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arXiv:cond-mat/9612140 · Physical Review Letters

Domain Walls and Phase Transitions in the Frustrated Two-Dimensional XY Model

Colin Denniston B, Chao Tang

Published Jul 21, 1997• 17 citations
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Abstract

We compare the critical properties of the two-dimensional (2D) $\mathrm{XY}$ model in a transverse magnetic field with filling factors $f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$ and $2/5$. To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for $f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5$. A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for $f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$, and for $f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5$ with disorder, are those of the pure 2D Ising model. Studying the possible domain walls in the system provides some explanations for our results.

DOI
10.1103/physrevlett.79.451

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We compare the critical properties of the two-dimensional (2D) XY\mathrm{XY}XY model in a transverse magnetic field with filling factors f=1/3f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3f=1/3 and 2/52/52/5. To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for f=2/5f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5f=2/5. A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f=1/3f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3f=1/3, and for f=2/5f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5f=2/5 with disorder, are those of the pure 2D Ising model. Studying the possible domain walls in the system provides some explanations for our results.
We study the critical properties of the two-dimensional (2D) XY\mathrm{XY}XY model in a transverse magnetic field with filling factors f=1/3f=1/3f=1/3 and 2/5.2/5.2/5. To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for f=2/5.f=2/5.f=2/5. A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f=1/3f=1/3f=1/3 and for f=2/5f=2/5f=2/5 with disorder are those of the pure 2D Ising model. The relevant low-energy excitations are domain walls, and we show that their properties determine the nature of the phase transition.
We present a detailed study of the critical properties of the two-dimensional (2D) XY model with maximal frustration in a square lattice. We use extensive Monte Carlo simulations to study the thermodynamics of the spin and chiral degrees of freedom, concentrating on their correlation functions. The gauge-invariant spin-spin correlation functions are calculated close to the critical point for lattice sizes up to 240\ifmmode\times\else\texttimes\fi{}240; the chiral correlation functions are studied on lattices up to 96\ifmmode\times\else\texttimes\fi{}96. We find that the critical exponents of the spin-phase transition are \ensuremath{\nu}=0.3069 and \ensuremath{\eta}=0.1915, which are to be compared with the unfrustrated XY model exponents \ensuremath{\nu}=1/2 and \ensuremath{\eta}=0.25. We also find that the critical exponents of the chiral transition are ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\chi}}}$=0.875, 2\ensuremath{\beta}=0.1936, 2\ensuremath{\gamma}=1.82, and 2\ensuremath{\gamma}'=1.025, which are different from the expected 2D Ising critical exponents. The spin-phase transition occurs at TU(1){\mathit{T}}_{\mathit{U}(1)}TU(1)​=0.446, which is about 7% above the estimated chiral critical temperature TZ2{\mathit{T}}_{\mathit{Z}2}TZ2​=0.4206. However, because of the size of the statistical errors, it is difficult to decide with certainty whether the transitions occur at the same or at slightly different temperatures. Finally, the jump in the helicity modulus in the fully frustrated system is found to be about 23% below the unfrustrated universal value. The most important consequence of these results is that the fully frustrated XY model appears to be in a novel universality class. Recent successful comparisons of some of these results with experimental data are also briefly discussed.
We study the critical behavior of the fully frustrated XY\mathrm{XY}XY model on a square lattice by means of extensive histogram Monte Carlo simulations. We find a single transition in contrast to the scenario of two distinct transitions. From a finite-size scaling we determine the critical exponents associated with the Ising-like order parameter and find ${T}_{c}(\ensuremath{\infty})=0.4552(2)$, $\ensuremath{\nu}=0.852(2),$ and $\ensuremath{\gamma}=1.531(3)$. These exponents, different from the pure two-dimensional Ising critical exponents, are in agreement with the ones obtained for a mixed XY\mathrm{XY}XY Ising system.
In this thesis I study the Frustrated XY Model on a square lattice A direct physical realization of such a model can be found in arrays of Josephson junctions In Chapter I examine the Frustrated XY Model on a ladder via a transfer matrix formalism The eigenvalues of the transfer matrix are found numerically giving a determination of the di erent phases of the ladder The spatial periodicity of the ground state exhibits a devil s staircase as a function of the magnetic ux ning factor f If the transverse Josephson coupling is varied a continuous superconducting normal transition in the transverse direction is observed analogous to the breakdown of the Kolomogorov Arnold Moser trajectories in dynamical systems The remainder of the thesis examines the Frustrated XY Model in two dimensions Chapter examines the ground states and low energy excitations of the model In Chapter I compare the critical properties of the model with lling factors f and To obtain a comparison with recent experiments I investigate the e ect of weak quenched bond disorder for f A nite size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f and for f with disorder are those of the pure D Ising model Studying the possible domain walls in the system provides some explanations for these results In order to examine the properties of the model at an incommensurate eld I
We numerically investigate the critical properties of the two-dimensional XY model with four-fold and eight-fold crystal fields. Using extensive Monte Carlo simulations, we compute the specific heat, magnetization, orientational magnetization, and correlation length ratio to examine how these anisotropies influence the nature of phase transitions. The critical temperature and the critical exponent are systematically mapped across both compatible and competing field regimes, spanning from weak to strong field strengths. The numerical results reveal that the critical exponent exhibits a linear dependence on the inverse of the four-fold field strength in the strong field regime, while remaining finite and non-divergent in the weak field regime. Furthermore, the orientational magnetization effectively identifies and characterizes Ising-type phase transitions in the competing field regime.
The XY\mathrm{XY}XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.