We discuss a cyclic chain of N identical sites, each with two possible quantum states, whose Hamiltonian may be written in terms of the spin operators for N spins S = 1/2, and transformed into the Fermi site excitation representation. We show that in the Fermion wave representation the requirement that the states transform according to irreducible representations of the translational symmetry group demands that the quasiparticle momenta are drawn from the representations of the double group. They must be all integral or all half-integral when the number of quasiparticles is odd or even respectively. The requirement that a quantum-mechanical operator has a definite symmetry type leads to very complicated conditions unless the operator conserves the odd or even particle number. If it does the symmetry operator can be defined in terms of Fermi wave operators and projection operators. The time reversal transformation is developed in detail and applied to problems of degeneracy in McConnell's model of singlet excitons and in the anisotropic transverse Heisenberg antiferromagnet. In the Bogoliubov-Valatin transformation time reversal shifts the quasiparticle momenta by ± π, converting integral into half-integral momenta when N is odd. The energies of the excited states are doubly-periodic functions of the momentum.
We discuss a cyclic chain of N identical sites, each with two possible quantum states, whose Hamiltonian may be written in terms of the spin operators for N spins S = 1/2, and transformed into the Fermi site excitation representation. We show that in the Fermion wave representation the requirement that the states transform according to irreducible representations of the translational symmetry group demands that the quasiparticle momenta are drawn from the representations of the double group. They must be all integral or all half-integral when the number of quasiparticles is odd or even respectively. The requirement that a quantum-mechanical operator has a definite symmetry type leads to very complicated conditions unless the operator conserves the odd or even particle number. If it does the symmetry operator can be defined in terms of Fermi wave operators and projection operators. The time reversal transformation is developed in detail and applied to problems of degeneracy in McConnell's model of singlet excitons and in the anisotropic transverse Heisenberg antiferromagnet. In the Bogoliubov-Valatin transformation time reversal shifts the quasiparticle momenta by ± π, converting integral into half-integral momenta when N is odd. The energies of the excited states are doubly-periodic functions of the momentum.
Wave functions are calculated for the ground state and low-lying excited states of a linear antiferromagnetic chain of N spins S = 1/2 with alternate strong and weak exchange integrals, using the Fermi operator method (N is even). There are two degenerate ground states with alternating spin distributions, formed by taking a Hartree-Fock state vector composed of ‘alternant spin waves’ with different amplitudes on the two sub-lattices. The normal Hartree-Fock state is unstable and collapses into one or other alternating state. One of these is equivalent to the pseudo-spin ground state of Soos as well as the wave functions used by Ruijgrok and Rodriguez, and Oguchi. The other is derived by the time-reversal transformation or by a translation of the regular chain, and is asymptotically orthogonal to the first when N is large. New states are constructed which have the full translational symmetry of the Hamiltonian, and there is a doubling of the low-lying spectrum of states in which the excitation density is of order 1/N, as compared with Soos' theory. This affects the magnetic susceptibility since the higher states are not doubled and do not remain orthogonal.
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.
A novel Bethe ansatz scheme is proposed to calculate physical properties of quantum integrable systems without U(1) symmetry. As an example, the antiperiodic XXZ spin chain, a typical correlated many-body system embedded in a topological manifold, is examined. Conserved ``momentum'' and ``charge'' operators are constructed despite the absence of translational invariance and U(1) symmetry. The ground state energy and elementary excitations are derived exactly. It is found that two intrinsic fractional (one half) zero modes accounting for the double degeneracy exist in the eigenstates. The elementary excitations show quite a different picture from that of a periodic chain. This method can be applied to other quantum integrable models either with or without U(1) symmetry.
We consider the Hubbard chain in a magnetic field and chemical potential. We introduce a pseudohole basis where all states are generated from a single reference vacuum. This allows the evaluation for all sectors of Hamiltonian symmetry of the model of the expression of the σ electron and hole operators at Fermi momentum ±kFσ and vanishing excitation energy in terms of pseudohole operators. In all sectors and to leading order in the excitation energy the electron and hole are constituted by one c pseudohole, one s pseudohole, and one topological momenton. These three quantum objects are confined in the electron or hole and cannot be separated. We find that the set of different pseudohole types which in pairs constitute the two electrons and two holes associated with the transitions from the (N↑,N↓) ground state to the (N↑+1,N↓), (N↑,N ↓ +1) and (N ↑ −1,N↓), (N↑,N ↓ −1) ground states, respectively, transform in the representation of the symmetry group of the Hamiltonian in the initial-ground-state sector of parameter space. We also find the pseudohole generators for the half-filling holon and zero-magnetic-field spinon. The pseudohole basis introduced in this paper is the only suitable for the extension of the present type of operator description to the whole Hilbert space. 1 I.
Properties and implications of the SO(2N+1) regular representation of a fermion system, which is a representation on the SO(2N+1) group proposed recently by us, are studied. By using the independent variables of the SO(2N+1) group, a representation of fermion annihilation-creation and pair operators is obtained which is an extension of the Dyson representation to cover both paired and unpaired modes. The structures of the SO(2N+1) spinor space and the SO(2N+1) wave function are determined. An integral and a differential Schrödinger equation on the SO(2N+1) spinor space are derived which are applicable to both even and odd fermion systems. The structures of the symmetry projected SO(2N+1) wave function and Schrödinger operator are determined. It is shown that odd fermion systems can be treated with an analogous way as even fermion systems by using a special SO(2N+1) canonical transformation which is very similar to the Bogoliubov transformation but corresponds to an improper rotation in the O(2N) group.
We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin S=1/2. We then introduce a systematic generalization of the approach for symmetric Z2 quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral Z2 states, we study the subset of U(1) phases variationally in the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{d}$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.