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arXiv:1901.07037 · Physical review. D/Physical review. D.

Local entropies across the Mott transition in an exactly solvable model

Luke Yeo, Philip Phillips

Published May 22, 2019• 18 citations
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Abstract

We study entanglement in the Hatsugai-Kohmoto model, which exhibits a continuous interaction-driven Mott transition. By virtue of the all-to-all nature of its center-of-mass-conserving interactions, the model lacks dynamical spectral weight transfer, which is the key to the intractability of the Hubbard model for $d>1$. In order to maintain a nontrivial Mott-like electron propagator, SU(2) symmetry is preserved in the Hamiltonian, leading to a ground state that is mixed on both sides of the phase transition. Because of this mixture, even the metal in this model is unentangled between any pair of sites, unlike free fermions whose ground state carries a filling-dependent site-site entanglement. We focus on the scaling behavior of the one- and two-site entropies ${s}_{1}$ and ${s}_{2}$, as well as the entropy density $s$, of the ground state near the Mott transition. At low temperatures in the two-dimensional Hubbard model, Walsh et al. [Phys. Rev. B 99, 075122 (2019)] observed numerically that ${s}_{1}$ and $s$ increase continuously into the metal, across a first-order Mott transition. In the Hatsugai-Kohmoto model, ${s}_{1}$ acquires the constant value ln 4 even at the Mott transition. On the other hand, ${s}_{2}$ and $s$ each act as a sharp signal of the Mott transition, in any dimension, by decreasing at the transition into the metal. Specifically, we find that in one dimension, ${s}_{2}$ and $s$ exhibit kinks at the transition while in two dimensions, only $s$ exhibits a kink.

DOI
10.1103/physrevd.99.094030

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We study entanglement in the Hatsugai-Kohmoto model, which exhibits a continuous interaction-driven Mott transition. By virtue of the all-to-all nature of its center-of-mass-conserving interactions, the model lacks dynamical spectral weight transfer, which is the key to the intractability of the Hubbard model for d>1d>1d>1. In order to maintain a nontrivial Mott-like electron propagator, SU(2) symmetry is preserved in the Hamiltonian, leading to a ground state that is mixed on both sides of the phase transition. Because of this mixture, even the metal in this model is unentangled between any pair of sites, unlike free fermions whose ground state carries a filling-dependent site-site entanglement. We focus on the scaling behavior of the one- and two-site entropies s1{s}_{1}s1​ and s2{s}_{2}s2​, as well as the entropy density sss, of the ground state near the Mott transition. At low temperatures in the two-dimensional Hubbard model, Walsh et al. [Phys. Rev. B 99, 075122 (2019)] observed numerically that s1{s}_{1}s1​ and sss increase continuously into the metal, across a first-order Mott transition. In the Hatsugai-Kohmoto model, s1{s}_{1}s1​ acquires the constant value ln 4 even at the Mott transition. On the other hand, s2{s}_{2}s2​ and sss each act as a sharp signal of the Mott transition, in any dimension, by decreasing at the transition into the metal. Specifically, we find that in one dimension, s2{s}_{2}s2​ and sss exhibit kinks at the transition while in two dimensions, only sss exhibits a kink.
At the Mott transition, electron-electron interaction changes a metal, in which electrons are itinerant, to an insulator, in which electrons are localized. This phenomenon is central to quantum materials. Here we contribute to its understanding by studying the two-dimensional Hubbard model at finite temperature with plaquette cellular dynamical mean-field theory. We provide an exhaustive thermodynamic description of the correlation-driven Mott transition of the half-filled model by calculating pressure, charge compressibility, entropy, kinetic energy, potential energy, and free energy across the first-order Mott transition and its high-temperature crossover (Widom line). The entropy is extracted from the Gibbs-Duhem relation and shows complex behavior near the transition, marked by discontinuous jumps at the first-order boundary, singular behavior at the Mott endpoint, and inflections marking sharp variations in the supercritical region. The free energy allows us to identify the thermodynamic phase boundary, to discuss phases stability and metastability, and to touch upon nucleation and spinodal decomposition mechanisms for the transition. We complement this thermodynamic description of the Mott transition by an information-theoretic description. We achieve this by calculating the local entropy, which is a measure of entanglement, and the single-site total mutual information, which quantifies quantum and classical correlations. These information-theoretic measures exhibit characteristic behaviors that allow us to identify the first-order coexistence regions, the Mott critical endpoint and the crossovers along the Widom line in the supercritical region.
Tools of quantum information theory offer a new perspective to characterize phases and phase transitions in interacting many-body quantum systems. The Hubbard model is the archetypal model of such systems and can explain rich phenomena of quantum matter with minimal assumptions. Recent measurements of entanglement-related properties of this model using ultracold atoms in optical lattices hint that entanglement could provide the key to understanding open questions of the doped Hubbard model, including the remarkable properties of the pseudogap phase. These experimental findings call for a theoretical framework and new predictions. Here we approach the doped Hubbard model in two dimensions from the perspective of quantum information theory. We study the local entropy and the total mutual information across the doping-driven Mott transition within plaquette cellular dynamical mean-field theory. We find that upon varying doping these two entanglement-related properties detect the Mott insulating phase, the strongly correlated pseudogap phase, and the metallic phase. Imprinted in the entanglement-related properties we also find the pseudogap to correlated metal first-order transition, its finite temperature critical endpoint, and its supercritical crossovers. Through this footprint we reveal an unexpected interplay of quantum and classical correlations. Our work shows that sharp variation in the entanglement-related properties and not broken symmetry phases characterizes the onset of the pseudogap phase at finite temperature.
In a conventional boson system, the ground state can either be an insulator or a superfluid (SF) due to the duality between particle number and phase. This paper reveals that the long-sought Bose metal (BM) state can be realized in an exactly solvable interacting bosonic model, i.e., the Bose-Hatsugai-Kohmoto (BHK) model, which acts as the nontrivial extension of Bose-Hubbard (BH) model. By tuning the parameters such as bandwidth WWW, chemical potential $\ensuremath{\mu}$, and interaction strength UUU, a BM state without any symmetry breaking can be accessed for a generic W/UW/UW/U ratio, while a Mott insulator (MI) with integer boson density is observed at small W/UW/UW/U. The quantum phase transition between the MI and BM states belongs to the universality class of the Lifshitz transition, which is further confirmed by analyzing the momentum-distribution function, the Drude weight, and the SF weight. Additionally, our investigation at finite temperature reveals similarities between the BM state and the Fermi liquid, such as a linear-TTT dependent heat capacity ($Cv\ensuremath{\sim}\ensuremath{\gamma}T$) and a saturated charge susceptibility (${\ensuremath{\chi}}_{c}\ensuremath{\sim}$ constant) as TTT approaches zero. Comparing the BM state with the SF state in the standard BH model, we find that the key feature of the BM state is a compressible total wavefunction accompanied by an incompressible zero-momentum component. Given that the BM state prevails over the SF state at any finite UUU in the BHK model, our paper suggests the possibility of realizing the BM state with on-site repulsion interactions in momentum space.
We investigate a nonmagnetic metal-insulator transition in the 1/5-depleted square lattice Hubbard model at half filling within the eight-site cellular dynamical mean field theory. We find that a metal-insulator transition without any signatures of a first-order transition, a continuous Mott transition, takes place in a certain range of parameters. The nature of the continuous Mott transition is merely a Lifshitz transition driven by the on-site Coulomb interaction. The renormalized matrix elements of hoppings and the spin-spin correlation functions reveal that the physics of this transition is a strong enhancement of the dimerization due to the nonlocal effects of the electron-electron interaction.
We show that the physics of the SU(NNN) Hubbard model can be realistically simulated with the recently developed orbital Hatsugai-Kohmoto model. In this approach, the momentum mixing absent from the band Hatsugai-Kohmoto model is included by grouping nnn-Hubbard atoms into a cluster. We take advantage of the rapid convergence of this scheme (1/n21/n^21/n2) and show that for n≈O(10)n\approx O(10)n≈O(10), quantitative agreement with the determinantal quantum Monte Carlo method arises for the double occupancy as well as qualitative agreement for the filling and compressibility across the Mott transition. Additional features of this work are that we can obtain low-temperature physics, which could be qualitatively different from its high-temperature counterpart, and dynamical quantities without resorting to analytical continuation, thereby establishing the orbital Hatsugai-Kohmoto model as a useful alternative perspective for studying strong correlations in SU(NNN) systems.
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.