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Journal de physique

Phase transitions in an extended hubbard model

François Brouers, Carlos Mauricio Chaves, A.A. Gomés, A. Troper

Published Jan 1, 1977• 5 citations
Publisher

Abstract

We derive the free energy functional of an extended Hubbard model Hamiltonian with interatomic interactions. One has to solve a four coupled fields (two vectorial spin fields and two scalar charge fields) problem. Two are potentially soft (one spin and one charge field). We show that as far as the ferromagnetic transition is concerned, universality holds and that introducing a short range interaction between neighbours yields the same free energy functional as for the simple Hubbard model if irrelevant coupling terms are discarded. For the charge ordered transition we show using renormalization group theory arguments that the spin-charge coupling is irrelevant and that the coupling between the two charge fields can give rise to a first or second order transition depending on the details of the metallic band structure and the value of the interaction energy parameters.

DOI
10.1051/jphys:0197700380100125700

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We derive the free energy functional of an extended Hubbard model Hamiltonian with interatomic interactions. One has to solve a four coupled fields (two vectorial spin fields and two scalar charge fields) problem. Two are potentially soft (one spin and one charge field). We show that as far as the ferromagnetic transition is concerned, universality holds and that introducing a short range interaction between neighbours yields the same free energy functional as for the simple Hubbard model if irrelevant coupling terms are discarded. For the charge ordered transition we show using renormalization group theory arguments that the spin-charge coupling is irrelevant and that the coupling between the two charge fields can give rise to a first or second order transition depending on the details of the metallic band structure and the value of the interaction energy parameters.
The authors study phase transitions in the isotropic non-degenerate Hubbard Hamiltonian with renormalization group techniques using the epsilon =4-d expansion to first order in epsilon . The functional obtained from the Hubbard Hamiltonian displays full rotational symmetry and describes two coupled fields: a vector spin field, with n components and a non-soft scalar charge field. This coupling is pure imaginary, which has interesting consequences for the critical properties of this coupled field system. The effect of simple constraints imposed on the charge field is considered. The relevance of the coupling between the fields in producing Fisher renormalisation of the critical exponents is discussed. The possible singularities introduced in the charge-charge correlation function by the coupling are also discussed.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.
The extended Hubbard model in one dimension is studied for half-filled bands using a continuum theory based on the Luttinger model. Interactions going beyond the Luttinger model are treated with a renormalization group. We compute correlation functions and use these to establish the phase diagram of the model. It compares favorably with earlier numerical work. Specifically, we identify additional interaction terms coupling charge and spin degrees of freedom; one of them is shown to mediate the crossover from a continuous to a first-order transition from spin- to charge-density waves at a finite value of U=2V. Our approach suggests viewing the model on this line as two weakly coupled interpenetrating systems of up-spin and down-spin electrons with strong backscattering interactions favoring charge-density waves separately in each of the subsystems. A first-order transition occurring for attractive interactions is interpreted as a characteristic instability of the Luttinger liquid towards phase separation. We finally discuss a possible equivalence to some recent studies of coupled two-dimensional XY models.
We present a generalization of the recently proposed variational cluster perturbation theory to extended Hubbard models at half-filling with repulsive nearest neighbor interaction. The method takes into account short-range correlations correctly by the exact diagonalization of clusters of finite size, whereas long-range order beyond the size of the clusters is treated on a mean-field level. For one dimension, we show that quantum Monte Carlo and density-matrix renormalization-group results can be reproduced with very good accuracy. Moreover we apply the method to the two-dimensional extended Hubbard model on a square lattice. In contrast to the one-dimensional case, a first order phase transition between spin density wave phase and charge density wave phase is found as function of the nearest-neighbor interaction at onsite interactions $U\ensuremath{\geqslant}3t$. The single-particle spectral function is calculated for both the one-dimensional and the two-dimensional system.
We consider a modified extended Hubbard model (EHM) which, in addition to the on-site repulsion U and nearest-neighbor repulsion V, includes polarization effects in second-order perturbation theory. The model is equivalent to an EHM with renormalized U plus a next-nearest-neighbor repulsion term. Using a method based on topological quantum numbers (charge and spin Berry phases), we generalize to finite hopping t the quantum phase diagram in one dimension constructed by van den Brink et al. [Phys. Rev. Lett. 75, 4658 (1995)]. At hopping t=0t=0t=0 there are two charge density-wave phases, one spin density-wave phase, and one intermediate phase with charge and spin ordering, depending on the parameter values. At $t\ensuremath{\ne}0$ the nature of each phase is confirmed by studying correlation functions. However, in addition to the strong-coupling phases, a small region with bond ordering appears. The region occupied by the intermediate phase first increases and then decreases with increasing t, until it finally disappears for t of the order but larger than U. For small t, the topological transitions agree with the results of second-order perturbation theory.