Dirac Systems İn Terms Of The Berry Gauge Fields And Effective Field Theory Of A Topological Insulator

Dirac systems in terms the of Berry gauge fields and the effective field theory of a time-reversal invariant topological insulator are investigated. Dirac systems or Dirac-like systems are non-relativistic systems, e.g. condensed matter systems, where the description of the physical system is given by either the massive or massless Dirac Hamiltonian. The Dirac systems investigated in this thesis are the time-reversal invariant topological insulators. A topological insulator is a bulk insulator with conducting edge states characterized by a topological number. The first theoretical model of the time-reversal invariant topological insulators is the Kane-Mele model of graphene where the intrinsic spin-orbit interaction and time-reversal symmetry is predicted to cause a quantized spin Hall current at the edges , leading to a quantized spin Hall conductivity given by the the topological Chern number. \\ As the theoretical background, the explicit derivation of $2+1$ dimensional massless Dirac Hamiltonian on graphene is given. The Berry gauge field and the corresponding Berry curvature are defined for massive free Dirac Hamiltonian in arbitrary dimensions employing the Foldy-Wouthuysen transformation of the Dirac Hamiltonian. The definitions of the first and second Chern numbers in terms of Berry curvature are given. \\ In the first part of the thesis, a semiclassical formulation of the quantum spin Hall effect for physical systems satisfying Dirac-like equation is introduced. Quantum spin Hall effect is essentially a phenomenon in two space dimensions. In the semiclassical formulation adopted in the thesis, the position and momenta are classical phase space variables, and spin is not considered as a dynamical degree of freedom. The derivation of the matrix-valued one-form lying at the heart of the semiclassical formulation adopted is made explictly using a wave-packet constructed from the positive energy eigenstates of free Dirac equation. Defining the symplectic two-form and employing Liouville equation, the semiclassical matrix-valued equations of motion are obtained. The phase space measure, $\tilde{w}_{1/2},$ and time evolutions of phase space variables, $\dot{\tilde x}_i\tilde{w}_{1/2}$ and $\dot{\tilde p}_i\tilde{w}_{1/2}$, are obtained in terms of the phase space variables. As an introductory example, the formalism is displayed through the anamolous Hall effect. The anamolous Hall conductivity is established from the term linear in the electric field and the Berry curvature in $\dot{\tilde x}_i\tilde{w}_{1/2}. $ The semiclassical formulation adopted is then illustrated within the Kane-Mele model of graphene in the absence and in the presence of the Rashba spin-orbit coupling term. The spin Hall current is defined with the aid of the equations for the time evolutions of phase space variables in terms of phase space variables. The spin Hall conductivity is established from the term linear in the elctric field and the Berry curvature in $\dot{\tilde x}_i\tilde{w}_{1/2}. $ It is shown that if one adopts the correct definition of the spin current in two space dimensions, the essential part of the spin Hall conductivity is always given by the spin Chern number whether the spin is conserved or not at the quantum level. In the absence of Rashba spin-orbit coupling, the third component of spin is conserved, and the definition of the spin Hall current is straightforward. In the presence of Rashba spin-orbit coupling, the third component of spin is not conserved so that a suitable base of spin eigenstates need to be employed to define spin Hall current. The anomalous velocity term survives in any $d+1$ spacetime dimension, since independent of the spacetime dimension and the origin of the Berry curvature in the time evolution of the coordinates there is always a term which is linear in both electric field and the Berry field strength. In the basis where a certain component of spin is diagonal this term will be diagonal. \\ In the second part of the thesis, a field theoretic investigation of topological insulators in $2+1$ and $4+1$ dimensions is presented using Chern-Simons theory and a method of dimensional reduction. Chern-Simons actions emerge as the effective field theories from the actions describing Dirac fermions in the presence of external gauge fields. A time-reversal invariant topological insulator model in $2+1$ dimensions is discussed and by means of a dimensional reduction the $1+1$ dimensional descendant is presented. The field strength of the Berry gauge field corresponding to the $4+1$ dimensional Dirac theory is explicitly derived through the Foldy-Wouthuysen transformation. Acquainted with it, the second Chern number is calculated for specific choices of the integration domain. The Foldy-Wouthuysen transformation which diagonalizes the Dirac Hamiltonian is proven to be a powerful tool to perform calculations in the effective field theory of the $4+1$ dimensional time-reversal invariant topological insulator. A method is proposed to obtain $3+1$ and $2+1$ dimensional descendants of the effective field theory of the $4+1$ dimensional time reversal invariant topological insulator. Inspired by the spin Hall effect in graphene, a hypothetical model of the time reversal invariant spin Hall insulator leading to a dissipationless spin current in $3+1$ dimensions is proposed. In terms of the explicit constructions presented in this thesis, one can discuss $\mathbb{Z}_2$ topological classification of TRI insulators in a tractable fashion. In principle, the approach presented can be generalized to interacting Dirac particles where the related Foldy-Wouthuysen transformation at least perturbatively exists.