scholarly journals Some exact results on the Potts model partition function in a magnetic field

2009 ◽  
Vol 42 (38) ◽  
pp. 385004 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Robert Shrock
Keyword(s):  
Magnetic Field ◽  
Partition Function ◽  
Potts Model ◽  
Exact Results

FORMAL CONSTRAINTS ON SERIES ANALYSIS ON THE POTTS MODEL

1987 ◽  
Vol 01 (04) ◽  
pp. 145-153 ◽  
Author(s):  
D. HANSEL ◽  
J.M. MAILLARD
Keyword(s):  
Magnetic Field ◽  
Partition Function ◽  
Low Temperature ◽  
Potts Model ◽  
Correlation Functions ◽  
Exact Results ◽  
Nearest Neighbour ◽  
Temperature Expansion ◽  
Higher Dimensional ◽  
Formal Constraints

It is shown that the low temperature expansion of the partition function, magnetization and nearest neighbour correlation functions of the q-state checkerboard Potts model in a magnetic field drastically simplify on a very simple algebraic variety. These four formal constraints on the expansions are also analysed in the framework of the resummed low temperature expansions and the large q expansions. These exact results are generalized straightforwardly to higher dimensional hypercubic lattices and also to some random problems.

Some exact results for the Ashkin-Teller model

1979 ◽  
Vol 365 (1722) ◽  
pp. 371-380 ◽  
Keyword(s):  
Renormalization Group ◽  
Partition Function ◽  
Potts Model ◽  
Critical Line ◽  
Square Lattice ◽  
Direct Connection ◽  
Exact Results ◽  
Renormalization Group Theory ◽  
Vertex Model ◽  
Special Cases

It is shown that various cases of the Ashkin-Teller model on the square, triangular and hexagonal lattices can be transformed by the dual and star-triangle transformations and, further, that these problems can be reduced to special cases of the eight vertex model on the Kagomé lattice. In general, we can only obtain the partition function of the Ashkin-Teller model if we are on its line of fixed points, and it then turns out that it is reducible to the six vertex model. Since the partition function of the q -state Potts model at its critical point can also be related to the six vertex model, a direct connection between the Ashkin-Teller model and the Potts model can be made. It turns out that moving along the critical line of the Ashkin-Teller model corresponds to varying q for the Potts model. For the square lattice comparison is made with renormalization group calculations, and the agreement found is a satisfactory check of renormalization group theory.

scholarly journals Spin models in complex magnetic fields: a hard sign problem

2018 ◽  
Vol 175 ◽  
pp. 07026 ◽  
Author(s):  
Philippe de Forcrand ◽  
Tobias Rindlisbacher
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Partition Function ◽  
External Field ◽  
Potts Model ◽  
Cluster Algorithm ◽  
Spin Models ◽  
Sign Problem ◽  
Special Cases ◽  
Severe Sign

Coupling spin models to complex external fields can give rise to interesting phenomena like zeroes of the partition function (Lee-Yang zeroes, edge singularities) or oscillating propagators. Unfortunately, it usually also leads to a severe sign problem that can be overcome only in special cases; if the partition function has zeroes, the sign problem is even representation-independent at these points. In this study, we couple the N-state Potts model in different ways to a complex external magnetic field and discuss the above mentioned phenomena and their relations based on analytic calculations (1D) and results obtained using a modified cluster algorithm (general D) that in many cases either cures or at least drastically reduces the sign-problem induced by the complex external field.

EXACT RESULTS FOR THE ISING MODEL ON A 3–12 LATTICE

1989 ◽  
Vol 03 (08) ◽  
pp. 1237-1245
Author(s):  
K.Y. LIN
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Ising Model ◽  
Partition Function ◽  
Functional Relation ◽  
Interaction Parameters ◽  
Zero Field ◽  
Exact Results ◽  
Kagome Lattice ◽  
Kagomé Lattice

We consider the Ising model on a 3–12 lattice with magnetic field. An exact functional relation is established for the partition function and our result is a generalization of Giacomini’s work on the Kagomé lattice. We calculate the zero-field magnetic susceptibility when an appropriate relation among the interaction parameters is satisfied.

MAGNETIZATION PLATEAUS ON THE ZIGZAG LADDER WITH TWO-, AND THREE-SITE EXCHANGES

2007 ◽  
Vol 21 (20) ◽  
pp. 3567-3579 ◽  
Author(s):  
V. V. HOVHANNISYAN ◽  
L. N. ANANIKYAN ◽  
N. S. ANANIKIAN
Keyword(s):  
Magnetic Field ◽  
Partition Function ◽  
Strong Magnetic Field ◽  
Low Temperatures ◽  
Heisenberg Model ◽  
Spin Exchange ◽  
Exchange Interactions ◽  
Exact Results ◽  
Recursion Method ◽  
Exchange Parameters

We consider the Heisenberg model with two-, and three-spin exchange interactions on a zigzag ladder in a strong magnetic field. Using the recursion method for the partition function in Ising approximation we have found exact results for the magnetization. We have shown the existence of magnetization plateaus in the case of mutual two-, and three-spin exchanges at low temperatures and different exchange parameters. The system exhibits different magnetic behaviors, depending on the values of the exchange parameters.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

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