Condensed Matter

Non-Hermiticity can destroy Anderson localization and lead to delocalization even in one dimension. However, the unified understanding of the non-Hermitian delocalization has yet to be established. Here, we develop a scaling theory of localization in non-Hermitian systems. We reveal that non-Hermiticity introduces a new scale and breaks down the one-parameter scaling, which is the central assumption of the conventional scaling theory of localization. Instead, we identify the origin of the unconventional non-Hermitian delocalization as the two-parameter scaling. Furthermore, we establish the threefold universality of non-Hermitian localization based on reciprocity; reciprocity forbids the delocalization without internal degrees of freedom, whereas symplectic reciprocity results in a new type of symmetry-protected delocalization.

Recently, progress has been made in the understanding of anomalous vibrational excitations in amorphous solids. In the lowest-frequency region, the vibrational spectrum follows a non-Debye quartic law, which persists up to zero frequency without any frequency gap. This gapless vibrational density of states (vDOS) suggests that glasses are on the verge of instability. This feature of marginal stability is now highlighted as a key concept in the theories of glasses. In particular, the elasticity theory based on marginal stability predicts the gapless vDOS. However, this theory yields a quadratic law and \textit{not} the quartic law. To address this inconsistency, we presented a new type of instability, which is different from the conventional one, and proposed that amorphous solids are marginally stable considering the new instability in the preceding study~[M. Shimada, H. Mizuno, and A. Ikeda, Soft Matter, {\bf 16}, 7279, 2020]. In this study, we further extend and detail the results for these instabilities. By analyzing various examples of disorder, we demonstrate that real glasses in finite spatial dimensions can be marginally stable by the proposed novel instability.

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of ℓ -bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only ±1 eigenvalues). While MBL and ℓ -bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary ℓ -bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the ℓ -bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality ℓ -bits at large disorder strength and rapid qualitative changes in the distributions of ℓ -bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models which further validates the evidence of MBL phenomenology in the other two and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.

The integer quantum Hall transition (IQHT) is one of the most mysterious members of the family of Anderson transitions. Since the 1980s, the scaling behavior near the IQHT has been vigorously studied in experiments and numerical simulations. Despite all efforts, it is notoriously difficult to pin down the precise values of critical exponents, which seem to vary with model details and thus challenge the principle of universality. Recently, M. Zirnbauer\citep{Zirnbauer2019} [Nucl. Phys. B \textbf{941}, 458 (2019)] has conjectured a conformal field theory for the transition, in which linear terms in the beta-functions vanish, leading to a very slow flow in the fixed point's vicinity which we term marginal scaling. In this work, we provide numerical evidence for such a scenario by using extensive simulations of various network models of the IQHT at unprecedented length scales. At criticality, we show that the finite-size scaling of the disorder averaged longitudinal Landauer conductance is consistent with its recently predicted fixed-point value and a third-order expansion of RG beta functions. In the future, our numerical findings can be checked with analytical results from the conformal field theory. Away from criticality we describe a mechanism that could account for the emergence of an \emph{effective} critical exponents ν eff , which is necessarily dependent on the parameters of the model. We further support this idea by numerical determination of ν eff in suitably chosen models.

We present a numerical model that simulates the current-voltage (I-V) characteristics of materials that exhibit percolation conduction. The model consists of a two dimensional grid of exponentially different resistors in the presence of an external electric field. We obtained exponentially non-ohmic I-V characteristics validating earlier analytical predictions and consistent with multiple experimental observations of the Poole-Frenkel laws in non-crystalline materials. The exponents are linear in voltage for samples smaller than the correlation length of percolation cluster L, and square root in voltage for samples larger than L.

We study the spectrum of generalized Wishart matrices, defined as F=(X Y ⊤ +Y X ⊤ )/2T , where X and Y are N×T matrices with zero mean, unit variance IID entries and such that E[ X it Y jt ]=c δ i,j . The limit c=1 corresponds to the Marcenko-Pastur problem. For a general c , we show that the Stietjes transform of F is the solution of a cubic equation. In the limit c=0 , T≫N the density of eigenvalues converges to the Wigner semi-circle.

The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a non-ergodic 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 wavefunctions, and a postulated correlated mini-band 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 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 mini-bands.

The growth of the spin-glass correlation length has been measured as a function of the waiting time t w on a single crystal of CuMn (6 at.\%), reaching values ξ∼150 nm, larger than any other glassy correlation-length measured to date. We find an aging rate dln t w /dlnξ larger than found in previous measurements, which evinces a dynamic slowing-down as ξ grows. Our measured aging rate is compared with simulation results by the Janus collaboration. After critical effects are taken into account, we find excellent agreement with the Janus data.

We consider the additive superimposition of an extensive number of independent Euclidean Random Matrices in the high-density regime. The resolvent is computed with techniques from free probability theory, as well as with the replica method of statistical physics of disordered systems. Results for the spectrum and eigenmodes are shown for a few applications relevant to computational neuroscience, and are corroborated by numerical simulations.

For many systems with quenched disorder the study of ground states can crucially contribute to a thorough understanding of the physics at play, be it for the critical behavior if that is governed by a zero-temperature fixed point or for uncovering properties of the ordered phase. While ground states can in principle be computed using general-purpose optimization algorithms such as simulated annealing or genetic algorithms, it is often much more efficient to use exact or approximate techniques specifically tailored to the problem at hand. For certain systems with discrete degrees of freedom such as the random-field Ising model, there are polynomial-time methods to compute exact ground states. But even as the number of states increases beyond two as in the random-field Potts model, the problem becomes NP hard and one cannot hope to find exact ground states for relevant system sizes. Here, we compare a number of approximate techniques for this problem and evaluate their performance.