Kibble-Zurek scaling from linear response theory

TL;DR

The paper shows that the Kibble-Zurek mechanism and linear response theory are consistent for quantum phase transitions, using the quantum Ising chain in a transverse field as a solvable testbed. By deriving the relaxation function $\Psi(t)$ and the associated relaxation time $\tau_R$, it identifies the energy gap with a relaxation rate, exhibiting critical slowing down near the quantum critical point. The excess work $W_{ex}$ computed from linear response displays the Kibble-Zurek scaling $W_{ex}\sim\tau^{\gamma_{KZ}}$ with $\gamma_{KZ}=-1$ for the Ising model (where $\Lambda=0$, $z=1$, $\nu=1$), agreeing with exact dynamics in the impulse regime. The results clarify the range of validity of linear response near quantum criticality while highlighting how linear-response insights can inform and corroborate KZ arguments in quantum dynamics.

Abstract

While quantum phase transitions share many characteristics with thermodynamic phase transitions, they are also markedly different as they occur at zero temperature. Hence, it is not immediately clear whether tools and frameworks that capture the properties of thermodynamic phase transitions also apply in the quantum case. Concerning the crossing of thermodynamic critical points and describing its non-equilibrium dynamics, the Kibble-Zurek mechanism and linear response theory have been demonstrated to be among the very successful approaches. In the present work, we show that these two approaches are consistent also in the description of quantum phase transitions, and that linear response theory can even inform arguments of the Kibble-Zurek mechanism. In particular, we show that the relaxation time provided by linear response theory gives a rigorous argument for why to identify the "gap" as a relaxation rate, and we verify that the excess work computed from linear response theory exhibits Kibble-Zurek scaling.

Figures (5)

• Figure S1: Illustration of the Kibble-Zurek mechanism. Far from the critical point, the dynamics of the system is essentially adiabatic, meaning that the system recovers from the defects of the driving faster than the inverse of the driving rate. Close to the critical point the situation changes dramatically. The healing capacity is lost and finite-size domains are "frozen" into the system.
• Figure S2: Magnetic susceptibility \ref{['eq:susc']} as a function of the external field $\Gamma_0$ for $J=1$.
• Figure S3: Effective relaxation time \ref{['eq:relax_env']} for $J=1$.
• Figure S4: Excess work \ref{['eq:ExcessWork2']} computed from linear response theory and exact numerics for protocols driving in the ferromagnetic ((a), (b), (c)) and paramagnetic ((d), (e), (f)) phase, and crossing the critical point ((g), (h), (i)). Figures (a), (d) and (g) depict situationsin which linear response theory and the exact result perfectly math. Figures (b), (e), (h) depict situations with large $N$. Figures (c), (f) and (i) depict situations with strong driving.
• Figure S5: Comparison between Kibble-Zurek scaling of the excess work \ref{['eq:ExcessWork2']} from exact dynamics and linear response theory.