2D Disordered Hubbard Model

• The two-dimensional disordered Hubbard model is a lattice Hamiltonian that combines nearest-neighbor hopping, random potential disorder, and on-site repulsion to study complex phase transitions.
• Modern computational techniques like DMFT, CPA, statDMFT, and semiclassical truncated-Wigner dynamics are used to analyze localization, inhomogeneous transport, and subdiffusive behavior.
• Results demonstrate that varying disorder geometry and interaction strength yields reproducible coexistence of metallic and insulating regions, with significant implications for many-body localization and superconducting tendencies.

Searching arXiv for the papers on arXiv and closely related work on the two-dimensional disordered Hubbard model. arxiv_search(query="two-dimensional disordered Hubbard model Anderson-Hubbard DMFT statDMFT CPA plaquette Hubbard localization", max_results=10) The two-dimensional disordered Hubbard model denotes a family of lattice Hamiltonians in which nearest-neighbor hopping competes with quenched spatial inhomogeneity and on-site repulsion on a two-dimensional lattice, most often the square lattice. In its standard fermionic form, disorder enters through random local potentials, while the Hubbard interaction penalizes double occupancy; the resulting problem interpolates between band-insulating, Anderson-like, correlated metallic, and Mott-insulating regimes, and in real space it can generate metallic regions inside an insulator and insulating regions inside a metal. Modern treatments range from coherent-potential and dynamical mean-field constructions to real-space statDMFT, nonequilibrium Green’s functions, semiclassical truncated-Wigner dynamics, and few-body transfer-matrix analyses, each resolving a different sector of the disorder–interaction competition (Lee et al., 2016, Suárez-Villagrán et al., 2019, Lev et al., 2015).

1. Hamiltonian structure and disorder architectures

A representative formulation is the half-filled square-lattice Hubbard Hamiltonian with random on-site energy,

$$H = -t \sum_{\langle i,j\rangle,\sigma} \big(c_{i\sigma}^\dagger c_{j\sigma} + \text{H.c.}\big) - \sum_{i,\sigma} (\mu - \epsilon_i)\, n_{i\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow},$$

with $t$ the nearest-neighbor hopping, $U$ the local repulsive Coulomb interaction, and $\epsilon_i$ a quenched random potential. In one widely used 2D variant, the disorder is uniform on an $N_c\times N_c$ plaquette, chosen once and then periodically repeated, so that translational invariance is restored only at the superlattice scale. At half-filling one sets $\mu=U/2$, and the model is often studied in a paramagnetic sector at finite temperature. This plaquette construction interpolates between the ionic Hubbard limit $N_c=\sqrt{2}$ and the Anderson limit $N_c\to\infty$, making it possible to follow how the non-interacting insulator changes from band-like to Anderson-like as the disorder pattern becomes less regular (Lee et al., 2016).

Other 2D realizations modify either the kinetic sector or the interaction sector. A statDMFT study used a square lattice with both nearest-neighbor hopping $t$ and purely imaginary next-nearest-neighbor hopping $t^*=0.5it$, together with box disorder $t$0 and the particle-hole-symmetric interaction $t$1 at average half-filling (Suárez-Villagrán et al., 2019). A dynamical study of long-range interacting fermions employed a square lattice with spin-independent random potentials $t$2 and power-law density-density interactions $t$3, including the short-range Hubbard limit and the infinite-range limit $t$4 (Sajna et al., 2020). Disorder can also be combined with spatially modulated hopping, as in the checkerboard Hubbard model, where intra-plaquette bonds have strength $t$5, inter-plaquette bonds have strength $t$6, and either on-site disorder or bond disorder is added on top of that inhomogeneous kinetic background (Smith et al., 2013).

2. Static phases at half-filling

In the plaquette-disordered model, the non-interacting phase depends strongly on disorder geometry. For strong disorder and small plaquette size $t$7, the repeated short-period potential produces a clear gap in the density of states and a band insulator. For large plaquette size $t$8, the density of states near the Fermi level becomes gapless and almost flat within CPA, and this is identified phenomenologically as a paramagnetic Anderson insulator. At weak disorder $t$9, by contrast, $U$0 retains a smeared van Hove singularity at $U$1 and behaves as a paramagnetic metal within CPA, whereas $U$2 already shows a band gap (Lee et al., 2016).

Turning on $U$3 reorganizes the low-energy spectrum in a way that depends on the nature of the non-interacting insulator. For weak disorder, the first-order paramagnetic metal–Mott insulator transition remains close to the clean single-site DMFT result: for $U$4, $U$5 and $U$6; for $U$7, $U$8 and $U$9, with only weak dependence on plaquette size. For strong disorder $\epsilon_i$0, the sequences diverge. At $\epsilon_i$1, increasing $\epsilon_i$2 drives

$\epsilon_i$3

with metallic onset around $\epsilon_i$4, a quasiparticle peak and Hubbard bands for $\epsilon_i$5, and Mott behavior for $\epsilon_i$6. At $\epsilon_i$7, the sequence is

$\epsilon_i$8

with the Anderson-like regime persisting up to $\epsilon_i$9, a narrow correlated-metal window around $N_c\times N_c$0, and vanishing low-energy spectral weight beyond that scale. The underlying mechanism is the disorder-driven accumulation of electrons in deep potential minima at small $N_c\times N_c$1, followed by interaction-driven repopulation of higher-energy states when double occupancy becomes costly; in the authors’ phrasing, electrons on the low-energy states “push each other” into high-energy states, and this repopulation activates new hopping channels (Lee et al., 2016).

This half-filled phenomenology distinguishes several notions of insulation. A band insulator is diagnosed by a clear spectral gap at $N_c\times N_c$2; a correlated metal by finite $N_c\times N_c$3, nonzero quasiparticle weight $N_c\times N_c$4, a quasiparticle peak, and Hubbard bands; a Mott insulator by $N_c\times N_c$5, vanishing $N_c\times N_c$6, and a correlation gap. The Anderson-insulating regime is more delicate: within arithmetic-average CPA it is inferred from a flat, gapless density of states without a quasiparticle peak, because CPA cannot describe localization properly (Lee et al., 2016).

3. Real-space Mott criticality and inhomogeneous transport

Real-space statDMFT recasts the disordered 2D Mott transition as a lattice of $N_c\times N_c$7 distinct impurity problems, each with its own local hybridization function and local self-energy $N_c\times N_c$8. In this framework the transition is followed by the local low-frequency Green’s function $N_c\times N_c$9 and the local self-energy $\mu=U/2$0. At finite size one still finds spinodal lines and hysteresis loops: sweeping $\mu=U/2$1 upward or downward produces bundles of local hysteresis curves rather than a single homogeneous jump, and the metallic and insulating solutions remain metastable over a coexistence window (Suárez-Villagrán et al., 2019).

The central real-space result is the appearance of mixed-phase textures near the transition. Starting from the metallic side and increasing $\mu=U/2$2, insulating bubbles nucleate inside a metallic background and then grow. Starting from the insulating side and decreasing $\mu=U/2$3, metallic bubbles nucleate inside an insulating matrix. At the same $\mu=U/2$4 inside the coexistence region, the two metastable branches correspond to distinct spatial mosaics. The local disorder fluctuations correlate with these patterns: in the analysis of three ranges of $\mu=U/2$5, insulating patches had $\mu=U/2$6, metallic patches had $\mu=U/2$7, and intermediate regions were close to unity. This directly realizes a random-field picture in which disorder locally biases the system toward metallic or insulating behavior (Suárez-Villagrán et al., 2019).

Finite-size scaling then changes the interpretation of the first-order transition. The study explicitly argues, by the same considerations used by Imry and Ma for the random-field Ising model, that disorder destroys the two-dimensional metal–insulator transition in the thermodynamic limit. Numerically, increasing $\mu=U/2$8 rounds the hysteresis and proliferates metallic and insulating bubbles, so the finite-size first-order line is smeared into a disorder-controlled crossover in the infinite system (Suárez-Villagrán et al., 2019).

The same statDMFT data support an incoherent transport description. Using $\mu=U/2$9 as a local scattering rate, the inelastic mean free path satisfies $N_c=\sqrt{2}$0. At $N_c=\sqrt{2}$1 and $N_c=\sqrt{2}$2, the computed range $N_c=\sqrt{2}$3 implies $N_c=\sqrt{2}$4, so transport is mapped to a classical random resistor network with bond resistance $N_c=\sqrt{2}$5. The resulting current maps are highly inhomogeneous: large-current paths wind through metallic backbones while insulating bubbles act as bottlenecks, and the total current drops sharply across the transition (Suárez-Villagrán et al., 2019).

4. Nonequilibrium dynamics, subdiffusion, and nonergodicity

Real-time dynamics in the 2D Anderson–Hubbard model have been studied by nonequilibrium self-consistent perturbation theory within the second-Born approximation, formulated through Kadanoff–Baym equations and reduced with the generalized Kadanoff–Baym ansatz to a quantum master equation for the equal-time density matrix. At $N_c=\sqrt{2}$6, half-filling, and infinite temperature, the method was benchmarked against exact diagonalization on a $N_c=\sqrt{2}$7 cluster and then extended to lattices as large as $N_c=\sqrt{2}$8, eliminating finite-size effects up to the times studied. Transport was diagnosed through the density-fluctuation mean-square displacement $N_c=\sqrt{2}$9 and the running exponent $N_c\to\infty$0, where $N_c\to\infty$1 corresponds to diffusion and $N_c\to\infty$2 to localization. The main result is that for sufficiently strong disorder the system becomes nonergodic, while for intermediate disorder strengths and for all accessible time scales transport is strictly subdiffusive. The authors argue that this is incompatible with a simple percolation picture but consistent with a heuristic random resistor network model in which subdiffusion can persist for long times before a crossover to diffusion (Lev et al., 2015).

A complementary dynamical program uses the semiclassical fermionic truncated Wigner approximation for disordered 2D Hubbard models with short-range and long-range interactions. In this formulation the choice of semiclassical Hamiltonian is crucial: a representation based on $N_c\to\infty$3 becomes asymptotically exact in the infinite-range limit, whereas the naive Weyl symbol of $N_c\to\infty$4 introduces spurious nonlinearities. Within this approach, different dynamical timescales of charge and spin are resolved. For weak and moderate disorder, charges show subdiffusive behavior while spins exhibit diffusive dynamics. At strong disorder, the quantum Fisher information grows logarithmically in time, with a slower increase for charges than for spins. Strong initial inhomogeneities such as domain walls can substantially slow thermalization, especially in the long-range model and even at weak disorder. The study explicitly states that it cannot make definite statements about the existence of a many-body localized phase, but it finds a very fast crossover as a function of disorder strength from rapidly thermalizing dynamics to a slow glassy-like regime (Sajna et al., 2020).

Taken together, these dynamical studies support a distinctly finite-time description of the 2D disordered Hubbard problem. The accessible regime is neither simply diffusive nor reducible to static Anderson localization. Instead, interactions generate long-lived subdiffusive transport, sector-selective relaxation of charge and spin, logarithmic growth of information-theoretic observables at strong disorder, and a nontrivial separation between experimentally relevant time windows and asymptotic transport classifications (Lev et al., 2015, Sajna et al., 2020).

5. Few-body and quasiparticle localization sectors

The two-particle limit provides a stringent test of interaction-induced delocalization claims. For two interacting particles moving on a 2D disordered lattice with contact interaction, the problem can be mapped to an effective single-particle eigenproblem with kernel $N_c\to\infty$5, and localization can then be studied through a reduced localization length $N_c\to\infty$6 extracted from transmission amplitudes on strip geometries. Earlier claims of an interaction-induced mobility edge at $N_c\to\infty$7 and $N_c\to\infty$8 are shown to result from severe finite-size effects: when the strip width is extended to $N_c\to\infty$9, all crossings of $t$0 disappear, $t$1 decreases monotonically with $t$2, and all pair states remain localized in the thermodynamic limit. The pair localization length is nevertheless strongly renormalized: at $t$3 and $t$4, the extracted unnormalized localization length $t$5 varies over more than three orders of magnitude, grows approximately exponentially for small $t$6, peaks around $t$7, and then decreases again. The conclusion is that 2D short-range interactions can enhance two-body propagation very strongly, but not generate a true two-body delocalization transition (Stellin et al., 2020).

Within the Mott regime, the localization problem can be reformulated in terms of doublon and holon quasiparticles moving on a disordered background. Using the hierarchy of correlations in 2D lattices, together with a comparison to strong-coupling perturbation theory, one finds a sharp distinction between two disorder channels. Charge disorder, implemented as binary random on-site potentials, splits holon and doublon bands into sub-bands and yields both energetic and spatial separation between localized and delocalized states: states tied to doped regions are predominantly localized, while states associated with undoped regions remain predominantly extended. Spin disorder, implemented as a frozen random spin background or random local Zeeman splitting, instead produces localized states throughout the quasiparticle bands, without a clean sub-band separation. This identifies a quasiparticle-level difference between scalar disorder and a disordered spin background in the Mott insulator phase (Knipšis et al., 10 Mar 2026).

These few-body and quasiparticle analyses constrain broader interpretations of the 2D disordered Hubbard model. The absence of a two-body mobility edge at zero density limits any construction of a many-body mobility edge from delocalized few-particle clusters, while the doublon/holon results show that localization properties in the Mott regime depend qualitatively on whether disorder acts in the charge sector or in the spin sector (Stellin et al., 2020, Knipšis et al., 10 Mar 2026).

6. Superconducting tendencies, materials contexts, and unresolved issues

Disorder does not act only on metal–insulator competition; it also reshapes pairing tendencies. In the checkerboard Hubbard model, exact diagonalization on 8-site and 12-site clusters with either potential disorder or hopping disorder shows that superconducting tendencies are more robust to disorder when hopping is inhomogeneous than in the uniform Hubbard model. The diagnostics are the pair binding energy $t$8, the spin gap $t$9, and a $t^*=0.5it$0-wave pairing matrix element $t^*=0.5it$1. At weak disorder, all three are maximal near intermediate inhomogeneity, around $t^*=0.5it$2–0.5 and intermediate $t^*=0.5it$3. As disorder increases, the region with positive disorder-averaged pair binding shrinks, but the surviving region remains centered around intermediate inhomogeneity rather than the uniform limit $t^*=0.5it$4. The same study analyzed all inequivalent staggered potentials on an 8-site cluster and identified specific dimerized and plaquette-staggered configurations that preserve positive pair binding to relatively large disorder (Smith et al., 2013).

The fermionic 2D disordered Hubbard framework is also used as a qualitative model for several experimental settings. The plaquette-disordered DMFT study specifically identifies disordered transition-metal oxides such as SrTi$t^*=0.5it$5Ru$t^*=0.5it$6O$t^*=0.5it$7 and Sr$t^*=0.5it$8Ir$t^*=0.5it$9Rh$t$00O$t$01, layered dichalcogenides such as 1T-TaS$t$02 with Cu intercalation, granular metallic films and nano-arrays, two-dimensional electron gases in Si MOSFETs, and cold-atom optical lattices realizing ionic-Hubbard-type setups as relevant contexts for disorder–interaction competition. The dynamical and semiclassical studies likewise emphasize cold-atom implementations, where density imbalance, domain-wall melting, and information-theoretic proxies can test slow relaxation, sector-selective transport, and glassy dynamics (Lee et al., 2016, Sajna et al., 2020).

Several issues remain open. statDMFT is local in the self-energy and neglects nonlocal magnetic and pairing correlations; CPA with arithmetic averaging cannot resolve Anderson localization properly; second-Born and semiclassical truncated-Wigner approaches are controlled only on finite times; and the zero-density two-particle result leaves finite-density many-body delocalization unresolved. Open problems stated explicitly in the recent literature include the fate of finite-density many-body localization in 2D, the role of true long-range Coulomb interactions, the effect of changing symmetry class through magnetic fields or spin-orbit coupling, and the question of whether quasiparticle states that appear extended in finite-size or quasiparticle-level treatments remain extended under full thermodynamic scaling (Suárez-Villagrán et al., 2019, Stellin et al., 2020, Knipšis et al., 10 Mar 2026).