Phase Locking in the Heisenberg Helimagnet

The commensurability energy ΔE is calculated for a Heisenberg helimagnet whose wavelength is three lattice constants at zero temperature with a small but nonzero uniform field applied in the plane of polarization of the spins. It is shown that ΔE=0 for classical spins but ΔE≠0 for quantum spins when spin‐wave interactions are considered. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/435 Phase locking in the Heisenberg helimagnet A. 8. Harris Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 The commensurability energy AE is calculated for a Heisenberg helimagnet whose wavelength is three lattice constants at zero temperature with a small but nonzero uniform field applied in the plane of polarization of the spins. It is shown that AE=O for classical spins but AE$O for quantum spins when spin-wave interactions are considered. In this paper it is shown that phase locking occurs in a broad class of Heisenberg helimagnets when a uniform field is applied in the plane of polarization. The model Hamiltonian I treat is’ z=f C Jpi’ Sj h x Six + E C 5’; v i i ==REx -Izz + ZA, (lb) where Si is a quantum spin of magnitude S on the ith site of a simple tetragonal lattice. I include a small easy plane anisotropy energy E whose effect is to orient the spins in the x-y plane but can otherwise be neglected in the limit of small h. Arbitrarily I take the x-y plane to coincide with the a-b plane. Also Jgz J(ri rj) is assumed to have the symmetry of the lattice. In the a-b plane interactions between first, second, and third nearest neighbors are, respectively, jt = 1, j,, and j,. For nearest neighbors in the c direction, Jir = JI > 0. All other interactions are neglected. For sufficiently negative values of jZ and/or j, the classical ground state of this model is’ a helix of wave vector Q # 0. For h =0 the value of Q is a continuous function of j, and js in contrast to the devil’s staircase (i.e., stepwise discontinuous) behavior for the axial nearest, next-nearest-neighbor Ising (ANNNI) model2 Since the Heisenberg helix (for h=O) has a circular cross section, the free energy is clearly independent of the phase of the helix and the commensurability energy vanishes. On the other hand, application of a field in the plane of polarization distorts the cross section of the helix into an ellipse and leads to my results for hf0 that the Heisenberg system exhibits an incomplete devil’s staircase. Early spin-wave calculations3” indicated the presence of a Goldstone mode for small h. This phason mode occurs if the energy is independent of the phase of the helix. This early work was based on linearized spin-wave theory and implied that Q varied continuously with j, and j,. Recently, it was shown6 that nonlinear spin-wave interactions modified this picture. For small h the pinning free energy (omitted in the early work) is of the form 8F= ,>TG Ap( T) hPS(pQ G) ~0s ~4, (2) where p is an integer, G a (big) reciprocal lattice vector, and 4 the phase of the helix. Minimization with respect to 4 yields SF= &(T)hplS(pQ -G), (3) and consequently Q will remain pinned at the (commensurate) value G/p for a range of values of j, andj, of order Aj, Aj3 hp”. This behavior is referred to as devil’s staircase behavior and has been extensively studied in the Frenkel-Kontorova model.7 Previously6 we found AP( T) to be nonzero for a classical model except for the special case p = 3, T = 0. Some time ago Elliott and Lange4 showed explicitly that A3( T=O) =0 for classical models within linear spin-wave theory. Our previous result for a classical model [A3( T=O) T] extended this result to include the effect of nonlinearity. In the present paper, however, it is shown that A3( T=O) +O when quantum spin-wave interactions are included. Thus even for p=3, Ap( T) = 0 should be viewed as defining a special isolated multicritical point. We now consider the calculation of /iZ( T=O) for a quantum spin system. Since this calculation is extremely complicated, it can only be summarized here. Following Ref. 8 one writes SF= sin(Q*ri + +)fl+ cos(Q*ri + #)S$, (4d Sf=cos(Q*ri+ +)A’: + sin(Q-ri+ +)S$, (4b)