Efekti konačne veličine u XY lancu

Abstract

The XY chain in a transverse field is a prototypical exactly solvable model with interesting and desirable properties: its two-dimensional phase diagram, characterized by an anisotropy parameter and the strength of an external magnetic field, hosts two different quantum phase transitions at zero temperature, while the model can always be mapped into a system of free (spin-less) fermions. One of the phase transitions is the universality of free fermions (c = 1 CFT), while the other is that of the 1D quantum Ising model (c = 1=2 CFT). As a matter of fact, the XY chain is a generalization of the Ising chain. While both lines of phase transitions can be described by Conformal Field Theory, the bi-critical point at which these lines meet is non-conformal, since it has a quadratic spectrum. The aim of this thesis is to observe the non-conformal nature of the bi-critical point in the XY chain in a numerical experiment trough quantities accessible also in non-exactly solvable models, such as the entanglement entropy and the eigenvalues of the reduced density matrix. In this way, we want to propose in the future numerical tests whether a multi-critical point in an arbitrary model is conformal or not. First we introduce the XY chain and using standard analytical methods find its microscopic solution. Like the Ising model, the XY chain has a phase of broken Z2 symmetry, where the model shows two degenerate ground states in the thermodynamic limit of infinite system length. Having noticed a scarcity of results on the exact energy degeneracy between two putative ground states, we examine the problem using numerical and analytical methods. The analytical method, employing complex analysis and Fourier series, has given a definite answer for the Ising model. Using this method for the more general XY chain we find the dependence of the exact degeneracy on the number of spins in the special case of zero magnetic field and show the absence of an exact degeneracy when the anisotropy is greater than 1 and the number of spins is even. Coming back to the main aim of the thesis, we derive the reduced density matrix eigenvalues and the entanglement entropy in the XY chain and construct a numerical algorithm for calculating them. We accomplish to discriminate the non-conformal nature of the bi-critical point by comparing the entanglement entropy and the largest reduced density matrix eigenvalue with the conformal field theory predictions near and far from the bi-critical point.