Physics Letters B 798 (2019) 134940
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A non-integrable quench from AdS/dCFT
Marius de Leeuw
a
, Charlotte Kristjansen
b,∗
, Kasper E. Vardinghus
b
a
School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Dublin, Ireland
b
Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
a r t i c l e i n f o a b s t r a c t
Article history:
Received
15 July 2019
Received
in revised form 3 September 2019
Accepted
11 September 2019
Available
online 17 September 2019
Editor:
N. Lambert
Keywords:
AdS/CFT
correspondence
Defect
CFT
Probe
branes
One-point
functions
Matrix
product states
Quantum
quenches
We study the matrix product state which appears as the boundary state of the AdS/dCFT set-up where a
probe D7 brane wraps two two-spheres stabilized by fluxes. The matrix product state plays a dual role,
on one hand acting as a tool for computing one-point functions in a domain wall version of N = 4SYM
and on the other hand acting as the initial state in the study of quantum quenches of the Heisenberg
spin chain. We derive a number of selection rules for the overlaps between the matrix product state and
the eigenstates of the Heisenberg spin chain and in particular demonstrate that the matrix product state
does not fulfil a recently proposed integrability criterion. Accordingly, we find that the overlaps can not
be expressed in the usual factorized determinant form. Nevertheless, we derive some exact results for
one-point functions of simple operators and present a closed formula for one-point functions of more
general operators in the limit of large spin-chain length.
© 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
Exact results for overlaps between states in integrable spin
chains have important applications in the calculation of correlation
functions in supersymmetric gauge theories as well as in the study
of quantum quenches in statistical physics. Recently, especially
overlaps between Bethe eigenstates and matrix product states have
attracted attention. From the point of view of the AdS/dCFT corre-
spondence,
overlaps between Bethe eigenstates and specific matrix
product states encode information about one-point functions in
domain wall versions of N = 4SYM theory [1–5] and in statis-
tical
physics the same matrix product states play the role of the
initial state of a quantum quench [6–8].
Interestingly,
all spin chain states | for which it has been pos-
sible
to write the overlap with the Bethe eigenstates in a closed
form have been characterized by being annihilated by the entire
tower of parity odd conserved charges of the chain. Furthermore,
for all of these cases the annihilation of the state by the odd
charges could be used to show that the overlaps with Bethe eigen-
states
were only non-vanishing for Bethe states with paired roots
1
*
Corresponding author.
E-mail
address: kristjan@nbi.dk (C. Kristjansen).
1
States with paired roots are states for which the roots take the form
{u
i
, −u
i
}
S
u
, where q
2n+1
(u) = 0for u ∈ S
u
. For the SU(2) Heisenberg spin chain
and finally the overlaps took a factorized form with the Gaudin
norm matrix, G [9,10], playing a prominent role. More precisely,
for Bethe states with paired roots the determinant of the Gaudin
matrix factorizes as
2
det G = det G
+
det G
−
, (1)
and the normalized overlap takes the (schematic) form
|u
u |u
1/2
=
i
f (u
i
)
det G
+
det G
−
. (2)
These observations lead the authors of [11]to suggest that ma-
trix
product states should be denoted as integrable when annihi-
lated
by all odd charges of the spin chain and in that case would
play a role analogous to that of the integrable boundary states
of Zamolodnikov for continuum quantum field theories [12]. Fur-
thermore,
in [13] integrable matrix product states were related
that we consider in the present letter, S
u
=∅, but for spin chains with nested Bethe
ansätze such as the SU(3) or the SO(6) spin chain there can be a single root a
zero [3,5].
2
For a detailed explanation of how this happens for a model with a nesting we
refer to [3].
https://doi.org/10.1016/j.physletb.2019.134940
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
.
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