Antiferromagnetic Potts Model Research Articles

Uniqueness of the Gibbs Measure for the Anti-ferromagnetic Potts Model on the Infinite Delta -Regular Tree for Large Delta

In this paper we prove that for any integer qge 5, the anti-ferromagnetic q-state Potts model on the infinite Delta -regular tree has a unique Gibbs measure for all edge interaction parameters win [1-q/Delta ,1), provided Delta is large enough. This confirms a longstanding folklore conjecture.

Antiferromagnitnaya model' Pottsa na ob\"emno-tsentrirovannoy kubicheskoy reshetke

We have studied phase transitions and thermodynamic properties of the antiferromagnetic Potts model with number of spin states q = 3 on a body-centered cubic lattice by the Monte Carlo method. Investigations have been performed with account for exchange interactions J1 and J2 between the first and second nearest neighbors. The phase transition order has been analyzed using the histogram method. It is found that in this model with J2 = 0, a second-order phase transition is observed. It is concluded that the inclusion of the interaction between the second nearest neighbors changes the type of the phase transition.

Machine-learning-assisted Monte Carlo fails at sampling computationally hard problems

Several strategies have been recently proposed in order to improve Monte Carlo sampling efficiency using machine learning tools. Here, we challenge these methods by considering a class of problems that are known to be exponentially hard to sample using conventional local Monte Carlo at low enough temperatures. In particular, we study the antiferromagnetic Potts model on a random graph, which reduces to the coloring of random graphs at zero temperature. We test several machine-learning-assisted Monte Carlo approaches, and we find that they all fail. Our work thus provides good benchmarks for future proposals for smart sampling algorithms.

Uniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular tree

AbstractWe show that the $4$ -state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$ -regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$ . This is tight since it is known that there are multiple Gibbs measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and $d\geq 4$ . We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$ -state Potts model on the $(d+1)$ -regular tree for $w\geq 1-\dfrac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$ .

Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $\Delta$-regular tree) implies strong spatial mixing on all graphs of maximum degree $\Delta$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $\Delta$ provided only that $q\ge 2\Delta$. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.

Surface criticality of the antiferromagnetic Potts model

We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get $y_{h1}=0.780(3)$, $\eta_\parallel=1.44(1)$, and $\eta_\perp=0.736(6)$; for the special transition, we get $y_s=0.59(1)$, $y_{h1}=1.693(2)$, $\eta_\parallel=-0.391(4)$, and $\eta_\perp=-0.179(5)$; in the extraordinary-log phase, the surface correlation function $C_\parallel(r)$ decays logarithmically, with decaying exponent $q=0.60(2)$, however, the correlation $C_\perp(r)$ still decays algebraically, with critical exponent $\eta_\perp=-0.442(5)$. If the ferromagnetic next nearest neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the $Z_6$ symmetry-breaking phase, with critical exponent $y_{\rm s}=0.41(2)$. The scaling behaviors of the second transition is very interesting, the surface spin correlation function $C_\parallel(r)$ and the surface squared staggered magnetization at this point decays logarithmically, with exponent $q=0.37(1)$; however, the surface structure factor with the smallest wave vector and the correlation function $C_\perp(r)$ satisfy power-law decaying, with critical exponents $\eta_\parallel=-0.69(1)$ and $\eta_\perp=-0.37(1)$, respectively.

Phase transitions in a two-dimensional antiferromagnetic Potts model on a kagome lattice

The phase transitions and thermodynamic properties of the two-dimensional antiferromagnetic Potts model with the number of spin states q=4 on the kagome lattice are studied by the Monte Carlo method, taking into account the exchange interactions of the first J1 and second J2 nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the interval -1.0≤ J_2≤0.0. An analysis of the character of phase transitions has been carried out. It is shown that in the interval -1.0≤ J_2≤-0.1, a second-order phase transition is observed, while at J_2=0.0, frustrations disturb the order in the system and lead to the disappearance of the phase transition Keywords: Frustration, phase transition, Monte Carlo method, Potts model.

Фазовые переходы в двумерной антиферромагнитной модели Поттса на решетке кагоме

The phase transitions and thermodynamic properties of the two-dimensional antiferromagnetic Potts model with the number of spin states q = 4 on the kagome lattice are studied by the Monte Carlo method, taking into account the exchange interactions of the first J1 and second J2 nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the interval -1.0 ≤ J2 ≤ 0.0. An analysis of the character of phase transitions has been carried out. It is shown that in the interval -1.0 ≤ J2 ≤ -0.1, a second-order phase transition is observed, while at J2 = 0.0, frustrations disturb the order in the system and lead to the disappearance of the phase transition.

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where $[k]:=\{1,\ldots,k\}$. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $\Delta\in \mathbb{N}$ and any $k\geq e\Delta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${\bf Z}(G;k,w)\neq 0$ for any $w\in U$ and any graph $G$ of maximum degree at most $\Delta$. (Here $e$ denotes the base of the natural logarithm.) For small values of $\Delta$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.

High pressure hydrogen and the Potts model on a triangular lattice

We present Monte Carlo studies and analysis of the frustrated antiferromagnetic Potts model of a triangular lattice. This Potts model shows a remarkably rich range of structures, and striking similarities to the high pressure phases of hydrogen which are typified by hexagonal close packed layered structures []. There are four known H2 molecular phases, all of which are isostructural to within the resolution of x-ray diffraction. Experimentally, the phase lines have been mapped by spectroscopy, which cannot reveal the structure. Study by density functional theory (DFT) has suggested a large number of candidate structures, based on the hexagonal-close packing of H2 molecules. The Potts model exhibits structures similar to DFT candidate hydrogen phases I, II and III: the range of different Potts model structures suggests that the hydrogen system in the ‘phase II’ region, may exhibit more than a single phase. It also suggests reorientational excitations which may be detectable in spectroscopy.

Periodic Gibbs measures for the antiferromagnetic Potts model on a Cayley tree of order k

Periodic Gibbs measures for the antiferromagnetic Potts model on a Cayley tree of order k

Phase Diagram of the Antiferromagnetic Potts Model with Number q = 4 of Spin States in the Hexagonal Lattice

Phase Diagram of the Antiferromagnetic Potts Model with Number q = 4 of Spin States in the Hexagonal Lattice

A comprehensive neural networks study of the phase transitions of Potts model

Using the techniques of neural networks (NN), we study the three-dimensional (3D) five-state ferromagnetic Potts model on the cubic lattice as well as the two-dimensional (2D) three-state antiferromagnetic Potts model on the square lattice. Unlike the conventional approach, here we follow the idea employed by Li et al (2018 Ann. Phys., NY 391 312–331). Specifically, instead of numerically generating numerous objects for the training, the whole or part of the theoretical ground state configurations of the studied models are considered as the training sets. Remarkably, our investigation of these two models provides convincing evidence for the effectiveness of the method of preparing training sets used in this study. In particular, the results of the 3D model obtained here imply that the NN approach is as efficient as the traditional method since the signal of a first order phase transition, namely tunneling between two channels, determined by the NN method is as strong as that calculated with the Monte Carlo technique. Furthermore, the outcomes associated with the considered 2D system indicate even little partial information of the ground states can lead to conclusive results regarding the studied phase transition. The achievements reached in our investigation demonstrate that the performance of NN, using certain amount of the theoretical ground state configurations as the training sets, is impressive.

Integrable boundary conditions in the antiferromagnetic Potts model

We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine {D}_2^2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the {D}_2^2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.

Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2, \u211d)/U(1) sigma model

We initiate a study of the boundary version of the square-lattice Q-state Potts antiferromagnet, with Q ∈ [0, 4] real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the SL(2, ℝ)k/U(1) Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the SL(2, ℝ)k/U(1) coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The CBC we find are of two types. The first is free boundary Potts spins, for which we confirm an old conjecture for the generating functions of conformal levels, and show them to be related to characters in a non-linear deformation of the W∞ algebra. The second type of CBC — which corresponds to restricting the values of the Potts spins to a subset of size Q1, or its complement of size Q − Q1, at alternating sites along the boundary — is new, and turns out to be conformal in the antiferromagnetic case only. In the following, we refer to these new boundary conditions as “alt” boundary conditions. Using algebraic and numerical techniques, we show that the corresponding spectrum generating functions produce all the characters of discrete representations for the coset CFT. The normalizability bounds of the associated discrete states in the coset CFT are found to have a simple interpretation in terms of boundary phase transitions in the lattice model. In the two-boundary case, with two distinct alt conditions, we obtain similar results, at least in the case when the corresponding boundary condition changing operator also inserts a number of defect lines. For sqrt{Q} = 2 cos frac{pi }{k} , with k ≥ 3 integer, we show also how our boundary conditions can be reformulated in terms of a RSOS height model. The spectrum generating functions are then identified with string functions of the compact SU(2)k−2/U(1) parafermion theory (with symmetry Zk−2). The new alt conditions are needed to cover all the string functions. We provide an algebraic proof that the two-boundary alt conditions correctly produce the fusion rules of string functions. We expose in detail the special case of Q = 3 and its link with three-colourings of the square lattice and a corresponding boundary six-vertex model. Finally, we discuss the case of an odd number of sites (in the loop model) and the relation with wired boundary conditions (in the spin model). In this case the RSOS restriction produces the disorder operators of the parafermion theory.

Translation Invariance of the Periodic Gibbs Measures for the Potts Model on the Cayley Tree

Work is a continuation of work TMPh, 175(2), 2013. We study the Potts model with zero external field on the Cayley tree. It is shown that for any values of the parameter all periodic Gibbs measures are translation-invariant for the q-state antiferromagnetic Potts model on the Cayley tree of order two and for the q-state ferromagnetic Potts model on the Cayley tree of order k.

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on $\mathbb{Z}^d$ for sufficiently large $d$. In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Kotecký conjecture.

The critical behavior of the two-dimensional three-state Potts model on a triangular lattice with quenched disorder

The critical behavior of the two-dimensional three-state antiferromagnetic Potts model with quenched disorder on a triangular lattice is investigated by the Monte Carlo method. Static critical exponents for the susceptibility γ, the magnetization β, the specific heat α, and the exponent of the correlation radius ν at spin concentrations p = 0.90; 0.80; 0.70; 0.65 are calculated on the basis of the finite-size scaling theory. The critical exponents are found to be increasing with a rise in disorder without violating the feasibility of scaling equation 2βν+γν=d, while relations γ/ν and β/ν remain unchanged. This behavior of the critical exponents we have come to associate with the weak universality of a critical behavior typical for disordered systems.

Computer simulation of the critical behavior of highly diluted low-dimensional antiferromagnetic systems on a triangular lattice

A computer simulation of the critical behavior of a two-dimensional highly diluted 3-state antiferromagnetic Potts model on a triangular lattice is performed. The calculations are done for systems with periodic boundary conditions at spin concentrations p equal to 0.70 and 0.65. Systems with linear dimensions L×L = N, L = 20–144 are considered. Based on the theory of finite-dimensional scaling, the static critical exponents of heat capacity α, susceptibility γ, order parameter β, and critical index ν for the correlation radius are calculated. It is numerically shown that the calculated critical exponents vary with changes in the spin concentration p, while the ratios β/ν and γ/ν remain unchanged within error, showing a weak universality of the critical behavior of disordered systems.

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.A conjecture of Mohar (2007) asserts that, for k≥3, all k-colourings of a k-regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither K4 nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each k≥4. We also report the implications of this result on the validity of the Wang–Swendsen–Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.