Study Of The Free Fermion And Free Parafermion Spin Chains

In this thesis,we study the physical properties of two one-dimensional models originating in the theory of phase transitions and critical phenomena in statistical physics and condensed matter physics.These models are the quantum Ising model in a transverse field and a more general quantum Z(N)model.Such quantum spin chains are realistic models for quantum magnetism.Both the exact solution and the physical properties of the spin-1/2(two-state)Ising model are described by the powerful concept of free fermions,as treated in detail in many textbooks.A very interesting N-state generalization of this model has been discovered which has a connection to free parafermions.Free parafermions are a natural generalization of free fermions.This research project aims to be the first to solve and study the model of free parafermions in detail.Just as for the quantum Ising model,which is a special case of the model of free parafermions,results from this project will be of relevance to various topics in statistical and condensed matter physics.The N-state Z(N)quantum spin chain central to this research project was discovered by R.J.Baxter in 1989.This model has open boundary conditions.The energy eigenspectrum of this model was shown recently to be described in terms of free parafermions by P.Fendley.However,no physical properties of Baxter's free parafermionic Z(N)quantum spin chain have been calculated.Moreover,there is no exact solution of the model in the familiar form of the solution of the quantum Ising chain.The concept of parafermions has been known in both the physics literature and the mathematics literature for over 60 years,but up until 2014 there was no known physical realization of free parafermions.This research topic is thus potentially very significant,from the purely academic point of view,and also more practically because parafermions have been studied recently in the context of topological phases and edge modes in condensed matter physics.Unlike the majority of models studied so far in physics,the free parafermion model is not conformally invariant at the critical point when the number of states N is greater than two.Nevertheless the model exhibits a form of anisotropic scaling,originating in the chiral nature of the model.More seriously the model is non-Hermitian,with a complex energy eigenspectrum above a real ground-state.The model of free parafermions will thus have some unexpected properties.Models of this kind describe the dynamics of physical systems that are not conservative.Several of the usual properties of Hermitian systems,such as insensitivity of bulk thermodynamic quantities to boundary conditions,can fail in the non-Hermitian case.In general non-Hermitian systems are a promising new and relatively unexplored research direction in physics.The purpose of this research project is first to provide an exact solution and second to calculate the physical properties of Baxter's Z(N)model of free parafermions.This includes calculating the behavior of the energy spectrum for both finite and infinite size systems.In this way key properties such as the critical exponents describing the behavior of the specific heat and correlation length can be derived.This project also aims to discuss the eigenstates of the model,from which other physical quantities,such as correlation functions can be obtained when N=2.And some ground state expectation values corresponding to the general N can be obtained via Hellmann-Feynman theorem.In chapter 1 we review the status,significance and origins of research on parafermions.In chapter 2,we describe the basic concepts and differences from classical phase transitions to quantum phase transitions,as well as the description of critical phenomena.In chapter 3,we introduce the concept of free fermions and a specific XYh model,which includes the quantum Ising model as a special case,and show how to transform such an interacting system into a system described by free fermions.We also recall that the quantum Ising chain is a special case of the more general Z(N)model of free parafermions,we use the free fermionization method to obtain some physical properties of the quantum Ising chain.In chapter 4,the exact solution for the energy spectrum of the free parafermion Z(N)chain is presented.This solution is then used to derive exact results for various physical quantities and critical exponents of the model.The exact results include a remarkably compact result for the ground-state energy per site in terms of a hypergeometric function.In chapter 5,we outline the formalism for the study of spin correlations of the quantum Ising chain subject to open boundary conditions.These correlations include nearest-neighbor spin correlations,end-to-end spin correlations and the surface magnetization.A new exact result is found for an end-to-end spin correlation.We also analyze the expectation value of some internal nearest neighbor physical quantities corresponding to the open boundary conditions of the free parafermion Z(N)chain and the numerical results associated with the endpoints ‘correlations'.Some new exact results are obtained for the correlations,some interesting new behavior is observed.The thesis concludes in chapter6 with a brief summary and an outlook pointing towards future research directions.