量子混沌系统中的随机矩阵行为起始时间研究

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本文主要探讨了加扰系统中的随机矩阵行为起源，特别是关注量子混沌系统中的能量谱精细结构。量子混沌系统通常被认为其能谱在一定程度上服从随机矩阵统计，这是基于其复杂动态下的无序性。系统中的基本尺度是决定这种随机矩阵行为持续存在的能量区间，这个尺度可以通过分析频谱形状因数中的线性增长区——斜坡区域开始的时间来定义，即所谓的“t斜坡”。作者研究了多种类型的加扰系统，包括具有几何局部性和守恒定律的系统，如一维量子比特链，以及更为复杂的k本地系统，如Sachdev-Ye-Kitaev (SYK) 模型。对于那些具有守恒定律的系统，例如一维链，研究发现t斜率与系统的扩散时间成正比，对于N个量子比特，时间尺度大约为N^2，这与单体局部混沌系统的观察结果相似。然而，对于k本地系统如SYK模型，时间尺度与N的关系有所不同，表现为log N阶。这种差异源于不同的扰动机制。在没有守恒定律的情况下，比如在通用随机量子电路中，研究发现t ramp的值与系统的连通性无关，呈现出一个与log N相关的趋势。文章利用数值模拟、解析估计以及对哈密顿系统的启发式分析方法，揭示了这些加扰系统中随机矩阵行为出现的时间尺度依赖于系统特性、相互作用强度和守恒定律的存在与否。这些发现不仅深化了我们对量子混沌行为的理解，也为未来的理论和实验研究提供了有价值的指导。此外，由于该研究是开放获取的，它对量子信息科学和量子热力学领域的发展具有重要意义，促进了学术界的交流和合作。

JHEP07(2018)124

Figure 2. [Left] Equal width slices of the energy spectrum marked with diﬀerent colors. [Right]

Ramp and plateau coming from two slices marked with the corresponding colors.

We call the constant region the “plateau” and the time at which it begins the “plateau

time”, which is of order L.

We now discuss the consequence of having an eigenvalue density ρ(

E) that varies

across the band. We divide the entire energy band up into a large number of regions of

equal energy width narrow enough so that ρ is roughly constant in each. Changing the

constant value ρ(

E) in (2.5) does not change the overall form of (3.9). It only changes

the plateau time and the height of the plateau (see ﬁgure 2). This means that summing

up the contributions from each region to compute the total spectral form factor is very

simple. Along the ramp one just adds up linear functions with equal coeﬃcients. The ﬁnal

plateau height is just L, the total number of eigenvalues. The only signiﬁcant change is a

rounding of the sharp transition to the plateau, because diﬀerent regions of the spectrum

have diﬀerent plateau times. The size of this eﬀect depends on the shape of ρ(E) and

depends on the system being studied.

This insensitivity of the ramp means that ﬂuctuations from sample to sample of the

eigenvalue density do not aﬀect the ramp structure, making it a robust diagnostic of

the Thouless time even without unfolding. This is in contrast to the number variance,

discussed above.

There is a problem, though, with using the spectral form factor to diagnose spectral

rigidity when it is very long ranged because the ramp time will be very short and the height

of the ramp will be very small compared to the plateau. So small eﬀects can hide the ramp.

The GUE ensemble oﬀers a simple example of this eﬀect when the variation of the spectral

density is included. The leading contribution at relatively early times to the spectral form

factor is just given by the Fourier transform of ρ

(E),

h|Z(t)|

i ∼ L



dEρ

(E)e

−iEt



= L



(2t)



∼ L

cos

(2t)

(3.10)

Note that from the deﬁnition (3.6) the presence of a plateau of height exactly L at suﬃciently large t

is manifest. The factor of 1/π here is an artifact of the approximations involving ρ.

– 9 –

JHEP07(2018)124

The

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

dip

∼ t

dip

, t

dip

∼ L

1/2

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)|

= h|Z(t)|

i − |hZ(t)i|

. (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|

. But ﬂuctuations in the edge position cancel

out in h|Z(t)|

i, leaving the 1/t

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)|



−αE

−itE



. (3.12)

In other words, we replace the energy spectrum ρ(E) with e

−αE

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.

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)|

/|Y (α, t = 0)|

. Note

h(0, t) = g(t).

In the GUE case at early time

|Y (α, t)|

∼ L

−t

/α

. (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.

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.

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.

– 10 –

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