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The Quantum Potts Model as an Exactly Solvable Model

Razieh Mohseninia, Vahid Karimipour

Published Nov 8, 2015• 0 citations
OpenAlex

Abstract

We generalize the classical one dimensional Potts model to the case where the symmetry group is a non-Abelian finite group. It turns out that this new model has a quantum nature in that its spectrum of energy eigenstates consists of entangled states. We determine the complete energy spectrum, i.e. the ground states and all the excited states with their degeneracy structure. We calculate the partition function by two different algebraic and combinatorial methods. We also determine the entanglement properties of its ground states.

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We generalize the classical one dimensional Potts model to the case where the symmetry group is a non-Abelian finite group. It turns out that this new model has a quantum nature in that its spectrum of energy eigenstates consists of entangled states. We determine the complete energy spectrum, i.e. the ground states and all the excited states with their degeneracy structure. We calculate the partition function by two different algebraic and combinatorial methods. We also determine the entanglement properties of its ground states.
We generalize the classical one-dimensional Potts model to the case where the symmetry group is a non-Abelian finite group. It turns out that this model has a quantum nature in that its spectrum of energy eigenstates consists of entangled states. We determine the complete energy spectrum, i.e., the ground states and all the excited states with their degeneracy structure. We calculate the partition function by two different algebraic and combinatorial methods. We also determine the entanglement properties of its ground states.
The Heisenberg model, a quantum-mechanical analogue of the Ising model, has a large ground-state degeneracy, due to the symmetry generated by the total spin. This symmetry is also responsible for degeneracies in the rest of the spectrum. We discuss the global structure of the spectrum of Heisenberg models with arbitrary couplings, using group-theoretical methods. The Hilbert space breaks up in blocks characterized by the quantum numbers of the total spin, S and M, and each block is shown to constitute the representation space of an explicitly given irreducible representation of the symmetric group SN{\mathit{S}}_{\mathit{N}}SN​, consisting of permutations of the N spins in the system. In the second part of the paper we consider, as a concrete application, the model where each spin is coupled to all the other spins with equal strength. Its partition function is written as a single integral, elucidating its N dependence. This provides a useful framework for studying finite-size effects. We give explicit results for the heat capacity, revealing interesting behavior just around the phase transition.
An exact correspondence between the staggered biquadratic spin-1 antiferromagnetic chain and the quantum Hamiltonian limit of the nine-state Potts model is established for finite chains with free ends. For uniform interactions this equivalence is used (via a further exact mapping to a Bethe ansatz soluble XXZ model) to calculate exactly the infinite lattice values of the ground-state energy per site and the (nonzero) gap to the lowest-energy excited state. Periodic boundary conditions and the nature of the ground state as a function of bond alternation are also discussed.
We construct topological defects in two-dimensional classical lattice models and quantum chains. The defects satisfy local commutation relations guaranteeing that the partition function is independent of their path. These relations and their solutions are extended to allow defect lines to fuse, branch and satisfy all the properties of a fusion category. We show how the two-dimensional classical lattice models and their topological defects are naturally described by boundary conditions of a Turaev-Viro-Barrett-Westbury partition function. These defects allow Kramers-Wannier duality to be generalized to a large class of models, explaining exact degeneracies between non-symmetry-related ground states as well as in the low-energy spectrum. They give a precise and general notion of twisted boundary conditions and the universal behaviour under Dehn twists. Gluing a topological defect to a boundary yields linear identities between partition functions with different boundary conditions, allowing ratios of the universal g-factor to be computed exactly on the lattice. We develop this construction in detail in a variety of examples, including the Potts, parafermion and height models.