热失序相关器与KMS条件的量子关联研究

171 浏览量更新于2024-07-16 收藏 793KB PDF 举报

"这篇研究论文深入探讨了热失序相关器、KMS关系以及光谱函数在量子系统中的重要角色。作者们展示了热相关函数如何通过KMS条件体现波动-耗散的关系，这是一种时间顺序相关的现象，特别是在考虑出时间顺序（Out-of-Time-Order, OTO）关联时更为显著。他们提出了一种方法，无需标准的KMS关系，而是直接利用OTO相关器来表述这些关系，从而避免了将热周期性和时间反转相结合的问题。此外，他们还构建了一个基于完全嵌套换向器的热n点函数的因果框架，并从置换的循环轨道中推导出一般性结果。文章以量子振荡器为例，具体阐述了这些抽象概念。" 这篇论文详细介绍了量子系统的热力学特性，特别是在非平衡态下的行为。热失序相关器（Out-of-Time-Order Correlator, OTOC）是一种测量量子系统中信息传播和混沌行为的重要工具，它能揭示系统内的相互作用和动力学。KMS条件（Kubo-Martin-Schwinger condition），则是统计力学中的一个基本原理，它在平衡态下连接了动态与静态性质，体现了波动-耗散关系。在热力学中，波动-耗散关系描述了系统中的波动（如能量的瞬时转移）如何导致耗散（如能量的损失或热化）。论文中提出的直接用OTO相关器来表示KMS关系的新方法，不仅简化了理论处理，也为理解和计算热力学系统的行为提供了新的途径。完全嵌套换向器的概念则帮助构建了一个因果结构，使得在不同时间顺序下的热n点函数可以被系统地分析。这种因果基础对于理解量子系统在非平衡态下的复杂动态至关重要。通过量子振荡器的实例，作者们展示了这些理论框架的实际应用，进一步巩固了这些概念的物理意义。量子振荡器是一个简单的模型系统，经常被用来研究量子力学的基本原理和现象，例如谐振子的量子化、能级结构和量子热力学行为。这篇论文对热失序相关器、KMS条件和光谱函数的综合研究，为理解和探索复杂量子系统的行为提供了新的见解和方法，对于理论物理学家和量子信息科学家来说具有很高的价值。通过这种方式，我们可以更深入地洞察量子世界中的非经典效应和热力学性质。

JHEP12(2017)154

Consider a Wightman correlation function speciﬁed by a given permutation σ ∈ S

We consider the action of the cyclic group Z

on the sequence σ(1)σ(2) ···σ(n). Denoting

by π an element of Z

, we would end up with map

σ(1)σ(2) ···σ(n) 7→ πσ(1)πσ(2) ···πσ(n) =

j=p+1

σ(j)

`=1

σ(`) . (2.17)

Taking a weighted average of the cyclic permutations we construct the master function

for the permutation σ ∈ S

(1, ··· , n) ≡

π∈Z

hπσ(1) πσ(2) ··· πσ(n)i

π∈Z

−

j ω

πσ(j)

, (2.18)

where π ∈ Z

is a cyclic permutation. Then the KMS relations can be stated in terms of

recovering a given Wightman correlator in frequency space directly from Ψ

. One ﬁnds:

hσ(1) ··· σ(n)i = Ψ

(1, ··· , n) e

−

j ω

σ(j)

. (2.19)

Note that the master function Ψ

is indexed by a permutation σ ∈ S

. Since we

are averaging over the Z

orbit of σ, we only need to consider (n − 1)! elements of S

Combined with the action of π we would end up covering all n! permutations.

There is a physically interesting way to write Ψ

. We claim that

(1, ··· , n) =

(ˆρ

)

σ(1) (ˆρ

)

σ(2) ··· (ˆρ

)

σ(n)

(2.20)

which makes the cyclic invariance manifest.

For n = 2 this spectral function is naturally interpreted in the thermoﬁeld double

construction with factors of ˆρ

appearing between the forward and backward evolutions.

Similarly, the chaos correlator in [20] was computed using an Euclidean regulator which

inserts powers of ˆρ

between the operators in a 4-point function. More generally, inspired

by these developments, [25] considered arbitrary combinations of commutators and anti-

commutators of two operators, [A, B] and {A, B}, to derive a set of bipartite ﬂuctuation-

dissipation relations. Their results follow from the statements mentioned above as the

reader can quickly infer.

The statement of (2.19) is that Ψ

is the natural object that symmetrizes the KMS

relations amongst the (n − 1)! independent Wightman functions, and should be viewed as

providing a suitable basis for the spectral functions.

Once again it is helpful to view these relations through some examples of low point

correlation functions.

Two-point functions: there being a single Z

orbit since Z

' S

, the unique 2-point

cyclic function Ψ is

Ψ(1, 2) =

h12i + h21i

−

+ e

−

. (2.21)

The KMS relations (2.14) can then be re-expressed as

h12i = Ψ(1, 2) e

, h21i = Ψ(1, 2) e

. (2.22)

This is analogous to a Bloch, or perhaps more appropriately, Floquet, basis for wavefunctions.

– 9 –

JHEP12(2017)154

Three-point functions: there are two non-trivial permutations of S

whose Z

orbits

help ﬁll out the six elements. Thus we need to consider two master Ψ functions indexed

by the two independent orderings. Consider ﬁrst σ = id and

(1, 2, 3) = Ψ(1, 2, 3) = h123i + h231i +

h312i

−

(ω

−ω

)

+ e

−

(ω

−ω

)

+ e

−

(ω

−ω

)

. (2.23)

In terms of this function we can write down three KMS relations in the Z

orbit of id as

h123i = Ψ(1, 2, 3)e

−

(ω

−ω

)

h231i = Ψ(1, 2, 3)e

−

(ω

−ω

)

h312i = Ψ(1, 2, 3)e

−

(ω

−ω

)

(2.24)

We are then left with the second choice σ = (23) (using the cycle notation for permu-

tations), which results in

(23)

(1, 2, 3) ≡ Ψ(1, 3, 2) =

h132i + h321i + h213i

−

(ω

−ω

)

+ e

−

(ω

−ω

)

+ e

−

(ω

−ω

)

, (2.25)

which then gives us three further relations

h132i = Ψ(1, 3, 2) e

−

(ω

−ω

)

h321i = Ψ(1, 3, 2) e

−

(ω

−ω

)

h213i = Ψ(1, 3, 2) e

−

(ω

−ω

)

(2.26)

3 Generalized ﬂuctuation-dissipation relations

While Wightman functions encode all the information inherent in the KMS relations quite

succinctly, it is often useful in physical applications to consider correlation functions of

various combinations involving nested sequence of commutators and anti-commutators.

This basis was referred to as the nested basis in [18] and its physical utility is that causal

response functions are determined in terms of nested commutators.

Unlike the Wightman basis the set of nested correlators has redundancies. For instance

even without including imaginary time shifted operators, it was noted in [18] that one

naively has 2

n−2

n! nested correlators, since each of the n! permutations allows for (n −1)

binary choices of either a commutator or an anti-commutator (and we only need to consider

alternating permutations owing to an overall sign ﬂip resulting from swapping order of

operators in a commutator). In this section we resolve this redundancy, leading to a

minimal set of (n − 1)! spectral functions. Generalized ﬂuctuation-dissipation relations

encode how to express other correlators (such as Wightman functions, or generic nested

correlators) in terms of these spectral functions.

3.1 Notation

To simplify some of the formulae that appear in the sequel, we introduce some new notation

which we collect here for quick reference:

– 10 –

剩余54页未读，继续阅读

weixin_38687968

粉丝: 7
资源: 969

热失序相关器与KMS条件的量子关联研究

SYK模型中的幂律失序相关函数

越控制越失序

Java数据报之失序和丢包.doc

基于UDP传输协议的包丢失和失序处理

探索无序相关器

有序保形域理论中的无序相关器和纯度

标准的秩序与区块链的失序.pdf

煤矿生产系统运行失序复杂性评价.pdf

医疗纠纷处理中失序现象的反思.pdf

太比特路由器分组调度中信元失序问题的解决.pdf

最新资源