Dynamic trapping near a quantum critical point

The study of dynamics in closed quantum systems has been revitalized by the emergence of experimental systems that are well-isolated from their environment. In this paper, we consider the closed-system dynamics of an archetypal model: spins driven across a second-order quantum critical point, which are traditionally described by the Kibble-Zurek mechanism. Imbuing the driving field with Newtonian dynamics, we find that the full closed system exhibits a robust new phenomenon—dynamic critical trapping—in which the system is self-trapped near the critical point due to efficient absorption of field kinetic energy by heating the quantum spins. We quantify limits in which this phenomenon can be observed and generalize these results by developing a Kibble-Zurek scaling theory that incorporates the dynamic field. Our findings can potentially be interesting in the context of early universe physics, where the role of the driving field is played by the inflaton or a modulus field.

Figure 1

Basic idea of the critical trapping phenomenon. The control field $\lambda$ is initialized in the disordered phase with initial velocity toward the QCP. (b) As the quantum degrees of freedom (e.g., spins) heat up, the dynamic field slows down, until (c) $\lambda$ can get trapped at or near the critical point.

Figure 2

Demonstration of critical trapping in the TFI model for $V\left(\lambda \right)=-E_{\mathrm{gs}}\left(\lambda \right)$, such that $v_c=v_{\mathrm{init}}$ (see main text). (a) As $v_{\mathrm{init}}$ is increased from 0.02 to 0.06, 0.1, and 0.14 at fixed $\mu =1$, the field undergoes a trapping/untrapping transition. (inset) The critical value of $v_{\mathrm{init}}$ for trapping (blue dots) scales as $1/\mu$ (red line) as predicted from a KZ analysis. (b) Scaling collapse of the dynamics at fixed $\mu v_{\mathrm{init}}=0.06$.

Figure 3

Trapping in the presence of an external potential. (a) Starting from rest, as the particle falls through the critical point, excitations act as an effective potential (red dashed line) drawing the system toward the QCP. Depending on the strength of this dressing, the system can be either trapped (upper) or untrapped (lower). Scaling arguments suggest that trapping should occur for $1\lesssim {\lambda }_{\mathrm{init}}/\alpha \lesssim 1/\left(\mu {\alpha }^2\right)$, which is encompassed in the phase diagram shown in (b). To visualize the transition, we show a one-dimensional cut across the phase boundary in (c), showing a field which is trapped (red), untrapped (blue), and near the transition (black). Data in (b) and (c) are shown for $\alpha =-3\times {10}^{-4}$, which numerically appears to approach the $\alpha \to 0$ scaling limit.

Figure 4

A simple experiment to illustrate critical trapping. As a permanent magnet moves down the hill (position $x)$, its field $h\left(x\right)$ drives a stationary quantum critical system through its phase transition. If the system parameters are in the trapped region of the phase diagram [Fig. 3], then the magnet is trapped very near this critical point $[x(t\to \infty )\approx x_c]$.

Figure 5

Higher-order corrections to the scaling theory. (a) Long-time behavior of the transverse field, where the dispersion relations are truncated at $n\mathrm{th}$ order in momentum. The Hamiltonians are given by Eq. (B1) for $n=1$ (linear), Eq. (B2) for $n=2$ (quadratic), and Eq. (A5) for $n=\infty$ (untruncated). The inset shows even longer times for $n=\infty$. (b) Excitation probability $p^{\mathrm{exc}}$ as a function of momentum for various times during the ramp truncated to linear order. The inset compares $p^{\mathrm{exc}}$ for $n=1$ and 2 truncation at late time $t=1000$. The characteristic momentum scale $k^{*}$ is labeled. (c) Log-log plot of initial kinetic energy vs $\lambda$ at late times, which is extrapolated from data similar to (a). The data show consistency with the expected scaling ${\lambda }_{\mathrm{final}}\sim K_{\mathrm{init}}$ (dashed line). Data in (a) and (b) are shown for $\mu =20$ and $v_{\mathrm{init}}={10}^{-2}$.

Figure 6

Illustration of KZ scaling of arbitrary field profiles. The KZ limit is taken by decreasing the initial velocity while rescaling the time and field axes by $t_{\mathrm{KZ}}$ and ${\lambda }_{\mathrm{KZ}}$ as given in Eq. (C2).

Figure 7

Generalized scaling in a linear potential. (a) For the TFI chain with a small slope $\alpha =3\times {10}^{-4}$ and small $\widehat{\mu }=\mu {\alpha }^2$, the critical point for trapping ${\left({\lambda }_{\mathrm{init}}\right)}_{\mathrm{crit}}$ scales as $1/\left(\mu \alpha \right)$ as predicted. (b) Proposed trapping phase diagram for general critical theories in the presence of a linear potential. (c) Scaling collapse at small $\mu {\alpha }^2$ of the dynamics when the system is initialized at the critical point in its ground state for the TFI chain, showing a lack of trapping. For $\mu =1$ and the largest values of $\alpha$ shown, deviations are seen at long times due to finite bare mass.

Figure 8

Thermal trapping/untrapping transition. As the initial energy is decreased, the entropy loses its local maximum and the system becomes untrapped. Data are shown for ${\lambda }_{\mathrm{init}}/\alpha =-20$ (trapped), $-10$ (untrapped), and $-15.3$ (marginal).