Field-Induced Kosterlitz-Thouless Transition in the Zero-Temperature Triangular Ising Antiferromagnet

We investigate the zero-temperature triangular Ising antiferromagnet in a magnetic field by means of transfer matrix and Monte Carlo methods. The finite-size results are compared with predictions obtained from a mapping to the Gaussian model. The results confirm the presence of a field-induced Kosterlitz-Thouless transition to a state with long-range order. The ground state of the antiferromagnetic triangular Ising model with only nearest-neighbor interactions is characterized by the absence of long-range order and has a nonzero entropy, i at least in the isotropic case. Since the correlation functions decay with power laws of the distance rather than exponentially, the ground state is critical. When a magnetic field B of sufBcient strength is applied to the ground state, the residual entropy is removed. The resulting ordered state has spins of one sign on two of the three sublattices of the triangular lattice, and spins of the opposite sign on the third sublattice. Here we investigate the nature of the associated phase transition, induced by a magnetic field of the order of the (vanishing) thermal energy kT Thus, we c. on-sider the T ~ 0 limit such that H = I3/kT remains finite, while the nearest-neighbor coupling K does

Field-induced Kosterlitz- We investigate the zero-temperature triangular Ising antiferromagnet in a magnetic field by means of transfer matrix and Monte Carlo methods. The finite-size results are compared with predictions obtained from a mapping to the Gaussian model. The results confirm the presence of a field-induced Kosterlitz-Thouless transition to a state with long-range order.
The ground state of the antiferromagnetic triangular Ising model with only nearest-neighbor interactions is characterized by the absence of long-range order and has a nonzero entropy, i at least in the isotropic case. Since the correlation functions decay with power laws of the distance rather than exponentially, the ground state is critical. When a magnetic field B of sufBcient strength is applied to the ground state, the residual entropy is removed. The resulting ordered state has spins of one sign on two of the three sublattices of the triangular lattice, and spins of the opposite sign on the third sublattice. Here we investigate the nature of the associated phase transition, induced by a magnetic field of the order of the (vanishing) thermal energy kT Thus, we c.onsider the T~0 limit such that H = I3/kT remains finite, while the nearest-neighbor coupling K does not: The net effect of this sum is, apart from contributing an infinite constant, the restriction that elementary triangles with three spins of the same sign are forbidden.
It was originally suspected that the reduced Geld 0 was relevant, i.e. , that the system would immediately enter the ordered state when H g 0. However, this was not supported by a later analysis4 which used a mapping to the Gaussian model and the Coulomb gas. 5 We summarize some of the main steps of this analysis. As which is equal to 4 for p = 2. This scaling field is irrelevant: If sufIiciently small, it does not destroy the algebraic correlations in the zero-field Ising ground state (at least, if our assumption concerning the existence of a mapping between the SOS and the Gaussian model was right). However, it is also to be expected that H will play the role of a temperaturelike parameter and will decrease the marginal Gaussian temperature field. At some value HKT the scaling dimension Xq will become marginal (TG = 16), and the system will undergo a Kosterlitz-Thouless (KT) transition. io Another consequence of this theory is that the smallest anomalous dimension is Xs& which is equal to 4 at H = 0; Xs is expected to decrease as a function of~H~and should reach the value 9 at H = HKT. For H & HKT the system enters a long-range ordered state which is threefold degenerate: the majority of the minus spins are located on one of the three sublattices of the triangular lattice. The anomalous dimension Xs is associated with a staggered field, e.g. , 2H on one sublattice and +H on the other two sublattices, and should govern the long-range behavior of the spin-spin correlation function g(r) ar In this paper we verify this scenario by two different approaches. Firstly, we determine the magnetic susceptibility of the zero-temperature Ising model as a function of the magnetic field for several finite L x L systems with toroidal boundaries. Finite-size scaling predicts that the susceptibility y(L) of a system with linear size L at a critical point scales as y(L) = g(0)+ AI, '"" '+ .
where A is a constant and the ellipsis stands for corrections to scaling and yH is the renormalization exponent associated with the field H. If the transition is indeed KT-like, yH --0 and the susceptibility will not diverge; it will behave qualitatively as the specific heat of the XY' model. However, if the transition is of the three-state Potts type, as might be suggested by the symmetry of the ordered state, we have y~-s, so that g(L) will diverge with L. L equal to powers of 2, ranging from 8 to 256. The simulations were performed at K = -12, which, after truncation of the Monte Carlo transition probabilities to machine precision, inhibits spin fiips that increase the nearest-neighbor interaction energy.
The way in which the initial states were prepared deserves some attention. Via the SOS representation, a T = 0 Ising configuration can be specified by means of a system of nonintersecting strings. s Since the number of strings is conserved under the Glauberiz dynamics used in the spin-updating algorithm, it should initially be set are shown for several I x I systems with periodic boundaries and I equal to powers of 2. In order to avoid crowding of symbols, the data are shown by smooth curves except for the largest system size (256) for which the data points are shown as &. The statistical error in the latter points is slightly larger than the size of the symbols. The errors in the other data do not exceed the thickness of the curves in most cases. The susceptibilities for system sizes I = 2 and 4 are zero because the initial state (described in the text) is frozen: no spin Sips are possible. These results shown here indicate that the susceptibility remains finite when the system enters the ordered phase. equal to 2I /3 which is appropriate for the state at H = 0 as well as the ordered state at 0 « H « -6I~. It does not seem likely that fluctuations in the number of strings, as occurring in the summation over alt Ising configurations in the partition sum, would alter universal aspects of the critical behavior. Since the DISP can only simulate systems with sizes equal to powers of 2, the aforementioned choice is not possible and we have chosen the number of strings equal to the nearest even integer to 2L/3 instead. Only configurations with an even number of strings correspond to Ising configurations with periodic boundaries. This choice leads to corrections to scaling in y proportional to 1/I, and to some alternation between results for system sizes that are odd and even powers of 2. While such corrections to scaling, and alternation, are much in line with the data shown in Fig. 1, there are no indications for a power-law divergence of y(L) with L.
As a further check of the Gaussian model description of the T = 0 Ising model, we have performed a finitesize scaling analysis using transfer matrix calculations. These calculations pertain to a lattice wrapped on an infinitely long cylinder, with one set of lattice edges perpendicular to the cylinder s axis. The finite-size parameter L is the circumference of the cylinder measured in lattice units.
The matrices TK+ and TK can be decomposed into L sparse matrices, which is convenient for computational purposes. is is This decomposition proceeds as for the honeycomb lattice, see, e.g. , Ref. 16. However, since we are now working in the limit I~-+oo, we have to be careful to share each bond strength Ix between adjacent triangles and to divide out an infinite factor exp( -K/2) from the Boltzmann weight of each triangle. The correlation length can be expressed in the two largest eigenvalues A1 and A2 of the transfer matrix as