JHEP02(2018)162
the scalar lives is now conformal to AdS
2
× I. The scalar field is subject to a non-trivial
potential in addition to a delta function at the center of the interval. This reproduces the
spectrum exactly. Furthermore, the three dimensional propagator whose points lie at the
center of I reproduce the arbitrary q SYK propagator up to a factor which depends only
on q. We also discuss the large q limit in this picture. In the SYK model the spectrum
becomes evenly spaced in this limit. However only one mode - the zero mode - contributes
to the propagator since the residues at the other poles vanish. In the three dimensional
picture, the different modes appear as KK modes. However in the large q limit we find
that the normalized wave function at the center of the interval is nonzero for only one of
these modes, in a way consistent with the SYK result.
In section 2 we describe the bilocal collective formulation of the model for arbitrary q.
In section 3 we describe our three dimensional model. Section 4 contains the comparison
of the propagator of the three dimensional model with the SYK bilocal propagator. In
section 5 we comment on the large q limit. Section 6 contains some concluding remarks.
2 q fermion SYK model
The model is defined by a hamiltonian, with any even q,
H = (i)
q
2
X
1≤i
1
<i
2
<···<i
q
≤N
j
i
1
i
2
···i
q
χ
i
1
χ
i
2
···χ
i
q
, (2.1)
where χ
i
are Majorana fermions, which satisfy {χ
i
, χ
j
} = δ
ij
. The random coupling has a
gaussian distribution with
< j
2
i
1
i
2
···i
q
>=
J
2
(q − 1)!
N
q−1
(2.2)
One way to perform the averaging is to use the replica trick. At large N, one does not
expect a spin glass state in this model [5] so that we can restrict to the replica diagonal
subspace [8]. The large N this model is efficiently solved by re-writing the theory in terms
of replica diagonal bilocal collective fields [8, 9].
Ψ(t
1
, t
2
) ≡
1
N
N
X
i=1
χ
i
(t
1
)χ
i
(t
2
) , (2.3)
where we have suppressed the replica index. The corresponding path-integral is
Z =
Z
Y
t
1
,t
2
DΨ(t
1
, t
2
) µ(Ψ) e
−S
col
[Ψ]
, (2.4)
where S
col
is the collective action:
S
col
[Ψ] =
N
2
Z
dt
h
∂
t
Ψ(t, t
0
)
i
t
0
=t
+
N
2
Tr log Ψ −
J
2
N
2q
Z
dt
1
dt
2
Ψ
q
(t
1
, t
2
) . (2.5)
Here the second term comes from a Jacobian factor due to the change of path-integral
variable, and the trace is taken over the bi-local time. One also has an appropriate order
– 3 –
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