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arXiv:cond-mat/0605690 · Physical Review B

Continuous phase transition of a fully frustrated XY model in three dimensions

Kwangmoo Kim, D. Stroud

Published Jun 6, 2006• 9 citations
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Abstract

We have used Monte Carlo simulations, combined with finite-size scaling and two different real-space renormalization group approaches, to study a fully frustrated three-dimensional $XY$ model on a simple cubic lattice. This model corresponds to a lattice of Josephson-coupled superconducting grains in an applied magnetic field $\mathbf{H}=({\ensuremath{\Phi}}_{0}∕{a}^{2})(1∕2,1∕2,1∕2)$. We find that the model has a continuous phase transition with critical temperature ${T}_{c}=0.681\phantom{\rule{0.3em}{0ex}}J∕{k}_{B}$, where $J$ is the $XY$ coupling constant, and critical exponents $\ensuremath{\alpha}∕\ensuremath{\nu}=0.87\ifmmode\pm\else\textpm\fi{}0.01$, $v∕\ensuremath{\nu}=0.82\ifmmode\pm\else\textpm\fi{}0.01$, and $\ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.07$, where $\ensuremath{\alpha}$, $v$, and $\ensuremath{\nu}$ describe the critical behavior of the specific heat, helicity modulus, and correlation length. We briefly compare our results with other studies of this model, and with a mean-field approximation.

DOI
10.1103/physrevb.73.224504

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We have used Monte Carlo simulations, combined with finite-size scaling and two different real-space renormalization group approaches, to study a fully frustrated three-dimensional XYXYXY model on a simple cubic lattice. This model corresponds to a lattice of Josephson-coupled superconducting grains in an applied magnetic field H=(\ensuremathΦ0∕a2)(1∕2,1∕2,1∕2)\mathbf{H}=({\ensuremath{\Phi}}_{0}∕{a}^{2})(1∕2,1∕2,1∕2)H=(\ensuremathΦ0​∕a2)(1∕2,1∕2,1∕2). We find that the model has a continuous phase transition with critical temperature Tc=0.681J∕kB{T}_{c}=0.681\phantom{\rule{0.3em}{0ex}}J∕{k}_{B}Tc​=0.681J∕kB​, where JJJ is the XYXYXY coupling constant, and critical exponents \ensuremathα∕\ensuremathν=0.87\ifmmode±\else\textpm\fi0.01\ensuremath{\alpha}∕\ensuremath{\nu}=0.87\ifmmode\pm\else\textpm\fi{}0.01\ensuremathα∕\ensuremathν=0.87\ifmmode±\else\textpm\fi0.01, v∕\ensuremathν=0.82\ifmmode±\else\textpm\fi0.01v∕\ensuremath{\nu}=0.82\ifmmode\pm\else\textpm\fi{}0.01v∕\ensuremathν=0.82\ifmmode±\else\textpm\fi0.01, and \ensuremathν=0.72\ifmmode±\else\textpm\fi0.07\ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.07\ensuremathν=0.72\ifmmode±\else\textpm\fi0.07, where \ensuremathα\ensuremath{\alpha}\ensuremathα, vvv, and \ensuremathν\ensuremath{\nu}\ensuremathν describe the critical behavior of the specific heat, helicity modulus, and correlation length. We briefly compare our results with other studies of this model, and with a mean-field approximation.
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.
We study the phase diagram of the two-dimensional fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising-XY model. We present Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the O(n)O(n)O(n) spin-vector model which describe numerous phenomena's. Here, nnn is the number of vector dimensions and for the XY model n=2n=2n=2. The O(n)O(n)O(n)-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order p=4p=4p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions 90o90^o90o apart. This symmetry is called C4C_{4}C4​, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for p<4p<4p<4 becomes dangerously irrelevant for pgeq4p\\geq4pgeq4. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature Tc=2.2019textnormalkT_c = 2.2019 \\ \\textnormal kTc​=2.2019textnormalk for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right TcT_cTc​ and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with ppp and an investigation of the hexagonal anisotropy p=6p=6p=6 has been started.
The fully frustrated XY model with Villain interaction on a square lattice is studied by means of Monte Carlo simulations. On the basis of the universal jump condition it is argued that there are two distinct transitions in the model, corresponding to the loss of XY order and Z2{\mathrm{Z}}_{2}Z2​ order, respectively. The Kosterlitz-Thouless (KT) transition is analyzed by finite-size scaling of the helicity modulus at lattices of size L=32 through 128, giving TKT{\mathrm{T}}_{\mathrm{KT}}TKT​\ensuremath{\approx}0.8108(1). The vorticity-vorticity correlation function is used to determine two different characteristic lengths, the Z2{\mathrm{Z}}_{2}Z2​ correlation length \ensuremath{\xi}, and the screening length \ensuremath{\lambda}, associated with the KT transition and free vortices. The temperature dependence of \ensuremath{\xi} is examined in order to determine Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and the correlation length exponent, \ensuremath{\nu}. The exponent is found to be consistent with the two-dimensional (2D) Ising value, \ensuremath{\nu}=1, and the obtained critical temperature is Tc{\mathrm{T}}_{\mathrm{c}}Tc​=0.8225(5). The determinations of both \ensuremath{\xi} and \ensuremath{\nu} are done carefully, first applying the techniques to the 2D Ising model, which serves as a convenient testing ground.
We study the vortex glass transition in disordered high temperature superconductors using Monte Carlo simulations. We use a random pinning model with strong point-correlated quenched disorder, a net applied magnetic field, long-range vortex interactions, and periodic boundary conditions. From a finite size scaling study of the helicity modulus, the rms current, and the resistivity, we obtain critical exponents at the phase transition. The new exponents differ substantially from those of the gauge glass model, but are close to those of the pure three-dimensional XY model.