Multifractal Analysis of the Non-Hermitian Skin Effect: From Many-Body to Tree Models

Abstract

The non-Hermitian skin effect is an anomalous localization phenomenon induced by nonreciprocal dissipation and has attracted considerable attention in recent years both theoretically and experimentally. In this article, we review the multifractal aspects of the non-Hermitian skin effect. In particular, we discuss how the many-body skin effect exhibits multifractality in many-body Hilbert space, unlike the trivial Hilbert-space occupation of the single-particle skin effect on crystalline lattices. We further highlight that the many-body skin effect can coexist with random-matrix spectral statistics, in contrast to the multifractality associated with many-body localization, which typically accompanies the absence of ergodicity. We also introduce a solvable model on a Cayley tree as an effective description of the many-body Hilbert space, in which the multifractal dimensions can be obtained analytically. This review provides a unified perspective on multifractal structures in the non-Hermitian skin effect across single-particle, many-body, and tree models, and clarifies their distinctive relation to ergodicity in open quantum systems.

Figures (6)

• Figure 1: Multifractal scaling of the non-Hermitian spin chain in Eq. \ref{['eq: Ham']} ($t=1/\sqrt{2}$, $J=1$, $g=\left( 5+\sqrt{5}\right)/8$, $h = \left( 1+\sqrt{5}\right)/4$). (a, b) Eigenvalues $\left( \mathrm{Re}\,E/L, \mathrm{Im}\,E/L \right)$ scaled by the system length $L=15$ under the (a) periodic boundary conditions (PBC) and (b) open boundary conditions (OBC) ($\gamma = 0.8$). The color bars show the multifractal dimension $D_2$ for each right eigenstate. (c, d) Multifractal dimensions $D_2$ of individual right eigenstates as a function of $\mathrm{Re}\,E/L$ for the different system lengths $L$ under (c) PBC and (d) OBC ($\gamma = 0.8$). (e) Multifractal dimension $\langle{D_2}\rangle$ averaged over all right eigenstates as functions of non-Hermiticity $\gamma$ under both PBC (red dots) and OBC (blue dots). (f) $q$-dependence of average multifractal dimension $\langle{D_q}\rangle$ under both PBC (red dots) and OBC (blue dots) ($\gamma=0.4$). Figure \ref{['fig: Hamiltonian']} adapted with permission from Ref. SH-Many. https://doi.org/10.1103/PhysRevB.111.035144
• Figure 2: Level-spacing-ratio statistics of the non-Hermitian spin chain in Eq. \ref{['eq: Ham']} under PBC (red dots) and OBC (blue dots) ($t = 1/\sqrt{2}$, $\gamma = 0.6$, $J=1$, $g = (5+\sqrt{5})/8$, $h=(1+\sqrt{5})/4$, $L=14$). All data are taken away from the spectral edges and the symmetry line, and are averaged over $50$ disorder realizations. (a) Spacing-ratio distribution $r$ of singular values. The averages are $\langle r \rangle = 0.5297$ for PBC and $\langle r \rangle = 0.5299$ for OBC. The black dashed curve shows the analytical result for small non-Hermitian random matrices in class AI, namely $p(r)=27(r+r^2)/[4(1+r+r^2)^{5/2}]$, with $\langle r \rangle = 4-2\sqrt{3} \simeq 0.5359$Kawabata-23SVD. (b, c) Level-spacing-ratio statistics of complex eigenvalues, shown for (b) the absolute value $|z|$ and (c) the argument $\arg z$. The averages are $\langle |z| \rangle = 0.7275$ and $\langle \cos \arg z \rangle = -0.1842$ for PBC, and $\langle |z| \rangle = 0.7365$ and $\langle \cos \arg z \rangle = -0.2355$ for OBC. For comparison, $10^4 \times 10^4$ non-Hermitian random matrices yield $\langle |z| \rangle = 0.7381$ and $\langle \cos \arg z \rangle = -0.2405$Sa-20. Figure \ref{['fig: SVD']} adapted with permission from Ref. SH-Many. https://doi.org/10.1103/PhysRevB.111.035144
• Figure 3: (a) Sketch of the model with nonreciprocal hopping on the Cayley tree for branching number $K=2$. The number of layers is $M=6$. The hopping amplitude from the boundary to the center (from the center to the boundary) is $t_{\rm L}$ ($t_{\rm R}$).(b) Phase diagram of the multifractal dimension $D_q$. For $K\ge 2$, the symmetric eigenstates are localized in the limit $\beta\to0$, delocalized for $\beta>\sqrt{K}$, and multifractal in the intermediate regime $0<\beta<\sqrt{K}$. (c) Sketch of the symmetric basis construction in the case $K=2$. The symmetric basis states are constructed by linear superposition of the position basis $\ket{l,j, m}$. Figure \ref{['fig:graph']} adapted with permission from Ref. SH-Cayley. https://doi.org/10.1103/PhysRevB.111.075162
• Figure 4: Dependence of the multifractal dimensions $D_q$ on $q$ in the three regimes: (a) $0<\beta<1$, (b) $1<\beta<\sqrt{K}$, and (c) $\beta>\sqrt{K}$. The curves are shown for $K=2$, with representative values $\beta=0.8$, $1.1$, and $2.0$ for panels (a), (b), and (c), respectively. Figure \ref{['fig:Dq']} adapted with permission from Ref. SH-Cayley. https://doi.org/10.1103/PhysRevB.111.075162
• Figure 5: The dependence on $q$ of the exponent $\tau_q$, and the dependence on $\alpha$ of the multifractal spectrum $f_\alpha$, displayed for $\beta = 0.8$ (a1, a2) and for $\beta = 1.1$ (b1, b2). The data are plotted for the case $K=2$. Figure \ref{['fig:fq']} adapted with permission from Ref. SH-Cayley. https://doi.org/10.1103/PhysRevB.111.075162