Non-equilibrium Phase Transitions In Isolated Quantum Systems

This thesis is the result of researching the steady-state in the transverse Ising model and the dynamical quantum phase transition in the Dicke model analytically and numerically.Quench is the way to change the equilibrium to non-equilibrium in isolated systems.The quenched quantum system will produce a different relaxation evolution mode than before.In the way of mean field theory,the transverse Ising model prepared in the ground state,producing three different evolution modes with respect to the order parameter magnetization after quenching.Two kinds of ever-lasting oscillated modes are divided by the critical modes of vanishing exponentially into ferro-magnetic phase and para-magnetic phase.After quenching,the transverse field Ising model will change from the ferromagnetic phase to the paramagnetic phase as the quenching magnetic field increases.By analytically solving the period of the magnetization in the ferromagnetic phase,we can obtain the expression of the elliptic integral of the average magnetization with respect to the initial magnetic field and the quenching magnetic field.Further calculations show that the accumulation of magnetization in one cycle is a constant.This makes the average magnetization in a period inversely proportional to the evolution period.The phase transition of the average magnetization is different from the quantum phase transition.First,the average magnetization has a significant deviation from the ground state magnetization.Moreover,the average magnetization vanishes in the way of inverse logarithm in the vicinity of critical point while the order parameter in the ground state vanishes in the way of power law.This shows that the non-equilibrium steady-state phase transition is different from the quantum phase transition,because the integrability of the transverse field Ising model prevents the system from returning to the thermal equilibrium state.Compared with the transverse Ising model,the Dicke model is different in its nonintegrability.The Dicke model undergoes a quantum phase transition from the normal phase to the super-radiative phase in the ground state as the interaction between light and matter increases.Minimizing the classical Hamiltonian means finding the fixed point of ordinary differential equations.We can obtain the fixed point coordinates of the phase space with respect to the magnetization and positions and their canonical momenta.These coordinates contain information about the quantum phase transition of the system when the magnetization and position are used as the order parameters.HolsteinPrimakoff approximation is used to transform the Dicke model into two coupling harmonic oscillators which can be solved exactly in the way of Bogliubov transformation.After quenching from the ground state,the Dicke model has two different evolution modes as to the order parameter.We research the dynamical free energy in the Dicke model then.The dynamical free energy is non-analytic,that is dynamical phase transition,when the order parameter turns over with time evolving.In the other dynamic phase,the order parameter always evolves in one direction,and the dynamical quantum phase transition does not occur.Finally,we display the picture in the space of frequency.After Fourier transform,the free energy density can be expressed in frequency space.There are two different frequency pictures in the frequency space corresponding to two different dynamics.In one picture,the local length of the dynamic characteristic function is non-zero.In another picture,the local length of the dynamic characteristic function is zero.Therefore,the local length can also mark the dynamical quantum phase transition.