Open Collections

A study of the Ising model on the hexagonal closed-packed lattice with competing interactions

University of British Columbia

A study is made of an Ising model on the hexagonal closed-packed lattice, with ferromagnetic interactions D between nearest-neighbor spins located in adjacent layers, and antiferromagnetic interactions J between nearest-neighbor spins located in the same layer. The ground states of the model are studied for different values of the parameter κ = —J/D. For κ < 1/2 the ground states are ferromagnetic and for κ > 1/2 the ground state spin configurations consist of stacked identical layers, such that each layer is obtained by stacking rows of alternating spins. At the point (κ — 1/2,t = 0), where t = T/D, there exists a multitude of degenerate ground state spin configurations which are not stable for κ ≠ 1/2. Mean-field theory and low temperature expansions are used to study the phase diagram at low temperatures. Mean-field theory predicts that (κ= 1/2, t = 0) is a multiphase point where an infinite sequence of modulated phases coincide. In the vicinity of the multiphase point, the mean-field phase diagram is found to be similar to the mean-field phase diagram of the three-dimensional ANNNI model near its multiphase point. Low temperature expansions are performed to second order in x, where x = e [sup -2/t], around the phase boundary between the ferromagnetic and the modulated phases. In contrast to standard low temperature expansions, the complete contribution, to order x², is obtained by grouping the contributions from excitations which contribute to arbitrarily high orders in x. The phase boundary between the ferromagnetic and the modulated phases is found to coincide, to order x², with the line onto which Domany mapped a kinetic Ising model on the honeycomb lattice. This strongly suggests that the Domany line is a phase boundary in three-dimensions. Mean-field theory shows that this Ising model contains a continuous minimum-energy surface. A renormalization group method which applies to models which contain continuous minimum-energy surfaces is used to analyze the phase transition between the paramagnetic and the modulated phases. The calculation is performed using a Landau-Ginzburg-Wilson Hamiltonian whose minimum-energy surface consists of a hexagon and which contains fourth-order invariants due to the lattice. The calculation shows that the Hamiltonian does not contain a stable fixed point. This suggests that the paramagnetic-modulated phase transition of this Ising model is a fluctuation-induced first-order transition.

Text

2010-08-09

For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

http://hdl.handle.net/2429/27226

Physics

University of British Columbia

Graduate