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Physical review. B, Condensed matter

t U expansion for the Hubbard model

Allan H. MacDonald, S. M. Girvin, D. Yoshioka

Published Jun 1, 1988• 495 citations
Publisher

Abstract

We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The $S$ matrix for the transformation, and the transformed Hamiltonian, ${H}^{\ensuremath{'}}$, are generated by an iterative procedure which results in an expansion in powers of the hopping integral $t$ divided by the on-site energy $U$. For a half-filled band and in the space with no doubly occupied sites, ${H}^{\ensuremath{'}}$ is equivalent to a spin Hamiltonian. We discuss the implications of our results for ${H}^{\ensuremath{'}}$ on theories of high-temperature superconductivity.

DOI
10.1103/physrevb.37.9753

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We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The SSS matrix for the transformation, and the transformed Hamiltonian, H\ensuremath′{H}^{\ensuremath{'}}H\ensuremath′, are generated by an iterative procedure which results in an expansion in powers of the hopping integral ttt divided by the on-site energy UUU. For a half-filled band and in the space with no doubly occupied sites, H\ensuremath′{H}^{\ensuremath{'}}H\ensuremath′ is equivalent to a spin Hamiltonian. We discuss the implications of our results for H\ensuremath′{H}^{\ensuremath{'}}H\ensuremath′ on theories of high-temperature superconductivity.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N\'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.
A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.
Using the slave-boson method in its spin-and charge-rotational invariant representation we determine stability of the superconducting state in the Hubbard model with correlated hopping.In general the term with correlated hopping violates electron-hole symmetry, however, in the special case when the correlated hopping integral X is equal to the uncorrelated hopping inte- gral t the electron-hole symmetry remains.We investigate this case and show that correlations induced by onsite Coulomb interactions U yield to the nor- mal state with exactly single occupied sites (corresponding to |U = oo)).The phase diagram of the superconducting state is determined for the different electron concentration n.At the half-filled band (n = 1) a direct transition from the Mott insulator to the superconducting state occurs.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
Perturbative description of high-TcT_cTc​ superconductivity in the Hubbard model via unitary transformation and classical spins
A unitary transformation is applied to the Hubbard model, which maps the Hubbard interaction to a single particle term. The resulting Hamiltonian consists of unconstrained fermions, which is then mapped to a Hamiltonian of spinless fermions coupled to pseudospins. The fermions are integrated out using second order perturbation theory in 1/U1/U1/U, resulting in an effective spin Hamiltonian. An order parameter is identified, stabilizing d-wave superconductivity. The groundstate energy of classical spin configurations is minimized at a finite value of this order parameter after a critical chemical potential, resulting in d-wave superconductivity at non-zero doping. This suggests, that the onset of high-TcT_cTc​ superconductivity is governed by the groundstate of a classical spin system.