Dynamic Phase Transitions in Periodically Driving 1D Ising Model

TL;DR

This work addresses how dynamical quantum phase transitions (DQPTs) arise in a one-dimensional transverse-field Ising model under a periodically modulated field. By mapping to free fermions and using time-dependent perturbation theory for a drive $\lambda(t)=\lambda+\lambda'\cos(\omega t)$, it computes the Loschmidt amplitude and rate function to identify nonanalytic cusps signaling DQPTs. It finds two routes to DQPTs: resonant intra-phase driving where $\omega$ matches energy-gap transitions with onset time $\tau \propto \sin^{-1} k_c\, \Delta_{k_c}\, \lambda'^{-1}$, and low-frequency inter-phase driving across the critical point due to ground-state degeneracy, while high-frequency driving suppresses these effects. The results illuminate nonequilibrium spin-chain dynamics and offer guidance for controlling DQPTs with periodic protocols in quantum simulations.

Abstract

This work investigates dynamical quantum phase transitions (DQPTs) in a one-dimensional Ising model subjected to a periodically modulated transverse field. In contrast to sudden quenches, we demonstrate that DQPTs can be induced in two distinct ways. First, when the system remains within a given phase--ferromagnetic (FM) or paramagnetic (PM), a resonant periodic drive can trigger a DQPT when its frequency matches the energy-level transition of the system. The timescale for the transition is governed by the perturbation strength $λ'$, the critical mode $k_c$, and its energy gap $Δ_{k_c}$, following the scaling relation $τ\propto \sin^{-1}k_c Δ_{k_c}λ'^{-1}$. Second, for drives across the critical point between the FM and PM phases, low frequencies can always induce DQPTs, regardless of resonance. This behavior stems from the degeneracy of the energy-level at the critical point, which ensures that any drive with a frequency lower than the system's intrinsic transition frequency will inevitably excite the system. However, in the high-frequency regime, such excitation will be strongly suppressed, thereby inhibiting the occurrence of DQPTs. This study provides deeper insight into the nonequilibrium dynamics of quantum spin chains.

Figures (5)

• Figure 1: (Color online) Time evolution of the rate function $r(t)$ and the winding number $\nu(t)$ under different driving frequencies. Panel (a1-a2) depicts the magnetic field initially fixed within the FM phase $\lambda = 0.5$, with a resonance frequency range of $[1, 3]$. Panel (b1-b2) depicts the magnetic field initially fixed within the PM phase $\lambda = 2$, with a resonance frequency range of $[2, 6]$. The driving strength is $\lambda'=0.1$ for all the cases.
• Figure 2: (Color online) Time evolution of the rate function $r(t)$ and the winding number $\nu(t)$ under different disturbance intensities. Panel (a1-a2) depicts the evolutionary process within the FM phase (where $\lambda = 0.5$, $\omega = 2$), with a resonance frequency range of $[1, 3]$. Panel (b1-b2) depicts the evolutionary process within the PM phase (where $\lambda = 2$, $\omega = 4$), with a resonance frequency range of $[2, 6]$.
• Figure 3: (Color online) The scaling of the evolution time and the driving strength with $\lambda = 0$, $\omega = 2$ and $\lambda = 2$, $\omega = 4$.
• Figure 4: (Color online) The time evolution of the rate function in the (a) FM ($\lambda=0$) and (b) PM ($\lambda=2$) phases for different driving intensities. The frequency is $\omega=0.5$ for all cases.
• Figure 5: (Color online) The time evolution of the rate function (a) and the winding number (b) at different drive frequencies when the transverse magnetic field crosses the quantum critical point, where $\lambda = 0.5$ and $\lambda'=2$.