Towards a 2d QFT analog of the SYK model
Received: June
Towards a 2d QFT analog of the SYK model
Gustavo J. Turiaci 0 2
Herman Verlinde 0 1 2
0 Washington Road, Princeton, NJ, 08544 U.S.A
1 Princeton Center for Theoretical Science, Princeton University , Jadwin Hall
2 Physics Department, Princeton University , Jadwin Hall
We propose a 2D QFT generalization of the Sachdev-Ye-Kitaev model, which we argue preserves most of its features. The UV limit of the model is described by N copies of a topological Ising CFT. The full interacting model exhibits conformal symmetry in the IR and an emergent pseudo-Goldstone mode that arises from broken reparametrization Liouville CFT. ories
AdS-CFT Correspondence; Conformal Field Theory; Topological Field The-
-
QFT
analog of the SYK
model
symmetry.
We
nd that the e ective action of the Goldstone mode matches with the
3D AdS gravity action, viewed as a functional of the boundary metric. We compute the
spectral density and show that the leading deviation from conformal invariance looks like
a T T deformation.
We comment on the relation between the IR e ective action and
1 Introduction
2
E ective action of the Goldstone mode
Double Schwarzian action
Free energy and spectral density
Relation with AdS3 gravity
5
Conclusion
A Topological RCFT B Two point function from path integral 1 2
fermions i with anti-commutation relations f i; j g = ij interacting via a homogeneous
non-linear potential with random couplings. The model is speci ed by the 1D action
SSYK =
Z
0
q
i 2
X
i1;:::;iq
1
Ji1:::iq i1 : : : iq A :
(1.1)
Here Ji1:::jq denotes a set of gaussian random couplings. We can split (1.1) as S = SUV+SIR.
Note that both terms exhibit reparametrization invariance, but that
transforms as a
scalar in the UV, but has scale dimension
= 1=q in the IR. The SYK model exhibits
1Similar models have been originally introduced in [8{12] to model strongly interacting matter systems
without quasiparticle excitations that realize non-Fermi liquid states.
{ 1 {
approximate conformal symmetry in the IR, and has been proposed to give a holographic
description of a 2D black hole space-time. The link with the gravity dual nds support
in the fact that both sides give rise to an e ective 1D Goldstone mode whose action is
described by the Schwarzian derivative [1{4, 7, 13{18].
In this note we propose a 2D QFT generalization2 of the SYK model (1.1), which
we argue preserves most of the desired features. In particular, via the same reasoning
that applies to 1D case, we will argue that the 2D model appears to exhibit conformal
symmetry in the IR and gives rise to an emergent Goldstone mode associated with broken
2D reparametrization invariance. We nd that the same e ective action of the Goldstone
mode can also be derived from the 3D AdS gravity action, viewed as a functional of the
2D CFT, and may thus provide new insight into the dynamical mechanism that underlies
AdS3/CFT2 duality.
Recently, Witten found an adaptation of a class of so-called tensor models that give
rise to the same large N diagrammatical rules as the SYK model [
22, 23
]. It would be
worthwhile to investigate whether our proposal can be adapted to this case.
This paper is organized as follows. In section 2, we specify our 2D model. We give
both a Lagrangian and Hamiltonian formulation. We give special attention to the UV
limit, which is described by a topological Ising CFT. In section 3 we study the
SchwingerDyson equations that capture the large N dynamics of the model. We describe the solution
of the SD equations in the conformal IR regime, and study the four point function. We
nd that the chiral spectrum of our 2D model coincides with that of the 1D SYK model.
In section 4, we analyze the dynamics of the pseudo-Goldstone mode, and show that its
action is given by the product of two Schwarzian derivatives and exhibit a connection with
2+1-D gravity. In section 5 we list some open questions. In appendix A we summarize
the properties of topological RCFTs, and discuss the 2-point function of the UV model in
appendix B.
2
The 2D model
In this section, we will give two characterizations of our 2D model. First we introduce the
model via its Lagrangian, and then we present a Hamiltonian formulation. We give some
special attention to the UV limit of our model.
2.1
Lagrangian formulation
A nave attempt to generalize the SYK model to 2D is to promote the
variables to
2D Majorana fermions with a standard kinetic term 2i @= . This choice assigns canonical
scale dimension [ ] = 1=2. The interaction term then has dimension q=2, which is at best
marginal. In the 1D action (1.1), on the other hand, the UV term assigns
scale dimension
[ ] = 0, so the interaction term is relevant and the model is strongly coupled in the IR.
To write the 2D generalization of (1.1) we introduce fermionic variables
with i = 1; : : : ; N . One can think of
+ and
as the two chiral components of a 2D
2Proposals for 2D generalizations of SYK with a discretized spatial dimension a (...truncated)