Duality and Fisher zeros in the two-dimensional Potts model on a square lattice

Astorino, Marco; Canfora, Fabrizio

doi:10.1103/physreve.81.051140

Cited by 3 publications

(4 citation statements)

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“…[93] the phase diagram of these vertex models with real exp(αQ) has been studied for x 1 x −1 2 = 1 (in our notations). Applying these results one can observe that when the self-duality condition is satisfied, the condition (56) leads to the horizontal line on a phase diagram of [93][94][95]. This line meets a critical "antiferromagnetic line" at Q = 2 (Q = 4 in the notations of [93]).…”

Section: B Floquet Models Related To the Row Transfer Matrixmentioning

confidence: 86%

Integrable Floquet dynamics

2017

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We discuss several classes of integrable Floquet systems, i.e. systems which do not exhibit chaotic behavior even under a time dependent perturbation. The first class is associated with finite-dimensional Lie groups and infinite-dimensional generalization thereof. The second class is related to the row transfer matrices of the 2D statistical mechanics models. The third class of models, called here "boost models", is constructed as a periodic interchange of two Hamiltonians -one is the integrable lattice model Hamiltonian, while the second is the boost operator. The latter for known cases coincides with the entanglement Hamiltonian and is closely related to the corner transfer matrix of the corresponding 2D statistical models. We present several explicit examples. As an interesting application of the boost models we discuss a possibility of generating periodically oscillating states with the period different from that of the driving field. In particular, one can realize an oscillating state by performing a static quench to a boost operator. We term this state a "Quantum Boost Clock". All analyzed setups can be readily realized experimentally, for example in cold atoms.

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Section: B Floquet Models Related To the Row Transfer Matrixmentioning

confidence: 86%

Integrable Floquet dynamics

2017

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“…The key idea to analyze the partition function of the Potts model on self-similar lattices is to exploit the recursive symmetry deriving a closed set of dynamical equations describing the possible phases of the system. The great importance of recursive symmetry in statistical systems can be recognized also in the cases of more usual lattices as in [36][37][38][39][40]. In these works, recursive symmetry has been used to get a phenomenological description of the Ising model in three dimensions in a quite good agreement with the available numerical data.…”

Section: J Stat Mech (2024) 083101mentioning

confidence: 97%

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Alvarez

2024

J. Stat. Mech.

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We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

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“…Such efforts have produced detailed studies for the square, triangular, and honeycomb lattices [18,14,15,16,19,20,17,21].In many cases it is useful to exploit the recursive structure which is present in lattices of physical interest. Even when the analytic solution is not available, using a simple ansatz which respects the recursive symmetry one can get an excellent agreement with the numerical data [22,23,24,25]. Motivated by these facts we develop a method that permits the calculation of the exact partition function for strips of layered lattices.The idea to use the dichromatic polynomial in the cases of recursive lattices has been proposed in [26,27,28], where the authors derive a scalar homogeneous recurrence equation of order higher than one, whose solution is the dichromatic polynomial of the corresponding graph.…”

mentioning

confidence: 99%

“…In many cases it is useful to exploit the recursive structure which is present in lattices of physical interest. Even when the analytic solution is not available, using a simple ansatz which respects the recursive symmetry one can get an excellent agreement with the numerical data [22,23,24,25]. Motivated by these facts we develop a method that permits the calculation of the exact partition function for strips of layered lattices.…”

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confidence: 99%

Potts model on recursive lattices: some new exact results

et al. 2012

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We compute the partition function of the Potts model with arbitrary values of q and temperature on some strip lattices. We consider strips of width Ly = 2, for three different lattices: square, diced and 'shortest-path' (to be defined in the text). We also get the exact solution for strips of the Kagome lattice for widths Ly = 2, 3, 4, 5.As further examples we consider two lattices with different type of regular symmetry: a strip with alternating layers of width Ly = 3 and Ly = m + 2, and a strip with variable width. Finally we make some remarks on the Fisher zeros for the Kagome lattice and their large q-limit.where the temperature variable v = e K − 1 has been introduced with K ≡ βJ. For the ferromagnetic case (J > 0) one has 0 ≤ v ≤ ∞ corresponding to the temperature interval ∞ ≥ T ≥ 0; for the antiferromagnetic Potts (J < 0) the interval −1 ≤ v ≤ 0 corresponds to 0 ≤ T ≤ ∞. It can be shown directly from (1.1) that the partition function admits also the following polynomial (Fortuin-Kasteleyn) representation [9,10],1.2) * alvarez AT cecs.cl † canfora AT cecs.cl ‡ sreyes AT fis.puc.cl arXiv:0912.4705v4 [cond-mat.stat-mech] 7 May 2012where G is a graph with the same vertex set V of G and an edge set E ⊆ E, with E being the edge set of G; k(G ) and e(G ) are respectively the number of connected components and the number of vertices of G . It is important to notice that using (1.2) one can extend q and v from their physical values (q ∈ Z + , v ∈ [−1, +∞)) to the whole complex plane.Unlike the case of the Ising model (q = 2), solved by Onsager [11], the exact solution for the Potts model in any infinite two dimensional lattice is still not available. Nevertheless, thanks to universality, the critical behavior of the ferromagnetic Potts model is fairly well understood. On the other hand, the antiferromagnetic Potts depends on the structural properties of the lattice and its critical properties can vary strongly from case to case. Thus, the search for tools to obtain analytical information on the free energy of the Potts model on generic lattices is a very important task.An approach that has proven fruitful in recent years is to solve the simpler problem of calculating the partition function for periodic strips of finite width (L y ) and arbitrary length (L x ). Previous work along these lines has relied basically on a transfer matrix method [12,13,14,15,16] to obtain exact results at arbitrary temperatures for strips of widths going up to L y = 7 (honeycomb lattice with free boundary conditions) [17]. Such efforts have produced detailed studies for the square, triangular, and honeycomb lattices [18,14,15,16,19,20,17,21].In many cases it is useful to exploit the recursive structure which is present in lattices of physical interest. Even when the analytic solution is not available, using a simple ansatz which respects the recursive symmetry one can get an excellent agreement with the numerical data [22,23,24,25]. Motivated by these facts we develop a method that permits the calculation of the exact partition function ...

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Duality and Fisher zeros in the two-dimensional Potts model on a square lattice

Cited by 3 publications

References 46 publications

Integrable Floquet dynamics

Integrable Floquet dynamics

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Potts model on recursive lattices: some new exact results

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