Three-dimensional Hubbard Model Research Articles

Universality of Mean-Field Antiferromagnetic Order in an Anisotropic 3D Hubbard Model at Half-Filling

We study Hartree–Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x- and y-directions and a possibly different hopping parameter tz in the z-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases tz=0 and tz=t, respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that t=1, we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, U, and on the hopping parameter tz. We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as tz→0. It is found that the asymptotic formulas are qualitatively different for tz=0 (the two-dimensional case) and tz>0 0$$\\end{document}]]> (the case of nonzero hopping in the z-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit tz→0 in which the three-dimensional model reduces to the two-dimensional model.

Recent Progress on Ultracold-atom Quantum Simulation of Fermi-Hubbard model

Fermi-Hubbard model is a fundamental lattice model for describing correlated electron systems in condensed matter physics, with a profound connection to high-temperature superconductivity. In recent years, quantum simulation with cold atoms has emerged as an important paradigm for studying the Fermi-Hubbard model, while advancements in quantum many-body computations have contributed to our understanding of its fundamental properties. Notably, a recent ultracold-atom experiment achieved the celebrated antiferromagnetic (AFM) phase transition in the three-dimensional (3D) Hubbard model, representing a crucial step in quantum simulation that establishes a foundation for exploring the connection between the quantum magnetism and high-temperature superconductivity. This paper reviews both the experimental and theoretical progress in studying the Fermi-Hubbard model, focusing primarily on 3D systems, discusses the history and current state of developments, and outlines future directions in this field.<br>The paper is organized as follows. We begin with a review of recent progress in observing AFM phase transitions in the 3D Hubbard model, focusing on an ultracold-atom experiment conducted by the research group at the University of Science and Technology of China (USTC). Next, we provide a theoretical introduction to the fundamental properties of the 3D Hubbard model. This includes a summary of prior theoretical studies, an overview of the current research landscape, and a discussion of unresolved or under-explored issues. In the third section, we discuss the quantum simulation of the Hubbard model using ultracold atoms in optical lattices, outlining the basic principle, historical developments and key challenges. The USTC team overcame these challenges with innovative techniques such as atom cooling, large-scale uniform box traps, and precise measurements of the AFM structure factor. Their work successfully confirmed the AFM phase transition via the critical scaling analysis. Finally, we highlight the significance of this achievement and discuss future research prospects for the 3D Hubbard model, including experimental studies on the doped regime and related theoretical benchmarks.

Evaluating Second-Order Phase Transitions with Diagrammatic Monte Carlo: Néel Transition in the Doped Three-Dimensional Hubbard Model

Diagrammatic MonteCarlo-the technique for the numerically exact summation of all Feynman diagrams to high orders-offers a unique unbiased probe of continuous phase transitions. Being formulated directly in the thermodynamic limit, the diagrammatic series is bound to diverge and is not resummable at the transition due to the nonanalyticity of physical observables. This enables the detection of the transition with controlled error bars from an analysis of the series coefficients alone, avoiding the challenge of evaluating physical observables near the transition. We demonstrate this technique by the example of the Néel transition in the 3D Hubbard model. At half filling and higher temperatures, the method matches the accuracy of state-of-the-art finite-size techniques, but surpasses it at low temperatures and allows us to map the phase diagram in the doped regime, where finite-size techniques struggle from the fermion sign problem. At low temperatures and sufficient doping, the transition to an incommensurate spin density wave state is observed.

Ferromagnetism in the Three-dimensional Hubbard Model with Generalized Type of Electron Hopping

Ferromagnetism in the Three-dimensional Hubbard Model with Generalized Type of Electron Hopping

Phase transitions in partial summation methods: Results from the three-dimensional Hubbard model

The accurate determination of magnetic phase transitions in electronic systems is an important task of solid state theory. While numerically exact results are readily available for model systems such as the half-filled 3D Hubbard model, the complexity of real materials requires additional approximations, such as the restriction to certain classes of diagrams in perturbation theory, that reduce the precision with which magnetic properties are described. In this work, we examine the description of magnetic properties in second order perturbation theory, GW, FLEX, and two TMatrix approximations to numerically exact CT-QMC reference data. We assess finite-size effects and compare periodic lattice simulations to cluster embedding. We find that embedding substantially improves finite size convergence. However, by analyzing different partial summation methods we find no systematic improvement in the description of magnetic properties, with most methods considered in this work predicting first-order instead of continuous transitions, leading us to the conclusion that non-perturbative methods are necessary for the accurate determination of magnetic properties and phase transitions.

Three-dimensional stacking of canted antiferromagnetism and pseudospin current in undoped Sr2IrO4 : Symmetry analysis and microscopic model realization

Recent optical second-harmonic generation experiments observed unexpected broken spatial symmetries in the undoped spin-orbit Mott insulator Sr$_2$IrO$_4$, leading to intensive debates on the nature of its ground state. We propose that it is a canted antiferromagnetism with a hidden order of circulating staggered pseudospin current. Symmetry analysis shows that a proper $c$-axis stacking of the canted antiferromagnetism and the pseudospin current lead to a magnetoelectric coexistence state that breaks the two-fold rotation, inversion, and time-reversal symmetries, consistent with experimental observations. We construct a three-dimensional Hubbard model with spin-orbit coupling for the five localized 5$d$ Wannier orbitals centered at Ir sites, and demonstrate the microscopic realization of the desired coexistence state in a wide range of band parameters via a combination of self-consistent Hartree-Fock and variational calculations.

Disorder-controlled relaxation in a three-dimensional Hubbard model quantum simulator

Understanding the collective behavior of strongly correlated electrons in materials remains a central problem in many-particle quantum physics. A minimal description of these systems is provided by the disordered Fermi-Hubbard model (DFHM), which incorporates the interplay of motion in a disordered lattice with local inter-particle interactions. Despite its minimal elements, many dynamical properties of the DFHM are not well understood, owing to the complexity of systems combining out-of-equilibrium behavior, interactions, and disorder in higher spatial dimensions. Here, we study the relaxation dynamics of doubly occupied lattice sites in the three-dimensional (3D) DFHM using interaction-quench measurements on a quantum simulator composed of fermionic atoms confined in an optical lattice. In addition to observing the widely studied effect of disorder inhibiting relaxation, we find that the cooperation between strong interactions and disorder also leads to the emergence of a dynamical regime characterized by \textit{disorder-enhanced} relaxation. To support these results, we develop an approximate numerical method and a phenomenological model that each capture the essential physics of the decay dynamics. Our results provide a theoretical framework for a previously inaccessible regime of the DFHM and demonstrate the ability of quantum simulators to enable understanding of complex many-body systems through minimal models.

Electronic and fluctuation dynamics following a quench to the superconducting phase

We investigate the dynamics of superconducting fluctuations in the attractive three-dimensional Hubbard model after a quench from the disordered phase to the ordered regime. While the long time evolution is well understood in terms of dissipative time-dependent Ginzburg-Landau models with unstable potentials, early times are more demanding due to the inseparable dynamics of the pairing fluctuations and the electronic quasiparticles. Our simulation using the time-dependent fluctuation exchange approximation treat both degrees of freedom on the same footing and reveal a non-thermal electronic regime causing a non-monotonous growth of the fluctuations. This feature is not directly captured from the Ginzburg-Landau theory, but nevertheless remains observable beyond the thermalization time of the electrons. We further explore how the growth of the order parameter fluctuations leads to an opening of a pseudo-gap in the electronic spectrum, and identify Andreev reflections as the dominant mechanism behind the gap opening.

Functional renormalization group for fermion lattice models in three dimensions: Application to the Hubbard model on the cubic lattice

The channel-decomposed functional renormalization group (FRG) approach, most recently in the variant of truncated-unity-(TU-)FRG, has so far been used for various two-dimensional model systems. Yet, for many interesting material systems the third spatial dimension is of clear relevance. Therefore FRG schemes working in three spatial dimensions (3D) are definitely on the wishlist. Here we demonstrate that a 3D TUFRG scheme can be set up in straightforward extension of previous 2D codes and gives physically sensible results with affordable numerical effort, both regarding the qualitative as well as the quantitative description. The computed phase diagram of the three-dimensional Hubbard model at half filling or perfect nesting shows a phase transition to a \((\pi,\pi,\pi)\)-ordered antiferromagnetic ground state for repulsive interactions at an energy scale that compares well with other numerical approaches in the literature. Furthermore, the method allowed us to detect a \(d\)-wave pairing and a concurring \((\pi,\pi,0)\) antiferromagnetic ground state in the hole doped Hubbard model.

Approximate expression for the ground-state energy of the two- and three-dimensional Hubbard model at arbitrary filling obtained from dimensional scaling

We generalize the linear discrete dimensional scaling approach for the repulsive Hubbard model to obtain a nonlinear scaling relation that yields accurate approximations to the ground-state energy in both two and three dimensions, as judged by comparison to auxiliary-field quantum Monte Carlo (QMC) data. Predictions are made for the per-site ground-state energies in two and three dimensions for n (filling factor) and U (Coulomb interaction) values for which QMC data are currently unavailable.

Efficient evaluation of the polarization function in dynamical mean-field theory

The dynamical susceptibility of strongly correlated electronic systems can be calculated within the framework of the dynamical mean-field theory (DMFT). The required measurement of the four-point vertex of the auxiliary impurity model is however costly and restricted to a finite grid of Matsubara frequencies, leading to a cutoff error. It is shown that the propagation of this error to the lattice response function can be minimized by virtue of an exact decomposition of the DMFT polarization function into local and nonlocal parts. The former is measured directly by the impurity solver, while the latter is given in terms of a ladder equation for the Hedin vertex that features an unprecedentedly fast decay of frequency summations compared to previous calculation schemes, such as the one of the dual boson approach. At strong coupling the local approximation of the TRILEX approach is viable, but vertex corrections to the polarization should be dropped on equal footing to recover the correct prefactor of the effective exchange. In finite dimensions the DMFT susceptibility exhibits spurious mean-field criticality, therefore, a two-particle self-consistent and frequency-dependent correction term is introduced, similar to the Moriya-$\lambda$ correction of the dynamical vertex approximation. Applications to the two- and three-dimensional Hubbard models on the square and cubic lattices show that the expected critical behavior near an antiferromagnetic instability is recovered.

Three-dimensional Hubbard model in the thermodynamic limit

We employ the numerical linked-cluster expansion to study finite-temperature properties of the uniform cubic lattice Hubbard model in the thermodynamic limit for a wide range of interaction strengths and densities. We carry out the expansion to the 9th order and find that the convergence of the series extends to lower temperatures as the strength of the interaction increases, giving us access to regions of the parameter space that are difficult to reach by most other numerical methods. We study the precise trends in the specific heat, the double occupancy, and magnetic correlations at temperatures as low as 0.2 of the hopping amplitude in the strong-coupling regime. We show that in this regime, accurate estimates for transition temperatures to the N\'{e}el ordered phase, in agreement with the predicted asymptotic behavior, can be deduced from the low-temperature magnetic structure factor. We also find evidence for possible instability to the magnetically ordered phase away from, but close to, half filling. Our results have important implications for parametrizing fermionic systems in optical lattice experiments and for benchmarking other numerical methods.

Cooling Atomic Gases With Disorder.

Cold atomic gases have proven capable of emulating a number of fundamental condensed matter phenomena including Bose-Einstein condensation, the Mott transition, Fulde-Ferrell-Larkin-Ovchinnikov pairing, and the quantum Hall effect. Cooling to a low enough temperature to explore magnetism and exotic superconductivity in lattices of fermionic atoms remains a challenge. We propose a method to produce a low temperature gas by preparing it in a disordered potential and following a constant entropy trajectory to deliver the gas into a nondisordered state which exhibits these incompletely understood phases. We show, using quantum MonteCarlo simulations, that we can approach the Néel temperature of the three-dimensional Hubbard model for experimentally achievable parameters. Recent experimental estimates suggest the randomness required lies in a regime where atom transport and equilibration are still robust.

Mechanisms of finite-temperature magnetism in the three-dimensional Hubbard model

We examine the nature of the transition to the antiferromagnetically ordered state in the half-filled three-dimensional Hubbard model using the dual-fermion multiscale approach. Consistent with analytics, in the weak-coupling regime we find that spin-flip excitations across the Fermi surface are important, and that the strong coupling regime is described by Heisenberg physics. In the intermediate interaction, strong correlation regime we find aspects of both local and non-local correlations. We analyze the critical exponents of the transition in the strong coupling regime and find them to be consistent with Heisenberg physics down to an interaction of $U/t=10$.

Local origin of the pseudogap in the attractive Hubbard model

We provide a fresh perspective on the pseudogap physics for attractive fermions as described by the three-dimensional Hubbard model. The pseudogap in the single-particle spectral function, which occurs for temperatures above the critical temperature ${T}_{c}$ of the superfluid transition, is often interpreted in terms of preformed uncondensed pairs. Here we show that the occurrence of pseudogap physics can be consistently understood as due to local excitations which lead to a splitting of the quasiparticle peak for sufficiently large interaction. This effect becomes prominent at intermediate and high temperatures when the quantum-mechanical hopping is incoherent. We demonstrate the absence of the conjectured pairing temperature below which pseudogap physics is expected to occur. Our results are based on approximating the physics of the three-dimensional Hubbard model by dynamical mean-field theory calculations and a momentum independent self-energy. Our predictions can be tested with ultracold atoms in optical lattices with currently available temperatures and spectroscopic techniques.

Collective charge excitations of strongly correlated electrons, vertex corrections, and gauge invariance

We consider the collective, long-wavelength charge excitations in correlated media in presence of short- and long-range forces. As an example for the case of a short-range interaction, we examine the two-dimensional Hubbard model within dynamical mean-field theory (DMFT). It is shown explicitly that the DMFT susceptibility including vertex corrections respects the Ward identity and yields a manifestly gauge-invariant response in finite dimensions. For computing the susceptibility, we use a different expression and establish its formal equivalence to the standard DMFT formula. It allows for a more stable analytical continuation. We find a zero-sound mode expected for short-range forces. The relation between the vertex corrections, gauge invariance, and the appearance of the collective modes is discussed. Long-range forces are treated within extended dynamical mean-field theory. In order to obtain a gauge-invariant response, it is necessary to additionally incorporate some non local vertex corrections into the polarization. In doing so, we obtain plasmons in the three-dimensional Hubbard model. The plasma frequency is determined by the (single-particle) density distribution as a consequence of gauge invariance. We compare this result with the plasma frequency extracted from the analytical continuation of the susceptibility. It is in good agreement with the prediction from the gauge-invariance condition.

Néel temperature and thermodynamics of the half-filled three-dimensional Hubbard model by diagrammatic determinant Monte Carlo

We study the thermodynamics of the three-dimensional Hubbard model at half filling on approach to the N\'eel transition by means of large-scale unbiased diagrammatic determinant Monte Carlo simulations. We obtain the transition temperature in the strongly correlated regime, as well as the temperature dependence of the energy, entropy, double occupancy, and nearest-neighbor spin correlation function. Our results improve the accuracy of previous unbiased studies and present accurate benchmarks in the ongoing effort to realize the antiferromagnetic state of matter with ultracold atoms in optical lattices.

Erratum: Spectral properties of the three-dimensional Hubbard model [Phys. Rev. B83, 235113 (2011)

Erratum: Spectral properties of the three-dimensional Hubbard model [Phys. Rev. B<b>83</b>, 235113 (2011)

Spectral properties of the three-dimensional Hubbard model

We present momentum resolved single-particle spectra for the three-dimensional Hubbard model for the paramagnetic and antiferromagnetically ordered phase obtained within the dynamical cluster approximation. The effective cluster problem is solved by continuous-time Quantum Monte Carlo simulations. The absence of a time discretization error and the ability to perform Monte Carlo measurements directly in Matsubara frequencies enable us to analytically continue the self-energies by maximum entropy, which is essential to obtain momentum resolved spectral functions for the N'eel state. We investigate the dependence on temperature and interaction strength and the effect of magnetic frustration introduced by a next-nearest neighbor hopping. One particular question we address here is the influence of the frustrating interaction on the metal insulator transition of the three-dimensional Hubbard model.

Submatrix updates for the continuous-time auxiliary-field algorithm

We present a submatrix update algorithm for the continuous-time auxiliary field method that allows the simulation of large lattice and impurity problems. The algorithm takes optimal advantage of modern CPU architectures by consistently using matrix instead of vector operations, resulting in a speedup of a factor of $\ensuremath{\approx}$8 and thereby allowing access to larger systems and lower temperature. We illustrate the power of our algorithm at the example of a cluster dynamical mean field simulation of the N\'eel transition in the three-dimensional Hubbard model, where we show momentum dependent self-energies for clusters with up to 100 sites.