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arXiv:cond-mat/0210356 · Physical Review B

Phase diagram of the three-dimensional axial next-nearest-neighbor Ising model

Andrej Gendiar, Tomotoshi Nishino

Published Jan 7, 2005• 25 citations
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Abstract

The three-dimensional axial next-nearest-neighbor Ising model is studied by a modified tensor product variational approach. A global phase diagram is constructed with numerous commensurate and incommensurate magnetic phases. The devil's stairs behavior for the model is confirmed. The wavelength of the spin modulated phases increases to infinity at the boundary with the ferromagnetic phase. Widths of the commensurate phases are considerably narrower than those calculated by mean-field approximations.

DOI
10.1103/physrevb.71.024404

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The three-dimensional axial next-nearest-neighbor Ising model is studied by a modified tensor product variational approach. A global phase diagram is constructed with numerous commensurate and incommensurate magnetic phases. The devil's stairs behavior for the model is confirmed. The wavelength of the spin modulated phases increases to infinity at the boundary with the ferromagnetic phase. Widths of the commensurate phases are considerably narrower than those calculated by mean-field approximations.
We consider an axial Ising model with competing interactions up to third neighbors within a low-temperature approximation in terms of interacting and nonreversing domain walls. Assuming small third-neighbor coupling, we find a close relation between the transfer matrix of this classical model and the Hamiltonian of an S=1/2 anisotropic Heisenberg chain in a uniform magnetic field. This connection allows us to obtain the phase boundaries of the ferromagnetic and other low-temperature modulated states. The ground-state energy for arbitrary magnetization values of the Heisenberg chain is shown to provide information about the density of walls or wave number. In contrast with the two-dimensional axial next-nearest-neighbor Ising model, this quantity displays plateaus and reentrant behavior for some relations among the coupling constants. An alternative way to evaluate correlation functions in the antiferromagnetic Heisenberg chain is presented.
Some features of the phase diagram of the two-dimensional axial next-nearest-neighbor Ising model have long been debated. The extended structural correlations and long relaxation times associated with its Kosterlitz-Thouless phase indeed result in analytical and numerical treatments making contradictory predictions. Here, we introduce a numerical transfer matrix approach that bypasses these problems and thus clears up various ambiguities. In particular, we confirm the transition temperatures and the order of the transition to the floating incommensurate phase. Our approach motivates considering transfer matrices for solving long-standing problems in related models.
We apply density-matrix renormalization-group methods to study the phase diagram of the quantum axial next-nearest-neighbor Ising model in the region of low frustration where the ferromagnetic coupling is larger than the next-nearest-neighbor antiferromagnetic one. By finite-size scaling on lattices with up to 80 sites we locate precisely the transition line from the ferromagnetic phase to a paramagnetic phase without spatial modulation. We then measure and analyze the spin-spin correlation function in order to determine the disorder transition line where a modulation appears. We give strong numerical support to the conjecture that the Peschel-Emery one-dimensional line actually coincides with the disorder line. We also show that the critical exponent governing the vanishing of the modulation parameter at the disorder transition is ${\ensuremath{\beta}}_{q}=1∕2$.