Non-Hermitian Many-Body Localization

• NHMBL is defined as the localization of many-body eigenstates in non-Hermitian operators, revealing a complex interplay between real-complex spectral transitions and dynamical stability.
• It employs diagnostics like entanglement entropy, inverse participation ratios, and memory observables to distinguish localized from ergodic phases in systems with non-unitary dynamics.
• Microscopic realizations, such as the interacting Hatano–Nelson and Aubry–André models, demonstrate NHMBL even in fully complex spectra through asymmetric hopping and gain/loss mechanisms.

Non-Hermitian many-body localization (NHMBL) denotes many-body localization in the spectrum of a non-Hermitian operator, most commonly a non-Hermitian Hamiltonian but, in open-system settings, also a Lindbladian super-operator. It extends the MBL paradigm to dynamics with asymmetric hopping, gain/loss, monitoring-induced no-jump evolution, quasiperiodic driving, and other non-unitary mechanisms. Across these settings, NHMBL is associated with non-ergodic many-body eigenstructures, area-law or strongly suppressed entanglement, Poisson or complex Poisson spectral statistics, and persistent memory in dynamics; in time-reversal-symmetry-preserving systems it can additionally suppress imaginary parts of eigenenergies and induce a real-complex spectral transition, whereas time-reversal-symmetry-breaking systems can remain fully complex while still localizing (Hamazaki et al., 2018, Hamazaki et al., 2022).

1. Conceptual definition and symmetry structure

NHMBL is defined operationally through the localization of right eigenstates or eigenoperators of a non-Hermitian generator, together with the suppression of ergodic hybridization under interactions and quenched structure. In the foundational TRS-preserving asymmetric-hopping chain, many-body localization suppresses imaginary parts of complex eigenenergies and produces a real-complex transition, while in non-Hermitian many-body systems with gain and/or loss that break time-reversal symmetry the real-complex transition is absent even though the many-body localization transition still persists (Hamazaki et al., 2018).

A central refinement is that localization and spectral “realification” need not coincide. In the interacting Hatano–Nelson chain, the phase diagram for mid-spectrum eigenstates is consistent with a two-step approach to the localized regime: real eigenstates become stable at a lower disorder than the disorder at which the fraction of complex eigenvalues changes its system-size trend, creating an intermediate regime where localized eigenstates coexist with a substantial fraction of complex eigenvalues (Mák et al., 2023). A related but sharper statement arises in a TRS-broken Aubry–André-type chain with non-reciprocal hopping and staggered gain/loss: the many-body spectrum remains fully complex for any $V\neq 0$, yet the entanglement entropy, IPR, and imbalance still diagnose NHMBL, and the real part of the energy expectation becomes remarkably stable only in the localized phase. This shows that localization, rather than spectral reality by itself, controls that dynamical stability (Cheng et al., 2023).

The symmetry distinction therefore organizes much of the field. TRS-preserving models frequently exhibit paired complex eigenvalues and a disorder- or field-driven real-complex transition. TRS-breaking models do not require such a transition, and NHMBL there is better understood as localization within a fully complex spectrum. This suggests that “NHMBL” is not a single spectral scenario but a family of localization phenomena whose detailed spectral expression depends on symmetry class and on how non-Hermiticity enters.

2. Microscopic realizations

A canonical NHMBL Hamiltonian is the interacting Hatano–Nelson chain of hard-core bosons or spinless fermions,

$$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$

with non-reciprocity controlled by $g$, interaction $U$, and $W_j$ taken as random, quasiperiodic, or Stark-like. This form underlies the original disordered TRS-preserving NHMBL model, the quasiperiodic hard-core-boson chain, and the disorder-free Stark chain; under periodic boundary conditions the asymmetry cannot be gauged away, whereas under open boundaries an imaginary gauge transformation can eliminate it in several settings, strongly affecting the spectrum (Hamazaki et al., 2018, Zhai et al., 2020, Liu et al., 2023).

A distinct TRS-breaking class supplements non-reciprocity with staggered gain/loss. In the Aubry–André-type model

$$\hat{H}=\sum_{j=1}^{L}\Big[-J\big(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\big) +U\hat{n}_{j}\hat{n}_{j+1}+V_j\hat{n}_{j}\Big],$$

with

$V_{j}=V\cos(2\pi\alpha j+\varphi)-i\gamma(-1)^j,$

non-Hermiticity is controlled jointly by non-reciprocal hoppings and on-site gain/loss, and the model realizes NHMBL in a fully complex spectrum (Cheng et al., 2023).

Disorder-free NHMBL can also arise from deterministic fields or interactions. In the non-Hermitian Stark system, the onsite term is a linear tilt plus weak quadratic curvature,

$$W_j=-\gamma (j-1)+\beta\left(\frac{j-1}{L-1}\right)^2,$$

and increasing $\gamma$ drives Ginibre-to-GOE-to-Poisson spectral crossovers together with a Stark MBL transition (Liu et al., 2023). In a clean long-range chain,

$$\hat{H} = \sum_{i=1}^L \left( J e^{-g} \hat{b}^\dagger_{i+1} \hat{b}_i + J e^{g} \hat{b}_i^\dagger \hat{b}_{i+1} \right) + V \sum_{i<j}^L d^{-\alpha}_{ij} \hat{n}_i \hat{n}_j,$$

localization is induced purely by long-range interactions, and neither the nearest-neighbor limit nor the all-to-all limit supports the transitions found at intermediate $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$0 (Wang et al., 9 Oct 2025).

Driven NHMBL introduces a Floquet layer. In the driven Fibonacci chain, the evolution alternates between a non-Hermitian interacting Hatano–Nelson segment $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$1 and an onsite quasiperiodic potential $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$2, with Floquet operator

$$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$3

For $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$4, the onsite term reaches the Fibonacci limit, and the resulting Floquet spectrum has complex quasienergies whose real-versus-complex character tracks localization (Banerjee et al., 18 Sep 2025).

3. Diagnostics and phase identification

NHMBL is diagnosed through a suite of spectral, eigenstate, and dynamical observables adapted to non-Hermitian many-body problems. Core diagnostics include the fraction of complex eigenvalues $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$5, half-chain entanglement entropy of right eigenstates, inverse participation ratios in Fock space, many-body inverse participation ratios built from local densities, level-spacing statistics in the complex plane, and memory observables such as density or spin imbalance. Delocalized phases show Ginibre or GOE-like spectral correlations, volume-law entanglement, and small IPR; localized phases show Poisson or complex Poisson statistics, area-law entanglement, finite IPR, and persistent memory of charge-density-wave or Néel initial states (Hamazaki et al., 2018, Cheng et al., 2023, Zhai et al., 2020).

The diagnostics themselves are model-dependent. In the TRS-preserving asymmetric-hopping chain, the fraction of complex eigenenergies $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$6 and the stability measure

$$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$7

separate a delocalized phase with unstable real eigenstates from a localized phase with stable eigenstates and suppressed coalescence (Hamazaki et al., 2018). In the TRS-broken quasiperiodic chain, NHMBL is instead diagnosed by the crossing of the half-chain entropy $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$8, by the saturation of the many-body IPR, and by a spectral-statistics transition from the Ginibre distribution to the complex Poisson distribution near $$\hat{H}=\sum_{j}\left[-J\left(e^{-g}\hat{c}^{\dagger}_{j+1}\hat{c}_{j}+e^{g}\hat{c}^{\dagger}_{j}\hat{c}_{j+1}\right)+U\hat{n}_{j}\hat{n}_{j+1}+W_j\hat{n}_j\right],$$9 (Cheng et al., 2023).

A major methodological development is the use of singular value decomposition. SVD replaces the complex spectrum by real singular values and uses singular vectors as orthonormal states, enabling Hermitian-style spacing ratios, singular form factors, IPR, and entanglement entropy. In the XXZ chain with random local dissipation, singular-value statistics evolve from GOE-like to Poisson-like as dissipation increases, while singular-vector IPR and entanglement switch from delocalized, volume-law behavior to localized, area-law behavior (Roccati et al., 2023). However, SVD is not generically reliable for locating NHMBL transitions in TRS-preserving disordered models: in nonreciprocal hard-core-boson chains with quasiperiodic or random potentials, ED gives mutually consistent transition estimates whereas SVD systematically shifts the inferred critical disorder strength to larger values and can change the phase assignment, even though in the clean Stark model ED and SVD agree quantitatively (You et al., 7 Feb 2026).

More recently, Krylov and spread-complexity diagnostics have been adapted to NHMBL. Singular-value spread complexity distinguishes ergodic and MBL phases from the presaturation peak height for both TRS and non-TRS models, while the saturation value of thermofield-double-state complexity detects the real-complex transition and differentiates symmetry classes; charge-density-wave complexity distinguishes the TRS model sharply, but behaves qualitatively differently without TRS (Ganguli, 2024). Collectively, these developments indicate that NHMBL diagnostics are plural rather than universal: right-eigenstate, biorthogonal, singular-vector, and Krylov constructions all capture different aspects of the same transition.

4. Boundary sensitivity, skin effects, and driven or quasiperiodic extensions

Boundary sensitivity is a defining complication of NHMBL. In the two-leg non-Hermitian ladder with non-reciprocal tunneling and disorder, one complex-real spectral transition is disorder-driven and present for both open and periodic boundaries, while a second transition exists only under open boundaries and is driven by inter-chain coupling at weak disorder. Right-eigenstate localization measures are contaminated by the non-Hermitian skin effect, but biorthogonal IPR and fractal dimension reveal that the skin effect is suppressed in the non-Hermitian MBL phase, where localization is disorder-induced rather than boundary-induced (Suthar et al., 2022).

The many-body skin effect is itself not equivalent to NHMBL. In a nonintegrable spin chain with asymmetric hopping, the many-body skin effect appears as multifractality in many-body Hilbert space, and this multifractality coexists with random-matrix spectral statistics under both periodic and open boundaries. The paper makes the contrast explicit: multifractality associated with many-body localization necessitates the absence of ergodicity, whereas multifractality caused by the many-body skin effect can live inside a spectrally ergodic phase. This result sharply limits the use of multifractality alone as an NHMBL diagnostic (Hamanaka et al., 2024).

Quasiperiodic and Stark systems sharpen the contrast between spectral and localization transitions. In the non-Hermitian Aubry–André chain of hard-core bosons, the imaginary parts of eigenenergies are suppressed by MBL, and the authors conjecture that the real-complex transition, topological winding-number transition, and quasiperiodic MBL transition occur at the same point in the thermodynamic limit (Zhai et al., 2020). In the disorder-free Stark chain, by contrast, the real-complex transition occurs at $g$0, the topological transition extrapolates to $g$1, and the Stark MBL transition sits much later at $g$2; under OBCs the real-complex transition is absent, but the Stark MBL transition is unchanged (Liu et al., 2023).

Periodic driving produces still richer behavior. In the driven non-Hermitian Fibonacci chain, increasing the drive period does not yield a single Floquet-heating-induced MBL-to-ergodic crossover. Instead, the system can delocalize, relocalize, and then undergo multiple localization-delocalization transitions as a function of $g$3. At strong quasiperiodicity, for example $g$4 and $g$5, a first delocalization occurs near $g$6, followed by re-entrant MBL and then additional extended “islands.” Under open boundaries, those extended islands become boundary-sensitive and develop a Floquet many-body skin effect (Banerjee et al., 18 Sep 2025).

5. Open-system, monitored, and stochastic formulations

NHMBL also appears beyond effective non-Hermitian Hamiltonians. In Lindbladian many-body localization, the non-Hermitian object is the Lindbladian super-operator $g$7, acting on operator space. Its mid-spectrum eigenoperators exhibit a delocalization-localization transition diagnosed by the complex spacing ratio $g$8, operator-space entanglement entropy, and eigenoperator statistics. At strong disorder, off-diagonal degrees of freedom localize in operator space, random-matrix universality breaks down, and the decay rate of coherence becomes rigid, with $g$9 for dephasing and $U$0 for damping (Hamazaki et al., 2022).

Continuous monitoring in the no-click limit provides a second open-system route. For the random-field Heisenberg chain subject to random continuous measurements, post-selection yields an effective non-Hermitian Hamiltonian with random imaginary potentials. Numerically, the model is localized for any finite amount of disorder, and the non-Hermitian QREM benchmark exhibits a parametrically stronger localization scale, with the transition changing from $U$1 in the Hermitian QREM to $U$2 in the non-Hermitian version with random gain-loss (Tomasi et al., 2023).

A complementary perspective maps non-Hermitian many-body Hamiltonians to Markov-chain Laplacians $U$3. In this framework, wavefunction amplitudes become stochastic many-body configuration probabilities, all state transition processes are preserved, and analogous non-Hermitian localization and state-space fragmentation survive in classical stochastic dynamics. The interacting many-body Hatano–Nelson model becomes a biased exclusion process with Fermi-Dirac-like steady-state density profiles, while correlated spin-flip models realize fragmentation into dynamically disconnected sectors labeled by conserved $U$4 or parity $U$5 (Hao et al., 5 Sep 2025). This suggests that some structural aspects of NHMBL—especially non-reciprocal state-space localization and fragmentation—admit a classical stochastic counterpart even when entanglement does not.

6. Controversies, limitations, and outlook

The main controversy concerns whether finite-size spectral indicators suffice to establish NHMBL. In the non-Hermitian disordered XXZ chain studied through steady-state spin transport, the steady-state current remains finite and decays exponentially with disorder strength, and there is no evidence of a transition up to disorder values far beyond the previously claimed critical point. Spectral indicators suggest localization, whereas transport indicates delocalization; the discrepancy is reinforced by a non-commutativity of limits in system size and time (Brighi et al., 3 Apr 2025). This directly challenges any definition of NHMBL that relies only on mid-spectrum spectral “realification” or finite-size level statistics.

More broadly, open problems recur across the literature. Exact diagonalization typically reaches only $U$6 or $U$7, and larger-scale studies are hindered by non-unitary growth and strong entanglement in extended phases. Several papers explicitly call for analytical frameworks that explain why re-entrant localization can occur in driven Fibonacci landscapes, why spectral real-complex transitions and localization transitions decouple in clean long-range systems, and how different interaction structures, fillings, higher dimensions, or full Lindblad jump processes reshape NHMBL (Banerjee et al., 18 Sep 2025, Wang et al., 9 Oct 2025). Proposed experimental platforms include ultracold atoms in optical lattices, photonic lattices, circuit QED, superconducting-qubit arrays, and post-selected digital quantum simulators (Cheng et al., 2023, Liu et al., 2023).

A plausible implication is that NHMBL is best regarded not as a single transition diagnosed by a single observable, but as a broader non-equilibrium localization framework in which symmetry, boundary conditions, driving protocol, and the physical meaning of non-Hermiticity determine which observables are reliable. The strongest current picture combines spectral statistics, right- or biorthogonal eigenstate structure, real-time memory and entanglement diagnostics, and transport or operator-space probes, while treating skin effects, SVD surrogates, and real-complex transitions as informative but not universally decisive.