Two-dimensional Hubbard Model Research Articles

Excited States Beyond Mott Gap in Half-Filled-Band Hubbard Model

In connection with recent experiments on excitation in which Mott insulators change to conductors, we study the properties of excited states beyond the Mott gap as quasi-stationary states for a two-dimensional Hubbard (t-t'-U) model at half filling. A variational Monte Carlo method is used with trial wave functions for paramagnetic or normal (PM), superconducting with dx2-y2-wave (d-SC), isotropic s-wave, and extended s-wave symmetries, and antiferromagnetic (AF) states. The excited states are generated by imposing a minimum number of doubly occupied sites (doublons) D_L on the lowest-energy states. For U>W (W: band width), d_L=D_L/Ns (Ns: number of sites) corresponds to the excitation intensity. It is found that the AF state is the most stable among the states we treated for d_L<0.14 and insulating. The PM and d-SC states become conductive over a threshold d_Lc, and the conduction is caused by unbound doublons and holons (empty sites). The PM state arises for d_L>0.14, but the d-SC state is always hidden by the AF state. The s-wave-type superconducting states are not stabilized for any parameter set.

Stripe correlations in the two-dimensional Hubbard-Holstein model

Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping {t}^{{prime} } or phonon energy Ω. Our results help elucidate how the e-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tc cuprates.

Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling

We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions.Graphic abstract

SU(2) gauge theory of the pseudogap phase in the two-dimensional Hubbard model

We present a SU(2) gauge theory of fluctuating magnetic order in the two-dimensional Hubbard model. The theory is based on a fractionalization of electrons in fermionic chargons and bosonic spinons. The chargons undergo N\'eel or spiral magnetic order below a density-dependent transition temperature ${T}^{*}$. Fluctuations of the spin orientation are described by a nonlinear sigma model obtained from a gradient expansion of the spinon action. The spin stiffnesses are computed from a renormalization group improved random phase approximation. Our approximations are designed for moderate, not for strong, Hubbard interactions. The stiffnesses are strongly doping dependent with discontinuities at half-filling and a pronounced electron-hole asymmetry. The spinon fluctuations prevent magnetic long-range order of the electrons at any finite temperature. The phase with magnetic chargon order shares characteristic features with the pseudogap regime in high-${T}_{c}$ cuprates: a strong reduction of charge carrier density, a spin gap, and Fermi arcs. A substantial fraction of the pseudogap regime exhibits electronic nematicity.

Superconducting correlations in the two-dimensional Hubbard model with long-range electron hopping

Projector quantum Monte Carlo calculations are used to examine superconducting correlations in the two-dimensional Hubbard model with exponentially decaying electron hopping amplitudes tij(q)∼−q|Ri−Rj|. It is shown that this type of hopping, which models more realistically (in comparison to the nearest-neighbor hopping) the situation in real transition-metal materials, stabilizes significantly the superconducting correlations in the d-wave channel for electron concentrations n above the quarter band filling, especially for q∼0.35. In this region the enhancement of the superconducting correlations at non-zero q, against the classical Hubbard limit (q=0), changes (depending on the electron concentration) from 2 in the worse case to 5 in the best case. Moreover, we have studied also the influence of the Coulomb interaction U and a disorder on superconducting correlations. We have found that the Coulomb interaction enhances significantly superconducting correlations in the d-wave channel, while a disorder suppresses them gradually, but they surprisingly persist up to the disorder strength comparable to the Coulomb interaction.

Optical conductivity of the two-dimensional Hubbard model: Vertex corrections, emergent Galilean invariance, and the accuracy of the single-site dynamical mean field approximation

We compute the frequency-dependent conductivity of the two-dimensional square lattice Hubbard model at zero temperature as a function of density to second order in the interaction strength, and compare the results to the predictions of single-site dynamical mean field theory computed at the same order. We find that despite the neglect of vertex corrections, the single-site dynamical mean field approximation produces semiquantitatively accurate results for most carrier concentrations, but fails qualitatively for the nearly empty or nearly filled band cases where the model exhibits an emergent Galilean invariance. The DMFT approximation also becomes qualitatively inaccurate very near half filling if nesting is important.

Mechanism of superconductivity in the Hubbard model at intermediate interaction strength

We study the fluctuations responsible for pairing in the d-wave superconducting state of the two-dimensional Hubbard model at intermediate coupling within a cluster dynamical mean-field theory with a numerically exact quantum impurity solver. By analyzing how momentum- and frequency-dependent fluctuations generate the d-wave superconducting state in different representations, we identify antiferromagnetic fluctuations as the pairing glue of superconductivity in both the underdoped and the overdoped regime. Nevertheless, in the intermediate coupling regime, the predominant magnetic fluctuations may differ significantly from those described by conventional spin fluctuation theory.

Systematic lowering of the scaling of Monte Carlo calculations by partitioning and subsampling

We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations-the scaling of the statistical fluctuations as a function of the number of particles N-is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.

One- and two-particle properties of the weakly interacting two-dimensional Hubbard model in proximity to the van Hove singularity

We study the weak-coupling limit of the $t-t^\prime-U$ Hubbard model on a two-dimensional square lattice using a direct perturbative approach. Aided by symbolic computational tools, we compute the longitudinal density-density correlation functions in the $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ basis from which we can obtain the dynamical spin and charge susceptibilities at arbitrary doping and temperature. We find that for non-zero $t^\prime$, the zero frequency commensurate $\mathbf{q} = (\pi, \pi)$ spin and charge excitations are each strongest at different densities and we observe a clear behavioral change that appears tied to the van Hove singularity of the non-interacting dispersion upon which the perturbative expansion is built. We find a strongly reduced compressibility in the vicinity of the van Hove singularity as well as a behavioral change in the double occupancy. For finite $t^\prime$, the observed van Hove singularity occurs away from half-filling leading us to conclude that that this reduction in compressibility is distinct from Mott insulating physics that one expects in the strong-coupling regime. We compute the full dynamical spin and charge excitations and observe distinct structure for electron and hole doped scenarios in agreement with experiments on cuprate materials. Finally, we observe a peculiar splitting in spin and charge excitations in the vicinity of the van Hove singularity, the origin of which is traced to a splitting near the bottom of the band.

Enhanced d-wave pairing in the two-dimensional Hubbard model with periodically modulated hopping amplitudes

Utilizing determinant quantum Monte Carlo algorithm, the evolution of the d-wave pairing in the Hubbard model on the square lattice tuned by the periodically modulated hopping amplitudes is studied. The hopping amplitudes are homogeneous in the -direction, while in the -direction the hopping amplitudes are modulated with period P, where , , and the modulation period P equals 2, 3 and 4 lattice spacings. The latter two modulation periods are motivated by the observation of period-3 and period-4 stripe order in cuprate superconductors. For all the periods P, we find that the modulated hopping inhomogeneity enhances the d-wave pairing and an optimal inhomogeneity exists.

Ferromagnetic diagonal stripe states in the two-dimensional Hubbard model with U≲∞

We have performed a variational Monte Carlo simulation to study the ground state of a two-dimensional Hubbard model on a square lattice in the strong coupling region. The energy gain of possible inhomogeneous electron states are computed as a function of U when the hole density ϵ=1/8 and next nearest-neighbor hopping t′/t=−0.30. The bond-centered ferromagnetic diagonal stripe state is stabilized in the strong coupling region (U/t≥16), which is due to the gain of both kinetic energy and on-site Coulomb interaction energy due to the holon moving over the ferromagnetic domain and the gain of kinetic-exchange-interaction energy at the antiferromagnetic domain wall.

Spin Seebeck coefficient and spin-thermal diffusion in the two-dimensional Hubbard model

We investigate the spin Seebeck coefficient $S_s$ in the square lattice Hubbard model at high temperatures of relevance to cold-atom measurements. We solve the model with the finite-temperature Lanczos and with the dynamical mean-field theory methods and find they give similar results in the considered regime. $S_s$ exceeds the atomic 'Heikes' estimates and the Kelvin entropic estimates drastically. We analyze the behavior in terms of a mapping onto the problem of a doped attractive model and derive an approximate expression that allows relating the enhancement of $S_s$ to distinct scattering of the spin-majority and the spin-minority excitations. Our analysis reveals the limitations of entropic interpretations of Seebeck coefficient even in the high-temperature regime. Large values of $S_s$ could be observed on optical lattices. We also calculate the full diffusion matrix. We quantify the spin-thermal diffusion, that is, the extent of the mixing between the spin and the thermal diffusion and discuss the results in the context of recent measurements of the spin-diffusion constant in cold atoms.

Multiplicity, localization, and domains in the Hartree–Fock ground state of the two-dimensional Hubbard model

We explore certain properties of the Hartree–Fock approximation to the ground state of the two-dimensional Hubbard model, emphasizing the fact that in the Hartree approach there is an enormous multiplicity of self-consistent solutions which are nearly degenerate in energy, reminiscent of a spin glass, but which may differ substantially in other bulk properties. It is argued that this multiplicity is physically relevant at low temperatures. We study the localization properties of the one-particle wavefunctions comprising the Hartree–Fock states, and find that these are unlocalized at small and moderate values of U/t, in particular in the stripe region, but become highly localized at values corresponding to strong repulsion. We also find rectangular domains as well as stripes in the stripe region of the phase diagram, and study pair correlations in the neighborhood of half-filling.

Two-dimensional attractive Hubbard model and the BCS-BEC crossover

Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature ${T}_{c}$ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength $U$ from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ ($t$ is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing ${T}_{p}$ and degeneracy ${T}_{d}$ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.

Network-Initialized Monte Carlo Based on Generative Neural Networks

We design generative neural networks that generate Monte Carlo configurations with complete absence of autocorrelation from which only short Markov chains are needed before making measurements for physical observables, irrespective of the system locating at the classical critical point, fermionic Mott insulator, Dirac semimetal, or quantum critical point. We further propose a network-initialized Monte Carlo scheme based on such neural networks, which provides independent samplings and can accelerate the Monte Carlo simulations by significantly reducing the thermalization process. We demonstrate the performance of our approach on the two-dimensional Ising and fermion Hubbard models, expect that it can systematically speed up the Monte Carlo simulations especially for the very challenging many-electron problems.

The plain and simple parquet approximation: single-and multi-boson exchange in the two-dimensional Hubbard model

The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16times 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation.Graphical abstract

Stripes and spin-density waves in the doped two-dimensional Hubbard model: Ground state phase diagram

We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near $1/8$ doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about $1/5$ doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.

Non-Fermi liquid phase and linear-in-temperature scattering rate in overdoped two-dimensional Hubbard model

SignificanceMost metals display an electron-scattering rate [Formula: see text] that follows [Formula: see text] at low temperatures, as prescribed by Fermi liquid theory. But there are important exceptions. One of the most prominent examples is the "strange" metal regime in overdoped cuprate supercondcutors, which exhibits a linear T dependence of the scattering rate [Formula: see text] that reaches a putative Planckian limit. Here, using cutting-edge computational approaches, we show that T-linear scattering rate can emerge from the overdoped Hubbard model at low temperatures. Our results agree with cuprate experiments in various aspects but challenge the Planckian limit. Finally, by identifying antiferromagnetic fluctuations as the physical origin of the T-linear scattering rate, we discover the microscopic mechanism of strange metallicity in cuprates.

Unraveling correlated material properties with noisy quantum computers: Natural orbitalized variational quantum eigensolving of extended impurity models within a slave-boson approach

We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a Noisy, Intermediate Scale Quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the Variational Quantum Eigensolver algorithm. This iterative scheme, dubbed Natural Orbitalization (NOization), gradually transforms the single-particle basis to the approximate Natural-Orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density-matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) ansatz circuits faster and more accurate, and we propose an ansatz, the Multireference Excitation Preserving (MREP) ansatz, that achieves great expressivity without requiring a prohibitive gate count. The one-impurity version of the ansatz has only one parameter, making the ground state preparation a trivial step, which supports the optimal character of our approach. Within a Rotationally Invariant Slave Boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.

Quantum Monte Carlo Study of Weyl Superconductivity

Using auxiliary-field quantum Monte Carlo we study superconductivity in a simple extension of the two-dimensional negative-U Hubbard model which hosts Weyl nodes near the Fermi surface. We discuss the model and our method as applied to the study of superconductivity in this system. We discuss the nature of pairing between Weyl quasiparticles which carry opposite spin and opposite topological charge leading to a spin-singlet pairing amplitude and to a mixture of pseudo-spin singlet and pseudo-spin triplet pairing.