Frictional work and entropy production in integrable and non-integrable spin chains

TL;DR

The paper addresses how coherence generated during finite-time driving reduces extractable work and how this friction can be quantified in spin chains. It shows that in non-integrable chains, frictional work is largely captured by diagonal entropy production with an effective temperature $T_A$ for slow driving, and by the quantum relative entropy when driving is fast; integrability breaks this single-temperature picture, requiring a sum of per-subspace temperatures. The work demonstrates, via a transverse-field Ising model with and without a longitudinal field, that integrability breaking can improve adiabatic work but hurt non-adiabatic work, revealing regime-dependent trade-offs. These results provide a thermodynamic interpretation of quantum coherence in many-body dynamics and suggest experimental probes in controllable spin-chain platforms to study friction and efficiency in quantum engines.

Abstract

The maximum work extractable from a quantum system is achieved when the system is driven adiabatically. Frictional work then quantifies the difference in work output between adiabatic and non-adiabatic driving. Here we show that frictional work in a non-integrable spin chain is well-described by the diagonal entropy production associated with the build up of quantum coherence. The relationship is characterized by an effective temperature of the final adiabatic state and holds for slow to moderate driving protocols. For fast protocols, the frictional work is instead described by the quantum relative entropy between the final non-adiabatic and adiabatic states. We compare our results to those obtained from an integrable spin chain, in which case the adiabatic state is no longer described by a single temperature. In this case, the frictional work is described by a sum of terms for each independent subspace of the spin chain, which are at different effective temperatures. We show how integrability breaking can enhance work extraction in the adiabatic limit, but degrade work extraction in sufficiently non-adiabatic regimes.

Figures (5)

• Figure 1: Schematic of work extraction in finite-time and adiabatic protocols. The system begins in a thermal state $\rho_i$, which is diagonal in the initial energy eigenbasis. Work is extracted by tuning $h$. Non-adiabatic extraction (duration $\tau$) results in coherences in the final state $\rho_\tau$ with respect to the final energy basis. In the adiabatic limit, the final state $\rho_A$ remains diagonal. The difference in work output between the non-adiabatic and adiabatic processes defines the frictional work $\langle W\rangle_\mathrm{fric}\equiv\langle W\rangle_\tau-\langle W\rangle_A$. A summary of definitions of important quantities is provided in Table \ref{['definitions']}.
• Figure 2: Frictional work (solid line) compared with diagonal entropy production Eq. \ref{['main']} (dash-dotted line) and quantum relative entropy Eq. \ref{['main1']} (dashed line). (a) For varying protocol duration $\tau$ ($T_i= 3 g$, $\Delta h=2g$) and (b) varying initial temperature $T_i$ ($\tau= g^{-1}$, $\Delta h=g$). For $\tau \gtrsim \Delta h/g^2$ and $T_i\gtrsim g$, frictional work is accurately described by Eq. \ref{['main']}, whereas for $\tau\ll \Delta h/g^2$, frictional work is better described by Eq. \ref{['main1']}. More precisely, $|\langle W\rangle_\mathrm{fric}-T_A\Delta S_\mathrm{d}|/\langle W\rangle_\mathrm{fric}<0.05$ to the right of the vertical dotted lines in (a) and (b). Black diamond in (b) is the frictional work calculated from the ground and the first excited state. Colored circles in (a),(b) correspond to distributions shown in Fig. \ref{['fig:fig3pop']}. (c) Frictional work for varying step size $\Delta h$ ($T_i=3g$). Main figure: Eq. \ref{['main']} remains valid for large $\Delta h$ with $\Delta h/(g^2\tau)\lesssim 1$ held fixed (here $\Delta h/(g^2\tau)=2$). Inset: frictional work deviates from Eq. \ref{['main']} for large $\Delta h/(g^2\tau)$ (here $\Delta h/(g^2\tau)= 10$) when $\Delta h\gtrsim g$. All results are for $N=8$, $L=g$ and $h_i=1.5g$.
• Figure 3: (a) Population distributions $p_n=\langle n_f|\rho| n_f\rangle$ in the final energy eigenbasis for $\rho_\tau^\mathrm{diag}$ (dots), $\rho_A$ (solid line) and $\rho_{T_A}^\mathrm{therm}$ (dashed line). (b) Cumulative contribution to the frictional work up to a given energy level [see Eq. \ref{['cumulative']}]. Columns are: (i) $\tau=1.75 g^{-1}$, $T_i = 3 g, \Delta h=2g$ [red circle in Fig. \ref{['fig:fig2new']}(a)], (ii) $\tau = 0$, $T_i = 3 g, \Delta h=2g$ [yellow circle in Fig. \ref{['fig:fig2new']}(a)], (iii) $\tau=g^{-1}$, $T_i = 0.5g, \Delta h=g$ [purple circle in Fig. \ref{['fig:fig2new']}(b)]. For energy levels where $|\langle w\rangle_\mathrm{fric}^n|$ is appreciable, $\rho_A$ closely follows $\rho_{T_A}^\mathrm{therm}$. (i) For slow to moderate work extraction and $T_i\gtrsim g$, $\rho_\tau^\mathrm{diag}$ is close to $\rho_A$ and frictional work is well approximated by Eq. \ref{['main']}. For rapid work extraction [(ii)] or low temperatures [(iii)], $\rho_\tau^\mathrm{diag}$ and $\rho_A$ differ substantially and frictional work is no longer described by Eq. \ref{['main']}. In (ii), deviations between $\rho_\tau$ and $\rho_A$ are much larger than deviations between $\rho_A$ and $\rho_{T_A}^\mathrm{therm}$, resulting in agreement with Eq. \ref{['main1']}. For low temperatures [(iii)] the frictional work predominantly arises from the two lowest energy levels, and hence Eq. \ref{['plastina']} can be applied. All results are for $N=8$ and $h_i=1.5g$.
• Figure 4: (a) Frictional work (solid line) for an integrable spin chain ($L=0$, $N=8$) is not described by Eq. \ref{['main']} (dash-dotted line) but is described by the modified expression Eq. \ref{['eq:WDeltaS']} (dotted line). (b) Population distributions $p_n$ in the final energy eigenbasis for $\rho_A$ (dots) and $\rho_{T_A}^\mathrm{therm}$ (line) for (i) $L=0$ and (ii) $L=g$, both with $N=8$. In the integrable case [(i)], $\rho_A$ is no longer accurately described by $\rho_{T_A}^\mathrm{therm}$. (c) The effective temperature $T_A^j$ for each fermion in a large ($N=5000$) spin chain with $L=0$. The temperature of each mode is plotted against the final fermion energy $\omega_j^f$, with the color indicating the contribution of frictional work from each mode. All results are for $T_i=3g$, $h_i=1.5g$, $\Delta h =2g$ and $\tau=g^{-1}$.
• Figure 5: (a) Work (solid lines) extracted from an integrable ($L=0$, green line) and non-integrable ($L=g$, red line) as a function of extraction duration $\tau$. Matching colored dotted lines are the corresponding optimal work outputs $\langle W \rangle_\mathrm{opt}$ as defined by Eq. \ref{['Wopt']}. For adiabatic extraction, integrability breaking reduces $\langle W\rangle_\tau-\langle W\rangle_\mathrm{opt}$, since $\rho_A$ is well approximated by $\rho_{T_A}^\mathrm{therm}$. For fast processes ($\tau\ll \Delta h/g^2$), integrability breaking gives rise to larger frictional work and hence increases $\langle W\rangle_\tau-\langle W\rangle_\mathrm{opt}$ compared to the integrable case. (b) Work (solid lines) as a function of the longitudinal field $L$ for different $\tau$ compared with $\langle W\rangle_\mathrm{opt}$ (gray dotted line). All results are for $T_i =2g$, $h_i= 1.5g$ and $\Delta h = 2g$.