![](https://csdnimg.cn/release/download_crawler_static/12265239/bgc.jpg)
The
L
2
t
3
behavior for t 1 is due to the sharp square root edge of the semicircle
distribution and dominates until the “dip time” t
dip
when it becomes smaller than the
ramp, i.e., at a time when L
2
/t
3
dip
∼ t
dip
, t
dip
∼ L
1/2
.
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This early time “slope” obscures
the ramp until after t
dip
, but the ramp time is typically much earlier than this. The
necessity of following the ramp underneath the slope in SYK was already emphasized in [8].
There are strategies to deal with this. In GUE one can just subtract off the discon-
nected part of the spectral form factor, computing
h|Z(t)|
2
i
c
= h|Z(t)|
2
i − |hZ(t)i|
2
. (3.11)
But this does not work in the SYK model, as explained in [8]. This model has a square
root edge in its eigenvalue density [8, 61, 62] but fluctuations in the edge position from
sample to sample are only suppressed by a power of N, the number of fermions, rather
than L (equal to 2
N/2
in SYK) as is the case in GUE. This fluctuation converts the power
law slope into a Gaussian falloff in |hZ(t)i|
2
. But fluctuations in the edge position cancel
out in h|Z(t)|
2
i, leaving the 1/t
3
slope.
In this paper we employ the alternate strategy, suggested to us by Douglas Stanford,
of using a type of microcanonical ensemble with a Gaussian window to suppress the con-
tribution of the edge. To see the onset of the ramp we introduce a variant of the spectral
form factor
|Y (α, t)|
2
=
X
i
e
−αE
2
i
−itE
i
2
. (3.12)
In other words, we replace the energy spectrum ρ(E) with e
−αE
2
ρ(E). This is in the spirit
of a microcanonical density of states centered around E = 0 but with smooth edges to the
energy window that will not create spurious signals in the time domain.
11
Although this
changes the global shape of the energy spectrum, locally it is just a multiplication by a
constant factor, and hence it does not spoil the RMT universality of the energy correlations.
When α is sufficiently large, any sharp edge of the spectrum is replaced with a Gaus-
sian. Then Y , which is essentially the Fourier transform of this ‘Gaussian-filtered energy
spectrum’, decays as a Gaussian at early time. Hence the non-universal part can be strongly
suppressed, while preserving the universal part.
We also introduce the normalized quantity h(α, t) ≡ |Y (α, t)|
2
/|Y (α, t = 0)|
2
. Note
h(0, t) = g(t).
In the GUE case at early time
|Y (α, t)|
2
∼ L
2
e
−t
2
/α
2
. (3.13)
This becomes of order one at t ∼
√
log L. So spectral rigidity can be probed out to this
scale. In fact t
ramp
∼ 1 here, so we cannot fully diagnose this system using this technique.
10
Many plots illustrating these phenomena are displayed in [8]. Another approach was discussed in [60],
where authors use a non-Gaussian potential for RMT ensemble to change the edge.
11
Of course we could choose to center this window around any energy we shoose, but to maximize the
number of eigenvalues we choose to center it around E = 0. We expect the same universal features to be
present as long as the center of the window is not extremely close to the edge of the spectrum.
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