Quantum Mpemba effect in long-range spin systems

Abstract

One of the manifestations of the quantum Mpemba effect (QME) is that a tilted ferromagnet exhibits faster restoration of the spin-rotational symmetry after a quantum quench when starting from a larger tilt angle. This phenomenon has recently been observed experimentally in an ion trap that simulates a long-range spin chain. However, the underlying mechanism of the QME in the presence of long-range interactions remains unclear. Using the time-dependent spin-wave theory, we investigate the dynamical restoration of the spin-rotational symmetry and the QME in generic long-range spin systems. We show that quantum fluctuations of the magnetization drive the restoration of symmetry by melting the initial ferromagnetic order and are responsible for the QME. We find that this effect occurs across a wide parameter range in long-range systems, in contrast to its absence in some short-range counterparts.

Figures (4)

• Figure 1: (a) Relation between spin operators ($\hat{S}_i^x,\hat{S}_i^y,\hat{S}_i^z$) in the laboratory frame and those ($\tilde{S}_i^x,\tilde{S}_i^y,\tilde{S}_i^z$) in the rotated frame. In the rotated frame, $\tilde{S}_i^x$ and $\tilde{S}_i^y$ correspond to the spin fluctuation components in the polar and azimuthal directions, while $\tilde{S}_i^z$ is the component in the direction of the precessing magnetization $\langle \hat{\mathbf{M}}(t)\rangle$. (b) Quasidistribution function $Q(\vartheta,\varphi)=|\bra{\Psi_t}\ket{\mathrm{TF}(\vartheta,\varphi)}|^2$ in spherical coordinates for the quench dynamics of a long-range spin-1/2 chain. We set $N=24$, $\Delta=1$, $\alpha=0$, and initial tilted angle $\theta=0.3\pi$.
• Figure 2: Time evolution of the entanglement asymmetry after a quench from different tilted ferromagnetic states in the long-range spin-$1/2$ chain \ref{['eq:XXZ']} with $J(|\mathbf{r}|)\propto|\mathbf{r}|^{-\alpha}$ and $h=0$. The parameters $(\alpha, \Delta)$ are set to $(0, 1)$, $(0.7, 1)$, $(0.7, 0.5)$, and $(0.7, -0.5)$ in panels (a), (b), (c), and (d), respectively. The symbols are the exact asymmetry obtained with ED. The solid curves are the prediction of the time-dependent spin-wave theory (Eq. \ref{['eq:EA_analytic']}) and the dotted curves (Eq. \ref{['eq:EA_long-range-limit']}) are the dominant contribution of the zero momentum spin-waves. The crossings between the curves for the different values of $\theta$ indicate the occurrence of the QME. We set $N=24$ and $N_A=6$ in all the plots.
• Figure sm-1: Color map of $\min_{k\neq0}[\omega_k^2]$ in Eq. \ref{['eq:w_k']} as a function of $\Delta \sin^2\theta$ and $\alpha$. The dotted curves are the boundary between the red and blue regions where $\Im[\omega_k]=0$ for all $k$ and $\Im[\omega_k]\neq0$ for certain $k$, respectively. The symbols correspond to the specific parameters used in Fig. 2 of the main text. We take system size $N=24$.
• Figure sm-2: Time evolution of the entanglement asymmetry from different initial tilted ferromagnetic states in the long-range spin-$1/2$ chain \ref{['eq:XXZ_chain_sp']}. (a) The symbols are the exact asymmetry calculated with ED using Eq. \ref{['eq:EA_exact']} and the dashed curves correspond to Eq. \ref{['SMeq:EA_analytic_full']}. (b)-(d) The symbols were obtained with the MPS-TDVP algorithm for maximal bond-dimension $D=16$ (triangles) and $D=32$ (circles). The solid lines are the analytic prediction \ref{['SMeq:EA_analytic_full']} of the spin-wave theory taking into account the contributions of the $\mathbf{k\neq0}$ modes, contained in the $X_A$ term. We set $N=128$ and $N_A=32$ in all the plots.