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Journal of Physics C Solid State Physics

Interface in Potts ferromagnet: phase diagram and critical exponents

L. R. da Silva, U.M.S. Costa, Constantino Tsallis

Published Nov 30, 1985• 4 citations
Publisher

Abstract

Within a real-space renormalisation group framework, the authors treat the q-state Potts ferromagnet in a simple cubic lattice consisting of two different semi-infinite bulks (respectively characterised by the coupling constants J1and J2) separated by a (1,0,0) interface (characterised by Js). The use of a rather sophisticated two-terminal cluster allows a quite reliable discussion of the phase diagram and its universality classes. Four physically different phases are obtained, namely the paramagnetic and the double-bulk, single-bulk and surface ferromagnetic ones. The critical surface contains a multicritical line (associated with J1not=J2) which in turn contains a special point associated with J1=J2.

DOI
10.1088/0022-3719/18/33/010

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Within a real-space renormalisation group framework, the authors treat the q-state Potts ferromagnet in a simple cubic lattice consisting of two different semi-infinite bulks (respectively characterised by the coupling constants J1and J2) separated by a (1,0,0) interface (characterised by Js). The use of a rather sophisticated two-terminal cluster allows a quite reliable discussion of the phase diagram and its universality classes. Four physically different phases are obtained, namely the paramagnetic and the double-bulk, single-bulk and surface ferromagnetic ones. The critical surface contains a multicritical line (associated with J1not=J2) which in turn contains a special point associated with J1=J2.
Within a real-space renormalisation group framework, the authors discuss the criticality of a system constituted by two (not necessarily equal) semi-infinite ferromagnetic q-state Potts bulks separated by an interface. This interface is a bond-diluted Potts ferromagnet with a coupling constant which is in general different from those of both bulks. The phase diagram presents four physically different phases, namely the paramagnetic one, and the surface, single-bulk and double-bulk ferromagnetic ones. These various phases determine a multicritical surface which contains a higher-order multicritical line. Particular attention is devoted to the discussion of the critical concentration p c . Here, p c is the concentration of the interface bonds above which surface magnetic ordering is possible even if the bulks are disordered. An interesting feature comes out which is that p c varies continuously with J 1 /J s and J 2 /J s . The standard two-dimensional percolation concentration is recovered for J 1 =J 2 =0. From the analysis of the various fixed points obtained within the present formalism, a very rich set of critical universality classes emerges.
Within a real space renormalisation group framework which uses rather sophisticated clusters, the authors discuss the phase diagram and the universality classes of a semi-infinite cubic-lattice q-state Potts ferromagnet. In particular, they study the influence, on the surface magnetism, of q and Delta identical to J S /J B -1 (where J S and J B are respectively the surface and bulk coupling constants). The exact d=2 critical temperature T c 2D is recovered for all values of q. The q-dependence of the value Delta c above which surface magnetic order can exist even if the bulk is disordered, is calculated and, through a convenient extrapolation, reliable results are obtained (for the Ising particular case, i.e. q=2, they obtain the extrapolated value Delta c =0.569, which compares satisfactorily with the series result 0.6+or-0.1 and the Monte Carlo one, 0.50+or-0.03). At the surface-bulk (SB) multicritical point they calculate the q-dependences of the critical amplitude A and the crossover exponent phi (defined in the Delta to Delta c +0 limit through (T c S ( Delta )/T c 3D -1) approximately A( Delta / Delta c -1) 1 phi /, where T c S ( Delta ) and T c 3D identical to T c S ( Delta c ) respectively are the surface and the bulk critical temperatures), as well as the critical exponent y t1 SB . For the Ising particular case they obtain phi equivalent to 0.641 (which compares satisfactorily with the epsilon -expansion (0.68) and the Monte Carlo (0.56+or-0.04) results), A equivalent to 0.4 and 1/y t1 SB equivalent to 1.623. (As far as we know, no series or Monte Carlo results are available in the literature for A or 1/y t1 SB .).
We consider two semi-infinite spin-1/2 Ising media (respectively characterized by the ferromagnetic coupling constants J1{J}_{1}J1​ and J2{J}_{2}J2​) separated by a spin-1/2 anisotropic Heisenberg interface (characterized by a ferromagnetic coupling constant JS{J}_{S}JS​ and by an anisotropy \ensuremath{\eta}\ensuremath{\in}[0,1] in spin space; \ensuremath{\eta}=1 and \ensuremath{\eta}=0, respectively, correspond to the Ising and isotropic Heisenberg limits). Within a real-space renormalization-group framework, we discuss the full phase diagram; this diagram exhibits, besides the paramagnetic phase, three physically different ordered phases, namely the double-bulk, single-bulk, and surface ferromagnetic ones. We also analyze the various universality classes appearing in this problem, and determine, in particular, the existence of a new high-order multicritical point associated with the J1{J}_{1}J1​=J2{J}_{2}J2​ case.
Using a recently introduced renormalisation group method, the authors study the behaviour of the spontaneous surface and bulk magnetisations as functions of the temperature for an Ising ferromagnet in a semi-infinite cubic lattice for various J S /J B ratios (J S and J B respectively are the surface and bulk coupling constants). In particular, they study the extraordinary transition where the surface maintains its magnetisation as the bulk disorders: the authors find a discontinuity in the first derivative of the surface magnetisation at the bulk transition temperature. The criticality of the system (universality classes, critical exponents and amplitudes) is discussed as well.
Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.