Non-Hermitian Hubbard Model Overview

• The non-Hermitian Hubbard model is defined by lattice Hamiltonians that incorporate asymmetric hopping, complex onsite potentials, or interactions, profoundly affecting spectral and many-body behavior.
• Key methodologies such as exact diagonalization, projector QMC, DMFT, and tensor-network approaches reveal unconventional correlation effects, pairing responses, and emergent phases.
• Its experimental relevance spans ultracold atoms, twisted moiré materials, and circuit-QED, demonstrating non-Hermiticity’s impact on magnetism, pairing enhancement, and topological transitions.

Searching arXiv for recent and foundational papers on the non-Hermitian Hubbard model to ground the article in current literature. arXiv_search query: "non-Hermitian Hubbard model" max_results: 10 The non-Hermitian Hubbard model denotes a family of interacting lattice models in which a Hubbard-type local interaction coexists with a non-Hermitian kinetic, potential, or interaction sector. In fermionic formulations this usually means spin-$\tfrac12$ particles with an onsite term $U\sum_i n_{i\uparrow}n_{i\downarrow}$, modified by asymmetric hopping, complex onsite potentials, complex-valued interactions, or effective gain–loss contributions; closely related Bose-Hubbard variants replace the fermionic interaction by the bosonic onsite nonlinearity. Across recent work, the subject has developed at the intersection of strong correlations, non-Hermitian skin effect, $\mathcal{PT}$-symmetry breaking, exceptional points, pairing physics, and open-system effective descriptions (Zhou et al., 18 Jun 2026, Takemori et al., 2024, Yu et al., 2023).

1. Model classes and microscopic realizations

There is no single canonical non-Hermitian Hubbard Hamiltonian. Instead, the literature contains several recurring constructions. A widely used class is the Hatano–Nelson-type Hubbard model with non-reciprocal hopping, where right- and left-going amplitudes differ, for example $t_r=t+\gamma$ and $t_l=t-\gamma$ in one dimension, or $t_{ij}^{\mathrm{eff}}=t\pm\gamma$ along a selected lattice direction in higher-dimensional lattices (Rangi et al., 25 Jul 2025, Zhou et al., 18 Jun 2026). In these models the non-Hermiticity is entirely in the one-body sector, while the onsite Hubbard term remains standard and Hermitian.

A second class introduces non-Hermiticity through complex onsite terms. One example is the non-Hermitian Aubry–André–Harper Hubbard ring, where the onsite modulation is $W\cos(2\pi b i+ih)$; here the interacting problem is a spinful Hubbard model on a finite Aharonov–Bohm ring, and the non-Hermitian part is diagonal rather than non-reciprocal (Sarkar et al., 18 Feb 2025). Another example is the spinless Haldane-Hubbard model with balanced staggered gain and loss $i\gamma\sum_l(-1)^l n_l$, which functions as an effective non-Hermitian description of an open system with sublattice-resolved gain and loss (Yi et al., 2 May 2025).

A third class places non-Hermiticity directly in the interaction sector. In the attractive Fermi-Hubbard setting, two-body loss produces a complex interaction $U=U_1+i\gamma/2$, while collective one-body loss generates asymmetric hopping; the resulting effective Hamiltonian is non-Hermitian both because of the kinetic asymmetry and because $U$ is complex (Takemori et al., 2024). Related constructions use purely imaginary onsite interactions, such as $U\sum_i n_{i\uparrow}n_{i\downarrow}$0 in effective pair-tunnelling descriptions, or complex onsite interactions $U\sum_i n_{i\uparrow}n_{i\downarrow}$1 in semiclassical spin-ladder reductions of the one-dimensional Hubbard model (Zhang et al., 2017, Sticlet et al., 2024).

A distinct but important usage appears in transcorrelated formulations of the Hermitian Hubbard model. There, a non-unitary similarity transformation $U\sum_i n_{i\uparrow}n_{i\downarrow}$2 with a Gutzwiller correlator produces a non-Hermitian Hamiltonian with exact correlated-hopping and three-body terms, while preserving the original many-body spectrum. In that setting the non-Hermiticity is computational rather than physical, but it has become part of the broader non-Hermitian Hubbard literature because it changes the left-right eigenvector structure and the numerical tractability of the problem (Dobrautz et al., 2018).

Non-Hermitian mechanism	Representative structure	Example papers
Asymmetric hopping	$U\sum_i n_{i\uparrow}n_{i\downarrow}$3, or $U\sum_i n_{i\uparrow}n_{i\downarrow}$4 on directed bonds	(Rangi et al., 25 Jul 2025, Zhou et al., 18 Jun 2026, Yu et al., 2023)
Complex onsite potential	$U\sum_i n_{i\uparrow}n_{i\downarrow}$5, or $U\sum_i n_{i\uparrow}n_{i\downarrow}$6	(Sarkar et al., 18 Feb 2025, Yi et al., 2 May 2025)
Complex interaction	$U\sum_i n_{i\uparrow}n_{i\downarrow}$7, or $U\sum_i n_{i\uparrow}n_{i\downarrow}$8	(Takemori et al., 2024, Zhang et al., 2017, Sticlet et al., 2024)
Effective impurity term in $U\sum_i n_{i\uparrow}n_{i\downarrow}$9-sector	$\mathcal{PT}$0	(Zhang et al., 2020)
Non-unitary similarity transform	$\mathcal{PT}$1	(Dobrautz et al., 2018)

2. Spectral structure, observables, and non-Hermitian state notions

Because $\mathcal{PT}$2, left and right eigenvectors generally differ. Several papers therefore define observables biorthogonally, for example $\mathcal{PT}$3 in non-Hermitian BCS theory or $\mathcal{PT}$4 in projector quantum Monte Carlo for the honeycomb Hubbard model (Takemori et al., 2024, Yu et al., 2023). Other works adopt right-eigenstate expectation values $\mathcal{PT}$5 and then compare them to biorthogonal values to test robustness; this is done explicitly in the moiré triangular-lattice study, which reports that the qualitative non-monotonic enhancement window survives the change of prescription (Zhou et al., 18 Jun 2026).

The notion of a many-body “ground state” is correspondingly model dependent. In several exact-diagonalization studies with complex spectra, the state with the smallest real part of the eigenvalue is used operationally as the ground state (Wang et al., 2023, Chen et al., 2021, Yi et al., 2 May 2025). Other works remain in a real-spectrum regime, as in the sign-problem-free honeycomb model for $\mathcal{PT}$6, where $\mathcal{PT}$7 symmetry is unbroken and unbiased projector QMC can be formulated directly (Yu et al., 2023). This suggests that the phrase “ground state” in non-Hermitian Hubbard physics is often a calculational convention rather than a universal thermodynamic object.

Observable choices reflect the physical questions being asked. Magnetic studies use antiferromagnetic structure factors and correlation ratios; pairing studies use double occupancy $\mathcal{PT}$8, pair-pair correlators $\mathcal{PT}$9, and susceptibilities $t_r=t+\gamma$0; NHSE studies emphasize nonlocal Green’s functions such as $t_r=t+\gamma$1 and $t_r=t+\gamma$2; topological studies use many-body Chern numbers under twisted boundary conditions; and entanglement-based work defines non-Hermitian reduced density matrices $t_r=t+\gamma$3 and edge entanglement entropies $t_r=t+\gamma$4 (Rangi et al., 25 Jul 2025, Zhou et al., 18 Jun 2026, Yi et al., 2 May 2025, Chen et al., 2021). In this literature, the observable prescription is therefore part of the model definition rather than a merely technical afterthought.

3. Analytical and numerical methods

Methodologically, the field is unusually heterogeneous. Exact diagonalization remains central for finite clusters and few-particle sectors. It is used for triangular-lattice moiré Hubbard clusters, non-Hermitian Haldane-Hubbard models, AAH-Hubbard rings, and two-particle Hatano–Nelson–Hubbard problems (Zhou et al., 18 Jun 2026, Wang et al., 2023, Sarkar et al., 18 Feb 2025, Longhi, 2023). In the latter case, exact analytical reduction to relative and center-of-mass coordinates yields closed-form scattering and doublon dispersions such as $t_r=t+\gamma$5 and $t_r=t+\gamma$6 on the infinite lattice (Longhi, 2023).

Unbiased Monte Carlo methods exist only for specially engineered sign-free models. A projector QMC algorithm was developed for the half-filled non-Hermitian honeycomb Hubbard model with spin-resolved asymmetric hopping, where an antiunitary symmetry after a partial particle-hole transformation eliminates the sign problem (Yu et al., 2023). A distinct determinant QMC construction on the square lattice uses conjugate asymmetric hoppings for the two spin components so that the Monte Carlo weight becomes the square of a real determinant at half filling on a bipartite lattice (Hayata et al., 2021). These models are not equivalent, and they lead to different many-body trends.

Tensor-network and dynamical mean-field methods have also entered the field. The moiré triangular-lattice study uses a biorthogonal non-Hermitian DMRG formulation on triangular cylinders, while the one-dimensional asymmetric-hopping Hubbard chain has been investigated by real-space dynamical mean-field theory with a local but site-dependent self-energy and an iterative perturbation theory impurity solver (Zhou et al., 18 Jun 2026, Rangi et al., 25 Jul 2025). Mean-field theory remains important on the attractive side: non-Hermitian BCS theory with biorthogonal left-right quasiparticles is used to derive gap equations, condensation energies, and phase diagrams in hypercubic lattices (Takemori et al., 2024).

Open-system formulations are equally prominent. Several papers start from Lindblad dynamics and then pass to effective non-Hermitian Hamiltonians by neglecting jump terms or by postselecting no-jump trajectories (Takemori et al., 2024, Wang et al., 2023, Yi et al., 2 May 2025). Others explicitly compare the effective non-Hermitian description with full quantum-trajectory simulations and show that the short-time or no-jump picture can differ qualitatively from the long-time open-system dynamics (Wang et al., 2023, Zhang et al., 2020). Taken together, these methodologies indicate that “non-Hermitian Hubbard model” can denote either a fundamental effective Hamiltonian or a reduced description of a broader dissipative problem.

4. Correlation effects: magnetism, pairing, currents, and $t_r=t+\gamma$7-pairing

The many-body consequences of non-Hermiticity are strongly channel dependent. In the half-filled non-Hermitian honeycomb Hubbard model with spin-dependent asymmetric hopping, projector QMC finds that non-Hermiticity enhances antiferromagnetism: the critical interaction decreases from $t_r=t+\gamma$8 at $t_r=t+\gamma$9 to $t_l=t-\gamma$0 at $t_l=t-\gamma$1, and the DSM-to-AFM transition shows critical exponents consistent with the Hermitian chiral-XY universality class, which the paper interprets as emergent Hermiticity at the quantum critical point (Yu et al., 2023). In a different square-lattice construction, however, determinant QMC finds that asymmetric non-Hermitian hopping suppresses antiferromagnetic order and estimates $t_l=t-\gamma$2 at $t_l=t-\gamma$3 for the disappearance of long-range order (Hayata et al., 2021). This suggests that there is no universal monotonic rule: the effect of non-Hermiticity on magnetism depends on how non-Hermiticity is embedded in spin and lattice structure.

On the pairing side, the triangular-lattice moiré study reports a non-monotonic “golden window” in non-reciprocity, $t_l=t-\gamma$4, where the non-Hermitian skin effect enhances finite-cluster pairing correlations. On the $t_l=t-\gamma$5 cluster at $t_l=t-\gamma$6, the double occupancy rises from $t_l=t-\gamma$7 to $t_l=t-\gamma$8 at $t_l=t-\gamma$9, a $t_{ij}^{\mathrm{eff}}=t\pm\gamma$0 increase, and the total pairing susceptibility increases from $t_{ij}^{\mathrm{eff}}=t\pm\gamma$1 to $t_{ij}^{\mathrm{eff}}=t\pm\gamma$2, i.e. by $t_{ij}^{\mathrm{eff}}=t\pm\gamma$3. The same work is explicit that it does not claim long-range superconducting order; its claim is enhancement of finite-cluster pairing correlations through NHSE-enhanced boundary local density of states and channel-selective suppression of competing antiferromagnetic correlations (Zhou et al., 18 Jun 2026).

The attractive non-Hermitian Fermi-Hubbard model with asymmetric hopping and complex attraction reaches a rather different conclusion at the mean-field level. There the asymmetric hopping contributes only through the imaginary part of the Bogoliubov–de Gennes matrix, so it does not modify the non-Hermitian BCS gap equation or effective density of states; by contrast, the complex interaction generated by two-body loss produces a dissipation-induced superfluid phase and reentrant normal–superfluid–dissipation-induced-superfluid behavior (Takemori et al., 2024). Taken together with the triangular-lattice results, this indicates that non-Hermiticity can enter pairing physics either through the kinetic sector, through the interaction sector, or through boundary amplification, with sharply different outcomes.

Several papers isolate exact or near-exact pairing states. In the attractive Hubbard model with purely imaginary hopping on a bipartite lattice, the bound-pair dispersion becomes $t_{ij}^{\mathrm{eff}}=t\pm\gamma$4, so the lowest bound state occurs at $t_{ij}^{\mathrm{eff}}=t\pm\gamma$5, where the two-particle ground state is the $t_{ij}^{\mathrm{eff}}=t\pm\gamma$6-pairing state. In the large negative-$t_{ij}^{\mathrm{eff}}=t\pm\gamma$7 limit the effective $t_{ij}^{\mathrm{eff}}=t\pm\gamma$8-spin Hamiltonian becomes ferromagnetic, and exact diagonalization indicates a transition from normal to $t_{ij}^{\mathrm{eff}}=t\pm\gamma$9-pairing ground states as imaginary hopping is increased (Zhang et al., 2020). A related but dynamically oriented construction adds a local non-Hermitian impurity $W\cos(2\pi b i+ih)$0 and shows that at $W\cos(2\pi b i+ih)$1 the maximal-$W\cos(2\pi b i+ih)$2 sector develops an exceptional point of order $W\cos(2\pi b i+ih)$3; the unique coalescing state has $W\cos(2\pi b i+ih)$4 for $W\cos(2\pi b i+ih)$5, and normalized long-time evolution from arbitrary initial states projects onto this ODLRO state (Zhang et al., 2020).

Other observables reveal still different correlation responses. In the non-Hermitian AAH-Hubbard ring threaded by flux, exact diagonalization finds enhancement of both real and imaginary parts of the persistent current with increasing non-Hermiticity, disorder strength, and moderate Hubbard interaction, followed by suppression at larger values (Sarkar et al., 18 Feb 2025). This suggests that non-Hermitian correlations need not manifest primarily through ordered phases; they can also reorganize mesoscopic response functions.

5. Boundary sensitivity, skin effect, topology, and exceptional points

Boundary conditions are often decisive. In the moiré triangular-lattice model, the pairing enhancement disappears under periodic boundary conditions, while open boundaries produce a non-monotonic dome in $W\cos(2\pi b i+ih)$6 and a clear NHSE-driven boundary mechanism (Zhou et al., 18 Jun 2026). In the one-dimensional asymmetric-hopping Hubbard chain treated by real-space DMFT, local spectral functions remain nearly symmetric between the two ends, but end-to-end Green’s functions sharply reveal directional amplification: in the noninteracting limit $W\cos(2\pi b i+ih)$7 grows exponentially with system size whereas $W\cos(2\pi b i+ih)$8 decays exponentially; Hubbard correlations suppress this effect at small and intermediate $W\cos(2\pi b i+ih)$9, but sufficiently strong asymmetric hopping restores amplification even at $i\gamma\sum_l(-1)^l n_l$0 (Rangi et al., 25 Jul 2025).

The two-particle Hatano–Nelson–Hubbard problem shows how profoundly boundary conditions can restructure the interacting spectrum. Under open boundary conditions a non-unitary gauge transformation maps the model to the Hermitian two-particle Hubbard equation, so the spectrum is real although eigenstates are skin localized. On the infinite lattice, by contrast, the scattering sector fills a two-dimensional area in the complex plane and the doublon band $i\gamma\sum_l(-1)^l n_l$1 undergoes an open-to-closed loop transition at $i\gamma\sum_l(-1)^l n_l$2, with detachment from the scattering continuum at $i\gamma\sum_l(-1)^l n_l$3. Dynamically, this model predicts bulk doublon dissociation and a burst edge revival when the particles reach the boundary (Longhi, 2023).

Topological variants add another layer of boundary sensitivity. In the non-Hermitian SSH-Hubbard chain, edge entanglement entropy $i\gamma\sum_l(-1)^l n_l$4 is used to track the breakdown of bulk-boundary correspondence. For $i\gamma\sum_l(-1)^l n_l$5, the open-boundary topological transition is at $i\gamma\sum_l(-1)^l n_l$6, while periodic-boundary Bloch gap closings occur at $i\gamma\sum_l(-1)^l n_l$7, yielding four phases. Increasing Hubbard $i\gamma\sum_l(-1)^l n_l$8 shrinks the intermediate non-Hermitian point-gap phases, and around $i\gamma\sum_l(-1)^l n_l$9 the bulk-boundary mismatch disappears, which the paper interprets as interaction-induced restoration of Hermitian-like behavior at half filling (Chen et al., 2021).

In non-Hermitian Haldane-Hubbard models, topology interacts with loss and gain in two distinct ways. With two-body loss, the effective interaction becomes $U=U_1+i\gamma/2$0, and exact diagonalization shows that the critical repulsion for charge ordering shifts to larger $U=U_1+i\gamma/2$1, stabilizing the topological regime against the CDW state (Wang et al., 2023). With balanced staggered single-particle gain and loss $U=U_1+i\gamma/2$2, the interacting phase diagram splits into a topologically gapped phase, a topological but real-gapless regime, and a trivial charge-ordered phase. In that model, $U=U_1+i\gamma/2$3-symmetry breaking in the low-lying spectrum marks the transition from the gapped topological regime to the real-gapless topological regime, while a further increase in $U=U_1+i\gamma/2$4 produces a first-order transition into the CDW phase with a level crossing in the imaginary part of the spectrum (Yi et al., 2 May 2025).

Exceptional points are therefore present, but they are not a universal organizing principle for all non-Hermitian Hubbard models. In some studies they are central, as in high-order $U=U_1+i\gamma/2$5-pairing coalescence (Zhang et al., 2020) and in the dissipative attractive Fermi-Hubbard model where the dissipation-induced-superfluid transition is accompanied by exceptional points in momentum space (Takemori et al., 2024). In others they are explicitly ruled out as the explanation of the main effect: the triangular-lattice pairing-enhancement work tracks the many-body gap and finds that it never closes in the “golden window,” so the enhancement is attributed to smooth NHSE physics rather than a spectral singularity (Zhou et al., 18 Jun 2026).

6. Open-system origins, experimental routes, and present scope

A large fraction of the literature treats non-Hermitian Hubbard models as effective descriptions of open quantum systems rather than as fundamental equilibrium Hamiltonians. Collective one-body loss and two-body loss lead to an effective attractive Fermi-Hubbard Hamiltonian with asymmetric hopping and complex interaction after postselection on no-jump trajectories (Takemori et al., 2024). Balanced sublattice gain and loss in the Haldane-Hubbard model likewise arise from a Lindblad problem with jump operators $U=U_1+i\gamma/2$6 on one sublattice and $U=U_1+i\gamma/2$7 on the other, while two-body loss in the spinless Haldane-Hubbard case yields a short-time effective non-Hermitian Hamiltonian with imaginary interaction $U=U_1+i\gamma/2$8 (Yi et al., 2 May 2025, Wang et al., 2023). These constructions are explicit reminders that the effective non-Hermitian Hamiltonian and the full dissipative evolution are not generally equivalent.

This distinction matters for interpretation. The moiré triangular-lattice study contrasts a coherent Floquet or lattice-modulation route, where a static Hatano–Nelson Hamiltonian is treated as the effective dressed description, with a more dissipative reservoir route described by a full Lindblad equation; in the latter, the NHSE-like density pileup survives but the sharp high-$U=U_1+i\gamma/2$9 downturn is smeared (Zhou et al., 18 Jun 2026). The Haldane-Hubbard work on two-body loss similarly shows that the effective non-Hermitian Hamiltonian can stabilize a topological regime at short times, while full quantum-trajectory simulations reveal eventual exponential melting of charge order under dissipation (Wang et al., 2023). This suggests that effective non-Hermitian phase diagrams should not be read automatically as asymptotic open-system steady-state diagrams.

Experimental proposals reflect the diversity of platforms. Candidate moiré realizations include twisted WSe$U$0, twisted MoTe$U$1, and MATBG, with asymmetric hopping interpreted as an engineered nonequilibrium control knob rather than an intrinsic equilibrium material parameter (Zhou et al., 18 Jun 2026). Ultracold-atom proposals use photoassociation to generate two-body loss and nonlocal Rabi coupling with local losses to engineer asymmetric hopping in the attractive Fermi-Hubbard model (Takemori et al., 2024). Driven-dissipative Bose-Hubbard extensions target superconducting circuits, where coherent pumping, phase gradients, Kerr nonlinearity, and photon loss generate effective non-Hermitian fluctuation matrices with point-gap topology (Rassaert et al., 2024). The term “non-Hermitian Hubbard model” therefore spans condensed-matter, cold-atom, photonic, and circuit-QED contexts.

The current scope of the subject remains uneven. Some results are numerically controlled in specific sign-free models (Yu et al., 2023, Hayata et al., 2021); others are confined to two-particle sectors (Longhi, 2023), exact-diagonalization clusters (Zhou et al., 18 Jun 2026), or mean-field descriptions (Takemori et al., 2024). Several papers are explicit that they do not establish thermodynamic long-range order or a full interacting topological classification (Zhou et al., 18 Jun 2026, Takemori et al., 2024). A plausible implication is that the non-Hermitian Hubbard model is presently best understood not as a single settled phase-diagram problem, but as a family of interacting non-Hermitian lattice theories whose many-body responses depend sensitively on how non-Hermiticity enters—through hopping, onsite potentials, interactions, impurities, or effective open-system reduction.