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Journal of the Physical Society of Japan

Exactly Solvable Models and New Link Polynomials. I. N-State Vertex Models

Yasuhiro Akutsu, Miki Wadati

Published Sep 1, 1987• 124 citations
Publisher

Abstract

Presented is a general method to construct representations of the braid group, a basic object in the knot theory, from the Boltzmann weights of the exactly solvable models in statistical mechanics at criticality. The method is applied to a class of N -state vertex models ( N =2, 3 and 4) to have explicit braid group representations. Furthermore, the Markov traces are introduced and a sequence of new link polynomials, topological invariants for knots and links, is constructed.

DOI
10.1143/jpsj.56.3039

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Presented is a general method to construct representations of the braid group, a basic object in the knot theory, from the Boltzmann weights of the exactly solvable models in statistical mechanics at criticality. The method is applied to a class of N -state vertex models ( N =2, 3 and 4) to have explicit braid group representations. Furthermore, the Markov traces are introduced and a sequence of new link polynomials, topological invariants for knots and links, is constructed.
It is shown that braid matrices and link polynomials can be systematically constructed from exactly solvable models in statistical mechanics. Through symmetry breaking transformations, different braid matrices are derived from a solvable model. By associating the Markov traces with multi-variable representations, multi-variable link polynomials are obtained. Infinitesimal operators for braid matrices are constructed. Connection of our approach to the conformal field theories and the topological quantum field theory is discussed.
Development in the theory of solvable (integrable) models is reviewed. It covers from basic knowledge on completely integrable systems to recent work by the authors. First, soliton theory is briefly summarized. Through the inverse scattering method and its quantum extension, a central concept, commuting transfer matrices, and a key relation, the Yang-Baxter relation, are introduced. Second, it is shown that there exists at least ∞ × ∞ number of solvable models in two-dimensional statistical mechanics. Third, quantum spin chains corresponding to solvable statistical mechanical models are discussed. In particular, finite temperature extension of Baxter's formula is given. Fourth, a new approach is presented to the classification problem of knots and links. It is shown that link polynomial, topological invariant for knots and links, can be associated with any solvable model in statistical mechanics. In the presentation, universality of the soliton picture in field theory, spin systems and statistical mechanics is observed.
Abstract The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.