The Hatsugai-Kohmoto (HK) model is an exactly solvable electronic lattice model where the interaction between electrons with opposite spin is diagonal in momentum space. We generalize the HK model by introducing momentum-mixing Landau interactions. Within a self-consistent mean-field analysis we find that the ground state of this model exhibits a partially flat energy band, in agreement with the fermion condensation scenario proposed by Khodel and Shaginyan [JETP Lett. 51, 553 (1990)]. Inspired by Andersons pseudospin formulation of BCS theory, we show that the HK model with Landau interactions can be mapped onto a generalized Ising model where each site of the reciprocal lattice hosts two Ising spins. In the pseudospin picture the emergence of a partially flat electronic band corresponds to the smoothing of a magnetic domain wall. Moreover, guided by the pseudospin picture, we propose an exactly solvable variant of the HK model which has a unique ground state for all densities.
The Hatsugai-Kohmoto (HK) model is an exactly solvable electronic lattice model where the interaction between electrons with opposite spin is diagonal in momentum space. We generalize the HK model by introducing momentum-mixing Landau interactions. Within a self-consistent mean-field analysis we find that the ground state of this model exhibits a partially flat energy band, in agreement with the fermion condensation scenario proposed by Khodel and Shaginyan [JETP Lett. 51, 553 (1990)]. Inspired by Andersons pseudospin formulation of BCS theory, we show that the HK model with Landau interactions can be mapped onto a generalized Ising model where each site of the reciprocal lattice hosts two Ising spins. In the pseudospin picture the emergence of a partially flat electronic band corresponds to the smoothing of a magnetic domain wall. Moreover, guided by the pseudospin picture, we propose an exactly solvable variant of the HK model which has a unique ground state for all densities.
The all-to-all momentum coupling of the Hubbard interaction makes interacting lattice models generically unsolvable. In many settings, however, from Peierls instabilities to Moiré superlattice physics, the low-energy behavior is dominated by scattering at a few characteristic wavevectors. We exploit this by constructing a momentum-space clustering scheme that retains only a chosen subset of interaction channels. Our scheme can be considered a generalization of twist-averaged boundary conditions. In proving this, we also prove that our scheme can be considered as a generalization of Hatsugai-Kohmoto (HK) models, and all versions of the HK model previously considered in the literature arise as special cases. This shows that the surprising phenomenological success of HK models arises from their correspondence to the finite-site Hubbard model. In particular, the recently introduced "Momentum-Mixing HK" model corresponds to a specific choice of clustering limit, which is equal to the original finite-site Hubbard model with twist-averaged boundary conditions. Our scheme becomes particularly powerful when a spatially varying potential selects the dominant momentum channels. We demonstrate this on the one-dimensional analogue of interacting moiré systems: the Aubry-André-Hubbard model. We show that for sufficiently strong onsite potential, clusters as small as two sites can recover the ground state energy to below 1% error relative to DMRG benchmarks. This establishes that physically motivated momentum-space truncations can yield accurate low-energy descriptions at feasible computational cost, opening a path toward tractable interacting models of Moiré systems in two dimensions. Code for reproducing all numerical results is available at https://github.com/chainik1125/decomposing-hubbard.
This pedagogic review aims to give a gentle introduction to an exactly solvable model, the Hatsugai-Kohmoto (HK) model, which has infinite-ranged interaction but conserves the center of mass. Although this model is invented in 1992, intensive studies on its properties ranging from unconventional superconductivity, topological ordered states to non-Fermi liquid behaviors are made since 2020. We focus on its emergent non-Fermi liquid behavior and provide discussion on its thermodynamics, single-particle and two-particle correlation functions. Perturbation around solvable limit has also been explored with the help of perturbation theory, renormalization group and exact diagonalization calculation. We hope the present review will be helpful for graduate students or researchers interested in HK-like models or more generic strongly correlated electron systems.
Motivated by the non-Fermi liquid (NFL) phase in solvable Hatsugai–Kohmoto (HK) model and ubiquitous quantum oscillation (QO) phenomena observed in strongly correlated electron systems, we have studied the QO in HK model in terms of a combination of analytical and numerical calculation. In the continuum limit, the analytical results indicate the existence of QO in NFL state and its properties can be described by Lifshitz–Kosevich-like formula. Furthermore, numerical calculations with Luttinger’s approximation on magnetic-field-dependent density of state, magnetization and particle’s density agree with the findings of analytical treatment. Although numerical simulation from exact diagonalization exhibits certain oscillation behavior, it is difficult to extract its oscillation period and amplitude. Therefore, more work (especially the large-scale numerical simulation) on this interesting issue is highly desirable and we expect the current HK model study to be helpful in understanding generic QO in correlated electron materials.
When non-magnetic impurity immerses in Fermi sea, a regular modulation of charge density around impurity will appear and such phenomena is called Friedel oscillation (FO). Although both Luttinger liquid and Landau Fermi liquid show such characteristic oscillation, FO in generic non-Fermi liquid (NFL) phase is still largely unknown. Here, we show that FO indeed exists in NFL state of an exactly solvable model, i.e. the Hatsugai-Kohmoto model which has been intensively explored in recent years. Combining T-matrix approximation and linear-response-theory, an interesting picture emerges, if two interaction-induced quasi-particles bands in NFL are partially occupied, FO in this situation is determined by a novel structure in momentum space, i.e. the 'average Fermi surface' (average over two quasi-particle Fermi surface), which highlights the inter-band particle-hole excitation. We hope our study here provides a counterintuitive example in which FO with Fermi surface coexists with NFL quasi-particle, and it may be useful to detect hidden 'average Fermi surface' structure in other correlated electron systems.
We introduce a generalized Hatsugai-Kohmoto multi-orbital model and study its phase diagram and physical properties in the additional presence of perturbations that lift any extensive ground-state degeneracies. The unperturbed, exactly solvable model already displays a rich set of spectral functions, including regimes reminiscent of unconventional magnets. We map the first-order study of additional spatially local multi-orbital Hubbard interactions to a Heisenberg model in momentum space, which leads to symmetry-breaking instabilities already at weak coupling. Interestingly, translational-symmetry breaking orders, such as antiferromagnetism, are excluded. Instead, in addition to ferromagnetism, unconventional p-wave and d-wave magnets occur, characterized by spin order on the bonds of the underlying square lattice. Adding another type of momentum-space interaction, which still allows to solve the model exactly, is shown to stabilize a non-degenerate singlet ground state that retains the spin splitting characteristic of unconventional magnets. We discuss its impact on the spin structure factor. Taken together, our findings show that Hatsugai-Kohmoto-like models provide a rich playground for unconventional magnetism.
Computing superconducting properties starting from an exactly solvable model for a doped Mott insulator stands as a grand challenge. We have recently shown that this can be done starting from the Hatsugai-Kohmoto (HK) model, which can be understood generally as the minimal model that breaks the nonlocal Z2 symmetry of a Fermi liquid, thereby constituting a new quartic fixed point for Mott physics [Phillips et al., Nat. Phys. 16, 1175 (2020); Huang et al., Nat. Phys. (2022)]. In the current paper, we compute the thermodynamics, condensation energy, and electronic properties such as the NMR relaxation rate 1/T1 and ultrasonic attenuation rate. Key differences arise with the standard BCS analysis from a Fermi liquid: (1) the free energy exhibits a local minimum at Tp where the pairing gap turns on discontinuously above a critical value of the repulsive HK interaction, thereby indicating a first-order transition; (2) a tricritical point emerges, thereby demarcating the boundary between the standard second-order superconducting transition and the novel first-order regime; (3) Mottness changes the sign of the quartic coefficient in the Landau-Ginzburg free-energy functional relative to that in BCS; (4) as this obtains in the strongly interacting regime, it is Mott physics that underlies the generic first-order transition; (5) the condensation energy exceeds that in BCS theory suggesting that multiple Mott bands might be a way of enhancing superconducting; (6) the heat-capacity jump is nonuniversal and increases with the Mott scale; (7) Mottness destroys the Hebel-Slichter peak in NMR; and (8) Mottness enhances the fall-off of the ultrasonic attenuation at the pairing temperature Tp. As several of these properties are observed in the cuprates, our analysis here points a way forward in computing superconducting properties of strongly correlated electron matter.