Models of High Temperature Superconductors

Abstract

A central problem in the theory of high-temperature superconductivity is the identification of the simplest model containing the essential physics on an energy scale of a few hundred degrees, where the main phenomena of magnetism, superconductivity and normal charge transport are to be found. A particular issue of importance for this workshop is the nature of the spin fluid probed by neutron scattering, NMR, or μSR. Much of the discussion has addressed the question of whether two spin fluids are required to fit the available data, or one spin fluid will suffice. In our view, neither description is quite accurate - it is better to think in terms of a composite spin fluid.

The widely adopted starting points for discussios of the effects of strong correlations are a single-band Hubbard model¹ or a three-band model involving copper d_x2-_y2 and oxygen p_x, p_y states.² These are not low-energy Hamiltonians since they involve parameters of the order 1 - 10 eV. They are relevant for various x-ray and UV spectroscopies which clearly show that the three band model is required. However, it is easier to focus on magnetism and superconductivity if high energy states are “integrated out” in order to obtain an effective Hamiltonian. It is then important to establish whether there are circumstances of relevance for high-temperature superconductivity in which the one-band and three-band models lead to the same effective Hamiltonian. In examining this question, it has been noticed by a number of people that there is a particularly strong superexchange interaction between holes on neighboring copper and oxygen sites and that it may be a good approximation to assume that they form singlets. Three representations have been used: copper-oxygen bond singlets,³ copper-centered singlets,⁴ and oxygen-centered singlets,^5,6 Once the translational motion of the oxygen hole has been taken into account, all three representations are equivalent and give same results if calculated completely. They differ in that further approximations are usually made in order to simplify the problem.

The physical picture is one of singlets moving through a background of antiferromagnetically correlated spins. The analogous representation of the single-band Hubbard model is known as the t-J model.¹ At this level, some of the strong interactions have been eliminated, but we have not yet arrived at a continuum limit of fixed point Hamiltonian. Zhang and Rice⁴ have argued that the copper-centered singlets behave in the same way as missing spins in the t-J model. Their essential point is that the number of degrees of freedom is the same and that therefore the two models may have the same effective Hamiltonian. However, we have shown that the models have different oxygen spin-spin correlation functions⁵ and that, in one dimension, the number of degrees of freedom are not the same.⁷ Thus we believe that the effective Hamiltonians are different, and more work is needed to determine whether or not their solution leads to the same physical properties for high-temperature superconductors.

In the three-band model considered here, the spins of copper and oxygen holes are strongly correlated and there is no sense in which they may be regarded as independent spin fluids. However, current neutron scattering⁸ and NMR data⁸ are not easily reconciled on the basis of a single “immutable” spin fluid. Proposed explanations⁹ of the difference between relaxation rates of copper and oxygen nuclear spins do not rest easily with the observed incommensurate position of antiferromagnetic fluctuations in k-space.¹⁰ Also the behavior of these relaxation rates below T_c is difficult to reconcile with the observation by neutron scattering⁸ of spin fluctuations below the superconducting gap. A complete understanding of these observations may well require us to take account of the composite nature of the spin fluid.