The consistency between eigenvalues calculated under open and periodic boundary conditions, named as {\it bulk-bulk correspondence} ($\mathcal{BBC}$), can be destroyed in systems with non-Hermitian skin effect (NHSE). In spite of the great success of the generalized Brillouin zone (GBZ) theory in clean non-Hermitian systems, the applicability of GBZ theory is questionable when the translational symmetry is broken. Thus, it is of great value to rebuild the $\mathcal{BBC}$ for disorder samples, which extends the application of GBZ theory in non-Hermitian systems. Here, we propose a scheme reconstructing $\mathcal{BBC}$, which can be regarded as the solution of an optimization problem. By solving this optimization problem analytically, we reconstruct the $\mathcal{BBC}$ and obtain the modified GBZ theory in several prototypical disordered non-Hermitian models. The modified GBZ theory gives a precise description of NHSE, which predicts the intriguing disorder-enhanced and disorder-irrelevant NHSEs.
The critical exponents of continuous phase transitions of a Hermitian system depend on and only on its dimensionality and symmetries. This is the celebrated notion of the universality of continuous phase transitions. Here we report the superuniversality notion of Anderson localization transitions in non-Hermitian two-dimensional (2D) systems with exceptional points in which the critical exponents do not depend on the symmetries. The Anderson localization transitions are numerically studied by using the finite-size scaling analysis of the participation ratios. At the exceptional points of eithe…
The correspondence between the real-complex transition in energy and delocalization-localization transition is well-established in a class of Aubry-Andr'e-Harper model with exponential non-Hermitian on-site potentials. In this paper, we study a generalized Aubry-Andr'e model with off-diagonal modulation and non-Hermitian on-site potential. We find that, when there exists an incommensurate off-diagonal modulation, the correspondence breaks down, although the extended phase is maintained in a wide parameter range of the strengths of the on-site potential and the off-diagonal hoppings. An additio…
We investigate the propagation of Gaussian beams through optical waveguide lattices characterized by correlated non-Hermitian disorder. In the framework of coupled mode theory, we demonstrate how the imaginary part of the refractive index needs to be adjusted to achieve perfect beam transmission, despite the presence of disorder. Remarkably, the effects of both diagonal and off-diagonal disorder in the waveguides and their couplings can be efficiently eliminated by our non-Hermitian design. Waveguide arrays thus provide an ideal platform for the experimental realization of non-Hermitian phenom…
We employ the fermionic and bosonic replicated nonlinear sigma models to treat Ginibre unitary, symplectic, and orthogonal ensembles of non-Hermitian random matrix Hamiltonians. Using saddle point approach combined with Borel resummation procedure we derive the exact large-N results for microscopic density of states in all three ensembles. We also obtain tails of the density of states as well the two-point function for the unitary ensemble.
Non-Hermiticity and dephasing, collaborating in an unusual wave packet dynamics, realizes unconventional entanglement evolution in a disordered, interacting and asymmetric (non-reciprocal) quantum medium. Taking the Hatano-Nelson model as a concrete example, we first consider how wave packet spreads in a non-Hermitian disordered system for demonstraing that it is very different from the Hermitian case. Interestingly, a cascade like wave packet spreading as in the Hermitian case is suppressed in the clean limit and at weak disorder, while it revives in the vicinity of the localization-delocaliz…
We show that Dirac fermion systems in two dimensions generally exhibit disorder-induced nodal arc replacing the nodal point and tilted Dirac cone, provided that the two components of the Dirac fermion correspond to two distinct orbitals unrelated by symmetry. This result is explicitly demonstrated using renormalization group analysis in a disordered Dirac model that we introduce, where the disorder potential acts differently on the two orbitals. As we show by numerical simulations and self-consistent Born approximation calculation, this drives the system into a new strongly disordered phase.
Asymptotically exact solution of the non-Hermitian disordered interacting Hatano-Nelson chain
Gleb Wataghin Physics Institute, University of Campinas (UNICAMP)（坎皮纳斯大学格莱布·瓦塔金物理研究所）；São Carlos Institute of Physics, University of São Paulo, IFSC – USP（圣保罗大学若昂·卡洛斯·塞斯研究所）
Unlike their Hermitian counterparts, non-Hermitian (NH) systems may display an exponential sensitivity to boundary conditions and an extensive number of edge-localized states in systems with open boundaries, a phenomena dubbed the "non-Hermitian skin effect." The NH skin effect is one of the primary challenges to defining a topological theory of NH Hamiltonians, as the sensitivity to boundary conditions invalidates the traditional bulk-boundary correspondence. The NH skin effect has recently been connected to the winding number, a topological invariant unique to NH systems. In this paper, we e…
Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two…
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…
Skin-Anderson localization transitions in disordered hybrid-nonreciprocal systems
Tianjin University（天津大学）；Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Joint School of National University of Singapore and Tianjin University, International Campus of Tianjin University（新加坡国立大学与天津大学联合学院，天津大学国际校区）
There is a common belief in the condensed matter community that bulk quantities become insensitive to the boundary condition in the infinite-volume limit. Here we reconsider this statement in terms of recent arguments of non-Hermitian skin effects, -strong dependence of spectra on boundary conditions for the non-Hermitian Hamiltonians-, in the traditional Green's function formalism. We find the criterion for quantities to be sensitive/insensitive against the boundary condition in Hermitian correlated/disordered systems, which is characterized by the residue theorem. We also discuss the uncerta…
A statistical field theory is developed to explore the density of states and spatial profile of `tail states' at the edge of the spectral support of a general class of disordered non-Hermitian operators. These states, which are identified with symmetry broken, instanton field configurations of the theory, are closely related to localized sub-gap states recently identified in disordered superconductors. By focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential, and a random imaginary vector potential, we discuss the methodology of our approa…
We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schroedinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of the information transmission through non-linear optical fibers.
Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields
Sanli Honors College, School of Physics and Electronics Engineering, Shanxi University（山西大学物理与电子工程学院三立荣誉学院）；Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所量子光学技术与器件国家重点实验室）；State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Laser Spectroscopy, Shanxi University（山西大学激光光谱研究所量子光学技术与器件国家重点实验室）
The nearest-neighbor level-spacing distributions are a fundamental quantity of disordered systems and universal. It is well-known that extended and localized states of random Hermitian systems follow the Wigner-Dyson and the Poison distributions, respectively, while the Ginibre distributions describe random non-Hermitian systems with complex eigenvalues. However, the level distribution of systems of neither Hermitian nor non-Hermitian with full complex eigenvalues is still unknown. Here we show a new class of universal level distributions in the vicinity of exceptional points of non-Hermitian …
Using the properties of random Möbius transformations, we investigate the statistical properties of the reflection coefficient in a random chain of lossy scatterers. We explicitly determine the support of the distribution and the condition for coherent perfect absorption to be possible. We show that at its boundaries the distribution has Lifshits-like tails, which we evaluate. We also obtain the extent of penetration of incoming waves into the medium via the Lyapunov exponent. Our results agree well when compared to numerical simulations in a specific random system.
Using exact diagonalization for non-interacting systems and density matrix renormalization group for interacting systems we show that Li and Haldane's conjecture on the correspondence between the low-lying many-particle excitation spectrum and the entanglement spectrum holds for disordered ballistic one-dimensional many-particle systems. In order to demonstrate the correspondence we develop a computational efficient way to calculate the ES of low-excitation of non-interacting systems. We observe and explain the presence of an unexpected shell structure in the excitation structure. The low-lyin…
The interplay among topology, disorder, and non-Hermiticity can induce some exotic topological and localization phenomena. Here we investigate this interplay in a two-dimensional non-Hermitian disordered Chern-insulator model with two typical kinds of non-Hermiticities, the nonreciprocal hopping and on-site gain-and-loss effects. The topological phase diagrams are obtained by numerically calculating two topological invariants in the real space, which are the disorder-averaged open-bulk Chern number and the generalized Bott index, respectively. We reveal that the nonreciprocal hopping (the gain…
In this work, we study the disorder effects on the bulk-boundary correspondence of two-dimensional higher-order topological insulators (HOTIs). We concentrate on two cases: (i) bulk-corner correspondence, (ii) edge-corner correspondence. For the bulk-corner correspondence case, we demonstrate the existence of the mobility gaps and clarify the related topological invariant that characterizes the mobility gap. Furthermore, we find that, while the system preserves the bulk-corner correspondence in the presence of disorder, the corner states are protected by the mobility gap instead of the bulk ga…
Non-Hermiticity enriches the 10-fold Altland-Zirnbauer symmetry class into the 38-fold symmetry class, where critical behavior of the Anderson transitions (ATs) has been extensively studied recently. Here, we propose a correspondence of the universality classes of the ATs between Hermitian and non-Hermitian systems. We illustrate that the critical exponents of the length scale in non-Hermitian systems coincide with the critical exponents in the corresponding Hermitian systems with additional chiral symmetry. A remarkable consequence of the correspondence is superuniversality, i.e., the ATs in …
Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational…
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 demonstrate a new design principle for unidirectionally invisible non-Hermitian structures that are not only invisible for one specific wavelength but rather for a broad frequency range. Our idea is based on the concept of constant-intensity waves, which can propagate even through highly disordered media without back-scattering or intensity variations. Contrary to already existing invisibility studies, our new design principle requires neither a specific symmetry (like $\mathcal{PT}$-symmetry) nor periodicity, and can thus be applied in a much wider context. This generality combined with br…
Non-Hermitian systems can exhibit unique topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems by studying a non-Hermitian disordered Su-Schrieffer-Heeger model with nonreciprocal hoppings. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing energy gaps. Moreover, we uncover a topological phase which emerges only under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge mode…
Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary cont…
Models of disorder with a direction (constant imaginary vector-potential) are considered. These non-Hermitian models can appear as a result of computation for models of statistical physics using transfer matrix technique or describe non-equilibrium processes. Eigenenergies of non-Hermitian Hamiltonians are not necessarily real and a joint probability density function of complex eigenvalues can characterize basic properties of the systems. This function is studied using the supersymmetry technique and a supermatrix $σ$-model is derived. The $σ$-model differs from already known by a new term. Th…
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 …
Dynamic and steady state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium, the Wigner time delay, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among the internal field, transmission, transmission time, dwell time, total excited energy, and the density of states in with …
In this paper, a one-dimensional non-Hermitian quasiperiodic $p$-wave superconductor without $\mathcal{PT}$-symmetry is studied. By analyzing the spectrum, we discovered there still exists real-complex energy transition even if the inexistence of $\mathcal{PT}$-symmetry breaking. By the inverse participation ratio, we constructed such a correspondence that pure real energies correspond to the extended states and complex energies correspond to the localized states, and this correspondence is precise and effective to detect the mobility edges. After investigating the topological properties, we a…
Recent studies of disorder or non-Hermiticity induced topological insulators inject new ingredients for engineering topological matter. Here we consider the effect of purely non-Hermitian disorders, a combination of these two ingredients, in a 1D coupled-cavity array with disordered gain and loss. Topological photonic states can be induced by increasing gain-loss disorder strength with topological invariants carried by localized states in the complex bulk spectra. The system showcases rich phase diagrams and distinct topological states from Hermitian disorders. The non-Hermitian critical behav…
Complex networks with directed, local interactions are ubiquitous in nature, and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context, and are key to describing statistical properties of the non-equilibrium dynamics that emerges from interactions within the network structure. Here, we study one-dimensional (1d) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point e…
