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arXiv:hep-th/9510010 · Journal of Physics A Mathematical and General

Two-state spectral-free solutions of the Frenkel-Moore simplex equation

L. C. Kwek, C. H. Oh, Kuldip Singh, Kye-Tae Wee

Published Dec 7, 1995• 0 citations
arXivHTMLPDF

Abstract

Whilst many solutions have been found for the quantum Yang-Baxter equation (QYBE), there are fewer known solutions available for its higher dimensional generalizations: Zamolodchikov`s tetrahedron equation (ZTE) and Frenkel and Moore`s simplex equation (FME). In this paper, we present families of solutions to the FME which may help us to understand more about higher dimensional generalization of the QYBE.

DOI
10.1088/0305-4470/28/23/028

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Whilst many solutions have been found for the quantum Yang-Baxter equation (QYBE), there are fewer known solutions available for its higher dimensional generalizations: Zamolodchikovs tetrahedron equation (ZTE) and Frenkel and Moores simplex equation (FME). In this paper, we present families of solutions to the FME which may help us to understand more about higher dimensional generalization of the QYBE.
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable statistical mechanics and their avatars are also based on YBE. Therefore, new solutions of the YBE could be used to construct new interesting 1D quantum or 2D classical systems with many other far-reaching applications. In this work, we attempt to find (almost) exhaustive set of solutions for the YBE in the lowest dimensions corresponding to a two-qubit case. We develop an algorithm, which can potentially be used for generating new higher-dimensional solutions of the YBE.
Multiple equations in math, physics, quantum information, and elsewhere are referred to as "the" Yang-Baxter equation, in spite of being a broad family of equations. Most of the equations are nonlinear matrix equations, where the unknown variable is a matrix. This is the case for the so called braided, algebraic, and generalized forms of "the" equation, which are the primary focus of this dissertation. Finding solutions to the various forms of these equations has been the subject of much research. The equations in all their forms are largely considered intractable in high dimensions, and only in dimension 2 have the solutions been fully classified. We begin with an introduction to quantum computation, with a focus on the topological model and its connection to the braid group. Next, we introduce the braided, algebraic, and generalized forms of "the" Yang-Baxter equation. We provide a full classification of diagonal solutions to each form. In particular, we show that any diagonal matrix is a solution to the algebraic form in any dimension, and each instance of the braided and generalized forms only have diagonal solutions that are scalar multiples of the identity. We exploit the relationship between the algebraic and braided forms to construct a solution in any dimension that is applicable to topological quantum computation as a universal gate. The generalized form of the equation is parameterized by three natural numbers, (d,m,l), and we show that the only invertible solutions when l ≥ m are scalar multiples of the identity. We completely classify all solutions arising from an X-shaped ansatz for five different choices of (d,m,l), and provide a complete classification of X-shaped solutions to every odd dimensional braided equation, where there are no X-shaped solutions in any dimension. We fully classify permutation solutions to each instance of the braided and algebraic equations that can be written as a product of 3 or fewer transpositions. We show that the problem of classifying all invertible upper triangular solutions to the 4-dimensional algebraic Yang-Baxter equation can be split into 48 cases, and fully classify one of the cases.