We study the nonequilibrium dynamics of the superconducting order parameter in the Hatsugai-Kohmoto (HK) model. In the absence of superconductivity, its ground state is a non-Fermi liquid, whose properties are controlled by the HK interaction. Our protocol involves quantum quenching the HK interaction but leaving the interaction responsible for superconductivity unchanged. We map out the nonequilibrium dynamical phase diagram of the interacting model which contains three phases where, at long times, the order parameter amplitude vanishes, approaches a constant value, or persistently oscillates. We also investigate the Loschmidt echo in searching for dynamical quantum phase transition and find that its nonanalytic temporal behavior is close but does not match exactly the vanishing of the order parameter. The momentum space entanglement entropy between positive and negative momentum modes, relevant for Cooper pairing, is calculated. Counterintuitively, this momentum space entanglement does not change significantly during the quench dynamics and its value remains reasonably large even for a vanishing superconducting order parameter. Nevertheless, its derivative with respect to the HK interaction signals the dynamical phase transition associated to the late time vanishing of superconductivity.
We study the nonequilibrium dynamics of the superconducting order parameter in the Hatsugai-Kohmoto (HK) model. In the absence of superconductivity, its ground state is a non-Fermi liquid, whose properties are controlled by the HK interaction. Our protocol involves quantum quenching the HK interaction but leaving the interaction responsible for superconductivity unchanged. We map out the nonequilibrium dynamical phase diagram of the interacting model which contains three phases where, at long times, the order parameter amplitude vanishes, approaches a constant value, or persistently oscillates. We also investigate the Loschmidt echo in searching for dynamical quantum phase transition and find that its nonanalytic temporal behavior is close but does not match exactly the vanishing of the order parameter. The momentum space entanglement entropy between positive and negative momentum modes, relevant for Cooper pairing, is calculated. Counterintuitively, this momentum space entanglement does not change significantly during the quench dynamics and its value remains reasonably large even for a vanishing superconducting order parameter. Nevertheless, its derivative with respect to the HK interaction signals the dynamical phase transition associated to the late time vanishing of superconductivity.
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.
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.
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.
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.
A nonequilibrium state shares macroscopic properties, such as conductivity and superconductivity, with a static state when the two states exhibit an identical average value of an observable over a period. We studied the quench dynamics of a Kitaev chain on the basis of two types of order parameters associated with two channels of pairing: local pairing in real space and Bardeen-Cooper-Schrieffer (BCS)-like pairing in momentum space. Based on the exact solution, we found that the two order parameters were identical for the ground state, which indicated a balance between the two pairing channels and would help determine the quantum phase diagram. However, for a nonequilibrium state obtained through time evolution from an initially prepared vacuum state, the two parameters varied, but both parameters could still help determine the phase diagram. In the region of nontrivial topological phases, nonequilibrium states favored BCS-like pairing. In this paper, we present an alternative approach to dynamically generate a superconducting state from a trivial empty state and illuminate the pairing mechanism.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.