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arXiv:2402.08557 · Physical review. B./Physical review. B

Real-space analysis of Hatsugai-Kohmoto interaction

Jan Skolimowski

Published Apr 16, 2024• 9 citations
arXivHTMLPDF

Abstract

The Hatsugai-Kohmoto interaction model has gained a lot of attention in recent years, due to the fact it is exactly solvable in momentum space in any dimension while capturing some key features of the Mott phase. Here a one-dimensional lattice model with this interaction is approached from the real-space perspective, to explore how breaking the translation invariance of a lattice affects the intuition built by studying the exact solution in $k$ space. The ground state properties of chains with periodic and open boundary conditions are calculated and compared with both the exact solution in momentum space as well as with analogous solutions of the Hubbard model. The results show that introducing hard edges enhances the ferromagnetic correlations and the system undergoes a magnetic transition before reaching the strong coupling limit. Understanding the impact of hard edges is a crucial step toward answering the looming question of the existence of edge states and other topological phenomena in systems with this type of interaction.

DOI
10.1103/physrevb.109.165129

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The Hatsugai-Kohmoto interaction model has gained a lot of attention in recent years, due to the fact it is exactly solvable in momentum space in any dimension while capturing some key features of the Mott phase. Here a one-dimensional lattice model with this interaction is approached from the real-space perspective, to explore how breaking the translation invariance of a lattice affects the intuition built by studying the exact solution in kkk space. The ground state properties of chains with periodic and open boundary conditions are calculated and compared with both the exact solution in momentum space as well as with analogous solutions of the Hubbard model. The results show that introducing hard edges enhances the ferromagnetic correlations and the system undergoes a magnetic transition before reaching the strong coupling limit. Understanding the impact of hard edges is a crucial step toward answering the looming question of the existence of edge states and other topological phenomena in systems with this type of interaction.
The exactly solvable ``H-K'' family of interacting electron models introduced by Hatsugai and Kohmoto have played an important role in recent studies of the effect of wave function topology on the properties of correlation-driven (``Mott'') insulators. The authors show here that the very features that enable the exact solution lead to pathological transport properties including electrical conduction in the notionally insulating gapped Mott state and to divergent responses to applied magnetic fields. Caution is required in interpreting the results of the exact solutions.
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.
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 ppp-wave and ddd-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.
The Hatsugai-Kohmoto (HK) model has recently been identified as a fixed-point description of Mottness that is stable against perturbing local interactions. This raises a natural question beyond the weak-perturbation regime: how does a finite local repulsion modify HK physics in real-space observables and local spectra? We address this question in a half-filled one-dimensional spinful fermion chain, where the HK interaction is supplemented by an onsite Hubbard repulsion. Using exact diagonalization in an open chain, we compute ground-state correlation functions and site-resolved local spectra. We find that the Hubbard term does not drive the system away from the HK-dominated regime. Instead, at moderate UHKU_{\rm HK}UHK​, a finite UHubU_{\rm Hub}UHub​ promotes the emergence of behavior characteristic of the pure HK chain at larger UHKU_{\rm HK}UHK​. This Hubbard-assisted strong-HK response is reflected consistently in the development of positive spin correlations, the suppression of single-particle coherence, the impurity-induced redistribution of local spectral weight, and the evolution of pairing correlations. As a control perturbation, we replace the onsite Hubbard interaction by a nearest-neighbor (NN) density interaction. In contrast to the Hubbard case, the HK and NN interactions display competing tendencies and do not reproduce the same strong-HK behavior. These results provide a nonperturbative real-space complement to the fixed-point stability of HK physics and show that the effect of an additional repulsive interaction depends sensitively on its spatial structure.
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.
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.