Separation of the Kibble-Zurek Mechanism from Quantum Criticality

TL;DR

By analyzing generalized compass, TFIMDM, and generalized XY models, the work demonstrates that defect-generation scaling is governed by the gap structure of the dynamically active quasiparticles, not solely by crossing a quantum critical point. The authors show that when the relevant quasiparticles remain massive at the critical point, defect suppression can outpace the Kibble-Zurek prediction, whereas quenches through non-critical points with massless modes can preserve $n_d \propto \tau^{-\beta}$ with $\beta = d\nu/(1+z\nu)$. The results distinguish quench dynamics from equilibrium criticality, clarifying conditions under which universal KZ scaling emerges. These insights have implications for designing adiabatic protocols and quantum simulations that aim to minimize non-adiabatic excitations.

Abstract

When a system is swept through a quantum critical point (QCP), the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp time scale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a non-critical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality.

Figures (4)

• Figure 1: The quasi-particle spectrum $\pm\varepsilon_{k}^{\alpha,\beta}$ of the generalized compass model versus $k$ at the critical point $\theta_c=\pi/2$ at (a) the isotropic point (IP) $J_o=J_e=1$, and at (b) the anisotropic point $J_o=1.5$, $J_e=1$. The quasi-particle spectrum versus $\theta$ at $k=\pm\pi$ at (c) the isotropic point (IP) $J_o=J_e=1$, and at (d) the anisotropic point $J_o=1.05$, $J_e=1$. (e) The density of excitations $n_d$ in the generalized compass model as a function of ramp time scale $\tau$ for quench from $\theta_i=0$ to $\theta_f=\pi$ for $J_e=1$ and different values of $J_o=1, 1.05, 1.1$ and $1.2$, for the system size $N=1024$. Inset represents the density of defects at the IP.
• Figure 2: Quasi-particle spectrum $\pm \varepsilon_{k}^{\pm}$ versus momentum $k$: (a) at the critical point $h_c=1$ for $D = 1/2$, (b) at $h=1$ and $D = 2$, and (c) at the critical point $h_c=2$ for $D = 2$. (d) Phase diagram of the transverse-field Ising model with DM interaction in the $h$--$D$ plane. (e) Defect density $n_d$ as a function of the ramp time scale $\tau$ in the TFIMDM. The system is quenched from $h_i = 8$ to $h_f = 1.5,\,0.5,\,-0.5,$ and $-1.5$, with system size $N = 1024$.
• Figure S1: (Color online) The transition probability of excitation to the upper level in the transverse field Ising model with Dzyaloshinsky-Moriya (DM) interaction for a quench from $h_i=8$ to the different values of quench end $h_f=1.5, 0.5, -0,5$ and $-1.5$, for (a) $D=0.5$, and (b) $D=2$. (c) The transition probability of excitation to the upper level in the generalized $XY$ model for different values of quench end $h_s^f=-0.5, -0.1, 0, 0.5$ and $5$, for a quench starting at $h_s^i=-5$.
• Figure S2: (Color online) (a) The phase diagram of generalized $XY$ model. (b) The transition probability of excitation to the upper level in the generalized $XY$ model for different values of quench end $h_s^f=-0.5, -0.1, 0, 0.5$ and $5$, for a quench starting at $h_s^i=-5$. Panel (c) depicts defects density as a function of ramp time scale $\tau$ in generalized $XY$ model for $J_2=2$ for a quench from $h^i_s=-5$ to the different values of quench field end $h_s^f=-0.5, -0.1, 0, 0.5$ and $5$ for the system size $N=1024$. The quasi-particle spectrum $\pm e_{k}^{\pm}$ versus $k$ (d) at the critical point $h_c=\pm1$ and (e) at the non-critical point $h=0$.