Damage spreading in two dimensional geometrically frustrated lattices: the triangular and kagome anisotropic Heisenberg model

The technique of damage spreading is used to study the phase diagram of the easy axis anisotropic Heisenberg antiferromagnet on two geometrically frustrated lattices. The triangular and kagome systems are built up from triangular units that either share edges or corners respectively. The triangular lattice undergoes two sequential Kosterlitz-Thouless transitions while the kagome lattice undergoes a glassy transition. In both cases, the phase boundaries obtained using damage spreading are in good agreement with those obtained from equilibrium Monte Carlo simulations. PACS numbers: 75.40.Cx, 75.40.Mg Submitted to: Journal of Physics A Damage spreading in two dimensional geometrically frustrated lattices 2 Frustrated antiferromagnets are particularly interesting because they allow novel kinds of low-temperature magnetic states to develop [1, 2] which are quite different from those observed in conventional magnets [3]. When the frustration is due entirely to the geometry of the lattice, exact results for the S = 1/2 Ising model on the triangular [4] and kagome [5] lattices have shown that there is no long range order at any temperature and the system has a macroscopic ground state degeneracy. In the triangular case, the spin-spin correlation function decays with a power law [6] whereas, in the kagome geometry, it decays exponentially at zero temperature [7, 8]. For isotropic vector spin models (XY or Heisenberg) on a triangular lattice [9, 10, 11] where the elementary triangles share edges, the frustration is partially relieved and a noncollinear planar ground state forms with neighbouring spins making angles of 120. Finite temperature transitions are evident in both cases (XY and Heisenberg) but are topological in nature. However, for the kagome lattice, the cornering sharing geometry leads to a disordered ground state for the Heisenberg model with the phenomenon of order by disorder occuring in the limit T → 0 with fluctuations favouring coplanar spin configurations [12] In order to explore the Heisenberg model on these geometrically frustrated lattices, various types of perturbations have been applied and have shown a strong effect on the ground state manifold [13]. We consider the following anisotropic Hamiltonian H = J ∑ i 1 represents an easy-axis anisotropy and we measure temperature and energy in units of J = 1. The limit A → 1 corresponds to the isotropic Heisenberg model whereas the limit A → ∞ corresponds to an infinite spin Ising model. Monte Carlo studies have shown for the triangular lattice the existence of two distinct KT types of defect-mediated phase transitions at finite temperature [14, 15] for A > 1. By examining the limit A → 1, the transition at the Heisenberg point also appears to be purely topological in character [16] but with exponentially decaying spin correlations in the low temperature phase. Studies of the same model on the kagome lattice [17, 18] indicate the existence of a finite, but very low, temperature ferromagnetic transition for intermediate values of A. Static results for the magnetization, susceptibility and specific heat indicate an Ising-like transition but with a nonuniversal critical behaviour [18] in which the values of the exponents vary with the strength of the anisotropy A. The individual spins do not exhibit any long ranged spatial order and resemble a glassy phase. An analysis of the time evolution of the double time spin auto-correlation function following a quench of the system from infinite temperature into a non-equilibrium state at low temperature displays the phenomenon Damage spreading in two dimensional geometrically frustrated lattices 3