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arXiv:1902.04421 · Scientific Reports

Dynamical Quantum Phase Transition and Quasi Particle Excitation

Rouhollah Jafari

Published Feb 27, 2019• 79 citations
arXivHTMLPDF

Abstract

Dynamical phase transitions (DPTs) are signaled by the non-analytical time evolution of the dynamical free energy after quenching some global parameters in quantum systems. The dynamical free energy is calculated from the overlap between the initial and the time evolved states (Loschmidt amplitude). In a recent study it was suggested that DPTs are related to the equilibrium phase transitions (EPTs) (Heyl, M. et al. Phys. Rev. Lett. 110, 135704 (2013)). We here study an exactly solvable model, the extended XY model, the Loschmidt amplitude of which provides a counterexample. We show analytically that the connection between the DPTs and the EPTs does not hold generally. Analysing also the general compass model as a second example, assists us to propound the physical condition under which the DPT occurs without crossing the equilibrium critical point, and also no DPT by crossing the equilibrium critical point.

DOI
10.1038/s41598-019-39595-3

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Dynamical phase transitions (DPTs) are signaled by the non-analytical time evolution of the dynamical free energy after quenching some global parameters in quantum systems. The dynamical free energy is calculated from the overlap between the initial and the time evolved states (Loschmidt amplitude). In a recent study it was suggested that DPTs are related to the equilibrium phase transitions (EPTs) (Heyl, M. et al. Phys. Rev. Lett. 110, 135704 (2013)). We here study an exactly solvable model, the extended XY model, the Loschmidt amplitude of which provides a counterexample. We show analytically that the connection between the DPTs and the EPTs does not hold generally. Analysing also the general compass model as a second example, assists us to propound the physical condition under which the DPT occurs without crossing the equilibrium critical point, and also no DPT by crossing the equilibrium critical point.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
Dynamical quantum phase transitions (DQPTs) are criticalities in the time evolution of quantum systems and their existence has been theoretically predicted and experimentally observed. However, how the system behaves in the vicinity of DQPT and its connection to physical observables remains an open question. In this work, we introduce the concept of the Loschmidt amplitude spectrum (LAS), which extends the Loscmidt amplitude - the detector of the transition - by considering the overlap of the initial state to all the eigenstates of the prequench Hamiltonian. By analyzing the LAS in the integrable transverse-field Ising model, we find that the system undergoes a population redistribution in the momentum space across DQPT. In the quasiparticle picture, collective excitations around lower-half kkk modes become dominant when the system approaches DQPT. The LAS is also applicable to study the dynamics of nonintegrable models where we have investigated the Ising model with next-nearest-neighbor interactions as an example. The time evolution of the system's magnetization is found to be connected to the products of the LAS and there exists a simultaneous overlap of the time-evolved state to pairs of eigenstates of the prequnech Hamiltonian that possess spin configurations of negative magnetization. Our findings provide a better understanding of the characteristics of the out-of-equilibrium system around DQPT.
Dynamical quantum phase transition is a nonstationary analog of the equilibrium quantum phase transitions that may occur after global quenches across a quantum phase transition in systems. In this work, we investigate the quench dynamics of single-particle tight-binding quasiperiodic lattices, where the quenching process originates and terminates within the same phases. Intriguingly, the system under quench dynamics undergoes an energy-dependent dynamical quantum phase transition, characterized by the singularities of the Loschmidt echo in time scale. Specifically, we find disentangling behavior between static and dynamical quantum phase transitions for eigenenergies excluding band edges. Remarkably, the connection between static and dynamical quantum phase transitions does not hold generally for all eigenenergies. Our findings suggest the existence of dynamical phase transitions inside the band energies without crossing the critical point under certain conditions. Our results pave the way to a deeper understanding of the energy-dependent nonequilibrium quench dynamics.
Abstract Dynamical quantum phase transitions (DQPTs) can occur following quenches in quantum systems when the rate function, a dynamical analogue of the free energy, becomes non-analytic at critical times. Here we exhaustively investigate in an exemplary model how the dynamically evolving state responds to a second quench. We demonstrate that for quenches where the initial and final Hamiltonian belong to different phases always result in DQPTs, irrespective of the intermediate quench and dynamics or the time of the second quench. However, if the initial and final Hamiltonian belong to the same equilibrium phase then the intermediate Hamiltonian must belong to a different phase. In this case, the second quench time in relation to the critical times of the first quench becomes crucial to the existence of DQPTs.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.
The (Loschmidt) overlap between the state at different times after a quantum quench is attracting increasing interest, as it was recently shown that in the thermodynamic limit its logarithm per unit of length has a non-analytic behavior if a Hamiltonian parameter is quenched across a critical point. This phenomenon was called a "dynamical phase transition" in analogy with the behavior of the canonical partition function at an equilibrium phase transition. We distinguish between local and nonlocal contributions to the aforementioned quantity and derive an analytic expression for the time evolution of the local part after quantum quenches in the XXZ spin-1/2 chain. The state that describes the stationary properties of (local) observables can be represented by a Gibbs ensemble of a generalized Hamiltonian; we reveal a deep connection between the appearance of singularities and the excitation energies of the generalized Hamiltonian.