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arXiv:1702.05396 · Physical review. B./Physical review. B

Complex semiclassical analysis of the Loschmidt amplitude and dynamical quantum phase transitions

Tomoyuki Obuchi, Sei Suzuki, Kazutaka Takahashi

Published May 30, 2017• 23 citations
arXivHTMLPDF

Abstract

We propose a computational method of the Loschmidt amplitude in a generic spin system on the basis of the complex semiclassical analysis on the spin-coherent state path integral. We demonstrate how the dynamical transitions emerge in the time evolution of the Loschmidt amplitude for the infinite-range transverse Ising model with a longitudinal field, exposed by a quantum quench of the transverse field $\mathrm{\ensuremath{\Gamma}}$ from $\ensuremath{\infty}$ or zero. For both initial conditions, we obtain the dynamical phase diagrams that show the presence or absence of the dynamical transition in the plane of transverse field after a quantum quench and the longitudinal field. The results of semiclassical analysis are verified by numerical experiments. Experimental observation of our findings on the dynamical transition is also discussed.

DOI
10.1103/physrevb.95.174305

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We propose a computational method of the Loschmidt amplitude in a generic spin system on the basis of the complex semiclassical analysis on the spin-coherent state path integral. We demonstrate how the dynamical transitions emerge in the time evolution of the Loschmidt amplitude for the infinite-range transverse Ising model with a longitudinal field, exposed by a quantum quench of the transverse field \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ from \ensuremath∞\ensuremath{\infty}\ensuremath∞ or zero. For both initial conditions, we obtain the dynamical phase diagrams that show the presence or absence of the dynamical transition in the plane of transverse field after a quantum quench and the longitudinal field. The results of semiclassical analysis are verified by numerical experiments. Experimental observation of our findings on the dynamical transition is also discussed.
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.
We study quenching dynamics of a one-dimensional transverse Ising chain with nearest neighbor antiferromagnetic interactions in the presence of a longitudinal field which renders the model nonintegrable. The dynamics of the spin chain is studied following a slow (characterized by a rate) or sudden quenches of the longitudinal field. Analyzing the temporal evolution of the Loschmidt overlap, we find different possibilities of the presence (or absence) of dynamical phase transitions (DPTs) manifested in the nonanalyticities of the rate function of the return probability. Even though the model is nonintegrable, there are periodic occurrences of DPTs when the system is slowly ramped across the quantum critical point (QCP) as opposed to the ferromagnetic version of the model; this numerical finding is qualitatively explained by mapping the original model to an effective integrable spin model which is appropriate for describing such slow quenches. Furthermore, concerning the sudden quenches, our numerical results show that in some cases, DPTs can be present even when the spin chain is quenched within the same phase or even to the QCP, while in some other situations they completely disappear even after quenching across the QCP. These observations lead us to the conclusion that it is the change in the nature of the ground state that determines the presence of DPTs following a sudden quench.
Abstract The dynamical properties of the Lipkin–Meshkov–Glick model with longitudinal and transverse fields are studied by considering quantum quenches. The model describes a many-body quantum system with long-range interactions. A semiclassical analysis is performed to obtain the critical points of quantum phase transitions. These critical points consist in the static separatrix, which is found by analyzing the stability structure of the semiclassical potential and is useful in the identification of ground state and excited states quantum phase transitions, and in the dynamical separatrix, which depends on initial conditions and is found by studying the classical equations of motion for the Hamiltonian. The static and dynamical separatrices divide the parameter space in regions of different dynamical behavior, and it is found that both separatrices play an important role in the determination of the occurrences of the dynamical quantum phase transitions. Graphical abstract
We study the response of a quantum Ising chain to transverse field oscillations in the asymptotic state attained after a quantum quench. We show that for quenches across a quantum phase transition, the dissipative part of the response at low frequencies is negative, corresponding to energy emission up to a critical frequency ${\ensuremath{\omega}}^{*}$. The latter is found to be connected to the time period t∗{t}^{*}t∗ of the singularities in the Loschmidt echo $({t}^{*}=2\ensuremath{\pi}/{\ensuremath{\omega}}^{*})$ signaling the presence of a dynamical quantum phase transition. This result suggests that a linear-response experiment can be used to detect this kind of phenomenon.
The recently discovered dynamical phase transition denotes nonanalytic behavior in the real-time evolution of quantum systems in the thermodynamic limit, and it has been shown to occur in different systems at zero temperature [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. In this paper, we extend the analysis to nonzero temperature by studying a generalized form of the Loschmidt echo, the work distribution function, of a quantum quench in the transverse field Ising model. Although the quantitative behavior at nonzero temperatures still displays features derived from the zero-temperature nonanalyticities, it is shown that in this model dynamical phase transitions do not exist if T>0T>0T>0. This is a consequence of the system being initialized in a thermal state. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski relation can be exploited as a symmetry relation for a global quench or to obtain the change of the equilibrium free-energy density.
We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.