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Physical review. E

Chebyshev polynomial approach to Loschmidt echo: Application to quench dynamics in two-dimensional quasicrystals

Niaz Ali Khan, Shihao Ye, Ziheng Zhou, Shujie Cheng, Gao Xianlong

Published Jun 25, 2024• 5 citations
Publisher

Abstract

The understanding of quantum phase transitions in disordered or quasicrystal media is a central issue in condensed matter physics. In this paper we investigate localization properties of the two-dimensional Aubry-André model. We find that the system exhibits self-duality for the transformation between position and momentum spaces at a critical quasiperiodic potential, leading to an energy-independent Anderson transition. Most importantly, we present the implementation of an efficient and accurate algorithm based on the Chebyshev polynomial expansion of the Loschmidt echo, which characterizes the nonequilibrium dynamics of quantum quenched quasiperiodic systems. We analytically prove that the system under quench dynamics displays dynamical quantum phase transitions and further provide numerical verification by computing the polynomial expansion of the Loschmidt echo. Our results may provide insight into the realization of electronic transport in experiments.

DOI
10.1103/physreve.109.065311

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The understanding of quantum phase transitions in disordered or quasicrystal media is a central issue in condensed matter physics. In this paper we investigate localization properties of the two-dimensional Aubry-André model. We find that the system exhibits self-duality for the transformation between position and momentum spaces at a critical quasiperiodic potential, leading to an energy-independent Anderson transition. Most importantly, we present the implementation of an efficient and accurate algorithm based on the Chebyshev polynomial expansion of the Loschmidt echo, which characterizes the nonequilibrium dynamics of quantum quenched quasiperiodic systems. We analytically prove that the system under quench dynamics displays dynamical quantum phase transitions and further provide numerical verification by computing the polynomial expansion of the Loschmidt echo. Our results may provide insight into the realization of electronic transport in experiments.
Recently the role of the mobility edge in localization transitions has been extensively studied for one-dimensional tight-binding quasiperiodic models. In this work we study the mobility edge in a family of quasiperiodic systems evolving far from equilibrium, such as quench dynamics. We report numerical simulations of the Loschmidt echo based on a polynomial expansion-based technique with a moderate computational cost. Remarkably, we obtain an identical energy dependence on the equilibrium and dynamical quantum phase transitions of quasiperiodic models. The self-dual energy-independent localization model under quench dynamics exhibits energy-independent dynamical quantum phase transitions. On the other hand, self-dual energy-dependent localization models undergo energy-dependent dynamical quantum phase transitions. The results provide insights into energy-dependent dynamical localization transitions in quasiperiodic systems relevant to experiments.
In this work we develop the theory of the Loschmidt echo and dynamical phase transitions in noninteracting strongly disordered Fermi systems after a quench. In finite systems the Loschmidt echo displays zeros in the complex time plane that depend on the random potential realization. Remarkably, the zeros coalesce to form a 2D manifold in the thermodynamic limit, atypical for 1D systems, crossing the real axis at a sharply defined critical time. We show that this dynamical phase transition can be understood as a transition in the distribution function of the smallest absolute value of the eigenvalues of the Loschmidt matrix and develop a finite-size scaling theory. Contrary to expectations, the notion of dynamical phase transitions in disordered systems becomes decoupled from the equilibrium Anderson localization transition. Our results highlight the striking qualitative differences of quench dynamics in disordered and nondisordered many-fermion systems.
The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p -wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.
We study the Loschmidt echo and the dynamical free energy of the Anderson model after a quench of the disorder strength. If the initial state is extended and the eigenstates of the postquench Hamiltonian are strongly localized, we argue that the Loschmidt echo exhibits zeros periodically with the period $2\ensuremath{\pi}/D$, where DDD is the width of spectra. At these zeros, the dynamical free energy diverges in a logarithmic way. We present numerical evidence of our argument in one- and three-dimensional Anderson models. Our findings connect the dynamical quantum phase transitions to the localization-delocalization phase transitions.
Dynamical quantum phase transitions reveal singularities in quench dynamics, characterized by the emergence of Loschmidt echo zeros at critical times, which usually exist only in the thermodynamic limit but are absent in finite-size quantum systems. In this Letter, we propose a theoretical scheme to probe Loschmidt echo zeros in finite-size systems by applying a two-step quenching protocol, which offers an experimentally feasible approach to study Loschmidt echo zeros. Using the transverse Ising model as a test bed, we identify that the exact Loschmidt echo zeros can be always accessed by tuning the quench rate, when the quench is across the phase transition point. The associated rate function displays divergence at critical times, accompanying with the change of the dynamical topological order parameter. The critical times are influenced by the quench rate, system size, and momentum modes, embodying the interplay between finite-size effects and critical dynamics. Moreover, the generality of these observations is further confirmed in the XY and Haldane models.
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.