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arXiv:2304.04787 · Physical review. B./Physical review. B

Proof of a stable fixed point for strongly correlated electron matter

Jinchao Zhao, Gabriele La Nave, Philip Phillips

Published Oct 19, 2023• 15 citations
arXivHTMLPDF

Abstract

We establish the Hatsugai-Kohmoto model as a stable quartic fixed point (distinct from Wilson-Fisher) by computing the $\ensuremath{\beta}$ function in the presence of perturbing local interactions. In vicinity of the half-filled doped Mott state, the $\ensuremath{\beta}$ function vanishes for all local interactions regardless of their sign. The only flow away from the HK model is through the superconducting channel which lifts the spin degeneracy as does any ordering tendency. The superconducting instability is identical to that established previously [Phillips et al., Nat. Phys. 16, 1175 (2020)]. A corollary of this work is that any system in which the spectral weight bifurcates into lower and upper bands such as the Hubbard model with repulsive interactions flows into the HK stable fixed point in the vicinity of half-filling. Consequently, although the HK model has all-to-all interactions, the bifurcation of the spectral weight is stable as nothing local destroys it. The consilience with Hubbard arises because both models break the ${Z}_{2}$ symmetry on a Fermi surface, the HK model being the simplest to do so. Indeed, the simplicity of the HK model belies its robustness and generality.

DOI
10.1103/physrevb.108.165135

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We establish the Hatsugai-Kohmoto model as a stable quartic fixed point (distinct from Wilson-Fisher) by computing the \ensuremathβ\ensuremath{\beta}\ensuremathβ function in the presence of perturbing local interactions. In vicinity of the half-filled doped Mott state, the \ensuremathβ\ensuremath{\beta}\ensuremathβ function vanishes for all local interactions regardless of their sign. The only flow away from the HK model is through the superconducting channel which lifts the spin degeneracy as does any ordering tendency. The superconducting instability is identical to that established previously [Phillips et al., Nat. Phys. 16, 1175 (2020)]. A corollary of this work is that any system in which the spectral weight bifurcates into lower and upper bands such as the Hubbard model with repulsive interactions flows into the HK stable fixed point in the vicinity of half-filling. Consequently, although the HK model has all-to-all interactions, the bifurcation of the spectral weight is stable as nothing local destroys it. The consilience with Hubbard arises because both models break the Z2{Z}_{2}Z2​ symmetry on a Fermi surface, the HK model being the simplest to do so. Indeed, the simplicity of the HK model belies its robustness and generality.
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{\mathbb{Z}}_{2}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/T11/{T}_{1}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{T}_{p}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{T}_{p}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.
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.
We investigate the Hatsugai-Kohmoto (HK) model on a square lattice, which describes both a Mott insulator at half-filling and a non-Fermi liquid phase on doping. Through the solution of this exactly solvable model with the inclusion of pairing interactions, we demonstrate the emergence of strong pair-density-wave (PDW) fluctuations associated with center-of-mass momentum $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$ at finite temperatures for low dopings up to a critical doping and intermediate UUU interaction. Furthermore, we also confirm that a superconducting instability appears in the model within a wide regime of interaction UUU and doping parameter xxx. In view of the fact that it has been recently put forward that the metal-insulator transition of the HK model belongs to the same universality class as the Mott transition of the paradigmatic Hubbard model [Huang et al., Nat. Phys. 18, 511 (2022)], our work may thus shed light on an interesting scenario regarding the emergence of a fluctuating PDW phase on doping a Mott insulator, which has been argued to be relevant for understanding the physics of the cuprate superconductors in the underdoped regime.
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.
Multiband superconductivity gives rise to a rich landscape of possible pairing states. Here we study superconductivity emerging from a half-filled normal state in a two-orbital extension of the Hatsugai-Kohmoto model, an exactly solvable model of correlated electrons with momentum-local interactions, which provides a minimal framework to explore the interplay of strong correlations, orbital structure and pairing symmetry. For a system with point-group symmetry D4h\rm D_{4h}D4h​, we classify the symmetry-allowed superconducting gap structures, taking into account spin, orbital and momentum degrees of freedom. We further compute the critical temperature and the superconducting order parameter for selected pairing channels as functions of interaction and pairing strength within a mean-field treatment. Our results provide a systematic framework for analyzing superconductivity in the orbital Hatsugai-Kohmoto model and extend symmetry-based approaches to correlated multiband settings.
Abstract Superconductivity in strongly correlated electrons can emerge out from a normal state that is beyond the Landau’s Fermi liquid paradigm, often dubbed as ‘non-Fermi liquid’. While the theory for non-Fermi liquid is still not yet conclusive, a recent study on the exactly-solvable Hatsugai–Kohomoto (HK) model has suggested a non-Fermi liquid ground state whose Green’s function resembles the Yang–Rice–Zhang ansatz for cuprates (2020 Phillips et al Nat. Phys. 16 1175). Similar to the effect of on-site Coulomb repulsion in the Hubbard model, the repulsive interaction in the HK model divides the momentum space into three parts: empty, single-occupied and double-occupied regions, that are separated from each other by two distinct Fermi surfaces. In the presence of an additional Bardeen–Cooper–Schrieffer-type pairing interaction of a moderate strength, we show that the system exhibits a ‘two-stage superconductivity’ feature as temperature decreases: a first-order superconducting transition occurs at a temperature T c that is followed by a sudden increase of the superconducting order parameter at a lower temperature T c ′ < T c . At the first stage, T c ′ < T < T c , the pairing function arises and the entropy is released only in the vicinity of the two Fermi surfaces; while at the second stage, T < T c ′ , the pairing function becomes significant and the entropy is further released in deep (single-occupied) region in the Fermi sea. The phase transitions are analyzed within the Ginzburg–Landau theory. Our work sheds new light on unconventional superconductivity in strongly correlated electrons.