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Physical review. B, Condensed matter

SU(2) gauge symmetry of the large- U limit of the Hubbard model

Ian K. Affleck, Z. Zou, Theodore C. Hsu, Philip Warren Anderson

Published Jul 1, 1988• 443 citations
Publisher

Abstract

We demonstrate explicitly the SU(2) gauge invariance of the large-$U$ limit of the Hubbard model, i.e., the Heisenberg model written in terms of electron operators with a constraint of one particle per site. We use this result to demonstrate the equivalence of two apparently very different mean-field theories.

DOI
10.1103/physrevb.38.745

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We demonstrate explicitly the SU(2) gauge invariance of the large-UUU limit of the Hubbard model, i.e., the Heisenberg model written in terms of electron operators with a constraint of one particle per site. We use this result to demonstrate the equivalence of two apparently very different mean-field theories.
Symmetries of the Hubbard model in the large-U limit are discussed. WE show that the global SU(2) symmetry of the full even-lattice Hubbard model generated by Yang's \ensuremath{\eta}-pairing operators becomes a local symmetry in the large-U limit and is applicable to even-odd mixed lattices. This extends the well-known SU(2) local symmetry of Anderson, for which we construct the generators. The associated local gauge theory of the extended symmetry can be similarly formulated and has interesting physical implications: new mean-field theories with parity-changed pairings. We point out the importance of searching for the parity-dependent Aharanov-Bohm-type of effects related to these local gauge symmetries.
The quantum Heisenberg antiferromagnet is expressed in terms of physical electron operators, with a constraint to one electron per site, as it emerges in the large-UUU limit of the Hubbard model. It is then possible to carry out a systematic 1n\frac{1}{n}n1​ expansion built on a translation-invariant singlet ground state. Such an expansion yields a Fermi-liquid picture for the elementary excitations which should be appropriate for low-dimensional antiferromagnets. Explicit calculations in one dimension show that the excitation spectrum consists of a gapless continuum in qualitative agreement with exact Bethe ansatz results.
To gain insight into the behavior of the Hubbard model, we define a SU(n)\mathrm{SU}(n)SU(n) invariant generalization of the Hubbard-Heisenberg model and, in the large-nnn limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term ttt, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "s+ids+ids+id" phase is the ground state at half filling for sufficiently large ttt or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-nnn problem to the high-Tc{T}_{c}Tc​ superconductors.
The spin-rotationally invariant SU(2) approach to the Hubbard model is extended to accommodate the charge degrees of freedom. Both U(1) and SU(2) gauge transformation are used to factorize the charge and spin contribution to the original electron operator in terms of the emergent gauge fields. It is shown that these fields play a similar role as phonons in the BCS theory; they provide the ``glue'' for fermion pairing. By tracing out gauge bosons the form of paired states is established and the role of antiferromagnetic correlations is explicated.