Quantum work statistics across a critical point: full crossover from sudden quench to the adiabatic limit

TL;DR

This paper addresses how quantum work statistics evolve when a system is driven across a quantum critical point at finite rate. It develops a framework that combines linear response theory, boundary conformal field theory, and numerical renormalization group to obtain exact scaling functions for the full crossover from sudden quench to adiabatic driving in critical quantum impurity problems. The results reveal universal Kibble-Zurek scaling in the dissipated work and higher cumulants across CMCK models, with special analytic control via the Emery-Kivelson solution for the two-channel case. The predictions are relevant for charge-Kondo quantum dot devices and potentially for realizing excitations such as Majorana fermions or Fibonacci anyons, offering a quantitative bridge between non-equilibrium thermodynamics and quantum criticality.

Abstract

When an external parameter drives a system across a quantum phase transition at a finite rate, work is performed on the system and entropy is dissipated, due to the creation of excitations via the Kibble-Zurek mechanism. Although both the adiabatic and sudden-quench limits have been studied in detail, the quantum work statistics along the crossover connecting these limits has largely been an open question. Here we obtain exact scaling functions for the work statistics along the full crossover from adiabatic to sudden-quench limits for critical quantum impurity problems, by combining linear response theory, conformal field theory, and the numerical renormalization group. These predictions can be tested in charge-multichannel Kondo quantum dot devices, where the dissipated work corresponds to the creation of nontrivial excitations such as Majorana fermions or Fibonacci anyons.

Figures (5)

• Figure 1: (a) System $\mathcal{S}$ and environment $\mathcal{E}$ are coupled, but share a conserved charge. In a quantum dot setup, the dot spin (red) can be flipped with a compensating spin-flip of an electron in the lead (blue). We consider the dissipated work, due to a weak perturbation $\lambda(t)$ such as a magnetic field, ramped in a finite time $\tau$. (b) Multichannel Kondo systems exhibit quantum critical physics and universality in the intermediate regime $T^*<T<T_K$, where $T_K$ is the Kondo temperature and $T^*\to 0$ at the QCP. (c) We obtain the full, universal crossover from sudden-quench to the adiabatic limit, including the intermediate KZ regime.
• Figure 2: Crossover in the dissipated work $\langle W_{\rm diss}\rangle$ due to a weak perturbation $\lambda(t)=At/\tau$ ramped over a finite time $\tau$, from sudden quench to adiabatic limits, for non-critical (a,b) and critical (c,d) quantum dot systems. (a) Spinless resonant level model describing a noninteracting quantum dot, subject to a ramp of the dot potential $\lambda(t)\equiv\epsilon_d(t)$, see inset. (b,c,d) Multichannel Kondo models with $M=1,2,3$ channels, respectively, subject to finite-time driving of the magnetic field $\lambda(t)\equiv B(t)$ (results obtained by NRG using $J=0.08D$). Such models describe charge-Kondo circuits with a metallic island coupled to $M$ leads, see insets. Dashed lines in (c,d) are the CFT scaling predictions derived from Eq. \ref{['eq:beta']}. Shown for different temperatures $T/\Lambda=10^{n}$ with $n=-5 ... +2$ for black, magenta, blue, brown, green, orange, purple and cyan lines ($\Lambda=\Gamma$ for RLM and $T_K$ for CMCK).
• Figure 3: Universal work statistics phase diagrams in the $(T/T_K,B/T_K)$ plane for the sudden-quench limit, comparing the C1CK model (non-critical, left panels) with the C2CK model (critical, right panels). The expected dissipated work $\langle W_{\rm diss}\rangle$ in the top row panels shows divergent behavior on approaching the QCP for $B \ll T_K$ and $T \ll T_K$ in the C2CK model (but saturation for C1CK) consistent with interpretation as a susceptibility. Middle row panels for $\partial_T\langle W_{\rm diss}^3\rangle$ show the appearance of the critical scale $T^*$ in the C2CK model, reflected also in the thermodynamic entropy $\Delta S$ shown for reference in the bottom row panels (regimes labelled by their RG fixed points). Note that $\langle W_{\rm diss}^3\rangle$ and $\Delta S$ are connected by a Maxwell relation. Results obtained by NRG for $J=0.08D$.
• Figure 4: In the sudden-quench limit $\tau\to 0$ the dissipated work $\langle W_{\rm diss} \rangle$ in LR is proportional to a susceptibility, see Eq. \ref{['eq:kappa1dndl']}. At low temperatures, $\langle W_{\rm diss} \rangle$ thus saturates to a finite constant for RLM and C1CK, but it diverges for critical systems like C2CK and C3CK in a way that is characteristic of the QCP. Here $\Lambda=\Gamma$ for RLM and $T_K$ for CMCK. We set $B=0$.
• Figure 5: Evolution of the third moment of the dissipated work $\langle W^3_{\rm diss}\rangle$ for the C2CK model, plotted as $\partial_T \langle W^3_{\rm diss}\rangle$ vs $T/\Lambda$ (left) and $\Lambda/T\times \partial_T \langle W^3_{\rm diss}\rangle$ vs $\tau T$ (right), where $\Lambda=T_K$ is the cut-off here. The CFT prediction (dashed/dotted lines, Eq. \ref{['eq:2ck_dhdt']}) agrees with the full EK solution (solid lines) in the scaling regime $\tau \gg 1/T_K$. Inset shows the temperature $T_p$ of the peak in $\partial_T \langle W^3_{\rm diss}\rangle$, which saturates at $T_p \sim T_K$ for $\tau \ll 1/T_K$. Plotted for $\tau \Lambda= 10^n$ with $n=4,2,0,-2,-4$ for black, blue, red, magenta and green lines in the left panel; and $T/\Lambda=10^m$ with $m=-3,-1.5,-1,-0.5,0$ for black, green, blue, magenta and orange lines in the right panel.