Effect of Antisymmetric Interactions on Critical Phenomena: A System with Helical Ground State

Abstract

A classical Heisenberg Hamiltonian, to which is added single-ion anisotropy and antisymmetric Dzialoshinsky-Moriya interactions is studied with use of Wilson theory. The phase diagram is obtained; it consists of ferromagnetic, spiral, and ferromagnetic-spiral regions, with critical exponents that are Ising-like, xy-like, and Heisenberg-like, respectively. These results lead to the conclusion that breaking of exchange symmetry does not change the nature of the phase transition. Crossover behavior is also discussed.Consider a system composed of three-dimensional classical spins situated on a ^-dimensional "hypercubical" lattice and described by the model Hamiltonian 3C=3C 1 + X 2 + 3C 3 . Herewhere spins Si = (S ixJ S iy , S ie ) and Sj interact only if lattice sites i, j are nearest neighbors, and the energy of a pair of parallel spins is -J. The second term in the Hamiltonian is the singleion anisotropy term, presumably due to interactions such as crystal fields:The third term is a Dzialoshinski-Moriya interaction, 1where the prime on the summation means that it is restricted to nearest-neighbor pairs of spins that are on adjacent "hyperplanes" (this means Yj-r { = z, where z is a nearest-neighbor vector in the z lattice direction). Although the most general Dzialoshinski-Moriya interaction is A •S { x Sj, we have set 'A=A 3 z in (lc) without loss of generality. The impetus for this study is as follows:(1) There are many magnetic materials 2 that display complex helical spin configurations in their ordered states, although there exists little analysis beyond mean-field theory. The Hamiltonian 3C is capable of describing systems with helical ordering [cf. Fig. 1].(2) The antisymmetric Dzialoshinski-Moriya interaction is of interest in its own right. From spin-orbit coupling theory, it is known to be the cause of "weak" ferromagnetism in certain materials (e.g., hematite, 3 a-Fe 2 0 3 ). Moreover, in a recent Letter, Melcher 4 has shown that the spin-wave dispersion relation includes a linear term, oo(q) =aq + bq 2 , where a and b are functions of J, D, and A 3 , and a = 0 unless A 3 ^0, leading one to suspect possible effects of nonzero A 3 on thermodynamic properties in general and on critical properties in particular.(3) The interaction 3C 3 provides an opportunity to test the effect of breaking exchange symmetry on critical properties, 5 " 6 since the antisymmetric interaction A^SjXSj changes sign upon interchange of S { and S i? whereas previously studied models of critical behavior are symmetric under this interchange.In this work I study the critical behavior of K using the Wilson renormalization group ap-\ V fW/ FIG. 1. Pictoral representation of the phase diagram of the Hamiltonian 3^ = 3^ + ^ +3C 3 . (a), (b), (e) The ordered states for the cases (i) D>A 3 2 , (ii) D<A Z 2 , and (iii) D=Az 2 , respectively, (d) The critical behavior as obtained from the e expansion.

The renormalization group and critical phenomena

“…More general multicritica1 points have been analyzed (see, for example, Fisher, 1974). Effects of dipolar forces Aharony and Fisher, 1973;Aharony, 1973aAharony, , 1973b, other long-range forces (Suzuki, 1972(Suzuki, , 1973aBaker and Golner, 1973;Suzuki et al , 1972;Sak, 1973), cubic perturbations, and anisotropies (Pfeuty and Fisher, 1972;Wegner, 1972b;Wallace, 1973;Ketley and Wallace, 1973;Aharony, 1973cAharony, , 1973dSuzuki, 1973b;Liu, 1973;Grover, 1973;Chang and Stanley, 1973"a recent reference is Blankschtein and Makamel, 1982) were pursued. The problems of dynamics of critical behavior were extensively studied.…”

Phase transitions and ordering structures of a model of a chiral helimagnet in three dimensions

“…While recent experiments [2,3] have reported the existence of the CSL state at finite temperatures in three dimensions, finite-dimensional effects beyond the meanfield theory on the nature of the finite-temperature phase transitions of the system are still less clear. In the absence of magnetic fields, renormalization-group approaches [13,14] predict that the system undergoes a continuous phase transition with critical exponents of the ferromagnetic XY model. Another theoretical study [15] also indicates that the system belongs to the same universality class of the ferromagnetic XY model.…”

Incommensurate-commensurate transitions in the monoaxial chiral helimagnet driven by the magnetic field

“…The case of easy-axis aniostropy (γ < 0), not analyzed in this work, is theoretically interesting-and perhaps also phenomenologically-as peculiar phenomena may take place in the crossover to Ising-like behavior as |γ | increases [17]. …”