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arXiv:1611.04710 · Physical review. B./Physical review. B

Universal Berezinskii-Kosterlitz-Thouless dynamic scaling in the intermediate time range in frustrated Heisenberg antiferromagnets on a triangular lattice

Ivan S Popov, Pavel Vladimirovich Prudnikov, A. N. Ignatenko, A. A. Katanin

Published Apr 21, 2017• 11 citations
arXivHTMLPDF

Abstract

We investigate nonequilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) near the temperature ${T}_{\mathrm{u}}=0.282J$ of the equilibrium ${\mathbb{Z}}_{2}$-vortex unbinding. We show that starting from the high-temperature initial state, due to the presence of ${\mathbb{Z}}_{2}$ vortices in the system being considered, in a broad temperature range $T<{T}_{\mathrm{u}}$ the dynamic properties in the intermediate time range are similar to those of the two-dimensional XY model below the Berezinskii-Kosterlitz-Thouless transition. The interaction of ${\mathbb{Z}}_{2}$ vortices with spin-wave degrees of freedom does not emerge until rather long times.

DOI
10.1103/physrevb.95.134437

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We investigate nonequilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) near the temperature Tu=0.282J{T}_{\mathrm{u}}=0.282JTu​=0.282J of the equilibrium Z2{\mathbb{Z}}_{2}Z2​-vortex unbinding. We show that starting from the high-temperature initial state, due to the presence of Z2{\mathbb{Z}}_{2}Z2​ vortices in the system being considered, in a broad temperature range T<TuT<{T}_{\mathrm{u}}T<Tu​ the dynamic properties in the intermediate time range are similar to those of the two-dimensional XY model below the Berezinskii-Kosterlitz-Thouless transition. The interaction of Z2{\mathbb{Z}}_{2}Z2​ vortices with spin-wave degrees of freedom does not emerge until rather long times.
We study by large-scale Monte Carlo simulation the RP3RP^3RP3 model, which can be regarded as an effective low-energy model of a triangular lattice Heisenberg antiferromagnet. Z2Z_2Z2​ vortices appear as elementary excitations in the triangular lattice Heisenberg antiferromagnet. Such Z2Z_2Z2​ vortices are ubiquitous in other frustrated Heisenberg spin systems that have noncollinear long-range orders. In this study, we investigate a possible topological phase transition driven by the binding--unbinding of Z2Z_2Z2​ vortices. By extracting important degrees of freedom, we map a frustrated spin system to an effective RP3RP^3RP3 model. From large-scale Monte Carlo simulation, we obtain an order parameter and a correlation length of up to L=16384L=16384L=16384. Concerning the existence of a Z2Z_2Z2​-vortex transition, by extrapolating the order parameter to the thermodynamics limit assuming the Z2Z_2Z2​-vortex transition, we obtain a finite transition temperature as Tv/J~≃0.25T_v/\tilde{J} \simeq 0.25Tv​/J~≃0.25. Our estimate of the correlation length at TvT_vTv​ is much larger than L=16384L=16384L=16384, which is beyond the previous estimate obtained with the triangular lattice Heisenberg model.
Classical XY spins on a two dimensional triangular lattice with antiferromagnetic interactions are reconsidered. We find that the Kosterlitz-Thouless transition associated to the U(1) symmetry appears at a temperature 0.0020(2) below the Ising transition at 0.5122(1) associated to the Z_2 symmetry. The Ising transition has critical exponents different from the standard ones. Using extensive Monte Carlo simulations for equilibrium and dynamical properties we show that the lack of universality observed in previous studies is due to finite size corrections not taken account. Likewise the Kosterlitz-Thouless transition has a critical exponent η\approx0.36 larger than the corresponding standard value 0.25. Also the helicity jump at the critical temperature is smaller than in the ferromagnetic case in disagreement with theoretical predictions. We try using the concept of an "quasi Universality class" to reconcile the standard critical behavior observable at higher temperatures with the different quasi universal one close to the critical region.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.
We study the antiferromagnetic {\it XY} model on a triangular lattice by extensive Monte Carlo simulations, focusing on its ordering and critical properties. Our result clearly shows that two separate transitions occur at two distinct temperatures, the one at a higher temperature is associated with a Z2Z_2Z2​-symmetry breaking driven by the chirality, and the one at a lower temperature is associated with the onset of the quasi-long-range order of the {\it XY} spin. We carefully examine the critical properties of each transition to find that the criticality of the chiral transition is consistent with the standard two-dimensional Ising universality class, whereas that of the spin transition might differ from the conventional Kosterlitz-Thouless (KT) one. The observed non-KT nature of the spin criticality is consistent with the most recent simulation result on the fully-frustrated {\it XY} model on a square lattice.
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength h = HTk B , as well as the Villain model on the square lattice. After a quench from a disordered initial state to T = 0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, L , we define and measure sample-dependent ‘first passage time’, t r , which is the number of Monte Carlo steps until the energy is relaxed to the ground state value. The distribution of t r , in particular its mean value, ⟨ t r ( L )⟩, is shown to obey the scaling relation, ⟨ t r ( L )⟩∼ L 2 ln( L / L 0 ), for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H = 0 and at the field-induced Kosterlitz–Thouless transition: however, the autocorrelation exponent is found to be H -dependent.