The Intrinsic Connection between Dynamical Phase Transitions and Magnetization in the 1D XY Model

TL;DR

This work addresses how the initial magnetization, encoded by a coherent Gibbs state with effective inverse temperature $\beta$ and phase $\phi$, governs dynamical quantum phase transitions (DQPTs) in a quenched 1D XY model. By mapping the model to free fermions and computing the Fisher zeros of the Loschmidt amplitude, the authors show that weaker initial magnetization (smaller $\beta$) generally facilitates DQPTs under intra-phase quenches, and they identify a $\beta_c$ line that marks the maximum quench amplitude allowing a DQPT, yielding a protocol-dependent 'DQPT area'. Importantly, quenches across critical lines can still exhibit DQPTs even with strong initial polarization, while certain intra-phase quenches require reduced magnetization or enhanced coherence to trigger dynamical criticality. The study provides mechanistic insight into the role of initial spin polarization in non-equilibrium dynamical phase transitions and points to experimental avenues for verifying and controlling DQPTs in tabletop quantum simulators.

Abstract

In this manuscript, we study the quench dynamics of a transverse-field XY model starting from coherent Gibbs states. The results reveal that the initial strength of magnetization plays a crucial role in the emergence of dynamical quantum phase transitions. In concrete terms, when quenching within the same phase, through the properties of observables such as Fisher zeros and magnetization, we show that the stronger the initial magnetization, the more difficult the emergence of dynamical quantum phase transitions. The underlying mechanism is that the strong initial magnetization provides a directional effect, which inhibits the spin flipping in the process of quantum quench, making the dynamical quantum phase transition difficult to emerge. Since dynamical quantum phase transitions can be experimentally realized in various artificial systems, we hope that the physics predicted here can be experimentally verified in tabletop platforms.

Figures (8)

• Figure 1: Phase diagram of the transverse-field XY model. Solid lines are the critical lines, which separate three gapped phases. Dashed arrows denote the different quench paths A-G.
• Figure 2: Fisher zeros for quenches of the transverse field along quench paths A ($\lambda=0\rightarrow0.5$), B ($\lambda=0.5\rightarrow1.5$), and C ($\lambda=1.5\rightarrow2.0$) with different $\beta$. Throughout, $\phi=-\pi/2$ and $\gamma=0.5$.
• Figure 3: Fisher zeros for anisotropy quench paths. The path D (E) satisfies $\gamma=0.8\rightarrow0.5$ ($\gamma=0.5\rightarrow-0.5$) with fixed $\lambda=0.2$. Path F (G) follows the identical $\gamma$ quench as path D (E), but at a different $\lambda$=1.5. Throughout, $\phi=-\pi/2$.
• Figure 4: Magnetization versus $\gamma$ for (a) $M_x$, $M_y$ with $\lambda=0.5$, and (b) $M_z$ with $\lambda=1.2$, respectively. Throughout, $\phi=-\pi/2$, and the values of $\beta$ are marked.
• Figure 5: Magnetization versus contour plots characterize the initial state $\psi(\lambda_0,\gamma_0,\beta)$, with the critical $\beta_c$ line (green dashed) marking the maximum intraphase quench amplitude. The hatched green region identifies the "DQPT area" where DQPT is accessible via the corresponding quench protocol. (a) and (b) display the results for transverse field quenches, where $\lambda$ is quenched from 0 to 0.999 in the FM$_x$ phase and from 1.2 to 2 in the PM phase, respectively. (c) and (d) show the corresponding behavior for anisotropy parameter quenches, with $\gamma$ quenched from 0.2 to 1 in the FM$_x$ phase and from 0.2 to $-1$ in the PM phase. Throughout, $\phi=-\pi/2$.