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arXiv 2310.03053quant-phnlin.CD
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封闭混沌多体量子系统的热化

Thermalization of closed chaotic many-body quantum systems

Hans A. Weidenmüller

发表信息：
J. Math. Phys. A: Math. Theor. 57 165001 (2024)
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AI总结：

该研究结合Hartree-Fock方法和BGS猜想，证明封闭混沌多体量子系统在半经典区域以时间尺度ħ/Δ热化至统计平衡，关联宽度Δ是核心参数。

摘要：

一个封闭量子系统发生热化，当且仅当在时间 $t \to \infty$ 时，函数 ${\rm Tr} (A ρ(t))$ 渐近趋向于 ${\rm Tr} (A ρ_{\rm eq})$。这里 $A$ 是表示可观测量的算符，$ρ(t)$ 是含时密度矩阵，$ρ_{\rm eq}$ 是其平衡值。我们通过结合 Hartree-Fock (HF) 方法和 Bohigas-Giannoni-Schmit (BGS) 猜想，研究了一个混沌多体量子系统的热化。HF 哈密顿量定义了一个可积系统以及谱的粗略特征。残余相互作用局域地混合了 HF 本征态。BGS 猜想意味着由此得到的本征值和本征函数的统计性质与随机矩阵预言一致。这样，系统的哈密顿量 $H$ 就获得了统计特征。统计性质与随机矩阵性质的吻合是局域的，即局限于一个区间 $Δ$（关联宽度）内。利用 $ρ(t) = \exp \{ - i t H / \hbar \} ρ(0) \exp \{ i H t / \hbar \}$，$H$ 的统计性质决定了 ${\rm Tr} (A ρ(t))$ 的统计性质。利用这些性质，我们证明在半经典区域，${\rm Tr} (A ρ(t))$ 以时间尺度 $\hbar / Δ$ 衰减至一个渐近值。如果系统的能量展宽量级为 $Δ$，则该值对应于统计平衡。关联宽度 $Δ$ 是我们方法的核心参数。它定义了谱涨落与随机矩阵预言一致的区间。它定义了允许热化的系统最大能量展宽。它还定义了 ${\rm Tr}(A ρ(t))$ 趋近于 ${\rm Tr}(A ρ_{\rm eq})$ 的时间尺度 $\hbar / Δ$。

英文摘要：

A closed quantum system thermalizes if for time $t \to \infty$, the function ${\rm Tr} (A ρ(t))$ tends asymptotically to ${\rm Tr} (A ρ_{\rm eq})$. Here $A$ is an operator that represents an observable, $ρ(t)$ is the time-dependent density matrix, and $ρ_{\rm eq}$ its equilibrium value. We investigate thermalization of a chaotic many-body quantum system by combining the Hartree-Fock (HF) approach and the Bohigas-Giannoni-Schmit (BGS) conjecture. The HF Hamiltonian defines an integrable system and the gross fatures of the spectrum. The residual interaction locally mixes the HF eigenstates. The BGS conjecture implies that the statistics of the resulting eigenvalues and eigenfunctions agrees with random-matrix predictions. In that way, the Hamiltonian $H$ of the system acquires statistical features. The agreement of the statistics with random-matrix properties is local, i.e, confined to an interval $Δ$ (the correlation width). With $ρ(t) = \exp \{ - i t H / \hbar \} ρ(0) \exp \{ i H t / \hbar \}$, the statistical properties of $H$ define the statistical properties of ${\rm Tr} (A ρ(t))$. Using these we show that in the semiclassical regime, ${\rm Tr} (A ρ(t))$ decays with time scale $\hbar / Δ$ towards an asymptotic value. If the energy spread of the system is of order $Δ$, that value corresponds to statistical equilibrium. The correlation width $Δ$ is the central parameter of our approach. It defines the interval within which the spectral fluctuations agree with random-matrix predictions. It defines the maximum energy spread of the system that permits thermalization. And it defines the time scale $\hbar / Δ$ within which ${\rm Tr}(A ρ(t))$ approaches the value ${\rm Tr}(A ρ_{\rm eq})$.

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