One challenge of studying the many-body localization transition is defining the length scale that diverges upon the transition to the ergodic phase. In this manuscript we explore the localization properties of a ring with onsite disorder subject to an imaginary magnetic flux. We connect the imaginary flux which delocalizes single-particle orbitals of an Anderson-localized ring with the localization length of an open chain. We thus identify the delocalizing imaginary flux per site with the inverse localization length. We put this intuition to use by exploring the phase diagram of a disordered interacting chain, and find that the inverse imaginary flux per bond provides an accessible description of the transition and its diverging localization length.
We show the emergence of light localization in arrays of coupled optical waveguides with randomness only in the imaginary part of their permittivity and develop a one-parameter scaling theory for the normalized participation number of the Floquet-Bloch modes. This localization introduces a new length scale in the decay of the autocorrelation function of a paraxial beam propagation. Our results are relevant to a vast family of systems with randomness in the dissipative part of their impedance spatial profile.
We take non-Hermitian Aubry-André-Harper models and quasiperiodic Kitaev chains as examples to demonstrate the equivalence and superposition of real and imaginary quasiperiodic potentials (QPs) on inducing localization of single-particle states. We prove this equivalence by analytically computing Lyapunov exponents (or inverse of localization lengths) for systems with purely real and purely imaginary QPs. Moreover, when superposed and with the same frequency, real and imaginary QPs are coherent on inducing the localization, under a way which is determined by the relative phase between them. Th…
In this and subsequent papers, one dimensional system of Dirac fermions with a random-varying mass is studied by the transfer-matrix methods which we developed recently. We investigate the effects of nonlocal correlation of the spatial-varying Dirac mass on the delocalization transition. Especially we numerically calculate both the "typical" and "mean" localization lengths as a function of energy and the correlation length of the random mass. To this end we introduce an imaginary vector potential as suggested by Hatano and Nelson and solve the eigenvalue problem. Numerical calculations are in …
Uncorrelated disorder potential in one-dimensional lattice definitely induces Anderson localization, while quasiperiodic potential can lead to both localized and extended phases, depending on the potential strength. We investigate the Anderson localization in one-dimensional lattice with nonHermitian complex disorder and quasiperiodic potential. We present a non-Hermitian SSH chain to demonstrate that a nonzero imaginary disorder on-site potential can induce the standard Anderson localization. We also show that the non-Hermitian Aubry-Andre (AA) model exhibits a transition in parameter space, …
We investigate the Anderson localization in non-Hermitian Aubry-André-Harper (AAH) models with imaginary potentials added to lattice sites to represent the physical gain and loss during the interacting processes between the system and environment. By checking the mean inverse participation ratio (MIPR) of the system, we find that different configurations of physical gain and loss have very different impacts on the localization phase transition in the system. In the case with balanced physical gain and loss added in an alternate way to the lattice sites, the critical region (in the case with p-…
When one applies a type of non-Hermitian effect, constant imaginary vector potential, to disordered systems, delocalization is induced even in two or lower dimension. By using the non-Hermitian induced transition as a probe, We propose a new procedure of estimating localization in arbitrary-dimensional systems. By examining numerically the two-dimensional non-Hermitian tight-biding model with onsite disorder, it is shown that the failure of absorbing the non-Hermitian effect, namely the breakdown of the imaginary gauge transformation, characterize the inverse localization length near the band …
For Anderson Localization models with multifractal eigenvectors on disordered samples containing $N$ sites, we analyze in a unified framework the consequences for the statistical properties of the Green function. We focus in particular on the imaginary part of the Green function at coinciding points $G^I_{xx}(E-i η)$ and study the scaling with the size $N$ of the moments of arbitrary indices $q$ when the broadening follows the scaling $η=\frac{c}{N^δ}$. For the standard scaling regime $δ=1$, we find in the two limits $c \ll 1$ and $c \gg 1$ that the moments are governed by the anomalous expone…
We study numerically the localization properties of eigenstates in a one-dimensional random lattice described by a non-Hermitian disordered Hamiltonian, where both the disorder and the non-Hermiticity are inserted simultaneously in the on-site potential. We calculate the averaged participation number, the Shannon entropy and the structural entropy as a function of other parameters. We show that, in the presence of an imaginary random potential, all eigenstates are localized in the thermodynamic limit and strong anomalous Anderson localization occurs at the band center. In contrast to the usual…
A class of one-dimensional lattice models with incommensurate complex potential $V(θ)=2[λ_r cos(θ)+i λ_i sin(θ)]$ is found to exhibit localization transition at $|λ_r|+|λ_i|=1$. This transition from extended to localized states manifests in the behavior of the complex eigenspectum. In the extended phase, states with real eigenenergies have finite measure and this measure goes to zero in the localized phase. Furthermore, all extended states exhibit real spectrum provided $|λ_r| \ge |λ_i|$. Another novel feature of the system is the fact that the imaginary part of the spectrum is sensitive to th…
We study localization properties of a one-dimensional disordered system characterized by a random non-hermitean hamiltonian where both the randomness and the non-hermiticity arises in the local site-potential; its real part being ordered (fixed), and a random imaginary part implying the presence of either a random absorption or amplification at each site. The transmittance (forward scattering) decays exponentially in either case. In contrast to the disorder in the real part of the potential (Anderson localization), the transmittance with the disordered imaginary part may decay slower than that…
In the previous paper, we studied the random-mass Dirac fermion in one dimension by using the transfer-matrix methods and by introducing an imaginary vector potential in order to calculate the localization lengths. Especially we considered effects of the nonlocal but short-range correlations of the random mass. In this paper, we shall study effects of the long-range correlations of the random mass especially on the delocalization transition. The results depend on how randomness is introduced in the Dirac mass.
We present a numerical study of Anderson localization in disordered non-Hermitian lattice models with flat bands. Specifically we consider one-dimensional stub and two-dimensional kagome lattices that have a random scalar potential and a uniform imaginary vector potential and calculate the spectra of the complex energy, the participation ratio, and the winding number as a function of the strength of the imaginary vector potential, $h$. The flat-band states are found to show a double transition from localized to delocalized and back to localized states with $h$, in contrast to the dispersive-ba…
We investigate localization properties in a two-coupled uniform chains with quasiperiodic modulation on interchain coupling strength. We demonstrate that this ladder is equivalent to a Aubry-Andre (AA) chain when two legs are symmetric. Analytical and numerical results indicate the appearance of mobility edges for asymmetric ladder. We also propose an easily engineered quasiperiodic ladder system which is a moiré superlattice system consisting of two-coupled uniform chains. An irrational lattice constant difference results in quasiperiodic structure. Numerical simulations show that such a syst…
Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this study we propose a finite-size scaling approach based on a convolutional neural network and analyze the critical behavior of a quantum Hall plateau transition. The localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We show that the general-purposed method can be used to extract critical e…
We study numerically the effects of short- and long-range correlations on the localization properties of the eigenstates in a one-dimensional disordered lattice characterized by a random non-Hermitian Hamiltonian, where the imaginary part of the on-site potential is random. We calculate participation number versus strengths of disorder and correlation. In the short-ranged case and when the correlation length is sufficiently small, we find that there exists a critical value of the disorder strength, below which enhancement and above which suppression of localization occurs as the correlation le…
The amplitude (Higgs) mode near the two-dimensional superfluid-Mott glass quantum phase transition is studied. We map the Bose-Hubbard Hamiltonian of disordered interacting bosons onto an equivalent classical XY model in (2+1) dimensions and compute the scalar susceptibility of the order parameter amplitude via Monte Carlo simulation. Analytic continuation of the scalar susceptibilities from imaginary to real frequency to obtain the spectral densities is performed by a modified maximum entropy technique. Our results show that the introduction of disorder into the system leads to unconventional…
Absence of localization is demonstrated analytically to leading order in weak disorder in a one-dimensional Anderson model of a ring threaded by an Aharonov-Bohm (A-B) flux. The result follows from adapting an earlier perturbation treatment of disorder in a superconducting ring subjected to an imaginary vector potential proportional to a depinning field for flux lines bound to random columnar defects parallel to the axis of the ring. The absence of localization in the ring threaded by an A-B flux for sufficiently weak disorder is compatible with large free electron type persistent current obta…
Many-body localization is shown to suppress imaginary parts of complex eigenenergies for general non-Hermitian Hamiltonians having time-reversal symmetry. We demonstrate that a real-complex transition, which we conjecture occurs upon many-body localization, profoundly affects the dynamical stability of non-Hermitian interacting systems with asymmetric hopping that respect time-reversal symmetry. Moreover, the real-complex transition is shown to be absent in non-Hermitian many-body systems with gain and/or loss that breaks time-reversal symmetry, even though the many-body localization transitio…
In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversal symmetry (TRS), will exhibit three types of phase transition: real-complex transition of eigenenergies, topological phase transition and many-body localization (MBL) phase transition. For thereal-complex transition, it is found that the imaginary parts of the eigenenergies are always suppressed by the MBL. Moreov…
We develop a novel and powerful method of exactly calculating various transport characteristics of waves in one-dimensional random media with (or without) coherent absorption or amplification. Using the method, we compute the probability densities of the reflectance and of the phase of the reflection coefficient, together with the localization length, of electromagnetic waves in sufficiently long random dielectric media. We find substantial differences between our exact results and the previous results obtained using the random phase approximation (RPA). The probabilty density of the phase of …
A recent development in studies of random non-Hermitian quantum systems is reviewed. Delocalization was found to occur under a sufficiently large constant imaginary vector potential even in one and two dimensions. The phenomenon has a physical realization as flux-line depinning in type-II superconductors. Relations between the delocalization transition and the complex energy spectrum of the non-Hermitian systems are described. Analytical and numerical results obtained for a non-Hermitian Anderson model are shown.
Arbitrarily Configurable Wavefunctions via Imaginary Gauge Phase Imprint in Non-Hermitian Lattices
South China Normal University（华南师范大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））；Hefei National Laboratory（合肥国家实验室）
We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions, and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imagina…
We study the Anderson model on the Bethe lattice by working directly with propagators at real energies $E$. We introduce a novel criterion for the localization-delocalization transition based on the stability of the population of the propagators, and show that it is consistent with the one obtained through the study of the imaginary part of the self-energy. We present an accurate numerical estimate of the transition point, as well as a concise proof of the asymptotic formula for the critical disorder on lattices of large connectivity, as given in [P.W. Anderson 1958]. We discuss how the forwar…
We study the influence of a strong imaginary vector potential on the quantum mechanics of particles confined to a two-dimensional plane and propagating in a random impurity potential. We show that the wavefunctions of the non-Hermitian operator can be obtained as the solution to a two-dimensional Dirac equation in the presence of a random gauge field. Consequences for the localization properties and the critical nature of the states are discussed.
We present a detailed discussion of scalar wave propagation and light intensity transport in three dimensional random dielectric media with optical gain. The intrinsic length and time scales of such amplifying systems are studied and comprehensively discussed as well as the threshold characteristics of single- and two-particle propagators. Our semi-analytical theory is based on a self-consistent Cooperon resummation, representing the repeated self-interference, and incorporates as well optical gain and absorption, modeled in a semi-analytical way by a finite imaginary part of the dielectric fu…
In the previous paper, we studied the random-mass Dirac fermion in one dimension by using the transfer-matrix methods. We furthermore employed the imaginary vector potential methods for calculating the localization lengths. Especially we investigated effects of the nonlocal but short-range correlations of the random mass. In this paper, we shall study effects of the long-range correlations of the random mass especially on the delocalization transition and singular behaviours at the band center. We calculate localization lengths and density of states for various nonlocally correlated random mas…
A key feature of the many-body localized phase is the breaking of ergodicity and consequently the emergence of local memory; revealed as the local preservation of information over time. As memory is necessarily a time dependent concept, it has been partially captured by a few extant studies of dynamical quantities. However, these quantities are neither optimal, nor democratic with respect to input state; and as such a fundamental and complete information theoretic understanding of local memory in the context of many-body localization remains elusive. We introduce the dynamical Holevo quantity …
We implement a recursive Green function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random onsite potential. We show that the typical value of the imaginary part of the local FS self-energy, Δ_t, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition; and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function o…
