Unconventional critical exponents at dynamical quantum phase transitions in a random Ising chain

Dynamical quantum phase transitions (DQPTs) feature singular temporal behavior in transient quantum states during nonequilibrium real-time evolution. In this work we show that DQPTs in random Ising chains exhibit critical behavior with nontrivial exponents that are not integer valued and not of mean-field type. By means of an exact renormalization group transformation we estimate the exponents with high accuracy eliminating largely any finite-size effects. We further discuss how the considered dynamical phenomena can be made accessible in current Rydberg atom platforms. In this context we explore signatures of the DQPTs in the statistics of spin configuration measurements available in such architectures. Specifically, we study the statistics of clusters of consecutively aligned spins and observe a marked influence of the DQPT on the corresponding distribution.

Figure 1

(a) Effective free energy $\lambda$ as a function of time for system size $N=2^{20}$, obtained with $h_z=0.25$. (b) Effective free energy $\lambda$ as a function of $\mathrm{\Delta }t=t-t_c$ in the vicinity of the critical time $t_c=4\pi$. The results are shown for three different system sizes: $N=2^{14},2^{16},2^{18}$. At the critical time $t_c, \lambda$ exhibits a nonanalytical pattern. (c) $ln\left[\mathrm{\Delta }\lambda \right]$ as a function of $ln\left[\mathrm{\Delta }t\right]$, where $\mathrm{\Delta }\lambda =\lambda \left(t\right)-\lambda \left(t_c\right)$. The resulting plot looks like a straight line, suggesting a power-law relation between $\mathrm{\Delta }\lambda$ and $\mathrm{\Delta }t$. (d) Logarithmic derivative of $\lambda$. It assumes an almost constant value which is the slope of the line in panel (c) and the value of the critical exponent $\alpha$. The red line is the best fit of the form $dln\left|\mathrm{\Delta }\lambda \right|/dln\left|\mathrm{\Delta }t\right|=a+{b\left|\mathrm{\Delta }t\right|}^c$ of the points obtained for $N=2^{18}$.

Figure 2

$\theta (M,t)=-\frac{1}{N}ln\left[\right|p(M,t)\left|\right]$ as a function of time and of cluster size $M$. Around $t_c$, and for large values of $M$, the underlying DQPT affects $\theta (M,t)$, which starts to assume relatively large numbers. This is a consequence of the fact that $\theta (M,t)$ tends to $\lambda$ in the limit of $M\sim N$ and the effective free energy $\lambda$ shows a cusp at $t_c$.

Figure 3

(a) Effective free energy $\lambda \left(t\right)$ in the vicinity of the critical time $t_c$ for different values of the longitudinal field $h$: $h=0,0.01,0.04,0.07,0.1$. The system size considered is $N=2^{10}$. In the limit $h=0$ the cusp is clearly visible, while increasing the value of $h$, the pattern becomes more and more smooth. (b) $\mathrm{\Delta }\lambda$ as a function of $\mathrm{\Delta }t$, rescaled by $h^b$ and $h^c$, respectively, for different values of $h$, where $\mathrm{\Delta }\lambda =\lambda \left(t\right)-\lambda \left(t_c\right)$ and $\mathrm{\Delta }t=t-t_c$. The curves for different $h$ collapse onto each other when the exponents are $b=-0.475$ and $c=-1.02$.