A general treatment of the symmetry for quantum spin systems is presented with the use of permutational operators. Symmetry operations which are commutable with a Hamiltonian form the symmetry group. Complete sets of spin functions which are specified by the total spin ( S tot ) are projected onto the irreducible representations of the symmetry group, using the character table. We show from symmetry consideration that a minimum dimension exists for the ground state of the Heisenberg Hamiltonian with finite N spins. The diagonalization of the Hamiltonian with the use of symmetry consideration can be effectively performed for the complete sets of S tot =0 by the method proposed in this paper. Numerical results of the ground state and all the excited states of S tot =0 are presented for spin systems up to N =20.
Abstract We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin
S=21 it is equivalent to the XXZ model, and in spin
S=1 to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin
S=21 and
S=1 we give an explicit formula for the free energy for all values of the two parameters, and for spin
S>1 for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin
S=21 and
S=1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.
Abstract We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.
We obtain the complete magnetic phase diagram of the two-dimensional antiferromagnetic Heisenberg model, , within the framework of the O ( N ) nonlinear sigma model. We find two magnetically ordered phases, one with Néel order, for , and another with collinear order, for , separated by a nonmagnetic region, for , where a gapped spin liquid is found. The transition at is of the second order while the one at is of the first order and the spin gaps cross at . Our results are exact at N → ∞ and agree with numerical results from different methods.