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Journal of Physics A Mathematical and General

Hamiltonian studies of the two-dimensional axial next-nearest neighbour Ising (ANNNI) model: II. Finite-lattice mass gap calculations

Philip M. Duxbury, Michael N. Barber

Published Apr 21, 1983• 2 citations
Publisher
DOI
10.1088/0305-4470/16/6/528

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We present a kinetic lattice system that evolves in general towards steady nonequilibrium states due to dynamical conflict between nearest- and next-nearest-neighbour interactions. Under two simple particular limits, the system would reach asymptotically the canonical equilibrium states for the ordinary Ising model and for the axial next-nearest-neighbour Ising (ANNNI) model, respectively. We find more generally that, independently of the lattice dimension, the steady state probability distribution for a given class of transition rates has a quasi-canonical structure with a short-range effective Hamiltonian. We solve exactly the one-dimensional version of this case, and compare its behaviour to the one for the ordinary ANNNI model. In particular, the system is shown to exhibit several spatially modulated phases and impure critical points. We also obtain some information on the phase diagram for the two-dimensional lattice.

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Finite-size calculations on the three-dimensional Ising model

Henk W. J. Blöte, G. Kamieniarz

Physica A Statistical Mechanics and its Applications•Jun 1, 1993•22 citations
The quantum Hamiltonian analogue of the two-dimensional ANNNI model is investigated by finite-lattice mass gap methods. By using lattice sizes capable of simulating systems of varying modulation, the authors are able to show the existence of a modulated phase between the paramagnetic and (2,2) antiphase regions. The modulation on the incommensurate to paramagnetic boundary is shown to vary and this variation is calculated as a function of the anisotropy. In addition, they find evidence for an XY-like transition from the incommensurate to the paramagnetic phase and perhaps a non-universal transition from the paramagnetic phase to the antiphase.
We present a kinetic lattice system that evolves in general towards steady nonequilibrium states due to dynamical conflict between nearest- and next-nearest-neighbour interactions. Under two simple particular limits, the system would reach asymptotically the canonical equilibrium states for the ordinary Ising model and for the axial next-nearest-neighbour Ising (ANNNI) model, respectively. We find more generally that, independently of the lattice dimension, the steady state probability distribution for a given class of transition rates has a quasi-canonical structure with a short-range effective Hamiltonian. We solve exactly the one-dimensional version of this case, and compare its behaviour to the one for the ordinary ANNNI model. In particular, the system is shown to exhibit several spatially modulated phases and impure critical points. We also obtain some information on the phase diagram for the two-dimensional lattice.