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Journal of Physics A Mathematical and General

Algebraic invariants of the O(2) gauge transformation

Jacques H. H. Perk, Fa Yueh Wu, X. N. Wu

Published Feb 21, 1990• 8 citations
Publisher

Abstract

The authors consider the O(2) gauge transformation for a two-state vertex model on a lattice, and derive its fundamental algebraic invariants, the minimal set of homogeneous polynomials of the vertex weights which are invariant under O(2) transformations. Explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p=2, 3, 4, 5, 6, generalising p=3 results obtained previously from more elaborate considerations.

DOI
10.1088/0305-4470/23/4/001

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The authors consider the O(2) gauge transformation for a two-state vertex model on a lattice, and derive its fundamental algebraic invariants, the minimal set of homogeneous polynomials of the vertex weights which are invariant under O(2) transformations. Explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p=2, 3, 4, 5, 6, generalising p=3 results obtained previously from more elaborate considerations.
Abstract We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the S L ( 2 ) × S L ( 2 ) symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field H = i π / ( 2 β ) , valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.
We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically with the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter eight-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.
We explore a geometric approach to generating local SU(2)\text{SU}(2)SU(2) and SL(2,C)\text{SL}(2,\mathbb{C})SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or ``gauge'' invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of ddd-level quantum systems. We give an operational interpretation of these invariants in terms of observables.
The mapping which produces a path between the p-state symmetric vertex model and the Ising model with spin [Formula: see text] on the honeycomb lattice is constructed. The mapping construction is based on a combination of decoration and gauge transformations. The formulation enables one to find the manifold in the vertex weights parameter space on which the mapping can be performed and to derive dual relations among the model parameters as well as the free energies of related symmetric vertex and Ising models. The extension of the mapping to arbitrary lattice coordination is outlined.