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arXiv:0807.3201 · Physical Review E

Finite-size-scaling ansatz for the helicity modulus of the triangular-lattice three-spin interaction model

Hiromi Otsuka

Published Jun 3, 2008• 3 citations
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Abstract

The Berezinskii-Kosterlitz-Thouless-type continuous phase transition observed in the three-spin interaction model is discussed. The relevant field theory describes the topological defects involved and enables us to perform the renormalization-group analysis. Based on it, we shall propose the finite-size-scaling ansatz for the helicity modulus, which exhibits the exponent nu=3/5 for the correlation length in the disordered phase. We perform the Monte Carlo simulations to confirm the ansatz. Also, we argue its relevance to the ground-state phase transition in the quantum spin chain.

DOI
10.1103/physreve.77.062101

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The Berezinskii-Kosterlitz-Thouless-type continuous phase transition observed in the three-spin interaction model is discussed. The relevant field theory describes the topological defects involved and enables us to perform the renormalization-group analysis. Based on it, we shall propose the finite-size-scaling ansatz for the helicity modulus, which exhibits the exponent nu=3/5 for the correlation length in the disordered phase. We perform the Monte Carlo simulations to confirm the ansatz. Also, we argue its relevance to the ground-state phase transition in the quantum spin chain.
We study a Berezinskii-Kosterlitz-Thouless-like phase transition in a triangular-lattice three-spin interaction model. The vector sine-Gordon field theory describes defects involved, and enables us to perform the renormalization-group analysis of the transition. Based on it, we discuss properties of the spin-spin correlation function, and also analyze the helicity modulus for the present model. To provide the data to support our analysis, we perform the Monte-Carlo simulations to calculate them. Further, we exhibit snapshots of some typical defect conflgurations, which are helpful to illustrate the defect-mediated phase transition.
We investigate infinite-order phase transitions like the Berezinskii-Kosterlitz-Thouless transition observed in a triangular-lattice three-spin interaction model. Based on a field theoretical description and the operator-production-expansion technique, we perform the renormalization-group analysis, and then clarify properties of marginal operators near the phase transition points. The results are utilized to establish criteria to determine the transition points and some universal relations among excitation levels to characterize the transitions. We verify these predictions via the numerical analysis on eigenvalue structures of the transfer matrix. Also, we discuss an enhancement of symmetry at the end points of a critical intermediate phase in connection with a transition observed in the ground state of the bilinear-biquadratic spin-1 chain.
Using Monte Carlo methods, we compute the finite-size scaling function of the helicity modulus \ensuremath{\Upsilon} of the two-dimensional O(3) model and compare it to the low-temperature expansion prediction. From this, we estimate the range of validity for the leading terms of the low-temperature expansion of the finite-size scaling function and for the low-temperature expansion of the correlation length. Our results strongly suggest that a Kosterlitz-Thouless transition at a temperature T>0 is extremely unlikely in this model.
A Monte Carlo study is carried out on the six-state clock model on the triangular lattice. Physical quantities which characterize the Kosterlitz-Thouless (KT) phase transition are calculated. The temperature dependence of the exponent of the spin-pair correlation function, η( T ), in the KT phase is obtained precisely. The exponential divergence of the correlation length, ξ∼ exp ( a t - b ), with the theoretical value, b =0.5, gives a good fit to the data by the present MC simulations by choosing a =1.6∼1.8, η=0.23∼0.25 at the upper critical temperature T U =1.46∼1.49. However, the choice of critical parameters, a ∼3.1, b ∼0.3 and η∼0.28 at T U ∼1.53 gives the most consistent result by a finite-size scaling analysis on several physical quantities. The lower critical temperature, T L , is estimated to be 1.1.
A systematic numerical study of the relevant finite-size effects of the helicity modulus for the two-dimensional classical XY model demonstrates the usefulness of this quantity as a means of obtaining information about Kosterlitz-Thouless type phase transitions. In this way, one does not have to compute correlation functions on finite lattices, and fit or compare the data thus obtained to theoretical scaling predictions, which only pertain to extremely large lattices. The approach may have many applications to systems of current interest that exhibit XY-like phase transitions.