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The generalized Loschmidt overlap amplitude and dynamical quantum phase transition in arbitrary quantum evolution

Haifeng Lang, Yixin Chen, Qiantan Hong, Heng Fan

Published Feb 27, 2018• 0 citations
OpenAlex

Abstract

Based on kinematic approach in defining geometric phase for a density matrix, we define the generalized Loschmidt overlap amplitude (GLOA) for an open system in arbitrary quantum evolution. GLOA reduces to Loschmidt overlap amplitude (LOA) with modified dynamic phase for unitary evolution of a pure state, and the argument of GLOA is the well-defined geometric phase. The rate function of GLOA exhibits non-analyticity at critical time, which corresponds to dynamical quantum phase transition. We observe that dynamical quantum phase transition related with GLOA will not be destroyed under finite temperature and weak enough dissipation. In particular, we find that a new type of dynamical quantum phase transition emerges in a dissipation system. The proposed GLOA provides a powerful tool in investigating quantum dynamical quantum phase transition in arbitrary quantum system.

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Based on kinematic approach in defining geometric phase for a density matrix, we define the generalized Loschmidt overlap amplitude (GLOA) for an open system in arbitrary quantum evolution. GLOA reduces to Loschmidt overlap amplitude (LOA) with modified dynamic phase for unitary evolution of a pure state, and the argument of GLOA is the well-defined geometric phase. The rate function of GLOA exhibits non-analyticity at critical time, which corresponds to dynamical quantum phase transition. We observe that dynamical quantum phase transition related with GLOA will not be destroyed under finite temperature and weak enough dissipation. In particular, we find that a new type of dynamical quantum phase transition emerges in a dissipation system. The proposed GLOA provides a powerful tool in investigating quantum dynamical quantum phase transition in arbitrary quantum system.
Based on a kinematic approach in defining a geometric phase for a density matrix, we define the generalized Loschmidt overlap amplitude (GLOA) for an open system for arbitrary quantum evolution. The GLOA reduces to the Loschmidt overlap amplitude (LOA) with a modified dynamic phase for unitary evolution of a pure state, with the argument of the GLOA well defined by the geometric phase, thus possessing a similar physical interpretation to that of the LOA. The rate function for the GLOA exhibits nonanalyticity at a critical time, which corresponds to the dynamical quantum phase transition. We observe that the dynamical quantum phase transition related to GLOA is not destroyed under a finite temperature and weak enough dissipation. In particular, we find that an alternate type of dynamical quantum phase transition emerges in a dissipation system. The proposed GLOA provides a powerful tool in the investigation of a dynamical quantum phase transition in an arbitrary quantum system, which not only can characterize the robustness of the dynamical quantum phase transition but also can be used to search for new transitions.
Preparing an integrable system in a mixed state described by a thermal density matrix, we subject it to a sudden quench and explore the subsequent unitary dynamics. To address the question of whether the nonanalyticities, namely, the dynamical quantum phase transitions (DQPTs), persist when the initial state is mixed, we consider two versions of the generalized Loschmidt overlap amplitude (GLOA). Our study shows that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature. On the other hand, a GLOA defined in the interferometric phase approach through the purifications of the time-evolved density matrix, indeed shows that nonanalyiticies in the corresponding ``dynamical free-energy density'' persist, thereby establishing the existence of mixed state dynamical quantum phase transitions (MSDQPTs). Our work provides a framework that perfectly reproduces both the nonanalyticities and also the emergent topological structure in the pure state limit. These claims are corroborated by analyzing the nonequilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.
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.
By deriving a general framework and analyzing concrete examples, we demonstrate a class of dynamical quantum phase transitions (DQPTs) in one-dimensional two-band systems going through double-quench processes. When this type of DQPT occurs, the Loschmidt amplitude vanishes and the rate function remains singular after the second quench, meaning the final state continually has no overlap with the initial state. This type of DQPT is named metamorphic DQPT to differentiate it from ordinary DQPTs that only exhibit zero Loschmidt amplitude and singular rate function at discrete time points. The metamorphic DQPTs occur at zero as well as finite temperatures. Our examples of the Su-Schrieffer-Heeger (SSH) model and Kitaev chain illustrate the conditions and behavior of the metamorphic DQPT. Since ordinary DQPTs have been experimentally realized in many systems, similar setups with double quenches will demonstrate the metamorphic DQPT. Our findings thus provide additional controls of dynamical evolution of quantum systems.
We present a unifying description of dynamical phase transitions in the unitary evolution of an isolated quantum particle and the dissipative relaxation of a classical Brownian particle, based on a dynamical generat- ing function. In the semiclassical and weak-noise limits, Fisher zeros of this function condense in the complex- time plane and reach the corresponding physical axes, producing dynamical phase transitions through a competition between return trajectories. This establishes a direct connection between quantum and classical (finite-time) dynamical phase transitions, where the semiclassical limit plays the role of the thermodynamic limit. In particular, the established link naturally provides a classical version of the Loschmidt amplitude and shows how dynamical quantum phase transitions, typically associated with isolated many-body systems, can arise in a single-particle quantum system. The dynamical phases are distinguished by a unifying, trajectory- based order parameter, realized as a classical correlation function and a sequential quantum weak value.
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.