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Modern Physics Letters B

ON THE TRANSLATIONAL SYMMETRY OF INFINITE U HUBBARD MODEL

Dorota Jakubczyk

Published Dec 13, 2011• 5 citations
Publisher

Abstract

We investigate two commonly used methods of obtaining the solution of the one-dimensional Hubard model, in the regime of infinite intrasite Coulomb repulsion U, that is the nested Bethe ansatz and the Gutzwiller projection operator approach. These two formalisms give rise to different kinds of the wavefunctions, received via the additional operator and as a general feature, in Gutzwiller and Bethe methods, respectively. We consider the finite system consisting of three particles on a four site closed chain. We investigate the consequences of the dissimilarities in the translational symmetry of the representations of the Hamiltonian for these two approaches. We look for the proof of the decoupling of the spin and charge degrees of freedom, known in context of high-temperature superconductors, in this case.

DOI
10.1142/s0217984911500059

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We investigate two commonly used methods of obtaining the solution of the one-dimensional Hubard model, in the regime of infinite intrasite Coulomb repulsion U, that is the nested Bethe ansatz and the Gutzwiller projection operator approach. These two formalisms give rise to different kinds of the wavefunctions, received via the additional operator and as a general feature, in Gutzwiller and Bethe methods, respectively. We consider the finite system consisting of three particles on a four site closed chain. We investigate the consequences of the dissimilarities in the translational symmetry of the representations of the Hamiltonian for these two approaches. We look for the proof of the decoupling of the spin and charge degrees of freedom, known in context of high-temperature superconductors, in this case.
Hubbard model plays a critical role in the study of strongly correlated electron systems. The only available exact solutions of 1-D Hubbard Hamiltonian, obtained through nested Bethe ansatz, show a spin charge decoupling for infinite U when there is no possibility of double occupancy of an electron orbital state. It had been shown that these features can be directly obtained by employing orthofermion algebra. In the present communication, the thermodynamic properties, namely, specific heat and entropy are provided for a 1-D chain and 2-D square lattice using orthofermion approach. It is shown that in 1-D, our results are identical to the known exact results. We have also compared the thermodynamic properties of free fermions and orthofermions in both 1-D and 2-D. Our thermodynamic results, based on orthofermion approach, can be used to ascertain the accuracy of the approximate methods employed in the context of infinite U Hubbard model.
Abstract We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.
We develop a bosonization approach for one-dimensional models based on Bethe ansatz equations. The operator formalism of the exact soluble models in the low energy limit provides a systematic method to calculate the asymptotic correlation functions. As examples with and without internal degrees of freedom, the Calogero-Sutherland (C-S) model and the repulsive Hubbard model are considered respectively. We verify that the low energy behavior of the C-S model is controlled by two classes of c=1 conformal field theories, depending on whether the C-S interactions are among bosons or among fermions. For the Hubbard model, we show the explicit charge-spin separation at low energy for arbitrary U>0. The low energy behavior of the system is described by the (semi-) direct product of two independent Virasoro algebras with c=1.
A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite NNN number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite N,N,N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
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.
A general method to construct basis functions for fermionic systems which account for the SU(2)SU(2)SU(2) symmetry and for the translational invariance of the Hamiltonian is presented. The method does not depend on the dimensionality of the system and it appears as a natural generalization of the Bethe Ansatz to the case of non integrable systems. As an example we present the block diagonalization of the Hubbard hamiltonian for finite number of sites in one and two dimensions.