Non-Hermitian Many-Body Localization

Observation of the non-Hermitian skin effect and Fermi skin on a digital quantum computer

Lately, the non-Hermitian skin effect (NHSE) has been demonstrated in various classical metamaterials and even ultracold atomic arrays. Yet, its interplay with many-body dynamics have never been experimentally investigated. Here, we report the observation of the NHSE and its many-fermion analog on a universal quantum processor. To implement NHSE accumulation on a quantum computer, the time-evolution circuit not only needs to be non-reciprocal and non-unitary, but must also contain sufficiently many lattice qubits. We demonstrate this by systematically post-selecting ancilla qubits, as demonstrated through two paradigmatic non-reciprocal models on noisy quantum processors, with clear signatures of asymmetric spatial propagation and many-body "Fermi skin" accumulation. To minimize errors from inevitable device noise, time evolution is performed using trainable, variationally optimized quantum circuits. Our demonstration represents an important step in the quantum simulation of non-Hermitian lattices on present-day quantum hardware, and can be readily generalized to more sophisticated many-body models.

Many-body localization in the age of classical computing. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2025;88:026502. [PMID: 39591655 DOI: 10.1088/1361-6633/ad9756] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBLregimegive rise to an MBLphase, in which the thermalization does not occur even in theasymptoticlimit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.

Breakdown of the Quantum Distinction of Regular and Chaotic Classical Dynamics in Dissipative Systems

Quantum chaos has recently received increasing attention due to its relationship with experimental and theoretical studies of nonequilibrium quantum dynamics, thermalization, and the scrambling of quantum information. In an isolated system, quantum chaos refers to properties of the spectrum that emerge when the classical counterpart of the system is chaotic. However, despite experimental progress leading to longer coherence times, interactions with an environment can never be neglected, which calls for a definition of quantum chaos in dissipative systems. Advances in this direction were brought by the Grobe-Haake-Sommers (GHS) conjecture, which connects chaos in a dissipative classical system with cubic repulsion of the eigenvalues of the quantum counterpart and regularity with linear level repulsion. Here, we show that the GHS conjecture does not hold for the open Dicke model, which is a spin-boson model of experimental interest. We show that the onset of cubic level repulsion in the open quantum model is not always related with chaotic structures in the classical limit. This result challenges the universality of the GHS conjecture and raises the question of what is the source of spectral correlations in open quantum systems.

Enhanced Many-Body Quantum Scars from the Non-Hermitian Fock Skin Effect

In contrast with extended Bloch waves, a single particle can become spatially localized due to the so-called skin effect originating from non-Hermitian pumping. Here we show that in kinetically constrained many-body systems, the skin effect can instead manifest as dynamical amplification within the Fock space, beyond the intuitively expected and previously studied particle localization and clustering. We exemplify this non-Hermitian Fock skin effect in an asymmetric version of the PXP model and show that it gives rise to ergodicity-breaking eigenstates-the non-Hermitian analogs of quantum many-body scars. A distinguishing feature of these non-Hermitian scars is their enhanced robustness against external disorders. We propose an experimental realization of the non-Hermitian scar enhancement in a tilted Bose-Hubbard optical lattice with laser-induced loss. Additionally, we implement digital simulations of such scar enhancement on the IBM quantum processor. Our results show that the Fock skin effect provides a powerful tool for creating robust nonergodic states in generic open quantum systems.

General Criterion for Non-Hermitian Skin Effects and Application: Fock Space Skin Effects in Many-Body Systems

Non-Hermiticity enables macroscopic accumulation of bulk states, named non-Hermitian skin effects. The non-Hermitian skin effects are well established for single-particle systems, but their proper characterization for general systems is elusive. Here, we propose a general criterion of non-Hermitian skin effects, which works for any finite-dimensional system evolved by a linear operator. The applicable systems include many-body systems and network systems. A system meeting the criterion exhibits enhanced non-normality of the evolution operator, accompanied by exceptional characteristics intrinsic to non-Hermitian systems. Applying the criterion, we discover a new type of non-Hermitian skin effect in many-body systems, which we dub the Fock space skin effect. We also discuss the Fock space skin effect-induced slow dynamics, which gives an experimental signal for the Fock space skin effect.

Dynamical Transition Due to Feedback-Induced Skin Effect

The traditional dynamical phase transition refers to the appearance of singularities in an observable with respect to a control parameter for a late-time state or singularities in the rate function of the Loschmidt echo with respect to time. Here, we study the many-body dynamics in a continuously monitored free fermion system with conditional feedback under open boundary conditions. We surprisingly find a novel dynamical transition from a logarithmic scaling of the entanglement entropy to an area-law scaling as time evolves. The transition, which is noticeably different from the conventional dynamical phase transition, arises from the competition between the bulk dynamics and boundary skin effects. In addition, we find that while quasidisorder or disorder cannot drive a transition for the steady state, a transition occurs for the maximum entanglement entropy during the time evolution, which agrees well with the entanglement transition for the steady state of the dynamics under periodic boundary conditions.

Space-time symmetry and nonreciprocal parametric resonance in mechanical systems

Linear mechanical systems with time-modulated parameters can harbor oscillations with amplitudes that grow or decay exponentially with time due to the phenomenon of parametric resonance. While the resonance properties of individual oscillators are well understood, those of systems of coupled oscillators remain challenging to characterize. Here we determine the parametric resonance conditions for time-modulated mechanical systems by exploiting the internal symmetries arising from the real-valued and symplectic nature of classical mechanics. We also determine how these conditions are further constrained when the system exhibits external symmetries. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the dynamics of the system than the Floquet operator does and use it to derive conditions for one-way amplification of traveling waves. Our exact theoretical framework based on symmetries enables the design of exotic responses such as nonreciprocal transport and one-way amplification in dynamic mechanical metamaterials and is generalizable to all physical systems that obey space-time symmetry.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Non-Hermitian Strongly Interacting Dirac Fermions

Exotic quantum phases and phase transition in the strongly interacting Dirac systems have attracted tremendous interests. On the other hand, non-Hermitian physics, usually associated with dissipation arising from the coupling to environment, emerges as a frontier of modern physics in recent years. In this Letter, we investigate the interplay between non-Hermitian physics and strong correlation in Dirac-fermion systems. We generalize the projector quantum Monte-Carlo (PQMC) algorithm to the non-Hermitian interacting fermionic systems. Employing PQMC simulation, we decipher the ground-state phase diagram of the honeycomb Hubbard model with spin resolved non-Hermitian asymmetric hopping processes. The antiferromagnetic (AFM) ordering induced by Hubbard interaction is enhanced by the non-Hermitian asymmetric hopping. Combining PQMC simulation and renormalization group analysis, we reveal that the quantum phase transition between Dirac semi-metal and AFM phases belongs to Hermitian chiral XY universality class, implying that a Hermitian Gross-Neveu transition is emergent at the quantum critical point although the model is non-Hermitian.

Analysis of chaos and regularity in the open Dicke model

We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.

Spectral crossovers in non-Hermitian spin chains: Comparison with random matrix theory

We present a systematic investigation of the short-range spectral fluctuation properties of three non-Hermitian spin-chain Hamiltonians using complex spacing ratios (CSRs). Specifically, we focus on the non-Hermitian variants of the standard one-dimensional anisotropic XY model having intrinsic rotation-time (RT) symmetry that has been explored analytically by Zhang and Song [Phys. Rev. A 87, 012114 (2013)1050-294710.1103/PhysRevA.87.012114]. The corresponding Hermitian counterpart is also exactly solvable and has been widely employed as a toy model in several condensed matter physics problems. We show that the presence of a random field along the x direction together with the one along the z direction facilitates integrability and RT-symmetry breaking, leading to the emergence of quantum chaotic behavior. This is evidenced by a spectral crossover closely resembling the transition from Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two phenomenological random matrix models in this paper to examine 1D Poisson to GinUE and 2D Poisson to GinUE crossovers and the associated signatures in CSRs. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively. These crossovers reasonably capture spectral fluctuations observed in the spin-chain systems within a certain range of parameters.

Non-Hermitian Floquet Topological Matter-A Review

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird's-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble

Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems, and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality, and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.

Delocalization of interacting directed polymers on a periodic substrate: Localization length and critical exponents from non-Hermitian spectra

We study a classical model of thermally fluctuating polymers confined to two dimensions, experiencing a grooved periodic potential, and subject to pulling forces both along and transverse to the grooves. The equilibrium polymer conformations are described by a mapping to a quantum system with a non-Hermitian Hamiltonian and with fermionic statistics generated by noncrossing interactions among polymers. Using molecular dynamics simulations and analytical calculations, we identify a localized and a delocalized phase of the polymer conformations, separated by a delocalization transition which corresponds (in the quantum description) to the breakdown of a band insulator when driven by an imaginary vector potential. We calculate the average tilt of the many-body system, at arbitrary shear values and filling density of polymer chains, in terms of the complex-valued non-Hermitian band structure. We find the critical shear value, the localization length, and the critical exponent by which the shear modulus diverges in terms of the branch points (exceptional points) in the band structure at which the bandgap closes. We also investigate the combined effects of non-Hermitian delocalization and localization due to both periodicity and disorder, uncovering preliminary evidence that while disorder favors localization at high values, it encourages delocalization at lower values.

Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos

We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and nonunitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy scale (and timescale). Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems, respectively. For dissipative quantum chaotic systems, we show that the DSFF exhibits an exact rotational symmetry in its complex time argument τ. Analogous to the spectral form factor (SFF) behavior for Gaussian unitary ensemble, the DSFF for GinUE shows a "dip-ramp-plateau" behavior in |τ|: the DSFF initially decreases, increases at intermediate timescales, and saturates after a generalized Heisenberg time, which scales as the inverse mean level spacing. Remarkably, for large matrix size, the "ramp" of the DSFF for GinUE increases quadratically in |τ|, in contrast to the linear ramp in the SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that the DSFF takes a constant value, except for a region in complex time whose size and behavior depend on the eigenvalue density. Numerically, we verify the above claims and additionally show that the DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behavior, except for a region in the complex time plane of measure zero in the limit of large matrix size. As a physical example, we consider the quantum kicked top model with dissipation and show that it falls under the Ginibre universality class and Poisson as the "kick" is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices or Markov chains and show that these models again fall under the Ginibre universality class.

Exploration of the topological properties in a non-Hermitian long-range system

The asymmetrical long-range hopping amplitudes have a rich influence on the topological properties. Here, a non-Hermitian model including the long-range hopping amplitudes is constructed to explore those properties. It can be found that an extra topological invariantW= 2 emerges as a consequence of the long-range hopping amplitudes. Furthermore, we find that the phaseW= 2 can be directly characterized by the generalized Brillouin zone (GBZ) itself through the concept of the argument principle. Meanwhile, a gapless phase dubbed as topological semimetal phase can be induced by the asymmetrical long-range hopping. Moreover, the physical origin of the topological semimetal phase can be explained by the solutions of eigen-equation. It is also shown that the skin modes exist as long as the GBZ and the Brillouin zone differ from each other. These interesting phases may be realized in an electrical-circuit simulator.

Exceptional dynamical quantum phase transitions in periodically driven systems

Extending notions of phase transitions to nonequilibrium realm is a fundamental problem for statistical mechanics. While it was discovered that critical transitions occur even for transient states before relaxation as the singularity of a dynamical version of free energy, their nature is yet to be elusive. Here, we show that spontaneous symmetry breaking can occur at a short-time regime and causes universal dynamical quantum phase transitions in periodically driven unitary dynamics. Unlike conventional phase transitions, the relevant symmetry is antiunitary: its breaking is accompanied by a many-body exceptional point of a nonunitary operator obtained by space-time duality. Using a stroboscopic Ising model, we demonstrate the existence of distinct phases and unconventional singularity of dynamical free energy, whose signature can be accessed through quasilocal operators. Our results open up research for hitherto unknown phases in short-time regimes, where time serves as another pivotal parameter, with their hidden connection to nonunitary physics.

Nonunitary Scaling Theory of Non-Hermitian Localization

Non-Hermiticity can destroy Anderson localization and lead to delocalization even in one dimension. However, a unified understanding of 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 unconventional non-Hermitian delocalization as the two-parameter scaling. Furthermore, we establish the threefold universality of non-Hermitian localization based on reciprocity; reciprocity forbids delocalization without internal degrees of freedom, whereas symplectic reciprocity results in a new type of symmetry-protected delocalization.

Controlling wave fronts with tunable disordered non-Hermitian multilayers

Unique and flexible properties of non-Hermitian photonic systems attract ever-increasing attention via delivering a whole bunch of novel optical effects and allowing for efficient tuning light-matter interactions on nano- and microscales. Together with an increasing demand for the fast and spatially compact methods of light governing, this peculiar approach paves a broad avenue to novel optical applications. Here, unifying the approaches of disordered metamaterials and non-Hermitian photonics, we propose a conceptually new and simple architecture driven by disordered loss-gain multilayers and, therefore, providing a powerful tool to control both the passage time and the wave-front shape of incident light with different switching times. For the first time we show the possibility to switch on and off kink formation by changing the level of disorder in the case of adiabatically raising wave fronts. At the same time, we deliver flexible tuning of the output intensity by using the nonlinear effect of loss and gain saturation. Since the disorder strength in our system can be conveniently controlled with the power of the external pump, our approach can be considered as a basis for different active photonic devices.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.

Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems

Contrary to the conventional wisdom in Hermitian systems, a continuous quantum phase transition between gapped phases is shown to occur without closing the energy gap Δ in non-Hermitian quantum many-body systems. Here, the relevant length scale ξ≃v_{LR}/Δ diverges because of the breakdown of the Lieb-Robinson bound on the velocity (i.e., unboundedness of v_{LR}) rather than vanishing of the energy gap Δ. The susceptibility to a change in the system parameter exhibits a singularity due to nonorthogonality of eigenstates. As an illustrative example, we present an exactly solvable model by generalizing Kitaev's toric-code model to a non-Hermitian regime.

Pseudo-Yang-Lee Edge Singularity Critical Behavior in a Non-Hermitian Ising Model

The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

We study the transition between integrable and chaotic behavior in dissipative open quantum systems, exemplified by a boundary driven quantum spin chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance s is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature β∈[0,2]. Here, β=0 yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and β=2 equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalize the results of Grobe, Haake, and Sommers, who derived a universal cubic level repulsion for small spacings s. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at β=2. It holds for all three Ginibre ensembles of random matrices with independent real, complex, or quaternion matrix elements.