XY Hamiltonian Research Articles

Novel results of extremely thin substrate-free liquid-crystal films obtained from calorimetric and computer simulation studies

Experimental results from extremely thin free-standing liquid-crystal films have been found to exhibit divergent heat-capacity anomalies in the vicinity of the smectic-A-hexatic-B (SmA-HexB) transition. In fact, the data can be successfully fit to a power-law expression yielding the critical exponent α=0.30±0.04. The results clearly indicate that the transition cannot be described solely in terms of hexatic order and that some additional molecular order must be created in the SmA-HexB transition in these liquid crystals. This transition is therefore not the simple liquid-hexatic transition as had been suggested previously. In light of x-ray diffraction studies indicating the existence of herringbone order in some liquid-crystal samples, Bruinsma and Aeppli formulated a coupled XY hamiltonian as a possible explanation for the anomalous experimental data obtained on the SmA-HexB transition in three dimensions (i.e. bulk samples). Based on our novel calorimetric and optical reflectivity results from thin liquid-crystal films, we have conducted extensive Monte Carlo simulations to explore this model in two-dimensional lattices. These studies have resulted in a number of novel discoveries that reveal important aspects of both the SmA-HexB transition in thin liquid-crystal films and two-dimensional melting phenomena in general.

The glassy state in arrays of superconducting junctions: the role of frustration and disorder in its formation

We have submitted to a close scrutiny the genesis of the glassy (or better the fully incommensurate) state in arrays of superconducting junctions at zero temperature. From our numerical simulations, using the XY hamiltonian, it comes out that frustration by itself is a necessary but not a sufficient condition. The other needed element is the disorder. However only the positional disorder is effective in the formation of a truly incommensurate state. The oscillation damping in the ground state energy and in the magnetic moment, as a function of the frustration, is not linear. The critical frustration (characteristic of the short range “order-disorder” crossover) decreases almost linearly with the root mean square of plaquette area, but not in a “universal” manner.

Free fermionic elliptic reflection matrices and quantum group invariance

Elliptic diagonal solutions for the reflection matrices associated to the elliptic $R$ matrix of the eight vertex free fermion model are presented. They lead through the second derivative of the open chain transfer matrix to an XY hamiltonian in a magnetic field which is invariant under a quantum deformed Clifford--Hopf algebra.

Integrability and virasoro symmetry of the noncritical baxter/ising model

The properties of a noncritical c = 1 lattice Virasoro algebra for the exactly solvable Baxter eight-vertex model are studied in the decoupling limit in which the Baxter model is equivalent to a staggered pair of Ising models. The Virasoro operator L 0 is given by the first moment of the XY spin chain density which is proportional to the logarithm of the corner transfer matrix (CTM). The eigenmodes of L 0 are explicitly constructed in terms of lattice fermion operators. They are related to the eigenmodes of the XY hamiltonian by harmonic analysis on the spectral (rapidity) torus, which arises as the CTM group manifold. At the critical point, the results reduce to a momentum space formulation of space-time conformal symmetry. Two related Virasoro algebras are constructed from the fermionic eigenmode operators of L 0. One is a generalization of the T = T c conformal algebra away from the critical point. The other represents diffeomorphisms of the real rapidity circle, which is compact due to the presence of the space-time lattice. Both are spectrum generating algebras of the corner transfer matrix. In the scaling limit, the Virasoro operators are expressed in terms of a massive, free Dirac field. For n ⩾ −1, the L n 's are given by integrals of local densities. We discuss the relation between these densities and the integrals of motion and higher spectral flows that characterize the integrability of the system.

Application of spin-wave theory to the ground state of XY quantum Hamiltonians.

Spin-wave theory is successfully applied to study the ground state of the quantum XY Hamiltonian. It is found that a judicious choice of the quantized spin axis removes difficulties of previous applications. Results for the simplest one-, two-, and three-dimensional lattices are shown to compare favorably with either exact or other approximate calculations.

Series expansions for a generalized XY hamiltonian : I. The XY model in a parallel magnetic field

Series expansions are derived for the free energy and the fluctuations of a generalized hamiltonian. These series are analysed in the special case of the XY hamiltonian on an f.c.c. lattice in an external parallel magnetic field. The critical indices seem to be independent of the field.

Series expansions for a generalized XY hamiltonian : II. Tricritical behaviour of the Takagi model

The high-temperature series expansions for the square of the fluctuation in the order parameter, for the specific heat, the concentration and the concentration susceptibility for the Takagi model on an f.c.c. lattice are analysed in the field variables. We obtain for the critical exponent γ = 4 3 and the specific heat is possibly logarithmically divergent. If we take the tricritical point to be determined by γ t = 1, the tricritical exponents α t, λ t and ω t are found to be consistent with 1 2 .

Baxter's quantum number in the XY model

In the particular case of the asymmetric XY model an explicit form for the operator corresponding to Baxter's quantum number n is obtained. It is shown that this operator is closely associated with a transformation of spin operators which converts the asymmetric XY hamiltonian into a symmetric XY hamiltonian with alternating nearest-neighbour interactions.

Baxter states in the XY model

The Baxter diagonalization of the XYZ hamiltonian is considered in the special case of the XY model. For this particular model the normalization properties of the basic Baxter states are obtained showing that the Baxter families form an overcomplete set of states. After choosing a simple subset of states which is complete, Bethe-type eigenstates of the XY hamiltonian are formed explicitly. By fixing a free parameter in Baxter's states an operator expression for these states is obtained. The XY hamiltonian is then diagonalized in terms of fermion operators closely related to the quasiparticle operators of the standard solution due to Lieb, Schultz, and Mattis. Finally relations are given expressing the Baxter states in terms of the standard quasiparticle states of this model.

Direct diagonalization of the transfer matrix of the zero-field free-fermion model

The transfer matrix of the zero-field free-fermion model is diagonalized by operator methods. The model is a special case of the eight-vertex model solved by Baxter, but it has the advantage that an easy survey of eigenvalues and eigenvectors is obtainable. The transfer matrix is expressed in terms of the fermion-number operators which also diagonalize the corresponding XY hamiltonian. A similar expression was previously conjectured by Jones and Valatin from a knowledge of the eigenvalues given by Baxter. The present derivation does not use this knowledge and provides a direct and independent diagonalization of the transfer matrix.