2D Hubbard Model Research Articles

Nonlocal Order Parameters for the 1D Hubbard Model

We characterize the Mott-insulator and Luther-Emery phases of the 1D Hubbard model through correlators that measure the parity of spin and charge strings along the chain. These nonlocal quantities order in the corresponding gapped phases and vanish at the critical point U(c)=0, thus configuring as hidden order parameters. The Mott insulator consists of bound doublon-holon pairs, which in the Luther-Emery phase turn into electron pairs with opposite spins, both unbinding at U(c). The behavior of the parity correlators is captured by an effective free spinless fermion model.

Truncated configuration interaction expansions as solvers for correlated quantum impurity models and dynamical mean-field theory

The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nano-science to the dynamical mean-field theory (DMFT). Here we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the 1D Hubbard model and $2\times 2$ cluster DMFT of the 2D Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including "difficult" regimes, such as in the pseudogap phase of the 2D Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the $2\times 2$ cluster DMFT using a 24 bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.

Kohn-Luttinger effect and anomalous pairing in repulsive Fermi-systems at low density (Review Article)

We demonstrate the possibility of triplet p-wave pairing at low electron density a large number of models such as 3D and 2D Fermi-gas models with hard-core repulsion, 3D and 2D Hubbard models, and the Shubin-Vonsovsky model. The critical temperature for p-wave pairing can be considerably higher in the spin-polarized case or even in a two-band situation at low density and can reach experimentally observable values of 1–5 K. We also discuss briefly the d-wave pairing and high-Tc superconductivity with Tc ∼ 100 K which arise in the extended Hubbard model and in the generalized t-J model when close to half-filling.

Spin-independent V-representability of Wigner crystal oscillations in one-dimensional Hubbard chains: The role of spin-charge separation

Electrons in one dimension display the unusual property of separating their spin and charge into two independent entities: The first, which derives from uncharged spin-$1/2$ electrons, can travel at different velocities when compared with the second, which is built from charged spinless electrons. Predicted theoretically in the early 1960s, spin-charge separation has attracted renewed attention since the first evidences of experimental observation, with usual mentions as a possible explanation for high-temperature superconductivity. In one-dimensional (1D) model systems, spin-charge separation causes the frequencies of Friedel oscillations to suffer a $2{k}_{F}\ensuremath{\rightarrow}4{k}_{F}$ crossover, mainly when dealing with strong correlations, where they are referred to as Wigner crystal oscillations. In nonmagnetized systems, the current density functionals that are applied to the 1D Hubbard model are not seen to reproduce this crossover, which leads to a more fundamental question: Are the Wigner crystal oscillations in 1D systems noninteracting V-representable? Or, is there a spin-independent Kohn-Sham potential that is able to yield spin-charge separation? Finding an appropriate answer to both questions is our main task here. By means of exact and density matrix renormalization group solutions, as well as an exchange-correlation potential introduced here, we show the answer to be positive. Specifically, the V-representable $4{k}_{F}$ oscillations emerge from attractive interactions mediated by positively charged spinless holes---the holons---as an additional contribution to the repulsive on-site Hubbard interaction.

Finite temperature properties of the 2D Hubbard model extended by next-nearest-neighbor hopping interaction

We present an exact diagonalization study on a two dimensional Hubbard model extended by next-nearest-neighbor (NNN) hopping for a small cluster. The thermodynamic properties of spin–spin correlation, local moment, double occupancy, entropy, specific heat are studied for different U and NNN hopping interactions (t′). The increase of NNN hopping interaction favors the formation of doublans and hence decreases local moment in the low temperature region. Transition temperature (Tc) region is identified from entropy curves. Characteristic double peak structure is observed in the specific heat curves with the inclusion of NNN hopping interaction.

Time and spatially resolved quench of the fermionic Hubbard model showing restricted equilibration

We investigate the quench of half-filled 1D and 2D fermionic Hubbard models to models without Coulomb interaction. Since the time propagation is gaussian we can use a variety of time-dependent quantum Monte Carlo methods to tackle this problem without generating a dynamical sign problem. Using a continuous time quantum Monte Carlo method (CTQMC) we achieve a system size of 128 sites in 1D, and using a Blankenbecler-Scalapino-Sugar (BSS) type algorithm we were able to simulate 20 x 20 square lattices. Applying these methods to study the dynamics after the quench, we observe that the final state of the system can be reasonably well described by a thermal single-particle density matrix that takes the initial single particle conservation laws into account. The characteristic decay towards this limit is found to be oscillatory with an additional power law decay that depends on the dimensionality. This numerically exact result is shown to compare favorable to mean-field approximations as well as to perturbation theory. Furthermore we observe the information propagation in the 1D-case in the charge charge and spin spin correlations and find that it is linear with a velocity of roughly v = 4 in units of the hopping amplitude.

T = 0 phase diagram of the 1D Hubbard model with magnetic interactions in the narrow band limit

AbstractIn this paper we study a generalization of the Hubbard model by considering spin-spin interactions described by the exchange constant J. An external magnetic field his also taken into account. In the narrowband limit and for the 1D case, we present the exact solution obtained in the framework of the Green’s function formalism, using the Composite Operator Method. We report the T = 0 phase diagram for both ferro (J > 0) and anti-ferro (J < 0) couplings. The competition of the different energy scales (U, J, and h; being U the local charge interaction) generates a variety of phases and different charge and spin orderings.

Role of the van Hove singularity in the quantum criticality of the Hubbard model

A quantum critical point (QCP), separating the non-Fermi liquid region from the Fermi liquid, exists in the phase diagram of the 2D Hubbard model [Vidhyadhiraja et. al, Phys. Rev. Lett. 102, 206407 (2009)]. Due to the vanishing of the critical temperature associated with a phase separation transition, the QCP is characterized by a vanishing quasiparticle weight. Near the QCP, the pairing is enhanced since the real part of the bare d-wave p-p susceptibility exhibits algebraic divergence with decreasing temperature, replacing the logarithmic divergence found in a Fermi liquid [Yang et. al, Phys. Rev. Lett. 106, 047004 (2011)]. In this paper we explore the single-particle and transport properties near the QCP. We focus mainly on a van Hove singularity (vHS) coming from the relatively flat dispersion that crosses the Fermi level near the quantum critical filling. The flat part of the dispersion orthogonal to the antinodal direction remains pinned near the Fermi level for a range of doping that increases when we include a negative next-near-neighbor hopping t' in the model. For comparison, we calculate the bare d-wave pairing susceptibility for non-interacting models with the usual two-dimensional tight binding dispersion and a hypothetical quartic dispersion. We find that neither model yields a vHS that completely describes the critical algebraic behavior of the bare d-wave pairing susceptibility. The resistivity, thermal conductivity, thermopower, and the Wiedemann-Franz Law are examined in the Fermi liquid, marginal Fermi liquid, and pseudo-gap doping regions. A negative next-near-neighbor hopping t' increases the doping region with marginal Fermi liquid character. Both T and negative t' are relevant variables for the QCP, and both the transport and the motion of the vHS with filling suggest that they are qualitatively similar in their effect.

Manifestation of the upper Hubbard band in the 2D Hubbard model at low electron density

We consider the 2D Hubbard model in the strong-coupling case (U ≫ W) and at low electron density (nd2 ≪ 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two holes below the bottom of the band for U ≫ W and low electron density. Both poles produce nontrivial corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D.

Classification of parameter-dependent quantum integrable models, their parameterization, exact solution and other properties

We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite N × N real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals. A type M family in our definition is formed by N–M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all type 1, 2 and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy-level crossings in type M integrable systems and analyze the taxonomy of types in the 1D Hubbard model.

Exact diagonalization study of 2D Hubbard model on honeycomb lattice: Semi-metal to insulator transition

Phase transition in a honeycomb lattice is studied by the means of the two-dimensional Hubbard model and the exact diagonalization dynamical mean field theory at zero temperature. At low energies, the dispersion relation is shown to be a linear function of the momentum. In the limit of weak interactions, the system is in the semi-metal phase. By increasing the on site interaction a semi-metal to insulator transition takes place in the paramagnetic phase. Calculation of double occupancy shows such a phase transition is of the second order. The respective phase transition point and critical on-site interaction are determined using renormalized Fermi velocity factor.

Expansion Dynamics in the One-Dimensional Fermi-Hubbard Model

Expansion dynamics of interacting fermions in a lattice is simulated within the one-dimensional (1D) Hubbard model, using the essentially exact time-evolving block decimation (TEBD) method. In particular, the expansion of an initial band-insulator state is considered. We analyze the simulation results based on the dynamics of a two-site two-particle system, the so-called Hubbard dimer. Our findings describe essential features of a recent experiment on the expansion of a Fermi gas in a two-dimensional lattice. We show that the Hubbard-dimer dynamics, combined with a two-fluid model for the paired and nonpaired components of the gas, gives an efficient description of the full dynamics. This should be useful for describing dynamical phenomena of strongly interacting Fermions in a lattice in general.

Doublon–holon-binding mechanism of Mott transition in 2D Hubbard model

The mechanism of nonmagnetic Mott transitions in the Hubbard model on the square lattice is studied, using a variational Monte Carlo method. A simple doublon (D)–holon (H) binding mechanism a previous study proposed [J. Phys. Soc. Jpn. 75 (2006) 114706] has to be modified, because even a wave function with completely bound D–H pairs brings about a Mott transition at a finite correlation strength. By introducing two characteristic lengths, D–H pair binding length, ξDH, and minimum inter-doublon distance, ξDD, we can properly describe the physics of Mott transitions, and determine the critical point by ξDD∼ξDH. This concept seems universal, because it is valid not only for newly introduced wave functions with long-range D–H and D–D (H–H) correlation factors discussed here, but for a wide range of wave functions with D–H binding factors.

Time-Dependent Density-Functional Theory Meets Dynamical Mean-Field Theory: Real-Time Dynamics for the 3D Hubbard Model

We introduce a new class of exchange-correlation potentials for a static and time-dependent density-functional theory of strongly correlated systems in 3D. The potentials are obtained via dynamical mean-field theory and, for strong enough interactions, exhibit a discontinuity at half-filling density, a signature of the Mott transition. For time-dependent perturbations, the dynamics is described in the adiabatic local density approximation. Results from the new scheme compare very favorably to exact ones in clusters. As an application, we study Bloch oscillations in the 3D Hubbard model.

Superconductivity enhanced by d-density wave: A weak-coupling theory

Making a revision of mistakes in Ref. [19], we present a detailed study of the competition and interplay between the d-density wave (DDW) and d-wave superconductivity (DSC) within the fluctuation-exchange (FLEX) approximation for the two-dimensional (2D) Hubbard model. In order to stabilize the DDW state with respect to phase separation at lower dopings a small nearest-neighbor Coulomb repulsion is included within the Hartree–Fock approximation. We solve the coupled gap equations for the DDW, DSC, and π-pairing as the possible order parameters, which are caused by exchange of spin fluctuations, together with calculating the spin fluctuation pairing interaction self-consistently within the FLEX approximation. We show that even when nesting of the Fermi surface is perfect, as in a square lattice with only nearest-neighbor hopping, there is coexistence of DSC and DDW in a large region of dopings close to the quantum critical point (QCP) at which the DDW state vanishes. In particular, we find that in the presence of DDW order the superconducting transition temperature T c can be much higher compared to pure superconductivity, since the pairing interaction is strongly enhanced due to the feedback effect on spin fluctuations of the DDW gap. π-pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small. In the present work, we have developed a weak-coupling theory of the competition between DDW and DSC in 2D Hubbard model, using the static spin fluctuation obtained within FLEX approximation and ignoring the self-energy effect of spin fluctuations. For our model calculations in the weak-coupling limit we have taken U/ t=3.4, since the antiferromagnetic instability occurs for higher values of U/ t.

Thermodynamics of the 3D Hubbard Model on Approaching the Néel Transition

We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Néel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S/N ≈ 0.65(6) at U/t = 8 is sufficient to achieve a Néel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.

Active-SpaceN-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

The ground-state energy of a system of fermions can be calculated by minimizing a linear functional of the two-particle reduced density matrix (2-RDM) if an accurate set of N-representability conditions is applied. In this Letter we introduce a class of linear N-representability conditions based on exact calculations on a reduced active space. Unlike wave-function-based approaches, the 2-RDM methodology allows us to combine information from calculations on different active spaces. By adding active-space constraints, we can iteratively improve our estimate for the ground-state energy. Applying our methodology to a 1D Hubbard model yields a significant improvement over traditional 2-positivity constraints with the same computational scaling.

The two dimensional Hubbard model: a theoretical tool for molecular electronics

When speaking about molecular electronics, the obvious question which occurs is how does one study it theoretically. The simplest theoretical model suitable for application in molecular electronics is the two dimensional Hubbard model. The aim of the present paper is to introduce this model, and give some examples of the systems which it can describe. After a short mathematically oriented discussion, it will be shown how to calculate the electrical conductivity of a particular planar system: a rectangular lattice with mutually independent conductivities along the two axes,but without using the 2D Hamiltonian. This system could find applications in high Tc studies. It will finally be shown that the electrical conductivity of graphene can be determined not by using the full formalism of the 2D Hubbard model, but by a slight reformulation of the Hamiltonian of the 1D Hubbard model.

Cluster perturbation theory in Hubbard model exactly taking into account the short-range magnetic order in 2 × 2 cluster

The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.

New superconductivity induced by d-density wave in high-temperature superconductors

We present a self-consistent theory of the interplay between the d-density wave (DDW) and superconductivity on the basis of the fluctuation-exchange (FLEX) approximation for the two-dimensional (2D) Hubbard model with the nearest-neighbor hopping t and on-site Coulomb repulsion U. In order to stabilize the DDW state with respect to phase separation at lower dopings a small nearest-neighbor Coulomb repulsion V is included within the Hartree–Fock approximation. The competition between the DDW and superconductivity can generically result in the appearance of a new superconducting state, which has the π-pairing order parameter. We solve the gap equations for competing DDW, BCS, and π-pairing order parameters with d-wave symmetry, together with calculating the FLEX interaction self-consistently, and calculating the corresponding three transition temperatures T ⁎, T c , and T c ⁎ as a function of hole doping. The calculated phase diagram has a quantum critical point (QCP) separating a pure BCS superconducting state at large dopings δ> δ 0 from a coexisting DDW and π-pairing state at lower dopings δ< δ 0. Remarkable is the much higher value of the T c ⁎ for the π-pairing superconductivity as compared with the pure BCS superconductivity.