arXiv:cond-mat/9803090 · Journal of the Physical Society of Japan

Kosterlitz-Thouless Transition of QuantumXYModel in Two Dimensions

Kenji Harada, Naoki Kawashima

Published Aug 15, 1998• 79 citations

Abstract

The two-dimensional $S=1/2$ XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to $128 \times 128$ near and below the critical temperature. The critical temperature is estimated as $T_{\rm KT} = 0.3427(2)J$. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

DOI: 10.1143/jpsj.67.2768

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arXiv:cond-mat/9803090 · Journal of the Physical Society of Japan

Kosterlitz-Thouless Transition of QuantumXYModel in Two Dimensions

Kenji Harada, Naoki Kawashima

Published Aug 15, 1998• 79 citations

arXiv HTML PDF

Abstract

The two-dimensional

S=1/2

XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to

128 \times 128

near and below the critical temperature. The critical temperature is estimated as

T_{\rm KT} = 0.3427(2)J

. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

DOI: 10.1143/jpsj.67.2768

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We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.

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Fu-Jiun Jiang

Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional $XY$ model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature $T_{\text{BKT,H}}$ is found to be ${1}/{\sqrt{2}}$ exactly. Surprisingly, the value of $T_{\text{BKT,H}}$ obtained from our NN calculations is 0.572(3), which deviates significantly from ${1}/{\sqrt{2}}$ . In addition, based on the helicity modulus, the $T_{\text{BKT,H}}$ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional $XY$ model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature $T_{\text{BKT,H}}$ is found to be ${1}/{\sqrt{2}}$ exactly. Surprisingly, the value of $T_{\text{BKT,H}}$ obtained from our NN calculations is 0.572(3), which deviates significantly from ${1}/{\sqrt{2}}$. In addition, based on the helicity modulus, the $T_{\text{BKT,H}}$ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.

Progress of Theoretical and Experimental PhysicsSep 26, 20244 citations

The two-dimensional

S=1/2

XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to

128 \times 128

near and below the critical temperature. The critical temperature is estimated as

T_{\rm KT} = 0.3427(2)J

{\mathrm{L}}_{0}

), which is inferred from the solution of the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

{\mathit{kT}}_{\mathit{c}}

/J=0.350(4). Correlation length and in-plane susceptibility diverge at

{\mathit{T}}_{\mathit{c}}

precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near

{\mathit{T}}_{\mathit{c}}

and exhibits a peak around kT/J=0.45.

{\mathit{kT}}_{\mathit{c}}

/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near

{\mathit{T}}_{\mathit{c}}

, and falls off as

{\mathit{T}}^{2}

near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.

{\mathit{T}}_{\mathit{c}}

. The critical temperature is determined to be

{\mathit{T}}_{\mathit{c}}

=0.892 13(10). The temperature dependence of the characteristic length right above

{\mathit{T}}_{\mathit{c}}

is also obtained and found to be in agreement with recent results from high-temperature series expansions.

Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional

XY

model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature

T_{\text{BKT,H}}

is found to be

{1}/{\sqrt{2}}

exactly. Surprisingly, the value of

T_{\text{BKT,H}}

obtained from our NN calculations is 0.572(3), which deviates significantly from

{1}/{\sqrt{2}}

. In addition, based on the helicity modulus, the

T_{\text{BKT,H}}

determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.