---
title: Non-Hermitian Hubbard Model Overview
url: https://www.emergentmind.com/topics/non-hermitian-hubbard-model
type: topic
---

# Non-Hermitian Hubbard Model Overview

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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 [2606.20425], [2406.16482], [2302.10115].

## 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 [2507.19471], [2606.20425]. 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 [2502.12805]. 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 [2505.00964].

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 [2406.16482]. Related constructions use purely imaginary onsite interactions, such as \( \frac{iU_\rho}{2}\sum_i n_{\rho,i}(n_{\rho,i}-1)\) in effective pair-tunnelling descriptions, or complex onsite interactions \({\cal U}e^{i\phi}\) in semiclassical spin-ladder reductions of the one-dimensional Hubbard model [1705.09493], [2408.08110].

A distinct but important usage appears in transcorrelated formulations of the Hermitian Hubbard model. There, a non-unitary similarity transformation \(e^{-\hat\tau}He^{\hat\tau}\) 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 [1811.03607].

| Non-Hermitian mechanism | Representative structure | Example papers |
|---|---|---|
| Asymmetric hopping | \(t_r\neq t_l\), or \(t\pm\gamma\) on directed bonds | [2507.19471], [2606.20425], [2302.10115] |
| Complex onsite potential | \(W\cos(2\pi b i+ih)\), or \(i\gamma(-1)^l n_l\) | [2502.12805], [2505.00964] |
| Complex interaction | \(U=U_1+i\gamma/2\), or \(iU\,n_\uparrow n_\downarrow\) | [2406.16482], [1705.09493], [2408.08110] |
| Effective impurity term in \(\eta\)-sector | \(\sum_j g_j(\lambda\eta_j^x+i\gamma\eta_j^z)\) | [2009.06167] |
| Non-unitary similarity transform | \(\bar H=e^{-\hat\tau}He^{\hat\tau}\) | [1811.03607] |

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

Because \(H\neq H^\dagger\), left and right eigenvectors generally differ. Several papers therefore define observables biorthogonally, for example \({}_L\!\langle \cdots\rangle_R\) in non-Hermitian BCS theory or \(\langle O\rangle_G=\langle \psi_{L,G}|\hat O|\psi_{R,G}\rangle/\langle \psi_{L,G}|\psi_{R,G}\rangle\) in projector quantum Monte Carlo for the honeycomb Hubbard model [2406.16482], [2302.10115]. Other works adopt right-eigenstate expectation values \(\langle O\rangle_R=\langle\psi_R|O|\psi_R\rangle/\langle\psi_R|\psi_R\rangle\) 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 [2606.20425].

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 [2305.18762], [2108.00607], [2505.00964]. Other works remain in a real-spectrum regime, as in the sign-problem-free honeycomb model for \(\delta<1\), where \(\mathcal{PT}\) symmetry is unbroken and unbiased projector QMC can be formulated directly [2302.10115]. 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 \(D\), pair-pair correlators \(P(i,j)\), and susceptibilities \(\chi_{\mathrm{SC}}\); NHSE studies emphasize nonlocal Green’s functions such as \(G_{1N}\) and \(G_{N1}\); topological studies use many-body Chern numbers under twisted boundary conditions; and entanglement-based work defines non-Hermitian reduced density matrices \(\rho_A^{RL}\) and edge entanglement entropies \(S_{\alpha,\mathrm{edge}}=S_{\alpha,\mathrm{OBC}}-\frac12 S_{\alpha,\mathrm{PBC}}\) [2507.19471], [2606.20425], [2505.00964], [2108.00607]. 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 [2606.20425], [2305.18762], [2502.12805], [2308.04505]. In the latter case, exact analytical reduction to relative and center-of-mass coordinates yields closed-form scattering and doublon dispersions such as \(E_1=-2J\cos(k_1-ih)-2J\cos(k_2-ih)\) and \(E_2=\sqrt{U^2+16J^2\cos^2(q-ih)}\) on the infinite lattice [2308.04505].

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 [2302.10115]. 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 [2106.06192]. 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 [2606.20425], [2507.19471]. 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 [2406.16482].

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 [2406.16482], [2305.18762], [2505.00964]. 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 [2305.18762], [2009.06167]. 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 \(\eta\)-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 \(U_c=3.87\) at \(\delta=0\) to \(U_c\approx 2.4\) at \(\delta=0.8\), 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 [2302.10115]. In a different square-lattice construction, however, determinant QMC finds that asymmetric non-Hermitian hopping suppresses antiferromagnetic order and estimates \(\kappa_c=0.0317\pm 0.0014\) at \(U/t=4\) for the disappearance of long-range order [2106.06192]. 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, \(\gamma\in[0.5,1.2]\,t\), where the non-Hermitian skin effect enhances finite-cluster pairing correlations. On the \(3\times3\) cluster at \(U=4t\), the double occupancy rises from \(D(0)=0.135\) to \(D_{\max}=0.164\) at \(\gamma^\ast\approx1.05t\), a \(21\%\) increase, and the total pairing susceptibility increases from \(0.44\) to \(0.87\), i.e. by \(98\%\). 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 [2606.20425].

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 [2406.16482]. 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 \(\epsilon_K=\mathrm{sgn}(U)\sqrt{U^2-16t^2\cos^2(K/2)}\), so the lowest bound state occurs at \(K=\pi\), where the two-particle ground state is the \(\eta\)-pairing state. In the large negative-\(U\) limit the effective \(\eta\)-spin Hamiltonian becomes ferromagnetic, and exact diagonalization indicates a transition from normal to \(\eta\)-pairing ground states as imaginary hopping is increased [2012.15577]. A related but dynamically oriented construction adds a local non-Hermitian impurity \(\sum_j g_j(\lambda\eta_j^x+i\gamma\eta_j^z)\) and shows that at \(|\lambda|=|\gamma|\) the maximal-\(\eta\) sector develops an exceptional point of order \(2N+1\); the unique coalescing state has \(\langle \Phi_c|\eta_i^+\eta_j^-|\Phi_c\rangle=\frac14\) for \(i\neq j\), and normalized long-time evolution from arbitrary initial states projects onto this ODLRO state [2009.06167].

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 [2502.12805]. 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 \(D(\gamma)\) and a clear NHSE-driven boundary mechanism [2606.20425]. 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 \(|G_{1N}(\omega_{\text{peak}})|\) grows exponentially with system size whereas \(|G_{N1}(\omega_{\text{peak}})|\) decays exponentially; Hubbard correlations suppress this effect at small and intermediate \(\gamma\), but sufficiently strong asymmetric hopping restores amplification even at \(U=5\) [2507.19471].

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 \(E_2=\sqrt{U^2+16J^2\cos^2(q-ih)}\) undergoes an open-to-closed loop transition at \(U_{c1}=4J\sinh h\), with detachment from the scattering continuum at \(U_{c2}=4J\sqrt{\cosh(2h)}\). Dynamically, this model predicts bulk doublon dissociation and a burst edge revival when the particles reach the boundary [2308.04505].

Topological variants add another layer of boundary sensitivity. In the non-Hermitian SSH-Hubbard chain, edge entanglement entropy \(S_{\alpha,\mathrm{edge}}=S_{\alpha,\mathrm{OBC}}-\frac12S_{\alpha,\mathrm{PBC}}\) is used to track the breakdown of bulk-boundary correspondence. For \(U=0\), the open-boundary topological transition is at \(\delta t=0\), while periodic-boundary Bloch gap closings occur at \(\delta t=\pm\gamma\), yielding four phases. Increasing Hubbard \(U\) shrinks the intermediate non-Hermitian point-gap phases, and around \(U\approx2.0\) the bulk-boundary mismatch disappears, which the paper interprets as interaction-induced restoration of Hermitian-like behavior at half filling [2108.00607].

In non-Hermitian Haldane-Hubbard models, topology interacts with loss and gain in two distinct ways. With two-body loss, the effective interaction becomes \(V_{\mathrm{eff}}=V-i\gamma_2/2\), and exact diagonalization shows that the critical repulsion for charge ordering shifts to larger \(V\), stabilizing the topological regime against the CDW state [2305.18762]. With balanced staggered single-particle gain and loss \(i\gamma\sum_l(-1)^l n_l\), 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, \({\cal PT}\)-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 \(\gamma\) produces a first-order transition into the CDW phase with a level crossing in the imaginary part of the spectrum [2505.00964].

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 \(\eta\)-pairing coalescence [2009.06167] and in the dissipative attractive Fermi-Hubbard model where the dissipation-induced-superfluid transition is accompanied by exceptional points in momentum space [2406.16482]. 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 [2606.20425].

## 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 [2406.16482]. Balanced sublattice gain and loss in the Haldane-Hubbard model likewise arise from a Lindblad problem with jump operators \(\hat c_l^\dagger\) on one sublattice and \(\hat c_l\) on the other, while two-body loss in the spinless Haldane-Hubbard case yields a short-time effective non-Hermitian Hamiltonian with imaginary interaction \(V-i\gamma_2/2\) [2505.00964], [2305.18762]. 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-\(\gamma\) downturn is smeared [2606.20425]. 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 [2305.18762]. 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\(_2\), twisted MoTe\(_2\), and MATBG, with asymmetric hopping interpreted as an engineered nonequilibrium control knob rather than an intrinsic equilibrium material parameter [2606.20425]. 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 [2406.16482]. 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 [2411.08965]. 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 [2302.10115], [2106.06192]; others are confined to two-particle sectors [2308.04505], exact-diagonalization clusters [2606.20425], or mean-field descriptions [2406.16482]. Several papers are explicit that they do not establish thermodynamic long-range order or a full interacting topological classification [2606.20425], [2406.16482]. 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.

Source: https://www.emergentmind.com/topics/non-hermitian-hubbard-model