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 and the SU(2) Bogoliubov transformation in BCS theory.
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral
t, on-site repulsion
U, and
Na2≫1 sites consistent with its exact global
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>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
c fermions, spin-1/2 spinons, and
η-spin-1/2
η-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the
η-spinon (and spinons) that are not invariant under that transformation have
η spin 1/2 (and spin 1/2) but are part of
η-spin-neutral (and spin-neutral)
2ν-
η-spinon (and
2ν-spinon) composite
ην fermions (and
sν fermions) where
ν=1,2,... is the number of
η-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=0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations
x>0.
We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimension and can be directly applied, among others, to the gauge groups
G=U(N) and
SU(2N), where $N\ensuremath{\in}\mathbb{N}$. For
SU(2N+1), one can also carry out the transformation after introducing an extra idle
Z2 gauge field, so that the resulting symmetry group trivially contains
Z2 as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.
Symmetries of the Hubbard model in the large-U limit are discussed. WE show that the global SU(2) symmetry of the full even-lattice Hubbard model generated by Yang's \ensuremath{\eta}-pairing operators becomes a local symmetry in the large-U limit and is applicable to even-odd mixed lattices. This extends the well-known SU(2) local symmetry of Anderson, for which we construct the generators. The associated local gauge theory of the extended symmetry can be similarly formulated and has interesting physical implications: new mean-field theories with parity-changed pairings. We point out the importance of searching for the parity-dependent Aharanov-Bohm-type of effects related to these local gauge symmetries.
We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single $U(1)\ifmmode\times\else\texttimes\fi{}SU(1|1)$ gauge theory, a group with larger symmetry than the product of two
U(1) gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund's interactions in multielectron atoms as a constraint. We show how this method can be combined with the
SP(N) group to yield a locally supersymmetric large-
N formulation of the $t\ensuremath{-}J$ model.
We analyze the structure of the group of (local) non-linear canonical transformations that exist in a system with n fermionic modes. To perform our study we develop an alternative framework to represent the generators of these canonical transformations; indeed we show how their definition, understanding and control is significantly improved using the Majorana fermion representation. These canonical transformations have the structure of a Lie group and we provide a representation for the elements of the Lie algebra that is very convenient both conceptually and practically (computationally): indeed our framework yields as side product an extremely effective tool to handle and work with SU(2n) Lie groups and algebras. Granting an enhanced control of the group of non-linear canonical transformation, our framework can be helpful in the study of strongly correlated electron systems, since it allows to easily identify fermionic degrees of freedom able to capture part of the correlations and thus may give a simpler representation of the Hamiltonian. Thanks to our analysis, also symmetry-based studies of the quantum Hamiltonians can be improved, since the simple representation of the generators of the canonical group permits to identify and understand otherwise hidden symmetries of difficult interpretation. The main aim of this work is to provide a comprehensive, general and scrupulous analysis of this framework, that we already applied in some circumstances. Therefore specific applications will not be presented in detail, since they can be found elsewhere, while only the formal and conceptual aspects will be developed thoroughly.