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arXiv:2005.00474 · Physical review. E

Complete density calculations of q -state Potts and clock models: Reentrance of interface densities under symmetry breaking

E. Can Artun, A. Nihat Berker

Published Dec 16, 2020• 14 citations
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Abstract

All local bond-state densities are calculated for q-state Potts and clock models in three spatial dimensions, d=3. The calculations are done by an exact renormalization group on a hierarchical lattice, including the density recursion relations, and simultaneously are the Migdal-Kadanoff approximation for the cubic lattice. Reentrant behavior is found in the interface densities under symmetry breaking, in the sense that upon lowering the temperature, the value of the density first increases and then decreases to its zero value at zero temperature. For this behavior, a physical mechanism is proposed. A contrast between the phase transition of the two models is found and explained by alignment and entropy, as the number of states q goes to infinity. For the clock models, the renormalization-group flows of up to 20 energies are used.

DOI
10.1103/physreve.102.062135

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Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
We consider the critical behavior of the random qqq-state Potts model in the large-qqq limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy P(I)P(I)P(I) which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.
The random-field XY model is studied in spatial dimensions d=3 and 4, and in between, as the limit q→∞ of the q-state clock models, by the exact renormalization-group solution of the hierarchical lattice or, equivalently, the Migdal-Kadanoff approximation to the hypercubic lattices. The lower critical dimension is determined between 3.81<d_{c}<4. When the random field is scaled with q, a line segment of zero-temperature criticality is found in d=3. When the random field is scaled with q^{2}, a universal phase diagram is found at intermediate temperatures in d=3.
Phase transitions in a two-dimensional q-state Potts model with long-range disorder in which the correlation decays as a power law approximately r -a for separation r are studied analytically within a Migdal-Kadanoff renormalization group. Correlation terms are introduced into the parameter space of the renormalization group and the recursion relations are derived. We expand q near q 0 , where the specific-heat exponent of the pure system alpha p vanishes, and a near d=2. For small alpha p and delta =2-a<<1, we find three fixed points: 'pure','short-range disorder' and 'long-range disorder'. In our calculation the disorder is relevant if av-2<0 for a<d, where the v are the correlation-length exponents of the fixed points. The correlation-length exponent for the 'long-range disorder' fixed point is v long =2/a. These are consistent with field-theoretic renormalization group results.
A spin system is studied with simultaneous permutation-symmetric Potts and spin-rotation-symmetric clock interactions in spatial dimensions d=2 and 3. The global phase diagram is calculated from the renormalization-group solution with the recently improved (spontaneous first-order detecting) Migdal-Kadanoff approximation or, equivalently, with hierarchical lattices with the inclusion of effective vacancies. Five different ordered phases are found: Conventionally ordered ferromagnetic, quadrupolar, antiferromagnetic phases and algebraically ordered antiferromagnetic, antiquadrupolar phases. These five different ordered phases and the disordered phase are mutually bounded by first- and second-order phase transitions, themselves delimited by multicritical points: Inverted bicritical, zero-temperature bicritical, tricritical, second-order bifurcation, and zero-temperature highly degenerate multicritical points. One rich phase diagram topology exhibits all of these phenomena.