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Register a new userMulticriticality of the (2+1)-dimensional gonihedric model: A realization of the (d,m)=(3,2) Lifshitz point
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Yoshihiro Nishiyama
Publication date
2007
fields
Physics
and research's language is
English
Authors
Yoshihiro Nishiyama
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Multicriticality of the gonihedric model in 2+1 dimensions is investigated numerically. The gonihedric model is a fully frustrated Ising magnet with the finely tuned plaquette-type (four-body and plaquette-diagonal) interactions, which cancel out the domain-wall surface tension. Because the quantum-mechanical fluctuation along the imaginary-time direction is simply ferromagnetic, the criticality of the (2+1)-dimensional gonihedric model should be an anisotropic one; that is, the respective critical indices of real-space (perp) and imaginary-time (parallel) sectors do not coincide. Extending the parameter space to control the domain-wall surface tension, we analyze the criticality in terms of the crossover (multicritical) scaling theory. By means of the numerical diagonalization for the clusters with Nle 28 spins, we obtained the correlation-length critical indices ( u_perp, u_parallel)=(0.45(10),1.04(27)), and the crossover exponent phi=0.7(2). Our results are comparable to ( u_{perp}, u_{parallel})=(0.482,1.230), and phi=0.688 obtained by Diehl and Shpot for the (d,m)=(3,2) Lifshitz point with the epsilon-expansion method up to O(epsilon^2).
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We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $epsilon=4-d+frac{m}{2}$ were presented for the critical exponents $ u_{{mathrm{L}}2}$, $eta_{{mathrm{L}}2}$ and $gamma_{{mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.
The existence of Neel order in the S=1/2 Heisenberg model on the square lattice at T=0 is shown using inequalities set up by Kennedy, Lieb and Shastry in combination with high precision Quantum Monte Carlo data.
The criticality of the (2+1)-dimensional XY model is investigated with the numerical diagonalization method. So far, it has been considered that the diagonalization method would not be very suitable for analyzing the criticality in large dimensions (d ge 3); in fact, the tractable system size with the diagonalization method is severely restricted. In this paper, we employ Novotnys method, which enables us to treat a variety of system sizes N=6,8,...,20 (N: the number of spins constituting a cluster). For that purpose, we develop an off-diagonal version of Novotnys method to adopt the off-diagonal (quantum-mechanical XY) interaction. Moreover, in order to improve the finite-size-scaling behavior, we tune the coupling-constant parameters to a scale-invariant point. As a result, we estimate the critical indices as u=0.675(20) and gamma/ u=1.97(10).
The universal critical point ratio $Q$ is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller (AT) model on the square lattice. A leading-order expansion of the ratio $Q$ in the presence of a non-vanishing thermal field is found from finite-size scaling and the corresponding expression is fitted to the accurate perturbative transfer-matrix data calculations for the $Ltimes L$ square clusters with $Lleq 9$.
The critical behaviour of d-dimensional n-vector models at m-axial Lifshitz points is considered for general values of m in the large-n limit. It is proven that the recently obtained large-N expansions [J. Phys.: Condens. Matter 17, S1947 (2005)] of the correlation exponents eta_{L2}, eta_{L4} and the related anisotropy exponent theta are fully consistent with the dimensionality expansions to second order in epsilon=4+m/2-d [Phys. Rev. B 62, 12338 (2000); Nucl. Phys. B 612, 340 (2001)] inasmuch as both expansions yield the same contributions of order epsilon^2/n.
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