We supplement (and critically overview) the existing extensive analysis of antiferromagnetic solution for the Hubbard model with a detailed discussion of two specific features, namely (i) the evolution of the magnetic (Slater) gap (here renormalized by the electronic correlations) into the Mott-Hubbard or atomic gap, and (ii) a rather weak renormalization of the effective mass by the correlations in the half-filled-band case, which contrasts with that for the paramagnetic case. The mass remains strongly enhanced in the non-half-filled-band case. We also stress the difference between magnetic and non-magnetic contributions to the gap. These results are discussed within the slave boson approach in the saddle-point approximation, in which there appears a non-linear staggered molecular field due to the electronic correlations that leads to the appearance of the magnetic gap. They reproduce correctly the ground-state energy in the limit of strong correlations. A brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly-degenerate-band case with one electron per atom is also made. 1 1
We supplement (and critically overview) the existing extensive analysis of antiferromagnetic solution for the Hubbard model with a detailed discussion of two specific features, namely (i) the evolution of the magnetic (Slater) gap (here renormalized by the electronic correlations) into the Mott-Hubbard or atomic gap, and (ii) a rather weak renormalization of the effective mass by the correlations in the half-filled-band case, which contrasts with that for the paramagnetic case. The mass remains strongly enhanced in the non-half-filled-band case. We also stress the difference between magnetic and non-magnetic contributions to the gap. These results are discussed within the slave boson approach in the saddle-point approximation, in which there appears a non-linear staggered molecular field due to the electronic correlations that leads to the appearance of the magnetic gap. They reproduce correctly the ground-state energy in the limit of strong correlations. A brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly-degenerate-band case with one electron per atom is also made. 1 1
PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength U being its parameters. At half-filling, a relation between the staggered magnetization M and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small U limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at UMott=10t. The interaction Uc for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than UMott at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.
We describe a low-temperature approach to the two-dimensional half-filled Hubbard model which allows us to study both antiferromagnetism and single-particle properties. This approach ignores amplitude fluctuations of the antiferromagnetic (AF) order parameter and is valid below a crossover temperature TX which marks the onset of AF short-range order. Directional fluctuations (spin waves) are described by a nonlinear sigma model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ that we derive from the Hubbard model. The parameters of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$---the spin stiffness and spin-wave velocity---are calculated as a function of the Coulomb repulsion U. The $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is solved by a saddle-point approximation within the CP1 representation where the N\'eel field is parametrized by two Schwinger bosons. At zero temperature, there is always Bose condensation of the Schwinger bosons, which signals AF long-range order for any value of the Coulomb repulsion. At finite temperature, the AF long-range order is suppressed (in agreement with the Mermin-Wagner theorem), but the AF correlation length remains exponentially large. In the CP1 representation, the fermion field is naturally expressed as the product of a Schwinger boson and a pseudofermion whose spin is quantized along the (fluctuating) N\'eel field. This allows us to write the fermion Green's function as the product (in direct space) of the Schwinger boson propagator (which is derived from the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ and the pseudofermion propagator. At zero temperature and weak coupling, our results are typical of a Slater antiferromagnet. The AF gap is exponentially small; there are well-defined Bogoliubov quasiparticles (QP's) (carrying most of the spectral weight) coexisting with a high-energy incoherent excitation background. As U increases, the Slater antiferromagnet progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands evolve into Mott-Hubbard bands separated by a large AF gap. A significant fraction of spectral weight is transferred from the Bogoliubov QP's to incoherent excitations. At finite temperature, there is a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling. Finally, we point out that our results straightforwardly translate to the half-filled attractive Hubbard model, where the $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge and q=0 pairing fluctuations combine to form an order parameter with SO(3) symmetry.
We introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.