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arXiv:0708.3773 · Physical Review B

Exact density matrix of the Gutzwiller wave function as the ground state of the inverse-square supersymmetric t−J model. II. Minority spin component

Onuttom Narayan, Yoshio Kuramoto, Mitsuhiro Arikawa

Published Jan 14, 2008• 0 citations
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Abstract

The density matrix, i.e., the Fourier transform of the momentum distribution, is obtained analytically for all values of the magnetization of the Gutzwiller wave function in one dimension with the exclusion of double occupancy per site. The present result complements the previous analytic derivation of the density matrix for the majority spin. The derivation makes use of a determinantal form of the squared wave function, and multiple integrals over particle coordinates are performed with the help of a diagrammatic representation. In the thermodynamic limit, the density matrix at distance $x$ is completely characterized by quantities ${v}_{c}x$ and ${v}_{s}x$, where ${v}_{s}$ and ${v}_{c}$ are spin and charge velocities in the supersymmetric $t\text{\ensuremath{-}}J$ model for which the Gutzwiller wave function gives the exact ground state. The present result then gives the exact density matrix of the $t\text{\ensuremath{-}}J$ model for all densities and all magnetization at zero temperature. Discontinuity, slope, and curvature singularities in the momentum distribution are identified. The momentum distribution obtained by numerical Fourier transform is in excellent agreement with existing results.

DOI
10.1103/physrevb.77.045114

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The density matrix, i.e., the Fourier transform of the momentum distribution, is obtained analytically for all values of the magnetization of the Gutzwiller wave function in one dimension with the exclusion of double occupancy per site. The present result complements the previous analytic derivation of the density matrix for the majority spin. The derivation makes use of a determinantal form of the squared wave function, and multiple integrals over particle coordinates are performed with the help of a diagrammatic representation. In the thermodynamic limit, the density matrix at distance xxx is completely characterized by quantities vcx{v}_{c}xvc​x and vsx{v}_{s}xvs​x, where vs{v}_{s}vs​ and vc{v}_{c}vc​ are spin and charge velocities in the supersymmetric t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model for which the Gutzwiller wave function gives the exact ground state. The present result then gives the exact density matrix of the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model for all densities and all magnetization at zero temperature. Discontinuity, slope, and curvature singularities in the momentum distribution are identified. The momentum distribution obtained by numerical Fourier transform is in excellent agreement with existing results.
The density matrix---i.e., the Fourier transform of the momentum distribution---is obtained analytically in closed form for the Gutzwiller wave function with exclusion of double occupancy per site. The density matrix for the majority spin is obtained for all magnetizations including the singlet case. Since the wave function gives the ground state of the supersymmetric $t\text{\ensuremath{-}}J$ model with the 1∕r21∕{r}^{2}1∕r2 exchange and transfer, the result gives the exact density matrix of the model at zero temperature. From the oscillating behavior of the density matrix, the discontinuity of the momentum distribution at the Fermi momentum kF{k}_{F}kF​ is identified. The form of the weaker singularity at 3kF3{k}_{F}3kF​ is also obtained; there is a discontinuity in the second derivative of the momentum distribution, whose magnitude is calculated analytically. The momentum distribution over the whole Brillouin zone is obtained numerically from the analytic solution of the density matrix. The result is in excellent agreement with previous results derived by different methods.
We present analytic results for ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function with nonzero values of the magnetization m. In dimension D=1D=1D=1 approximation-free evaluations are made possible by appropriate canonical transformations and an analysis of umklapp processes. We calculate the double occupation and the momentum distribution, as well as its discontinuity at the Fermi surface, for arbitrary values of the interaction parameter g, density n, and magnetization m. These quantities determine the expectation value of the one-dimensional Hubbard Hamiltonian for any symmetric, monotonically increasing dispersion ${\ensuremath{\epsilon}}_{k}.$ In particular for nearest-neighbor hopping and densities away from half filling the Gutzwiller wave function is found to predict ferromagnetic behavior for sufficiently large interaction U.
The Gutzwiller wave function is shown to be the exact solution of a supersymmetric t-J-type model. The model realizes a Fermi-liquid state in one dimension with a discontinuity in the momentum distribution. Analytic results are obtained for spin and charge susceptibilities, and the specific-heat coefficient with the help of the Luttinger-liquid theory. In the high-density limit the model exhibits a Mott-Hubbard gap and reduces to an antiferromagnetic spin chain with long-range exchange solved by Haldane and Shastry.
The Gutzwiller wave function is shown to be the exact solution of a supersymmetric t-J-type model. The model realizes a Fermi-liquid state in one dimension with a discontinuity in the momentum distribution. Analytic results are obtained for spin and charge susceptibilities, and the specific-heat coefficient with the help of the Luttinger-liquid theory. In the high-density limit the model exhibits a Mott-Hubbard gap and reduces to an antiferromagnetic spin chain with long-range exchange solved by Haldane and Shastry.