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Register a new userCoarsening Dynamics of an Antiferromagnetic XY model on the Kagome Lattice: Breakdown of the Critical Dynamic Scaling
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We find a breakdown of the critical dynamic scaling in the coarsening dynamics of an antiferromagnetic {em XY} model on the kagome lattice when the system is quenched from disordered states into the Kosterlitz-Thouless ({em KT}) phases at low temperatures. There exist multiple growing length scales: the length scales of the average separation between fractional vortices are found to be {em not} proportional to the length scales of the quasi-ordered domains. They are instead related through a nontrivial power-law relation. The length scale of the quasi-ordered domains (as determined from optimal collapse of the correlation functions for the order parameter $exp[3 i theta (r)]$) does not follow a simple power law growth but exhibits an anomalous growth with time-dependent effective growth exponent. The breakdown of the critical dynamic scaling is accompanied by unusual relaxation dynamics in the decay of fractional ($3theta$) vortices, where the decay of the vortex numbers is characterized by an exponential function of logarithmic powers in time.
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Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B of G; we call B a basis of G. We introduce a two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = exp(K)-1, defined on G. This curve predicts the phase diagram both in the ferromagnetic (v>0) and antiferromagnetic (v<0) regions. For larger bases B the approximations become increasingly accurate, and we conjecture that P_B(q,v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P_B(q,v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker-Kadanoff phase. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wus approach. The polynomial predictions are in excellent agreement with numerical computations. For v>0 the accuracy of the predicted critical coupling v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to 10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).
We investigate the coarsening dynamics in the two-dimensional Hamiltonian XY model on a square lattice, beginning with a random state with a specified potential energy and zero kinetic energy. Coarsening of the system proceeds via an increase in the kinetic energy and a decrease in the potential energy, with the total energy being conserved. We find that the coarsening dynamics exhibits a consistently superdiffusive growth of a characteristic length scale with 1/z > 1/2 (ranging from 0.54 to 0.57). Also, the number of point defects (vortices and antivortices) decreases with exponents ranging between 1.0 and 1.1. On the other hand, the excess potential energy decays with a typical exponent of 0.88, which shows deviations from the energy-scaling relation. The spin autocorrelation function exhibits a peculiar time dependence with non-power law behavior that can be fitted well by an exponential of logarithmic power in time. We argue that the conservation of the total Josephson (angular) momentum plays a crucial role for these novel features of coarsening in the Hamiltonian XY model.
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