论文标题
多纹状体符合纠缠：非共性扩展状态的关系
Multifractality meets entanglement: relation for non-ergodic extended states
论文作者
论文摘要
在这项工作中，我们通过推广Don N. Page [Phys [Phys。莱特牧师。 71，1291] {\ it稀疏}随机纯状态（S-RP）的情况。这些生活在$ n $的希尔伯特空间中的S-RP定义为归一化向量，只有$ n^d $（$ 0 \ le d \ le 1 $）随机非零元素。对于$ d = 1 $，页面使用的这些状态在无限温度下代表厄戈迪克状态。但是，以$ 0 <d <1 $的价格，S-RPS是非凝治和分形的，因为它们以整个希尔伯特空间的消失比率$ n^d/n $限制。从分析和数字上，我们都表明，平均纠缠熵$ {\ MATHCAL {s} _1}（a）子系统$ a $，带有Hilbert Space Dimension $ n_a $，将其缩放为$ \ overline {\ Mathcal {\ Mathcal {s} _1} $ sim $ sim $ n $ n. n_a $。值得注意的是，$ \ OVERLINE {\ MATHCAL {s} _1}（a）$在无限温度下的热（页）值饱和，$ \ overline {\ Mathcal {\ Mathcal {s} _1}（a）\ sim \ sim \ sim \ sim \ sim \ ln n_a $在较大的$ d $中。因此，即使波动函数高度非共进，我们也提供一个例子，即使纠缠熵具有沿着沿着的值。最后，我们将结果推广到Renyi内属$ \ Mathcal {s} _Q（a）$，$ q>> 1 $，并为真实的多重法状状态概括，并且还表明它们的波动在Ergodic State的较窄范围内具有甲基殖民地的行为，$ d = 1 $。
In this work we establish a relation between entanglement entropy and fractal dimension $D$ of generic many-body wave functions, by generalizing the result of Don N. Page [Phys. Rev. Lett. 71, 1291] to the case of {\it sparse} random pure states (S-RPS). These S-RPS living in a Hilbert space of size $N$ are defined as normalized vectors with only $N^D$ ($0 \le D \le 1$) random non-zero elements. For $D=1$ these states used by Page represent ergodic states at infinite temperature. However, for $0<D<1$ the S-RPS are non-ergodic and fractal as they are confined in a vanishing ratio $N^D/N$ of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy ${\mathcal{S}_1}(A)$ of a subsystem $A$, with Hilbert space dimension $N_A$, scales as $\overline{\mathcal{S}_1}(A)\sim D\ln N$ for small fractal dimensions $D$, $N^D< N_A$. Remarkably, $\overline{\mathcal{S}_1}(A)$ saturates at its thermal (Page) value at infinite temperature, $\overline{\mathcal{S}_1}(A)\sim \ln N_A$ at larger $D$. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly non-ergodic. Finally, we generalize our results to Renyi entropies $\mathcal{S}_q(A)$ with $q>1$ and to genuine multifractal states and also show that their fluctuations have ergodic behavior in narrower vicinity of the ergodic state, $D=1$.
