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Journal of the Physical Society of Japan

A Devil's Staircase and the Ground State Phase Diagram in an Ising Chain with Spin-Lattice Coupling

Norikazu Ishimura

Published Dec 15, 1985• 15 citations
Publisher

Abstract

A one-dimensional Ising chain with spin-lattice coupling is considered. If the lattice is harmonic but has no translational symmetry, there appears effectively through the spin-lattice coupling a four-spin interaction which decays exponentially with distance. It is shown rigorously that this spin system in the ground state exhibits a devil's staircase as a function of the exchange parameter J because of the long range interaction. The ground state phase diagram of the spin system under an external magnetic field is speculated.

DOI
10.1143/jpsj.54.4752

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Journal of the Physical Society of Japan•Nov 15, 1999•2 citations
A one-dimensional Ising chain with spin-lattice coupling is considered. If the lattice is harmonic but has no translational symmetry, there appears effectively through the spin-lattice coupling a four-spin interaction which decays exponentially with distance. It is shown rigorously that this spin system in the ground state exhibits a devil's staircase as a function of the exchange parameter J because of the long range interaction. The ground state phase diagram of the spin system under an external magnetic field is speculated.
A one-dimensional antiferromagnetic Ising chain with spin-lattice coupling is considered. If the underlying lattice is harmonic but has no translational symmetry, there appears at zero temperature an effective interaction between down spins, which is repulsive and decays exponentially with distance. It is shown rigorotusly that this system exhibits in the ground state a devil's staircase as a function of an external magnetic field on account of the effective long range interaction. The ground state phase diagram is also given.
An Ising spin system which is situated on a two-dimensional square or a three-dimensional simple cubic lattice is considered. If there is a spin-lattice coupling and the lattice has no translational symmetry, there appears effectively a long range four-spin interaction in which the four spins are on a same axis. The ground state of the spin system is represented by using the ground state of a one-dimensional system where only the interactions along one axis are considered. The devil's staircase which has recently been shown to exist in such a one-dimensional system is wholly reproduced in a system on a two-dimensional square or a three-dimensional simple cubic lattice.
We present and analyze spin models with long-range interactions whose ground state features a so-called devil's staircase and where plateaus of the staircase are accessed by varying two-body interactions. This is in contrast to the canonical devil's staircase, for example, occurring in the one-dimensional Ising model with long-range interactions, where typically a single-body chemical potential is varied to scan through the plateaus. These systems, moreover, typically feature a particle-hole symmetry which trivially connects the hole part of the staircase (filling fraction $f\ensuremath{\ge}1/2$) to its particle part ($f\ensuremath{\le}1/2$). Such symmetry is absent in our models and hence the particle sector and the hole sector can be separately controlled, resulting in exotic hybrid staircases.
A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.
The devil's staircase is a fractal structure that characterizes the ground state of one-dimensional classical lattice gases with long-range repulsive convex interactions. Its plateaus mark regions of stability for specific filling fractions which are controlled by a chemical potential. Typically, such a staircase has an explicit particle-hole symmetry; i.e., the staircase at more than half filling can be trivially extracted from the one at less than half filling by exchanging the roles of holes and particles. Here, we introduce a quantum spin chain with competing short-range attractive and long-range repulsive interactions, i.e., a nonconvex potential. In the classical limit the ground state features generalized Wigner crystals that--depending on the filling fraction--are composed of either dimer particles or dimer holes, which results in an emergent complete devil's staircase without explicit particle-hole symmetry of the underlying microscopic model. In our system the particle-hole symmetry is lifted due to the fact that the staircase is controlled through a two-body interaction rather than a one-body chemical potential. The introduction of quantum fluctuations through a transverse field melts the staircase and ultimately makes the system enter a paramagnetic phase. For intermediate transverse field strengths, however, we identify a region where the density-density correlations suggest the emergence of quasi-long-range order. We discuss how this physics can be explored with Rydberg-dressed atoms held in a lattice.
The ground-state phase diagram is analyzed for a one-dimensional Ising model with long range interaction in the presence of uniform and staggered magnetic fields; the interaction is antiferromagnetic and its strength decays exponentially with distance. The exact renormalization formulas for the interactions between interfaces are derived, which is used to calculate the phase diagram hierarchically. The phase diagram is self-similar, and it contains upsilon points and devil's staircases; this agrees with the result of Aubry, Axel and Vallet [J. Phys. C 18 (1985) 753] for an equivalent model obtained by a different method. A simple “phase rule” presenting what kind of phases appear and how they are located in the phase diagram is derived. The application of this rule to the phase diagram of betaine calcium chloride dihydrate is discussed.