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Journal de physique

Discussion on the onset of antiferromagnetism in the hubbard model

François Brouers, F. Ducastelle, J. Giner

Published Jan 1, 1976• 16 citations
Publisher

Abstract

We calculate the response to a periodical external magnetic field of the magnetically disordered phase of the Hubbard model in the alloy analogy approximation. We investigate the behaviour of the q-dependent susceptibility and in particular we look for a possible divergence of the susceptibility for a wave-vector corresponding to an antiferromagnetic ordering. We do not find any instability for a half-filled or nearly half-filled band.

DOI
10.1051/jphys:0197600370120142700

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arXiv:cond-mat/9309009•Physical review. B, Condensed matter•Feb 1, 1994•4 citations
We calculate the response to a periodical external magnetic field of the magnetically disordered phase of the Hubbard model in the alloy analogy approximation. We investigate the behaviour of the q-dependent susceptibility and in particular we look for a possible divergence of the susceptibility for a wave-vector corresponding to an antiferromagnetic ordering. We do not find any instability for a half-filled or nearly half-filled band.
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We present results of Monte Carlo simulations of the three-dimensional half-filled Hubbard model on simple-cubic 43{4}^{3}43 lattices and some 63{6}^{3}63 lattices with periodic boundary conditions, as well as of small clusters embedded in an effective medium. Magnetic properties were studied for values of the Hubbard repulsion U ranging from 0 to the bandwidth W=12t (t is the hopping matrix element). Our results show that antiferromagnetism is enhanced up to U\ensuremath{\sim}(5/6)W and then suppressed. The q=0 magnetic susceptibility is enhanced by U, although much less than predicted by the random-phase approximation. The transition temperature to an antiferromagnetic state is found to have a broad maximum around U\ensuremath{\sim}(5/6)W, with a maximum value of approximately W/18. The implications of our results for superconductivity in the attractive Hubbard model are also 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 study the gap for one-particle excitations as a function of the on-site repulsion U and the disorder W in the disordered Hubbard model in two and three dimensions. We consider the half-filled band case in the paramagnetic phase of this model. The analysis is done within the alloy-analogy approximation using the localized-eigenstates approximation to calculate the density of states. We find that there is a critical Uc{\mathit{U}}_{\mathit{c}}Uc​ for a gap opening at the Fermi level in both two and three dimensions. This allows us to determine a phase diagram of the model that separates a Mott-Hubbard insulating phase and an unspecified gapless phase. The critical Uc{\mathit{U}}_{\mathit{c}}Uc​ line grows with disorder and tends to be linear for large enough W. Our results are consistent with previous real-space renormalization-group studies.
The observed magnetic spin susceptibility of high-temperature superconductors such as ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​ increases when x increases from zero, i.e., as one dopes away from half-filling. Recent Monte Carlo simulations of A. Moreo [Phys. Rev. B 48, 3380 (1993)] suggest that this behavior can be reproduced by the two-dimensional Hubbard model only at large coupling, namely, U/t of order 10. Using longer runs, our Monte Carlo simulations show that the same behavior as for U/t=10 is obtained even in the intermediate coupling regime (U/t=4), as long as the temperature is low enough (T=t/6) that strong antiferromagnetic correlations are building up at half-filling. These results are consistent with the fact that in two dimensions, the generalized random phase approximation should fail in the parameter range where it predicts a magnetic phase transition.