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arXiv:2209.10895 · Physical Review Research

Theory of the Loschmidt echo and dynamical quantum phase transitions in disordered Fermi systems

Tuomas I. Vanhala, Teemu Ojanen

Published Sep 11, 2023• 13 citations
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Abstract

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.

DOI
10.1103/physrevresearch.5.033178

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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.
A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices, we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples---both in the integrable and the nonintegrable cases---where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.
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.
We study exact zeros of Loschmidt echo and quantum speed limit time for dynamical quantum phase transition in finite size systems. Our results illustrate that exact zeros of Loschmidt echo exist even in finite size quantum systems when the postquench parameter takes some discrete values in regions with the corresponding equilibrium phase different from the initial phase. As the system size increases and tends to infinity, the discrete parameters distribute continuously in the parameter regions. We further analyze the time for the appearance of the first exact zero of Loschmidt echo which is known as the quantum speed limit time ${\ensuremath{\tau}}_{\text{QSL}}$. We demonstrate that the maximal value of ${\ensuremath{\tau}}_{\text{QSL}}$ is proportional to LLL and approaches infinity in the thermodynamical limit, when we quench the initial noncritical state to the critical phase. We also calculate the minimal value of ${\ensuremath{\tau}}_{\text{QSL}}$ and find that its behavior is dependent on the phase of initial state.
The nonanalyticity of the Loschmidt echo at critical times in quantum quenched systems is termed as the dynamical quantum phase transition, extending the notion of quantum criticality to a nonequilibrium scenario. In this paper, we establish a new paradigm of dynamical phase transitions driven by a sudden change in the internal spatial correlations of the disorder potential in a low-dimensional disordered system. The quench dynamics between prequenched pure and postquenched random system Hamiltonian reveals an anomalous dynamical quantum phase transition triggered by an infinite disorder correlation in the modulation potential. The physical origin of the anomalous phenomenon is associated with the overlap between the two distinctly different extended states. Furthermore, we explore the quench dynamics between the prequenched random and postquenched pure system Hamiltonian. Interestingly, the quenched system undergoes dynamical quantum phase transitions for the prequench white-noise potential in the thermodynamic limit. In addition, the quench dynamics also shows a clear signature of the delocalization phase transition in the correlated Anderson model.
We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.
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.