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arXiv:cond-mat/9701220 · Physical review. B, Condensed matter

Representation of the t−J model via spin-charge variables

Evgueny Kochetov, Vladimir Sergeevich Yarunin

Published Aug 1, 1997• 8 citations
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Abstract

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 $f$ 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$(2|1)$ superalgebra in the degenerate fundamental representation and ensuing SU$(2|1)$ path-integral representation of the partition function ${Z}_{t\ensuremath{-}J}.$

DOI
10.1103/physrevb.56.2703

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

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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.

Physical review. B, Condensed matter•Jul 7, 2003•1 citation

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

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We show that the phase diagram and the odd p-wave quasiparticle gap for spinless fermions obtained in the paper by Kochetov and Mierzejewski for the $t\ensuremath{-}J$ model are artifacts of the adopted spin-charge separation treated in mean field approximation, while the critical remarks concerning the theory of exchange superconducting pairing derived within the Hubbard operator approach are misleading.

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

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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 $\mathrm{SU}(2|1)$ superalgebra in the fundamental representation and employing the ensuing ${\mathrm{CP}}^{1|1}$ parametrization for dynamical variables. As a result, an improved phase diagram ${T}_{c}(\ensuremath{\delta})$ in the paramagnetic phase is obtained.

arXiv:cond-mat/0002090•Physical review. B, Condensed matter•Jan 1, 2000•4 citations

Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [ SU ( 2 ) × SU ( 2 ) × U ( 1 ) ]/ Z 2 2 symmetry

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Large- N expansion based on the Hubbard operator path integral representation and its application to the t−J model

A. Foussats, A. Greco

In the present work we have developed a large-$N$ expansion for the $t\ensuremath{-}J$ model based on the path integral formulation for Hubbard operators. Our large-$N$ expansion formulation contains diagrammatic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-$N$ formulation we have calculated, for $J=0,$ the renormalized $O(1/N)$ boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order $1/N.$ We have compared our diagram technique and results with existing ones in the literature.

arXiv:cond-mat/0203258•Physical review. B, Condensed matter•Apr 19, 2002•25 citations
We show that the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J Hamiltonian is not in general reduced to Ht\ensuremath−J=H(S\ensuremath→,f){H}_{t\ensuremath{-}J}=H(S\ensuremath{\rightarrow},f)Ht\ensuremath−J​=H(S\ensuremath→,f), where S\ensuremath→S\ensuremath{\rightarrow}S\ensuremath→ and fff stand for independent ([S\ensuremath→,f]=0)([S\ensuremath{\rightarrow},f]=0)([S\ensuremath→,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 Zt\ensuremath−J.{Z}_{t\ensuremath{-}J}.Zt\ensuremath−J​.
We show that the SU(2∣1)\mathrm{SU}(2|1)SU(2∣1)-superalgebra path-integral representation for the partition function of the $t\ensuremath{-}J$ model follows from the conventional slave-fermion path-integral representation under a change of variables that explicitly resolves the constraint of no double occupancy.
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.
Most studies on the one-dimensional (1D) Hubbard model with transfer integral ttt and on-site repulsion UUU have assumed that for u=U/4t>0u=U/4t>0u=U/4t>0 and $h=\ensuremath{\mu}=0$, where hhh denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the U>0U>0U>0 Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond SO(4)\mathrm{SO}(4)SO(4) is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the h>0h>0h>0 spin and $\ensuremath{\mu}>0$ charge stiffnesses and the h=0h=0h=0 spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.
We review various ways to express the partition function of the single-band Hubard model as a path integral. The emphasis is made on the derivation of the action in the integrand of the path integral and the results obtained from this approach are discussed only briefly. Since the single-band Hubbard model is a pure fermionic model on the lattice and its Hamiltonian is a polynomial in creation and annihilation fermionic operators, with the help of the fermionic coherent states of holomorphic representions the partition function of the model can be straightforwardly expressed as a path integral with respect to Grassmann fields. Then the question arises how to calculate this path integral specially in the most interesting case of the strong coupling, which is described by the term quartic in the fermion fields. The standard way to get rid of this quartic term is to use so-called Hubbard-Stratonovich transformation by introducing auxilary boson field (this procedure is usually called bosonization). Since there are several equivalent forms of the interaction term in the single-band Hubbard model the bosonization procedure is not unique. In the first section of the article we discuss two types of bosoniztions one with two real scalar fields (charge and spin fields) and another with a real vector field. The second section is devoted to a spin-rotation-invariant bosonization procedure invented by Schulz and Weng, Ting and Lee, where an arbitary quantization axis is introduced at each point in space and time and an integration over all possible orientations enters the path integral measure. Spin rotation invariant formulation is necessary to study e.g. spiral magnetization. In the third section we present the contstruction of a path integral for the partition function when the single-band Hubbard Hamiltonian is written in terms of the Hubbard operators, which are elements of the doubly graded semi-simple algebra. In this case one should use generalized coherent states for superalgebra in order to get a path integral formulation. We follow the procedure proposed recently by T ngler and Kopp, to investigate the case of strong coupling.
In the present work we have developed a large-NNN expansion for the $t\ensuremath{-}J$ model based on the path integral formulation for Hubbard operators. Our large-NNN expansion formulation contains diagrammatic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-NNN formulation we have calculated, for J=0,J=0,J=0, the renormalized O(1/N)O(1/N)O(1/N) boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order 1/N.1/N.1/N. We have compared our diagram technique and results with existing ones in the literature.