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Physical Review Letters

Tricritical Transitions, Interface Roughening, and the Classical XY Model

Victor J. Emery, Robert H. Swendsen

Published Nov 28, 1977• 33 citations
Publisher

Abstract

Space renormalization group calculations are performed for a class of spin-$S$ Ising systems, using a modification of Migdal's recursion relation. Interface-roughening models are obtained when $S\ensuremath{\rightarrow}\ensuremath{\infty}$ and tricritical points and critical lines become the roughening temperature and rough phase. The latter corresponds to the low-temperature phases of the classical Coulomb gas and two-dimensional $\mathrm{XY}$ model and implications for the $\mathrm{XY}$-model exponent $\ensuremath{\eta}$ are discussed.

DOI
10.1103/physrevlett.39.1414

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Space renormalization group calculations are performed for a class of spin-SSS Ising systems, using a modification of Migdal's recursion relation. Interface-roughening models are obtained when S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞ and tricritical points and critical lines become the roughening temperature and rough phase. The latter corresponds to the low-temperature phases of the classical Coulomb gas and two-dimensional XY\mathrm{XY}XY model and implications for the XY\mathrm{XY}XY-model exponent \ensuremathη\ensuremath{\eta}\ensuremathη are discussed.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
The global phase diagram of a vectorial generalization of the Blume-Emery-Griffiths model is obtained using Migdal’s approximate renormalization procedure. Classical two-component spins and non-magnetic impurities populate a triangular lattice, with nearest-neighbor interactions. The phase diagram in thermodynamic field space is divided into magnetic and impurity-rich phases by a first-order surface of discontinuous impurity concentrations, terminating in an Ising-type critical line. The magnetic region is further divided into a high-temperature paramagnetic phase and a low-temperature Kosterlitz-Thouless phase. The exponent η=1/4 of the pure system is preserved at the higher-order surface separating these two phases. This surface terminates in a line of critical end-points on the first-order surface, and, consequently, no tricritical point occurs for any values of the model parameters. However, the Ising critical line and the line of critical end-points approach each other in a certain limit, yielding an effective tricritical phase diagram. Within the Kosterlitz-Thouless phase, lines of constant η bunch together as the effective tricritical point is approached, in apparent agreement with tricritical scaling. This is also a model for superfluidity and phase separation in helium films.
We study the roughening transition of the dual of the two-dimensional (2D) XY model, of the discrete Gaussian model, of the absolute value solid-on-solid model and of the interface in an Ising model on a three-dimensional (3D) simple cubic lattice. The investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago. The basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable body-centred solid-on-solid (BCSOS) model. Our estimates for the critical couplings are , and for the XY model, the discrete Gaussian model and the absolute value solid-on-solid model, respectively. For the inverse roughening temperature of the Ising interface we find . To the best of our knowledge, these are the most precise estimates for these parameters published so far.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.