分数统计与量子混沌的关联探索
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"分数统计和蝴蝶效应 - Yingfei Gu and Xiao-Liang Qi - JHEP08(2016)129 - Open Access - 物理学 - 量子混沌 - 分数统计 - 有理保形场理论 - 拓扑有序状态 - 混沌定量度量 - 拓扑缠结熵"
本文深入探讨了量子信息的非局部存储与分数统计及蝴蝶效应之间的关系，这两个概念在量子物理中都扮演着重要角色。蝴蝶效应通常被描述为在一个混沌系统中，初始条件的微小变化可能导致最终结果的显著不同，而在量子混沌中，这一概念涉及到量子态的敏感性。分数统计则是在（2+1）维拓扑有序状态中出现的一种现象，其中粒子之间的交换不遵循传统的整数统计，而是表现为分数的统计权重。
文章指出，（1+1）维的有理保形场理论（RCFT）中的蝴蝶效应与（2+1）维拓扑有序状态的分数统计存在关联。这种联系基于失序相关器（out-of-time-order correlator, OTOC）的新颖应用，OTOC是用于刻画系统混沌程度的工具。作者表明，失序相关器在长时间行为上受到RCFT的普遍特性如模块化S矩阵和共形自旋的影响。
模块化S矩阵是RCFT中的关键对象，它反映了在不同的模空间坐标下的态之间的关系，而共形自旋则与场的旋转性质相关。通过（1+1）维和（2+1）维理论之间的体边界对应原理，作者证明了失序相关器的后期演化本质上与拓扑有序状态中的分数统计行为相关联。
此外，文章还提出了一种在有理保形场理论中的混沌定量度量方法，该度量由相应的拓扑有序状态的拓扑缠结熵来确定。拓扑缠结熵是衡量拓扑秩序的一个关键属性，它揭示了系统中拓扑保护的纠缠性质。
这项工作为理解和研究量子系统中的混沌和非局域性提供了新的视角，尤其是在理解（2+1）维量子系统中的分数统计行为以及其与（1+1）维系统混沌的联系方面。这项研究对于量子计算、量子信息处理和高级量子材料的理论设计具有重要意义，因为它揭示了这些复杂物理现象间的深层次联系。
JHEP08(2016)129
As a measure of the difference between |xi and |yi, it is convenient to use the normalized
value:
e
f(t) :=
hy|xi
p
hx|xihy|yi
(2.2)
Previous studies [2, 3, 7, 18] of such correlation function focused on the early time β <
t < t
scr
, i.e., between the dissipation time and the scrambling time, and found interesting
“Lyapunov behavior”
4
for those systems that can be holographically described by Einstein
gravity. However, for a generic system that does not have a large separation between
dissipation time t
d
∼ β and scrambling time t
scr
, the “Lyapunov behavior” is not well-
defined. Instead, one can focus on the later time regime t t
scr
, which characterized the
residue part of the system that “survived” under the butterfly effect.
For later convenience, we denote the inner product hx|xi = hV
†
W
†
(t)W (t)V i
β
by
g(t). In the regime t β, such a four point function generically factorizes to g(t) ∼
hV
†
V i
β
hW
†
W i
β
, which represents a normalization for operators W and V . The same
applies to hy|yi. In the following section, we will study the late time behavior of
e
f(t) in
the context of RCFTs.
2.2 An overview of existing results
As a starting point of our discussion, we briefly review the calculation of OTOC in (1+1)-d
CFT in ref. [18].
To setup the CFT computation, it is essential to use complexified time t
c
= t − iτ,
where t stands for the Minkowski time, and τ for the Euclidean time. The strategy is
to begin with a pure Euclidean computation with t = 0 and then analytically continue
it to the desired Minkowski time t. Such strategy enables us to manipulate the order
of operators by tuning the auxiliary small imaginary part τ = . After the conformal
mapping z = exp (2πw/β) , z = exp (2πw/β), with w = x − t
c
and w = x + t
c
,
5
the
thermal expectation value is mapped to the vacuum expectation value. In the end, it is
essential to consider the vacuum expectation value of the four point function which has a
general decomposition in terms of conformal blocks [18, 21, 22]
hW
†
(z
1
, z
1
)W (z
2
, z
2
)V
†
(z
3
, z
3
)V (z
4
, z
4
)i =
1
z
2h
w
12
z
2h
v
34
1
z
2h
w
12
z
2h
v
34
X
p,p
g
p,p
F
p
(η) F
p
(η) (2.3)
where z
ij
= z
i
−z
j
, and η =
z
12
z
34
z
13
z
24
, η =
z
12
z
34
z
13
z
24
are cross ratios, g
p,p
is the pairing coefficient
for holomorphic block p and anti-holomorphic block p. It is important to note that the
summation is only over conformal families labeled by {p, p} rather than all the primaries
and descendants individually. For the parameter regime t β that we are interested in,
4
Here Lyapunov behavior refers to the functional dependence of f (t) on t at early time:
f(t) ∼ 1 −
1
N
e
λ
L
t
+ . . .
where N is some big number in the model, and the exponent λ
L
is recognized as Lyapunov exponent.
Physically, the Lyapunov behavior characterizes how fast the chaos develops in the system.
5
We use a different sign convention comparing to ref. [18] for later convenience.
– 4 –
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