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Physical review. B, Condensed matter

Antiferromagnetic correlation in the half-filled strongly correlated electron models at nonzero temperature: A rigorous result

Guang-Shan Tian

Published May 22, 2001• 9 citations
Publisher

Abstract

As an extension of our previous rigorous investigation on the spin correlations in the ground states of the half-filled Hubbard model and the periodic Anderson model [G. S. Tian, Phys. Rev. B 50, 6246 (1994)], in the present paper we study the behavior of these correlation functions at finite temperature. We show rigorously that, at any $T\ensuremath{\ne}0,$ the predominant spin correlations in these systems are antiferromagnetic. Furthermore, based on this result, we also show that a quasi-one (or two)-dimensional itinerant electron ferrimagnet must have a gapless branch of ferromagnetic excitations. This conclusion is consistent with the previous results derived by the spin-wave theory.

DOI
10.1103/physrevb.63.224413

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As an extension of our previous rigorous investigation on the spin correlations in the ground states of the half-filled Hubbard model and the periodic Anderson model [G. S. Tian, Phys. Rev. B 50, 6246 (1994)], in the present paper we study the behavior of these correlation functions at finite temperature. We show rigorously that, at any T\ensuremath≠0,T\ensuremath{\ne}0,T\ensuremath=0, the predominant spin correlations in these systems are antiferromagnetic. Furthermore, based on this result, we also show that a quasi-one (or two)-dimensional itinerant electron ferrimagnet must have a gapless branch of ferromagnetic excitations. This conclusion is consistent with the previous results derived by the spin-wave theory.
We extend a previous result of ours [G.S. Tian, Phys. Rev. B63 (2001) 224413] on the antiferromagnetic spin correlations in the half-filled Hubbard model at finite temperature to the double-exchange model. To overcome the mathematical difficulty caused by the localized spin freedom in this model, we apply both Zener's argument and the finite-temperature spin-reflection-positivity method to show rigorously that, at any temperature , the spin correlations in the half-filled double-exchange model are predominantly antiferromagnetic. This conclusion is completely consistent with the experimental observations and the previous theoretical results by approximate methods.
We extend a previous result of ours [G.S. Tian, Phys. Rev. B63 (2001) 224413] on the antiferromagneticspin correlations in the half-filled Hubbard model at finite temperature to the double-exchange model. To overcome themathematical difficulty caused by the S = 3/2 localized spin freedom in this model, we apply both Zener's argumentand the finite-temperature spin-reflection-positivity method to show rigorously that, at any temperature T, the spincorrelations in the half-filled double-exchange model are predominantly antiferromagnetic. This conclusion is completelyconsistent with the experimental observations and the previous theoretical results by approximate methods.
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\ensuremath{\Lambda}=\mathcal{A}\ensuremath{\cup}\mathcal{B}$ with $\mathcal{A}\ensuremath{\cap}\mathcal{B}=\ensuremath{\emptyset},$ and bonds connecting any element of A\mathcal{A}A with all the elements of B.\mathcal{B}.B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the A\mathcal{A}A sites and/or of the B\mathcal{B}B sites. This is the Hubbard model defined on the so-called (NA,NB){(N}_{A}{,N}_{B})(NA​,NB​) complete bipartite graph, NA(NB){N}_{A}{(N}_{B})NA​(NB​) being the number of elements in A(B).\mathcal{A}(\mathcal{B}).A(B). In this paper, we analytically find the exact ground state for NA=NB=N{N}_{A}{=N}_{B}=NNA​=NB​=N at half filling for any N; the repulsion has a maximum at a critical N-dependent value of the on-site Hubbard U. The wave function and the energy of the unique, singlet ground state assume a particularly elegant form for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We also calculate the spin-spin correlation function and show that the ground state exhibits an antiferromagnetic order for any nonzero U even in the thermodynamic limit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a Hubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles, and an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature. On the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramagnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this fact can be made explicit by the analytic solutions.
The relationship between charged and spin-excitation gaps of the half-filled Hubbard model, the symmetric periodic Anderson model, and the Kondo lattice model is considered for a general ddd-dimensional bipartite lattice. In a previous paper [G. S. Tian, Phys. Rev. B 58, 7612 (1998)], it was shown that the quasiparticle gaps of these models at half filling are always larger than their spin excitation gaps. In the present paper, we establish a theorem, which states that the charged gap is also greater than the corresponding spin gap in these models. This conclusion, which has been reached previously for one-dimensional systems based on variational and the density-matrix renormalization-group numerical calculations, is thus put on a rigorous and more general footing.
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.