A generalization of Sachdev-Ye-Kitaev
Received: December
Published for SISSA by Springer
Santa Barbara 0 1
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include f 0 1
0 Open Access , c The Authors
1 Kavli Institute for Theoretical Physics, University of California
The SYK model: fermions with a q-body, Gaussian-random, all-to-all interaction, is the rst of a fascinating new class of solvable large N models. We generalize SYK to avors of fermions, each occupying Na sites and appearing with a qa order in the interaction. Like SYK, this entire class of models generically has an infrared We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global O(N1) ArXiv ePrint: 1610.01569
1/N Expansion; AdS-CFT Correspondence; Conformal Field Theory; Matrix
-
generalization of Sachdev-Ye-Kitaev
O(Nf ) symmetry: a subgroup of
Contents
1 Introduction 2 Two-point function 2.1
Large qk
A generalization of SYK
Graphical solution
Spectral function
E ective action
Four-point function
Dimensions of composite operators
Four-point function
Generalized model
Equal qa, a
Equal qa, a
A E ective action
A.1 Fluctuations
Model with a scalar
C Random mass matrix fermions
C.1 In nite N
C.2 Finite N
Dirac fermion
Majorana fermion
Introduction
maximal chaos.
While there are models that contain some of these properties, SYK is
seminars [2].
Matrix models are closely related to string theories [3{8], with the most
calculations for strong coupling. SYK is such a model.
understanding both quantum
eld theories and string theories. One may hope that the
study of SYK-like models will also prove productive.
The SYK model consists of N
1 Majorana fermions i, with a q-body Hamiltonian
largely applicable.
with quenched disorder,
H =
The model has qualitatively similar properties for any choice of even q
4. The couplings
es a simple integral equation which can be explicitly solved near the infrared
xed point.
The fermions start with dimension 0 in the UV, and ow to dimension
= 1=q in the IR.
operators, singlets under O(N ), are schematically PN
i=1 i @2n+1 i. In the UV, these
opsmall n, and approach 2
+ 2n + 1 asymptotically for large n. The standard AdS/CFT
SYK is, and the extent to which it is nonlocal, remains an open problem.
H =
of the interaction, qa, can depend on the
avor a, as long as Na=N remains
nite as
N = P
the fermions, and
nd that the model (1.2) generically has an IR
xed point. While the
IR dimension for SYK (1.1) was
determine the dimensions
a. In the limit of large qa, these have simple analytic solutions.
the generalized model (1.2) has an O(N1)
O(Nf ) symmetry. The singlet
bilinear operators are PNa
same properties hold for the generalized model.
appears with a degeneracy of f
1. Indeed, this model is similar to SYK with N fermions
singlet operators exist, allowing for a richer model.
4. This appendix can be read independently of the rest of the paper.
Two-point function
S =
q! i1;:::;iq=1
a Gaussian ensemble. The two-point function of the Ji1;:::;iq is taken to be,
i2;:::;iq=1
hJi1i2:::iq Ji1i2:::iq i = J 2 :
hJi1i2:::iq Ji1i2:::iq i = (q
At zero coupling, the Euclidean two-point function hT i( ) j (0)i
G( ) ij is given by,
G0( ) =
sgn( ) ;
G0(!) =
> 0 and sgn( ) =
< 0) accounts for
for the two-point function drastically simplify and are given by,
G(!) 1 = G0(!) 1
(!) =
( ) = J 2G( )q 1
of the one-particle irreducible self-energy
(!). The second equation, which is written in
position space, is a special feature of SYK (see
however, an analytic solution to this equation is not known. At strong coupling, jJ j
(equivalently, the infrared limit), one can drop the i! in (2.5), to get,
with a lled circle is the two-point function. The dashed line is the disorder.
One can verify that,
is a solution to (2.7) provided one takes,
G(!) (!) =
( ) = J 2G( )q 1
G( ) = b
bq =
The Fourier transform of G( ), given in (2.8), is useful in verifying this,
where we de ned,
G(!) =
d ei! G( ) = b ( )J 2
1sgn(!);
2 ) = 2ip
reparameterization of time,
2) ! f 0( 1) f 0( 2) G(f ( 1) f ( 2)).
Theretwo-point function by mapping the real line to a circle [2, 35{37].
A generalization of SYK
The model we introduce is a generalization of SYK (2.1). It contains f
avors of fermions,
with Na fermions of
avor a, each appearing qa times in the interaction, so that the
Hamiltonian couples q = Pf
avor a 2 f1; : : : ; f g. Explicitly, the action is,
S =
a=1 i=1
Qfa=1 qa! I
X JI
Gaussian distribution,
where the disorder average hJI JI i is given by
It will be convenient to make the following de nitions,
P [JI ] / exp
hJI JI i = J
a=1 Na Y(qa
k =
a6=k
The class of models (2.12) for large N is characterized by f
1 independent continuous
paalized model (2.12) the symmetry is (...truncated)