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Open Physics

On the SU(2)×SU(2) symmetry in the Hubbard model

Dorota Jakubczyk, Paweł Jakubczyk

Published May 30, 2012• 7 citations
Publisher

Abstract

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.

DOI
10.2478/s11534-012-0055-6

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Physical review. B, Condensed matter•Nov 1, 1996•22 citations
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.
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 consider the one-dimensional Hubbard model at half filling. We show that both excitation spectrum and S matrix are determined by the SO(4) symmetry of the model. The complete set of excitations is given by the scattering states of four elementary excitations, which form the fundamental representation of SU(2)\ifmmode\times\else\texttimes\fi{}SU(2). We evaluate the exact S matrix, which satisfies the Yang-Baxter relation. The results for the repulsive and attractive Hubbard model are related by an interchange of spin and charge degrees of freedom.
A description of the Hubbard model on a square lattice consistent with its global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral ttt, on-site repulsion UUU, and Na2≫1N_a^2\gg 1Na2​≫1 sites consistent with its exact global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t>0U/4t>0U/4t>0 the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge ccc fermions, spin-1/2 spinons, and ηηη-spin-1/2 ηηη-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the ηηη-spinon (and spinons) that are not invariant under that transformation have ηηη spin 1/2 (and spin 1/2) but are part of ηηη-spin-neutral (and spin-neutral) 2ν2ν2ν-ηηη-spinon (and 2ν2ν2ν-spinon) composite ηνηνην fermions (and sνsνsν fermions) where ν=1,2,...ν=1,2,...ν=1,2,... is the number of ηηη-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x=0x=0x=0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x>0x>0x>0.
In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
Recently an exact SU(2)\ensuremath{\bigotimes}SU(2) symmetry for the half-filled Hubbard model has been elucidated but has not yet been properly incorporated in many analyses of this model. We compute the irreducible representations of the symmetry group, a necessary step for any consistent mean-field analysis. A proper mean-field theory valid for both negative- and positive-U Hubbard models is then presented. A by-product of the description is a systematic enumeration of the Lie group SU(8) of unitary canonical transformations that is a direct generalization of the SU(4) transformation in the theory of superfluid 3He^{3}\mathrm{He}3He and the SU(2) Bogoliubov transformation in BCS theory.
We present a method to compute eigenstates and eigenvalues of strongly correlated models on clusters. The calculation is performed applying the SU(2) symmetry, which allows a considerable reduction of the dimension of Hilbert space of the system. We illustrate the method classifying the states of the four-sites Hubbard model on a chain. This classification has been made using also the translational symmetry and the S\ifmmode \tilde{}\else \~{}\fi{}\ifmmode \tilde{U}\else \~{U}\fi{}(2) pseudospin symmetry. \textcopyright{} 1996 The American Physical Society.