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Journal of Mathematical Physics

Hidden local gauge invariance in the one-dimensional Hubbard model and its equivalent coupled spin model

Huan-Qiang Zhou, Lin-Jie Jiang, Ping‐Feng Wu

Published Jun 1, 1990• 7 citations
Publisher

Abstract

Hidden local gauge invariance in the one-dimensional (1-D) Hubbard model and its equivalent coupled spin model is studied. It is found that Abelian U(1)⊗U(1) gauge transformations appear in both cases. Furthermore, it is shown that the energy spectrum is gauge invariant whereas the eigenvectors are explicitly gauge dependent. However, this result relies heavily on Shastry’s conjecture about the eigenvalue of the transfer matrix for the 1-D Hubbard model. Lastly, there is also a discrete symmetry associated to Z2⊗Z2. Once this symmetry is broken, one immediately obtains another nontrivial solution to the Yang–Baxter relations. UFaipxr

DOI
10.1063/1.529023

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Hidden local gauge invariance in the one-dimensional (1-D) Hubbard model and its equivalent coupled spin model is studied. It is found that Abelian U(1)⊗U(1) gauge transformations appear in both cases. Furthermore, it is shown that the energy spectrum is gauge invariant whereas the eigenvectors are explicitly gauge dependent. However, this result relies heavily on Shastry’s conjecture about the eigenvalue of the transfer matrix for the 1-D Hubbard model. Lastly, there is also a discrete symmetry associated to Z2⊗Z2. Once this symmetry is broken, one immediately obtains another nontrivial solution to the Yang–Baxter relations. UFaipxr
The SO (4) invariance of the transfer matrix for the one-dimensional Hubbard model is clarified from the viewpoint of the quantum inverse scattering method. We demonstrate the SO (4) symmetry by means of the fermionic L -operator and the fermionic R -matrix, which satisfy the graded Yang-Baxter relation. The transformation law of the fermionic L -operator under the SO (4) rotation is identified with a kind of gauge transformation, which determines the corresponding transformation of the fermionic creation and annihilation operators under the SO (4) rotation. The transfer matrix is confirmed to be invariant under the SO (4) rotation, which ensures the SO (4) invariance of the conserved currents including the Hamiltonian. Furthermore, we show that the representation of the higher conserved currents in terms of the Clifford algebra gives manifestly SO (4) invariant forms.
We investigate the possibility of hidden non-Abelian Local Phase symmetries in large-U doped planar Hubbard antiferromagnets, believed to simulate the physics of two-dimensional (magnetic) superconductors. We present a spin-charge separation ansatz, appropriate to incorporate holon spin flip, which allows for such a hidden local gauge symmetry to emerge in the effective action. The group is of the form SU(2) ⊗ US(1) ⊗ UE(1), where SU(2) is a local non-Abelian group associated with the spin degrees of freedom, UE(1) is that of ordinary electromagnetism, associated with the electric charge of the holes, and US(1) is a ‘statistical’ Abelian gauge group pertaining to the fractional statistics of holes on the spatial plane. In a certain regime of the parameters of the model, namely strong US(1) and weak SU(2), there is the possibility of dynamical formation of a holon condensate. This leads to a dynamical breaking of SU(2) → U(1). The resulting Abelian effective theory is closely related to an earlier model proposed as the continuum limit of large-spin planar doped antiferromagnets, which lead to an unconventional scenario for two-dimensional parity-invariant superconductivity.