Solvable eight-vertex model on an arbitrary planar lattice

1978 ◽  
Vol 289 (1359) ◽  
pp. 315-346 ◽  
Keyword(s):  
Statistical Mechanics ◽  
Solvable Model ◽  
Ising Models ◽  
Square Lattice ◽  
Vertex Model ◽  
Temperature Dependent ◽  
Planar Lattice ◽  
Special Cases ◽  
Local Properties ◽  
Straight Lines

Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.

General Square Lattice Vertex Models

2019 ◽  
pp. 430-453
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Square Lattice ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Spectral Parameters ◽  
Vertex Models ◽  
Integrable Quantum ◽  
Special Cases ◽  
Mechanical Models ◽  
The One ◽  
Ice Model

Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.

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 Cylindric Hecke Characters and Gromov–Witten Invariants via the Asymmetric Six-Vertex Model

2020 ◽  
Author(s):  
Christian Korff
Keyword(s):  
Hecke Algebras ◽  
Square Lattice ◽  
Vertex Operators ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Genus Zero ◽  
Planar Lattice ◽  
Infinite Dimensional ◽  
Structure Constants ◽  
Restriction Functor

AbstractWe construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram’s formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called ‘ice configurations’ on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing’s vertex operators of Hall–Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.

scholarly journals QUANTUM GROUP APPROACH TO A SOLUBLE VERTEX MODEL WITH GENERALIZED ICE RULE

1996 ◽  
Vol 11 (10) ◽  
pp. 1747-1761
Author(s):  
C.L. SOW ◽  
T.T. TRUONG
Keyword(s):  
Free Energy ◽  
Quantum Mechanics ◽  
Positive Integer ◽  
Quantum Group ◽  
Algebraic Structure ◽  
Square Lattice ◽  
Free Fermion ◽  
Vertex Model ◽  
Group Approach ◽  
Operator Structure

Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

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