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arXiv:2112.11910 · Physical review. E

Nonergodic extended states in the β ensemble

Adway Kumar Das, Anandamohan Ghosh

Published May 12, 2022• 31 citations
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Abstract

Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the $\ensuremath{\beta}$ ensemble, known for its structural simplicity. However, eigenvector properties of the $\ensuremath{\beta}$ ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the $\ensuremath{\beta}$ ensemble and find that the Anderson transition occurs at $\ensuremath{\gamma}=1$ and ergodicity breaks down at $\ensuremath{\gamma}=0$ if we express the repulsion parameter as $\ensuremath{\beta}={N}^{\ensuremath{-}\ensuremath{\gamma}}$. Thus other than the Rosenzweig-Porter ensemble (RPE), the $\ensuremath{\beta}$ ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values $(0<\ensuremath{\gamma}<1)$. We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the $\ensuremath{\beta}$ ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the $\ensuremath{\beta}$ ensemble behave differently than the latter models.

DOI
10.1103/physreve.105.054121

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Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble, known for its structural simplicity. However, eigenvector properties of the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble and find that the Anderson transition occurs at \ensuremathγ=1\ensuremath{\gamma}=1\ensuremathγ=1 and ergodicity breaks down at \ensuremathγ=0\ensuremath{\gamma}=0\ensuremathγ=0 if we express the repulsion parameter as \ensuremathβ=N\ensuremath−\ensuremathγ\ensuremath{\beta}={N}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremathβ=N\ensuremath−\ensuremathγ. Thus other than the Rosenzweig-Porter ensemble (RPE), the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values (0<\ensuremathγ<1)(0<\ensuremath{\gamma}<1)(0<\ensuremathγ<1). We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the \ensuremathβ\ensuremath{\beta}\ensuremathβ ensemble behave differently than the latter models.
Motivated by the problem of Many-Body Localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two localization transitions as the parameter γγγ of the model varies from 0 to ∞\infty∞. One of them is the Anderson transition from the localized to the extended states that happens at γ=2γ=2γ=2. The other one at γ=1γ=1γ=1 is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality f(α)f(α)f(α) and the wave function overlap which all show the transitions at γ=1γ=1γ=1 and γ=2γ=2γ=2.
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a nonergodic extended phase between the ergodic extended metallic phase and the localized phase. Such a phase is relevant to many physical models from the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity to the interacting many-body localization in condensed-matter physics and quantum computing. This phase is characterized by fractal behavior of the wave functions and a postulated correlated miniband structure of the energy spectrum. Here we will seek evidence for the latter in the spectrum. Since this behavior is expected on intermediate energy scales, spectral rigidity or number variance is a natural way to tease it out. Nevertheless, due to the Thouless energy and ambiguities in the unfolding procedure, the results are inconclusive. On the other hand, by using the singular-value decomposition method, clear evidence for a super-Poissonian behavior in this regime emerges, consistent with a picture of correlated minibands.
In this paper, we consider an extension of the Rosenzweig-Porter model, the L\'evy-RP (L-RP) model, in which the off-diagonal matrix elements are broadly distributed, providing a more realistic benchmark to develop an effective description of nonergodic extended (NEE) states in interacting many-body disordered systems. We put forward a simple, general, and intuitive argument that allows one to unveil the multifractal structure of the minibands in the local spectrum when hybridization is due to anomalously large transition amplitudes in the tails of the distribution. The idea is that the energy spreading of the minibands can be determined self-consistently by requiring that the maximal hybridization rate Hij{\mathcal{H}}_{ij}Hij​ between a site iii and the other ND1{N}^{{D}_{1}}ND1​ sites of the support set is of the same order of the Thouless energy itself ${N}^{{D}_{1}\ensuremath{-}1}$. This argument yields the fractal dimensions that characterize the statistics of the multifractal wave functions in the NEE phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions are confirmed both analytically, by a thorough investigation of the self-consistent equation for the local density of states obtained using the cavity approach, and numerically, via extensive exact diagonalizations.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
In various chaotic quantum many-body systems, the ground states show nontrivial athermal behavior despite the bulk states exhibiting thermalization. Such athermal states play a crucial role in quantum information theory and its applications. Moreover, any generic quantum many-body system in the Krylov basis is represented by a tridiagonal Lanczos Hamiltonian, which is analogous to the matrices from the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$, a well-studied random-matrix model with level repulsion tunable via the parameter $\ensuremath{\beta}$. Motivated by this, here we focus on the localization properties of the ground and anti-ground states of the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$. Both analytically and numerically we show that both the edge states demonstrate nonergodic (fractal) properties for $\ensuremath{\beta}\ensuremath{\sim}\mathcal{O}(1)$, while the typical bulk states are ergodic. Surprisingly, the fractal dimension of the edge states remains three times smaller than that of the bulk states irrespective of the global phase of the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$. In addition to the fractal dimensions, we also consider the distribution of the localization centers of the spectral edge states, their mutual separation, as well as the spatial and correlation properties of the first excited states.