Large Deviations beyond the Kibble-Zurek Mechanism

The Kibble-Zurek mechanism (KZM) predicts that the average number of topological defects generated upon crossing a continuous or quantum phase transition obeys a universal scaling law with the quench time. Fluctuations in the defect number near equilibrium are approximately of Gaussian form, in agreement with the central limit theorem. Using large deviations theory, we characterize the universality of fluctuations beyond the KZM and report the exact form of the rate function in the transverse-field quantum Ising model. In addition, we characterize the scaling of large deviations in an arbitrary continuous phase transition, building on recent evidence establishing the universality of the defect number distribution.

Figure 1

Comparison of the scaled rate function $\overline{I}(\overline{\rho })=I(\rho )/{\rho }_{\mathrm{KZM}}$ derived analytically with the numerically exact computation for a finite ${\tau }_Q$ and $N$. A TFQI chain, initialized in its ground state, is driven by varying $g(t)$ from $g(-3{\tau }_Q)=4g_c$ to time ${\tau }_Q$, when $g({\tau }_Q)=0$. The cumulant generating function $\lambda (\theta )$ is computed in the final nonequilibrium state using Eq. (7) for finite $N$, from which the scaled rate function $\overline{I}$ is found with a Legendre-Fenchel transform. As the quench time increases, the agreement between numerics and the analytical solution based on the LZ approximation improves visibly, while the agreement with the central limit theorem (CLT) prediction, obtained by matching the first and second cumulants, does not. The value at the origin $\overline{I}(0)=\zeta (3/2)$ follows from Eq. (14), while the minimum $\overline{I}=0$ is attained at $\overline{\rho }=1$ (diamond). Finite-size analysis reveals the convergence of the numerically evaluated rate function $\overline{I}$ to the thermodynamic limit for $N=1000$, which is used in this figure.