Loschmidt echo zeros in finite-size quantum systems with linear quench

TL;DR

This work addresses observing Loschmidt echo zeros (LEZs) and dynamical quantum phase transitions in finite-size quantum systems, where LEZs are typically restricted to the thermodynamic limit. It proposes a two-step quench protocol: a linear ramp of the transverse field from $h_i$ to $h_f$ over time $\tau$ (rate $v=(h_f-h_i)/\tau$), followed by evolution under the final Hamiltonian. In the quantum Ising chain, LEZs can be accessed by tuning the ramp rate when the quench crosses the phase transition, with exact LEZs satisfying $|A|=|B|$, and the critical times $t_c$ associated with divergence of the rate function; the rate function is $lambda = -(1/N) \ln|G(t_f)|^2$. The approach extends to XY and Haldane models, and reveals finite-size scaling of the maximum quench time with system size $N$, while remaining experimentally feasible without analytic continuation.

Abstract

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.

Figures (6)

• Figure 1: (a) Illustration of the time-dependent transverse field $h_{t}$ during and after the linear quench. The field linearly varies from $h_{i}$ to $h_{f}$ over the quench duration $\tau$, and remains constant at $h_{f}$ afterward.(b) Rate function $|\lambda|$ versus $t_f$ for system with $N=50$ and the quench time $\tau=1$ and $200$, respectively.
• Figure 2: Panels (a1)-(a3) and (b1)-(b3) show the behavior of the complex coefficients $|A_{k,\tau,i,f}|$ (blue dots) and $|B_{k,\tau,i,f}|$ (orange dots) under varying quench times with $N=50$. The quench time is set to $\tau=0.01$ in (a1), (b1), $\tau=1$ in (a2), (b2), and $\tau=100$ in (a3), (b3). The initial transverse field strength is $h_{i}=0.5$, while the final transverse field strength is $h_{f}=1.5$ for (a1)-(a3) and $h_{f}=0.9$ for (b1)-(b3). The vertical green dashed line indicates the critical $k_{s}$.
• Figure 3: Time evolution of the rate function $\lambda$ for the quantum Ising model for different values of $k$ , highlighting the non-analytic line (gray dashed line) predicted by Eq.(\ref{['ct']}). Panels (a)-(d) correspond to $k = \frac{7\pi}{50}, \frac{5\pi}{50}, \frac{3\pi}{50} ,$ and $\frac{\pi}{50} ,$ with $N=50$. Panels (e)-(f) correspond to $k=\frac{\pi}{8}$ with $N=8$.
• Figure 4: Finite size scaling of the quench time $\tau$ and system size $N$ with (a) $h_{i}=0.5, h_{f}=1.5$ and (b) $h_{i}=0.25, h_{f}=2.25$. The blue points are the data by solving Eq.(\ref{['cba']}) and the orange dashed lines are the fit data.
• Figure 5: Time evolution of the rate function $\lambda$ for the XY model for different values of $k$, with the non-analytic line (gray dashed) predicted by Eq.(\ref{['ct']}). Panels (a)-(d) show results for $k = \frac{13\pi}{50}, \frac{9\pi}{50}, \frac{5\pi}{50},$ and $\frac{\pi}{50}$, respectively.