Universal Statistics of Topological Defects Formed in a Quantum Phase Transition

When a quantum phase transition is crossed in finite time, critical slowing down leads to the breakdown of adiabatic dynamics and the formation of topological defects. The average density of defects scales with the quench rate following a universal power law predicted by the Kibble-Zurek mechanism. We analyze the full counting statistics of kinks and report the exact kink number distribution in the transverse-field quantum Ising model. Kink statistics is described by the Poisson binomial distribution with all cumulants exhibiting a universal power law scaling with the quench rate. In the absence of finite-size effects, the distribution approaches a normal one, a feature that is expected to apply broadly in systems described by the Kibble-Zurek mechanism.

Figure 1

Cumulants ${\kappa }_q$ of the kink number distribution. From top to bottom, universal scaling of the mean density of defects ($q=1$), the corresponding variance ($q=2$) and the third cumulant ($q=3$) of the kink number distribution as a function of the quench time ${\tau }_Q$ in which the phase transition is crossed ($N=400$). Symbols represent numerical data while solid lines describe the analytical approximation derived in the scaling limit. The mean density ($q=1$) is predicted by the KZM, see Eq. (18), and was numerically confirmed in Refs. [10, 12]. For slow quenches, all cumulants exhibit a universal scaling with the quench time. The universality of critical dynamics thus extends beyond the scope of the KZM and governs the full distribution of topological defects. Deviations from the scaling limit due to finite-size effects and the onset of adiabatic dynamics are first signaled by high-order cumulants.

Figure 2

Kink number distribution. Dependence of the kink number distribution on the quench time ${\tau }_Q$ at which a 1D quantum Ising chain is driven through the quantum phase transition from the paramagnetic to the ferromagnetic phase. From right to left, ${\tau }_Q=10$, 100, 1000 ($N=400$). In spite of the nonzero cumulants ${\kappa }_q$ with $q>2$ the distribution in the scaling limit is well approximated by the normal distribution in Eq. (21) with ${\kappa }_1=Nd$ and ${\kappa }_2=3⟨n⟩/{\pi }^2$ (solid lines). However, deviations become apparent at the onset of the adiabatic dynamics.