Dynamical Critical Scaling of Long-Range Interacting Quantum Magnets

Slow quenches of the magnetic field across the paramagnetic-ferromagnetic phase transition of spin systems produce heat. In systems with short-range interactions the heat exhibits universal power-law scaling as a function of the quench rate, known as Kibble-Zurek scaling. In this work we analyze slow quenches of the magnetic field in the Lipkin-Meshkov-Glick (LMG) model, which describes fully connected quantum spins. We analytically determine the quantum contribution to the residual heat as a function of the quench rate $\delta$ by means of a Holstein-Primakoff expansion about the mean-field value. Unlike in the case of short-range interactions, scaling laws in the LMG model are only found for a ramp starting or ending at the critical point. If instead the ramp is symmetric, as in the typical Kibble-Zurek scenario, then the number of excitations exhibits a crossover behavior as a function of $\delta$ and tends to a constant in the thermodynamic limit. Previous, and seemingly contradictory, theoretical studies are identified as specific limits of this dynamics. Our results can be tested on several experimental platforms, including quantum gases and trapped ions.

Figure 1

Left: The dynamics of a spin-$1/2$ chain is analyzed when the strength of the magnetic field $h$ is slowly varied across the paramagnetic-ferromagnetic transition. The lines connecting the sites illustrate that each spin interacts with equal strength $J$ with the rest of the chain. Right: The energy gap $\mathrm{\Delta }(h)$ between the ground and the first excited state of the chain is displayed as a function of $h=h_c-\delta t$ (solid red line). The dashed black line shows the rate ${\gamma }_h=|\dot{h}|/|h-h_c|$ with which the magnetic field is varied in time. The freezing time $t_f$ is defined such that ${\gamma }_h(\pm t_f)=\mathrm{\Delta }[h(\pm t_f)]$. We determine the scaling with $\delta$ of the quantum heat generated by the quench.

Figure 2

(a) Heat $Q$ generated by the quench, in units of $J$, as a function of time $t$, in units of $1/(J\delta )$. The heat is determined from Eq. (8) using Eqs. (6) and (7). The plot shows different values of $\mathrm{\Lambda }=N\delta$ (indicated in the legend), chosen to be the same as in Ref. [49]. Solid lines correspond to $N=2^9$, dashed lines to $N=2^{12}$. The inset reports the corresponding fidelity $f(t)$. (b) The average number of excitations $n_{\mathrm{exc}}(t)$ and the fidelity $f(t)$ are reported as functions of the rescaled time $s={\delta }^{1/3}t$ for $N=2^{12}$ and $\delta =4\times {10}^{-5}$, ${10}^{-3}$, 1.2, corresponding to $\mathrm{\Lambda }=2\times {10}^{-1}$, 6, $6\times {10}^3$, respectively. (c) The number of excitations at the end of the quench, $n_{\mathrm{exc}}(t_0)$, is reported as a function of $\delta$ for $N=500$ (the behavior for a different system size $N^{\prime }$ is obtained by rescaling the $\delta$ axis by the factor $N^{\prime }/N$). The horizontal dashed line indicates the constant value $n_{\mathrm{exc}}(t_0)=0.35$ of the thermodynamic limit. The inset shows $n_{\mathrm{exc}}(t_0)$ on a logarithmic scale. The dotted line represents the KZ scaling prediction ${\delta }^{1/3}$. For the full ramp, there is only an accidental match in the crossover regime and no actual KZ scaling is found.