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arXiv:cond-mat/0203258 · Physical review. B, Condensed matter

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

A. Foussats, A. Greco

Published Apr 19, 2002• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.65.195107

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We have introduced a new perturbative approach for $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model where Hubbard operators are treated as fundamental objects. Using our vertices and propagators we have developed a controllable large-$N$ expansion to calculate different correlation functions. We have investigated charge density-density response and the phase diagram of the model. The charge correlations functions are not very sensitive to the value of $J$ and they show collective peaks (or zero sound) which are more pronounced when they are well separated (in energy) from the particle-hole continuum. For a given $J$ a Fermi liquid state is found to be stable for doping $\ensuremath{\delta}$ larger than a critical doping ${\ensuremath{\delta}}_{c}$. ${\ensuremath{\delta}}_{c}$ decreases with decreasing $J$. For the physical region of the parameters and, for $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}$, the system enters in an incommensurate flux or DDW phase. The inclusion of the nearest-neighbor Coulomb repulsion $V$ leads to a charge density wave phase when $V$ is larger than a critical value ${V}_{c}$. The dependence of ${V}_{c}$ with $\ensuremath{\delta}$ and $J$ is shown. We have compared the results with other ones in the literature.

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•Apr 1, 1997•0 citations
In the present work we have developed a large-NNN expansion for the t\ensuremath−Jt\ensuremath{-}Jt\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.
We have introduced a new perturbative approach for $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model where Hubbard operators are treated as fundamental objects. Using our vertices and propagators we have developed a controllable large-NNN expansion to calculate different correlation functions. We have investigated charge density-density response and the phase diagram of the model. The charge correlations functions are not very sensitive to the value of JJJ and they show collective peaks (or zero sound) which are more pronounced when they are well separated (in energy) from the particle-hole continuum. For a given JJJ a Fermi liquid state is found to be stable for doping $\ensuremath{\delta}$ larger than a critical doping ${\ensuremath{\delta}}_{c}$. ${\ensuremath{\delta}}_{c}$ decreases with decreasing JJJ. For the physical region of the parameters and, for $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}$, the system enters in an incommensurate flux or DDW phase. The inclusion of the nearest-neighbor Coulomb repulsion VVV leads to a charge density wave phase when VVV is larger than a critical value Vc{V}_{c}Vc​. The dependence of Vc{V}_{c}Vc​ with $\ensuremath{\delta}$ and JJJ is shown. We have compared the results with other ones in the literature.
The novel slave-free (i.e. without slave bosons and fermions) approach to the problem of strong correlations is used in studying the Hubbard (t-J) model, where the self-energy is expressed in terms of charge and spin vertex functions of Hubbard operators. A systematic 1/N expansion (N is the number of spin components) of the self-energy is carried out. In O(1) it gives the same quasiparticle spectrum as the slave boson theory. The next order O(1/N) of the self-energy Σ(k, ω) is calculated exactly by solving analytically the integral equations for the charge and spin vertices. As an example for the feasibility of the proposed method explicit results for Σ(k, ω) are given in the case J=0. Dealing only with observable quantities our method is transparent and easily applicable to models which exclude double occupancy on lattice sites, such as the Anderson model and the Emery model for high temperature superconductors.
Abstract We consider the large‐spin‐degeneracy (N) version of the U = ∞ one‐band Hubbard model and present a systematic 1/N expansion method which is based on Hubbard's X‐operator formulation and a Baym‐Kadanoff expansion. We compare this alternative X‐operator approach with the customary time‐continuum slave‐boson (SB) functional‐integral treatment, which has recently been criticized. In particular, we give a detailed derivation of the density Green's function within the new approach and compare the leading‐order results with those as obtained with the commonly used SB treatment, finding total agreement.
A path-integral formalism for the one-dimensional Hubbard model in the strong-coupling regime, which is equivalent to the t-J model in t/U expansion but without any explicit constraint, is developed. Based on this formalism, the zero-temperature properties of the Hubbard chain are systematically studied. In the infinite-U limit, the charge and spin degrees of freedom are shown to be completely separated. Such a separation at U=\ensuremath{\infty} provides a controllable perturbative scheme to study the strong-coupling case. In the large-U regime, the spin degree of freedom is represented by a ``squeezed'' Heisenberg chain. We solve the (squeezed) Heisenberg chain by introducing a soliton representation. Both the charge and spin excitations are found to agree well with the exact solution. The bare electron (hole) is shown to be a composite particle of these basic excitations, i.e., holon and spinon, together with a nonlocal string field. This string field makes the electron behave like a ``semion'' and plays an important role in determining various correlation functions. We analytically derive the asymptotic forms of the spin-spin and density-density correlation functions as well as the single-electron and the pair-electron Green's functions. The implications of the present work to the two-dimensional model are discussed.
By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.
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.