the coupling and show that the model exhibits chaos. At strong coupling the growth
exponent violates the Maldacena-Shenker-Stanford bound. We also consider the mass deformed versions
of the six dimensional honeycomb theories, which can also be solved in the planar limit. The honeycomb
theory shows a very similar behavior to that exhibited by the fishnet theory.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Chaos in many-body quantum systems can be probed using
non-time-ordered four point functions[1–7]. In this language, the
butterfly effect is the statement that for rather general operators
W and V , the thermal expectation value of the square of the com-
mutator
c(t) =[W (t), V ][W (t), V ]
†
β
(1.1)
becomes large at late time. For simple operators in large N sys-
tems,
there is a long period of exponential growth[5] and generi-
cally
we expect
c(t) ∝
1
N
2
e
λ
L
t
(1.2)
The rate λ
L
defined by this exponential quantifies the strength of
chaos and remarkably, it is governed by a bound λ
L
≤
2π
β
[9].
The exquisite paper [7]evaluated λ
L
in a weakly coupled large
N quantum field theory, at finite temperature. Our study is heav-
ily
influenced by [7] and we are repeating the same analysis for a
much simpler class of theories. For this reason it is useful to review
salient parts of [7]. The model considered is of a single hermitian
matrix
ab
(
x, t) of mass m interacting through a Tr(
4
) self cou-
pling.
The observable considered is a color and spatially-averaged
*
Corresponding author.
E-mail
addresses: robert@neo.phys.wits.ac.za (R. de Mello Koch),
wliming@scnu.edu.cn (W. LiMing), hjrvanzyl@gmail.com (H.J.R. Van Zyl),
Joao.Rodrigues@wits.ac.za (J.P. Rodrigues).
version of the squared commutator (we indicate the color sums ex-
plicitly
and use Tr for the thermal average computed with thermal
density matrix ρ at inverse temperature β)
C(t) =
1
N
4
aba
b
d
3
xTr
√
ρ
ab
(t,
x),
a
,b
(0,
0)
×
√
ρ
ab
(t,
x),
a
b
(0,
0)
†
(1.3)
By splitting ρ into two square root factors, we place the two com-
mutators
on opposite sides of the thermal circle, which nicely
regulates some divergences. The regularization dependence of λ
L
has been discussed in [8]. The computation simplifies at large N
since
only the planar diagrams contribute. Expanding, the two
commutators gives four terms with each computed by a particu-
lar
analytic continuation of the Euclidean correlator. Each term can
also be represented with path integral contours in complex time,
where some real-time folds are appended to the Euclidean ther-
mal
circle. For each term we generate the perturbation series by
expanding the interaction vertex, integrating each vertex along the
contour and then applying Wick’s Theorem with contour-ordered
propagators [7]. In [7]the region of integration for the interaction
vertices is restricted to the real-time folds. The rationale for this
restriction is that the integral over the thermal circle corrects the
thermal state. Although these corrections are important for getting
the exact C (t), the claim is that they do not affect the spectrum of
growth exponents. This is intuitively convincing and further, these
two simplifications are valid at any coupling. We will employ the
same simplifications in our study.
https://doi.org/10.1016/j.physletb.2019.04.044
0370-2693/
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.