Half-filled Two-dimensional Hubbard Model Research Articles

Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model

The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes. Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations, and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime. We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system.

Effects of frustration on the nonequilibrium dynamics of photoexcited lattice systems

We theoretically investigate the effects of the lattice geometry on the nonequilibrium dynamics of photo-excited carriers in a half-filled two-dimensional Hubbard model. Using a nonequilibrium generalization of the dynamical cluster approximation, we compare the relaxation dynamics in lattices which interpolate between the triangular lattice and square lattice configuration and thus reveal the role of the geometric frustration in these strongly correlated nonequilibrium systems. In particular, we show that the cooling effect resulting from the disordering of the spin background is less effective in the triangular case because of the frustration. This manifests itself in a longer relaxation time of the photo-doped population, as measured by the time-resolved photo-emission signal, and a higher effective temperature of the photo-doped carriers in the non-thermal steady state after the intra-Hubbard-band thermalization.

Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures

We present a highly parallelisable scheme for treating functional Renormalisation Group equations which incorporates a quasi-particle-based feedback on the flow and provides direct access to real-frequency self-energy data. This allows to map out the boundaries of Fermi-liquid regimes and to study the effect of quasi-particle degradation near Fermi liquid instabilities. As a first application, selected results for the two-dimensional half-filled perfectly nested Hubbard model are shown.

Magnetic and charge susceptibilities in the half-filled triangular lattice Hubbard model

We study magnetic and charge susceptibilities in the half-filled two-dimensional triangular Hubbard model within the dual fermion approximation in the metallic, Mott insulating, and crossover regions of parameter space. In the \textcolor{black}{insulating state}, we find strong spin fluctuations at the K point at low energy corresponding to the \textcolor{black}{120$^{\circ}$} antiferromagnetic order. These spin fluctuations persist into the metallic phase and move to higher energy. We also present data for simulated neutron spectroscopy and \textcolor{black}{spin-lattice} relaxation times, and perform direct comparisons to inelastic neutron spectroscopy experiments on the triangular material Ba$_8$CoNb$_6$O$_{24}$ and to the relaxation times on $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$. Finally, we present charge susceptibilities in different areas of parameter space, which should correspond to momentum-resolved electron-loss spectroscopy measurements on triangular compounds.

The antiferromagnetic phase of the Floquet-driven Hubbard model

A saddle point plus fluctuations analysis of the periodically driven half-filled two-dimensional Hubbard model is performed. For drive frequencies below the equilibrium gap, we find discontinuous transitions to time-dependent solutions. A highly excited, generically non-thermal distribution of magnons occurs even for drive frequencies far above the gap. Above a critical drive amplitude, the low-energy magnon distribution diverges as the frequency tends to zero and antiferromagnetism is destroyed, revealing the generic importance of collective mode excitations arising from a non-equilibrium drive.

Coupled Cluster as an Impurity Solver for Green's Function Embedding Methods.

We investigate the performance of Green's function coupled cluster singles and doubles (CCSD) method as a solver for Green's function embedding methods. To develop an efficient CC solver, we construct the one-particle Green's function from the coupled cluster (CC) wave function based on the non-Hermitian Lanczos algorithm. The major advantage of this method is that its scaling does not depend on the number of frequency points. We have tested the applicability of the CC Green's function solver in the weakly to strongly correlated regimes by employing it for a half-filled one-dimensional (1D) Hubbard model projected onto a single-site impurity problem and a half-filled two-dimensional (2D) Hubbard model projected onto a four-site impurity problem. For the 1D Hubbard model, for all interaction strengths, we observe an excellent agreement with the full configuration interaction (FCI) technique, both for the self-energy and spectral function. For the 2D Hubbard, we have employed an open-shell version of the current implementation and observed some discrepancies from FCI in the strongly correlated regime. Finally, on an example of a small ammonia cluster, we analyze the performance of Green's function CCSD solver within the Self-Energy Embedding Theory (SEET) with Hartree-Fock (HF) and Green's Function second order (GF2) for the treatment of the environment.

Local Entanglement Entropy and Mutual Information across the Mott Transition in the Two-Dimensional Hubbard Model.

Entanglement and information are powerful lenses to probe phases transitions in many-body systems. Motivated by recent cold atom experiments, which are now able to measure the corresponding information-theoretic quantities, we study the Mott transition in the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory, and focus on two key measures of quantum correlations: entanglement entropy and a measure of total mutual information. We show that they detect the first-order nature of the transition, the universality class of the end point, and the crossover emanating from the end point.

Signatures of the Mott transition in the antiferromagnetic state of the two-dimensional Hubbard model

The properties of a phase with large correlation length can be strongly influenced by the underlying normal phase. We illustrate this by studying the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory with continuous-time quantum Monte Carlo. Sharp crossovers in the mechanism that favors antiferromagnetic correlations and in the corresponding local density of states are observed. These crossovers occur at values of the interaction strength $U$ and temperature $T$ that are controlled by the underlying normal-state Mott transition.

Continuum of many-particle states near the metal-insulator transition in the Hubbard model

The strong coupling diagram technique is used for investigating states near the metal-insulator transition in the half-filled two-dimensional repulsive Hubbard model. The nonlocal third-order term is included in the irreducible part along with local terms of lower orders. Derived equations for the electron Green's function are solved by iteration for moderate Hubbard repulsions and temperatures. Starting iteration from Green's functions of the Hubbard-I approximation with various distances of poles from the real frequency axis continua of different metallic and insulating solutions are obtained. The insulating solutions vary in the width of the Mott gap, while the metallic solutions differ in the shape of the spectral function in the vicinity of the Fermi level. Besides, different scenarios of the metal-insulator transition -- with a sudden onset of a band of mobile states near the Fermi level and with gradual closure of the Mott gap -- are observed with a change in temperature. In spite of these dissimilarities, all solutions have a common curve separating metallic and insulating states in the phase diagram. Near this curve metallic and insulating solutions coexist. For moderate Hubbard repulsions metallic solutions are not Fermi liquids.

Magnetism, transport, and thermodynamics in two-dimensional half-filled Hubbard superlattices

We study magnetic, transport and thermodynamic properties of the half-filled two-dimensional ($2D$) Hubbard model with layered distributed repulsive interactions using unbiased finite temperature quantum Monte Carlo simulations. Antiferromagnetic long-ranged correlations at $T=0$ are confirmed by means of the magnetic structure factor and the onset of short-ranged ones is at a minimum temperature which can be obtained by peaks in susceptibility and specific heat following an RPA prediction. We also show that transport is affected in the large interaction limit and is enhanced in the non-repulsive layers suggesting a change of dimensionality induced by increased interactions. Lastly, we show that by adiabatically switching the interactions in layered distributed patterns reduces the overall temperature of the system with a potential application in cooling protocols in cold atoms systems.

Quantum Phase Transition and Local Entanglement in Extended Hubbard Model on Anisotropic Triangular Lattices**Supported by National Natural Science Foundation of China under Grant Nos.11274255, 11475027 and 11305132, Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20136203110001, and Technology of Northwest Normal University, China under Grants No. NWNU-LKQN-11-26

Using a mean-field theory based upon Hartree—Fock approximation, we theoretically investigate the competition between the metallic conductivity, spin order and charge order phases in a two-dimensional half-filled extended Hubbard model on anisotropic triangular lattice. Bond order, double occupancy, spin and charge structure factor are calculated, and the phase diagram of the extended Hubbard model is presented. It is found that the interplay of strong interaction and geometric frustration leads to exotic phases, the charge fluctuation is enhanced and three kinds of charge orders appear with the introduction of the nearest-neighbor interaction. Moreover, for different frustrations, it is also found that the antiferromagnetic insulating phase and nonmagnetic insulating phase are rapidly suppressed, and eventually disappeared as the ratio between the nearest-neighbor interaction and on-site interaction increases. This indicates that spin order is also sensitive to the nearest-neighbor interaction. Finally, the single-site entanglement is calculated and it is found that a clear discontinuous of the single-site entanglement appears at the critical points of the phase transition.

Phase Separation Induced by Symmetric Monocycle Optical Pulse in Extended Hubbard Models

Many-electron dynamics induced by a symmetric monocycle electric-field pulse of large amplitude is theoretically investigated in one- and two-dimensional half-filled extended Hubbard models on regular lattices (i.e., without dimerization) using the exact diagonalization method for small systems and the Hartree-Fock approximation for large systems. The formation of a negative-temperature state and the change from repulsive interactions to effective attractive interactions are shown to be realized for a wide region of the field amplitude and the excitation energy. For a nonnegligible intersite repulsive interaction, the numerical results are consistent with the fact that the phase separation between charge-rich and charge-poor regions is caused by the corresponding effective attraction.

Momentum-dependent pseudogaps in the half-filled two-dimensional Hubbard model

We compute unbiased spectral functions of the two-dimensional Hubbard model by extrapolating Green functions, obtained from determinantal quantum Monte Carlo simulations, to the thermodynamic and continuous time limits. Our results clearly resolve the pseudogap at weak to intermediate coupling, originating from a momentum selective opening of the charge gap. A characteristic pseudogap temperature T*, determined consistently from the spectra and from the momentum dependence of the imaginary-time Green functions, is found to match the dynamical mean-field critical temperature, below which antiferromagnetic fluctuations become dominant. Our results identify a regime where pseudogap physics is within reach of experiments with cold fermions in optical lattices.

Superconducting Phase and Pairing Fluctuations in the Half-Filled Two-Dimensional Hubbard Model

The two-dimensional Hubbard model exhibits superconductivity with d-wave symmetry even at half-filling in the presence of a next-nearest neighbor hopping. Using plaquette cluster dynamical mean-field theory with a continuous-time quantum Monte Carlo impurity solver, we reveal the non-Fermi liquid character of the metallic phase in proximity to the superconducting state. Specifically, the low-frequency scattering rate for momenta near (π, 0) varies nonmonotonically at low temperatures, and the dc conductivity is T linear at elevated temperatures with an upturn upon cooling. Evidence is provided that pairing fluctuations dominate the normal-conducting state even considerably above the superconducting transition temperature.

Conditions for magnetically induced singletd-wave superconductivity on the square lattice

It is expected that at weak to intermediate coupling, $d$-wave superconductivity can be induced by antiferromagnetic fluctuations. However, one needs to clarify the role of Fermi surface topology, density of states, pseudogap, and wave vector of the magnetic fluctuations on the nature and strength of the induced $d$-wave state. To this end, we study the generalized phase diagram of the two-dimensional half-filled Hubbard model as a function of interaction strength $U∕t$, frustration induced by second-order hopping ${t}^{\ensuremath{'}}∕t$, and temperature $T∕t$. In experiment, $U∕t$ and ${t}^{\ensuremath{'}}∕t$ can be controlled by pressure. We use the two-particle self-consistent approach, valid from weak to intermediate coupling. We first calculate as a function of ${t}^{\ensuremath{'}}∕t$ and $U∕t$ the temperature and wave vector at which the spin response function begins to grow exponentially. $d$-wave superconductivity in a half-filled band can be induced by such magnetic fluctuations at weak to intermediate coupling, but only if they are near commensurate wave vectors and not too close to perfect nesting conditions where the pseudogap becomes detrimental to superconductivity. For given $U∕t$, there is thus an optimal value of frustration ${t}^{\ensuremath{'}}∕t$ where the superconducting ${T}_{c}$ is maximum. The noninteracting density of states plays little role. The symmetry ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ vs ${d}_{xy}$ of the superconducting order parameter depends on the wave vector of the underlying magnetic fluctuations in a way that can be understood qualitatively from simple arguments.

Phonon-mediated tuning of instabilities in the Hubbard model at half-filling

We obtain the phase diagram of the half-filled two-dimensional Hubbard model on a square lattice in the presence of Einstein phonons. We find that the interplay between the instantaneous electron-electron repulsion and electron-phonon interaction leads to new phases. In particular, a d$_{x^2-y^2}$-wave superconducting phase emerges when both anisotropic phonons and repulsive Hubbard interaction are present. For large electron-phonon couplings, charge-density-wave and s-wave superconducting regions also appear in the phase diagram, and the widths of these regions are strongly dependent on the phonon frequency, indicating that retardation effects play an important role. Since at half-filling the Fermi surface is nested, spin-density-wave is recovered when the repulsive interaction dominates. We employ a functional multiscale renormalization-group method that includes both electron-electron and electron-phonon interactions, and take retardation effects fully into account.

Quantum Monte Carlo study of strongly correlated electrons: Cellular dynamical mean-field theory

We study the Hubbard model using the Cellular Dynamical Mean-Field Theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half-filled Hubbard model to gauge the performance of CDMFT+QMC particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one- and two-dimensions.

Magnon and hole excitations in the two-dimensional half-filled Hubbard model

Spin and hole excitation spectra and spectral weights are calculated for the half-filled Hubbard model, as a function of $t/U$. We find that the high energy spin spectra are sensitive to charge fluctuations. The energy difference $\Delta(\pi,0)- \Delta(\pi/2,\pi/2)$, which is negative for the Heisenberg model, changes sign at a fairly small $t/U\approx 0.053(5)$. The hole bandwidth is proportional to $J$, and considerably larger than in the t-J models. It has a minimum at ($\pi/2,\pi/2$) and a very weak dispersion along the antiferromagnetic zone boundary. A good fit to the measured spin spectra in La$_2$CuO$_4$ at $T=10K$ is obtained with the parameter values $U=3.1{\rm eV}$, $t=0.35{\rm eV}$.

Theoretical aspects of photoemission spectroscopy on strongly correlated electron systems

We will clarify, from the theoretical viewpoint, how electron correlations affect the density of states (DOS) and the angle-resolved photoemission spectra (ARPES). The two-dimensional half-filled Hubbard model is employed as a prototype of the interacting electron system. In the weak interaction regime, the DOS has a multi-peak structure, which consists of a sharp one-body coherent component and a broad many-body incoherent component. Accordingly, the ARPES has a sharp coherent peak near the Fermi level. We will show that a short-range spin correlation causes the multi-peak structure in the DOS. On the other hand, as the electronic interaction increases, the DOS and APRES are dominated by the many-body incoherent component even near the Fermi level. It will be shown that the strong coupling between the collective spin excitations and photo-generated hole causes the incoherent component.

Antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model: Slater vs. Mott?Heisenberg

We study antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model at low temperature. Collective spin fluctuations are governed by a non-linear sigma model that we derive from the Hubbard model for any value of the Coulomb repulsion. As the Coulomb repulsion increases, the ground state progressively evolves from a Slater to a Mott–Heisenberg antiferromagnet. At finite temperature, we find a metal–insulator transition between a pseudogap phase at weak coupling and a Mott–Hubbard insulator at strong coupling.