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Physical review. B, Condensed matter

Reply to “Comment on ‘BCS-type mean-field theory for the t−J model in the SU(2|1) superalgebra representation’ ”

Evgueny Kochetov, Marcin Mierzejewski

Published Jul 7, 2003• 1 citation
Publisher

Abstract

We show that the $\mathrm{SU}(2|1)$-superalgebra path-integral representation for the partition function of the $t\ensuremath{-}J$ model follows from the conventional slave-fermion path-integral representation under a change of variables that explicitly resolves the constraint of no double occupancy.

DOI
10.1103/physrevb.68.016502

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Comment on “BCS-type mean-field theory for the t−J model in the SU(2|1) superalgebra representation”

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Asymptotically exact mean-field theory for the Anderson model including double occupancy

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We show that the SU(2∣1)\mathrm{SU}(2|1)SU(2∣1)-superalgebra path-integral representation for the partition function of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model follows from the conventional slave-fermion path-integral representation under a change of variables that explicitly resolves the constraint of no double occupancy.
We show that the $t\ensuremath{-}J$ Hamiltonian is not in general reduced to ${H}_{t\ensuremath{-}J}=H(S\ensuremath{\rightarrow},f)$, where $S\ensuremath{\rightarrow}$ and fff stand for independent $([S\ensuremath{\rightarrow},f]=0)$ SU(2) (spin) generators and spinless fermionic (holon) fields, respectively. The proof is based upon an identification of the Hubbard operators with the generators of the su(2∣1)(2|1)(2∣1) superalgebra in the degenerate fundamental representation and ensuing SU(2∣1)(2|1)(2∣1) path-integral representation of the partition function ${Z}_{t\ensuremath{-}J}.$
A simple version of the Bardeen-Cooper-Schrieffer (BCS)-type mean-field theory for the $t\ensuremath{-}J$ model is developed. The present approach rigorously treats the constraint of no doubly occupied states and invokes two local order parameters to implement spontaneous breaking of the global $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry. This is achieved by identifying the Hubbard operators with generators of the SU(2∣1)\mathrm{SU}(2|1)SU(2∣1) superalgebra in the fundamental representation and employing the ensuing CP1∣1{\mathrm{CP}}^{1|1}CP1∣1 parametrization for dynamical variables. As a result, an improved phase diagram ${T}_{c}(\ensuremath{\delta})$ in the paramagnetic phase is obtained.
We show that the physical subspace in the Z2{\mathbb{Z}}_{2}Z2​-slave-spin theory is conserved under the time evolution of the system. Thus, when restricted to the physical subspace, this representation gives a complete and consistent description of the original problem. In addition, we review two known examples from the existing literature in which the projection onto the physical subspace can be relaxed: (i) the noninteracting limit in any dimension at half-filling and (ii) the interacting model in the infinite-dimensional limit at half-filling. In both cases, physical observables are correctly obtained without explicit treatment of the constraints which define the physical subspace. In these examples, correct results are obtained, despite the fact that unphysical states enter the solution.
The Anderson impurity model for finite values of the Coulomb repulsion U is studied using a slave-boson representation for the empty and doubly occupied f level. In order to avoid well-known problems with a naive mean-field theory for the boson fields, we use the coherent-state-path integral representation to first integrate out the double-occupancy slave bosons. The resulting effective action is linearized using two-time auxiliary fields. After integration over the fermionic degrees of freedom, one obtains an effective action suitable for a 1/Nf{\mathit{N}}_{\mathit{f}}Nf​ expansion. The same problem remains concerning the constraint as in the infinite U case. For T\ensuremath{\rightarrow}0 and Nf{\mathit{N}}_{\mathit{f}}Nf​\ensuremath{\rightarrow}\ensuremath{\infty}, exact results for the ground-state properties are recovered in the saddle-point approximation. Numerical solutions of the saddle-point equations show that even in the spin-degenerate case Nf{\mathit{N}}_{\mathit{f}}Nf​=2 the results are quite good.