Physics Letters B 735 (2014) 191–194
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A note on the Lee–Yang singularity coupled to 2d quantum gravity
J. Ambjørn
a,b,∗
, A. Görlich
a,c
, A.C. Ipsen
a
, H.-G. Zhang
c
a
The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
b
Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
c
Institute of Physics, Jagiellonian University, Reymonta 4, PL 30-059 Krakow, Poland
a r t i c l e i n f o a b s t r a c t
Article history:
Received
9 June 2014
Accepted
11 June 2014
Available
online 16 June 2014
Editor:
L. Alvarez-Gaumé
We show how to obtain the critical exponent of magnetization in the Lee–Yang edge singularity model
coupled to two-dimensional quantum gravity.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
1. Introduction
Two-dimensional quantum Liouville gravity and the theory of
random triangulations (or matrix models) most likely describe the
same theory, two-dimensional quantum gravity coupled to confor-
mal
field theories with a central charge c ≤1. The two realizations
are sufficiently different that the “proof” that they describe the
same theory is basically by comparing the result of calculations
of certain “observables”. The major problem of such a compari-
son
has been to identify the observables to be compared in the
two formulations. This problem has to a large extent been solved
in [2] for one and two-point correlation functions and in [3] for
three- and four-points correlation functions. Here we will address
an observable, the so-called “magnetization” at the Lee–Yang edge
singularity. We will show how the general assumptions of operator
mixing put forward in [1–3] allow us to obtain agreement between
the critical exponent of the Lee–Yang “magnetization” calculated in
quantum Liouville gravity and using matrix models.
The
rest of this article is organized as follows: in the next sec-
tion
we recapture how to calculate the magnetization exponent σ
in the Ising model and at the Lee–Yang edge singularity using stan-
dard
conformal field theory. In Section 3 we then show how to
reconcile Liouville and matrix model results.
2. Ising models and dimer models
The Ising model on an arbitrary connected graph G
V
with V
vertices
and L links is defined by
Z
G
V
(β, H) =
{σ
i
}
exp
β
L
ij=1
σ
i
σ
j
+ H
V
i=1
σ
i
,
(1)
*
Corresponding author.
E-mail
addresses: ambjorn@nbi.dk (J. Ambjørn), goerlich@nbi.dk (A. Görlich),
asgercro@nbi.dk (A.C. Ipsen), zhang@th.if.uj.edu.pl (H.-G. Zhang).
where the Ising spin σ
i
(which can take values ±1) is located at
vertex i, ij symbolizes that vertices i and j are neighbors in G
V
,
and β and H signify inverse the temperature and a magnetic field,
respectively.
If
G
V
is a regular two-dimensional lattice, e.g. a square lattice,
the partition function Z
G
V
(β, H = 0) has a second order phase
transition for a certain value β
c
in the limit V →∞. Let us cal-
culate
e
H
i
σ
i
β=β
c
, H=0
=e
−F
G
V
(H)
, (2)
using the partition function Z
G
V
(β
c
, 0). For large V the free energy
F
G
V
(H) becomes extensive and the magnetization m is given by
F
G
V
(H) = f (H)V
1 +o(V )
, m =−
df
dH
∼|H|
σ
,
σ =
1
15
, (3)
for small H.
The two-dimensional Ising model at its critical point β
c
is a
conformal field theory with central charge c = 1/2. Let us recall
how the above result is derived using conformal field theory. Con-
sider
a conformal field theory and let Φ be a primary operator
with scaling dimension
0
, i.e. Φ(
√
λx) = λ
−
0
Φ(x) (we consider
Φ to be the product of its holomorphic and anti-holomorphic
parts, i.e. real). Under a scaling x →
√
λx we thus have
A =
d
2
x → λA, D
0
=
d
2
xΦ(x) →λ
1−
0
D
0
. (4)
We can study a “deformation” away from the conformal point by
adding the term
δD
0
=δ
d
2
xΦ, [δ]=[A]
0
−1
(5)
http://dx.doi.org/10.1016/j.physletb.2014.06.025
0370-2693/
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.
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