论文标题
本征态热假说和本征态至元素的波动
Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations
论文作者
论文摘要
我们研究了本征态热假说〜（ETH）在不可融合和可集成的自旋$ 1/2 $ xxz链中有效或违反的程度。我们对矩阵元素的统计属性进行了能量分解分析$ \ {o_ {γα} \} $在能量特征基础上可观察的$ \ hat {o} $的$ \ hat {o} $。 Hilbert空间被粗粒固定在宽度$δ_e$的能量外壳中，通过它可以定义一个块subbatrix $ \ tilde {o}^{（b，a）} $，由$ a $ a $ a $ a $ a $ a $ a和$ b $ th shells组成的元素组成。每个块submatrix的特征是$ e_ {γα} =（e_γ+e_α）/2 \ simeq \ tilde {e} $和$ω__{γα} =（e_γ-e_α）\ simeqω$最高$δ_e$。我们将证明，在不可融合的情况下，一个块内的所有矩阵元素在统计学上彼此相等。它们的分布的特征是$ \ bar {e} $和$ω$，并遵循ETH的预测。与之形成鲜明对比的是，在可整合的情况下，本征态到元素的波动持续存在。因此，矩阵元素$ o_ {γα} $不能以能量参数为特征$ e_ {γα} $和$ω__{γα} $。我们的结果解释了可集成系统中波动耗散定理的崩溃的来源。特征态至欧根属态波动为ETH的含义带来了新的启示。
We investigate the extent to which the eigenstate thermalization hypothesis~(ETH) is valid or violated in the non-integrable and the integrable spin-$1/2$ XXZ chain. We perform the energy-resolved analysis of the statistical properties of matrix elements $\{O_{γα}\}$ of an observable $\hat{O}$ in the energy eigenstate basis. The Hilbert space is coarse-grained into energy shells of width $Δ_E$, with which one can define a block submatrix $\tilde{O}^{(b,a)}$ consisting of elements between eigenstates in the $a$th and $b$th shells. Each block submatrix is characterized by constant values of $E_{γα}=(E_γ+E_α)/2 \simeq \tilde{E}$ and $ω_{γα}= (E_γ-E_α) \simeq ω$ up to $Δ_E$. We will show that all matrix elements within a block are statistically equivalent to each other in the non-integrable case. Their distribution is characterized by $\bar{E}$ and $ω$, and follows the prediction of the ETH. In stark contrast, eigenstate-to-eigenstate fluctuations persist in the integrable case. Consequently, matrix elements $O_{γα}$ cannot be characterized by the energy parameters $E_{γα}$ and $ω_{γα}$ only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.
