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arXiv:cond-mat/9305005 · Journal of Physics Condensed Matter

Antiferromagnetic symmetry breaking in the half-filled Hubbard model in infinite dimensions

Peter Kopietz

Published Jun 27, 1994• 1 citation
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Abstract

We study the half-filled Hubbard model on a hypercubic lattice in infinite dimensions in the presence of a staggered magnetic field. Using the analyticity of the Anderson impurity model and assuming that for small enough on-site repulsion U the high-temperature phase is a Fermi liquid, we show that at weak coupling the nesting property of the non-interacting band-structure necessarily leads to antiferromagnetic symmetry breaking at a finite temperature.

DOI
10.1088/0953-8984/6/26/010

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We study the half-filled Hubbard model on a hypercubic lattice in infinite dimensions in the presence of a staggered magnetic field. Using the analyticity of the Anderson impurity model and assuming that for small enough on-site repulsion U the high-temperature phase is a Fermi liquid, we show that at weak coupling the nesting property of the non-interacting band-structure necessarily leads to antiferromagnetic symmetry breaking at a finite temperature.
We present a detailed, quantitative study of the competition between interaction- and disorder-induced effects in electronic systems. For this the Hubbard model with diagonal disorder (Anderson-Hubbard model) is investigated analytically and numerically in the limit of infinite spatial dimensions, i.e., within a dynamical mean-field theory, at half-filling. Numerical results are obtained for three different disorder distributions by employing quantum Monte Carlo techniques, which provide an explicit finite-temperature solution of the model in this limit. The magnetic phase diagram is constructed from the zeros of the inverse averaged staggered susceptibility. We find that at low enough temperatures and sufficiently strong interaction there always exists a phase with antiferromagnetic long-range order. A strong coupling anomaly, i.e., an increase of the N\'eel temperature for increasing disorder, is discovered. An explicit explanation is given, which shows that in the case of diagonal disorder this is a generic effect. The existence of metal-insulator transitions is studied by evaluating the averaged compressibility both in the paramagnetic and antiferromagnetic phases. A rich transition scenario, involving metal-insulator and magnetic transitions, is found and its dependence on the choice of the disorder distribution is discussed.
We present an analytic solution of the self consistent random phase approximation equations for the half-filled Hubbard model, in the strong coupling limit. In the half-filled limit the Hubbard model shows antiferromagnetic ordering. The self energy is calculated by considering processes whereby the electrons emit and absorb low frequency spin waves, which are the Goldstone modes of the broken symmetry phase. We examine the electronic spectrum, which is dominated by the broad incoherent portions of the spectrum associated with the upper and lower Hubbard bands. The quasi-particle spectrum has a divergent wave function renormalization, indicating a divergent renormalization of the quasi-particle mass and zero quasi-particle weight, as in a non-Fermi liquid. The self consistency equation determined from the Goldstone modes in the susceptibility shows that the antiferromagnetism is prohibited in one and two dimensions but not in three. The magnetic ordering temperature is determined by dimensionality and the magnitude of the Anderson kinetic exchange of the effective Heisenberg Hamiltonian.
The influence of local disorder on the thermodynamics of interacting electrons is studied within the infinite-dimensional disordered Hubbard model.Using a finite-temperature quantum Monte Carlo method, the averaged local moment and staggered susceptibility are calculated and the magnetic phase diagram at half-filling is constructed.From the averaged compressibility in the paramagnetic and antiferromagnetic phase we determine the metalinsulator transitions of the system.A rich transition scenario is revealed.Quite unexpectedly the disorder is found to stabilize the magnetic order in the strong-coupling limit.Randomness can significantly influence the low-temperature behaviour of interacting systems.In particular, it may lead to new phases which have no analogue in non-random systems [l].In the case of electrons, the simplest lattice model including both interactions and random potentials is the disordered Hubbard model.It is now known that even in the .mean-field.limit d + CQ [2,3] the Hubbard interaction remains dynamical [4] and leads to a highly non-trivial single-site problem with infinitely many coupled quantum degrees of freedom [5].This problem is, in fact, equivalent to an Anderson impurity model complemented by a self-consistency condition [6] and is thus amenable to numerical investigations [7] within a finite-temperature quantum Monte Carlo approach [81.In the absence of disorder this technique was already used by several groups to investigate the magnetic phase diagram [7,9], the Mott-Hubbard transition [lo, 1 1 1 and lately also superconductivity in a two-band version [12] of the Hubbard model in d = C Q ; thereby important new insight was gained.In this letter we wish to concentrate on the combined effect of disorder and interaction.We will study the effect of diagonal disorder on the low-temperature properties of the Hubbard model in d = CQ , in particular the competition between correlation-induced
We present and study the infinite-dimensional limit of the Hubbard model on a class of non-nested bipartite lattices which generalize the two-dimensional honeycomb and three-dimensional diamond lattice, and are characterized by a semimetallic noninteracting density of states. The infinite-dimensional limit is studied by the well-known mapping onto a self-consistent one-impurity problem. This is solved using quantum Monte Carlo and second-order perturbation theory. The (U,T) phase diagram at half-filling shows a nonmagnetic semimetallic region and an antiferromagnetic insulating phase with a critical value of U for the transition at T=0 which is strictly positive, Uc{\mathit{U}}_{\mathit{c}}Uc​/t\ensuremath{\approxeq}2.3, in contrast with the hypercubic lattice, where antiferromagnetic order sets in at Uc{\mathit{U}}_{\mathit{c}}Uc​=0.
The antiferromagnetic Ne\'el temperature TN{T}_{N}TN​ is calculated as a function of the Coulomb interaction U for a half-filled 3D Hubbard model. Numerical results are given for the specific heat and staggered magnetic suspectibility on periodic lattices ranging from 43{4}^{3}43--103{10}^{3}103 sites, and used to determine TN{T}_{N}TN​. The size dependence of the results suggests that the transition is weakly first order at small to intermediate values of U.