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Physical review. B, Condensed matter

Magnetic properties of the two-dimensionaln=0vector model

Hubert Saleur

Published Mar 1, 1987• 51 citations
Publisher

Abstract

We determine the magnetic properties of the two-dimensional n=0 vector model using transfer matrix calculations and finite-size scaling. In the high-temperature region, we obtain results similar to the mean-field picture. In the low-temperature region, we find a very different behavior with a susceptibility which is always negative and an infinite spontaneous magnetization. This is shown to be in agreementn with some works of Nienhuis. The physical interpretation is discussed.

DOI
10.1103/physrevb.35.3657

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We determine the magnetic properties of the two-dimensional n=0 vector model using transfer matrix calculations and finite-size scaling. In the high-temperature region, we obtain results similar to the mean-field picture. In the low-temperature region, we find a very different behavior with a susceptibility which is always negative and an infinite spontaneous magnetization. This is shown to be in agreementn with some works of Nienhuis. The physical interpretation is discussed.
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The thermodynamic properties of the two-dimensional Ising model with random (+or-J) nearest-neighbour interactions are determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The central result is that the spin-glass correlation length diverges as T- nu at low temperature T with nu =2.59+or-0.13. The heat capacity is similar to the Schottky anomaly. At zero temperature the entropy per spin is 0.0701+or-0.0005 and the internal energy per spin is -(1.4024+or-0.0012)J.
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The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.
We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
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