Nonlinear quantum Kibble-Zurek ramps in open systems at finite temperature

TL;DR

This work extends quantum Kibble-Zurek physics to open, finite-temperature systems by introducing nonlinear two-parameter ramps in which temperature and a Hamiltonian control parameter are tuned toward a quantum critical point. By formulating an open Kitaev chain with a thermal bath and deriving slow-ramp scaling laws, the authors demonstrate that the universal exponents $ν$ and $z$ govern the out-of-equilibrium excitation density $$ even at finite $T$, provided the ramp path is chosen to reveal both coherent and incoherent dynamics. They classify ramps into three classes (A,B,C) based on the ratio of ramp exponents and derive explicit scaling exponents $zet_A$, $zet_B$, and $zet_C$, validated by exact numerics and local-approximation analyses for the Kitaev chain. The results offer a practical framework to extract quantum-critical exponents from dynamical measurements in realistic, finite-temperature environments, with guidance on ramp shapes that suppress subleading corrections and maximize scaling visibility.

Abstract

We analyze quantum systems under a broad class of protocols in which the temperature and a Hamiltonian control parameter are ramped simultaneously and, in general, in a nonlinear fashion toward a quantum critical point. Using an open-system version of a Kitaev quantum wire as an example, we show that, unlike finite-temperature protocols at fixed temperature, these protocols allow us to probe, in an out-of-equilibrium situation and at finite temperature, the universality class (characterized by the critical exponents $ν$ and $z$) of an equilibrium quantum phase transition at zero temperature. Key to this is the identification of ramps in which both coherent and incoherent parts of the open-system dynamics affect the excitation density in a non-negligible way. We also identify the specific ramps for which subleading corrections to the asymptotic scaling laws are suppressed, which serves as a guide to dynamically probing quantum critical exponents in experimentally realistic finite-temperature situations.

Figures (6)

• Figure 1: A schematic representation of the $\delta\mu$--$T$ plane showing examples of the three classes of ramps described in Sec. \ref{['sec:Ramps']}. Here we chose $\nu z=1$, as is the case for the Kitaev chain introduced in Sec. \ref{['sec:KitaevDefinition']}.
• Figure 2: Top: Final excitation density $\mathcal{E}(t_f)$ for nine ramps, with ramp parameters as defined in the legend, obtained from exact numerical solutions of the dynamical equations \ref{['eq:FullDynamics']} for the Kitaev model defined in Sec. \ref{['sec:KitaevDefinition']}. The corresponding ramp paths are shown in Fig. \ref{['fig:ramp-paths']}, and have starting points parametrized by $\theta$ as $(\delta\mu_i,T_i)=(-\cos(\theta),\sin(\theta))$. The linear behavior observed for slow ramps, i.e., for large $t_f$, reflects the scaling given schematically in Eq. \ref{['eq:scalingsummary']}, and in detail for the three ramp classes in Eqs. \ref{['eq:ClassA_ED']}, \ref{['eq:ClassB_ED']} and \ref{['eq:ClassC_ED']}. The predicted values of the scaling exponent $\zeta$ are given in the legend. For ramps $A_1$ and $B_2$, black lines with slopes corresponding to the predicted exponents $\zeta=1/2$ and $\zeta=9/14$ are shown for illustration. Bottom: An illustration of the data collapse resulting from the scaling relation in Eq. \ref{['eq:scalingsummary']} satisfied by $\mathcal{E}(t_f)$ for slow ramp velocities. Data for all nine ramps collapse to the same universal line, with unit slope, after an appropriate rescaling of $\mathcal{E}(t_f)$ and its logarithm.
• Figure 3: (a) Numerical results for the logarithm of the excitation density $\mathcal{E}=\mathcal{E}(t_f)$ at the end of a ramp toward the critical point, shown as a function of $\log(1/t_f)$. Results were obtained using Eq. \ref{['eq:ExcitationDensity']} and solving the dynamical equations in \ref{['eq:FullDynamics']} numerically for the Kitaev chain. Results for the three ramp classes A, B, and C appear in different colours. For class A we chose $(\alpha,\beta)=(1/2,1)$, for class B $(\alpha,\beta)=(1,1)$, and for class C $(\alpha,\beta)=(1,3/4)$. Solid lines indicate results obtained using the exact expressions for $\lambda(\delta\mu,k)$ and $\beta(\delta\mu,k)$ in Eq. \ref{['eq:KitaevLambdaBeta']}. Dashed lines are results obtained using the local approximations in Eq. \ref{['eq:KitaevLocalApproximations']}. Solid black lines have slopes corresponding to the predicted value of $\zeta$ for each of the three ramps. Parameter values are $J=\Delta_p=1$, $s=\kappa=\delta=1$, $T_i=\delta\mu_i=0.28$, $\gamma=0.05$. (b) As in (a), but for a $\mu$-only ramp at $T=0$, starting from $\mu_i=-5$ and ending at $\mu_c=-1$. The bath is sub-ohmic with spectral parameter $s=1/4$. Here the local approximations in Eq. \ref{['eq:KitaevLocalApproximations']} were used, and we set $\alpha=\beta=1$. Also shown is data obtained by restricting to purely coherent ($\gamma=0$) and incoherent ($\kappa=0$) dynamics. The orange and blue dashed lines have slopes corresponding to the scaling exponents ${\zeta_{C}}=1/2$ and ${\zeta_{\rm coh}}=4/5$, respectively. Remaining parameters are $J=\Delta_p=1$, $\gamma=0.02$, and $\delta=J^{1-s}$.
• Figure 4: A comparison of the scaling exponent $\zeta$ calculated in Sec. \ref{['sec:ScalingLaws']} (solid black line) with the exponent $\zeta^{\rm est}(t_f)$ extracted numerically using Eq. \ref{['eq:zetaest']} from data for the Kitaev chain. Results for $t_f=10^6$ (blue dots) and $t_f=10^{4.6}$ (orange circles) are shown for a range of $\beta$ values, with $\alpha=1$ fixed. The ramp starts at $(\delta\mu_i,T_i)=(-4\cos(\theta),4\sin(\theta))$, and results for five values of $\theta$ are shown. Other parameters are $J=\Delta_p=1$, $s=\kappa=\delta=1$ and $\gamma=1/15$.
• Figure 5: The relative difference $(\zeta^{\rm est}(t_f)-\zeta)/\zeta$ between the scaling exponent $\zeta$ derived in Sec. \ref{['sec:ScalingLaws']} and the estimate $\zeta^{\rm est}(t_f)$ calculated numerically for the Kitaev chain using Eq. \ref{['eq:zetaest']} for $41$ values of $\beta$ and of $\theta$. Here $\nu=z=1$ and $t_f=\exp(10.7)$. Regions bounded by black lines correspond to data points for different ramp classes, with the central rectangular region, where $\beta=1$, corresponding to class B ramps. Note that the size of the latter region, and that of class A ramps with $\beta\leq1$ and class C ramps with $\beta\geq1$ appear exaggerated in the figure due to the discrete set of $\theta$ and $\beta$ values used in the calculation. See the caption of Fig. \ref{['fig:zetavsbeta']} for parameter settings and the definition of $\theta$.