Ergotropy of quantum many-body scars

Abstract

Quantum many-body scars break ergodicity and evade thermalization, resulting in area law entanglement entropy even with high energy density. While their quantum correlations and entanglement have been elaborated previously, their capacity in storing extractable energy, quantified by the notion ergotropy, remains an open question. Here we focus on the representative PXP model, and unveil the extensive ergotropy scaling of a family of states interpolating between quantum many-body scars and thermal states, the latter of which are known to be passive with vanishing ergotropy. A phenomenological relation between ergotropy and entanglement is uncovered, which generalizes the existing free fermion integrable results to an interacting scenario. The ergotropy in a dynamical protocol shows that a reset with a global uniform coherent rotation can inject extractable energy, as a proof of principle way to charge a quantum "battery". Our protocol is tailored for near term Rydberg neutral atoms array, while also being feasible for other quantum processors. Our results establish that quantum many-body scars, despite the tiny fraction of the Hilbert space they occupy, can be efficiently exploited for storing extractable energy, and "scarring" a many-body system as a promising route for engineering quantum many-body battery.

Figures (8)

• Figure 1: Schematics. (a) PXP model dynamics, realized in a blockaded Rydberg atom arrays with pulse that globally rotates the qubits. (b) Eigenstates ergotropy and entanglement scaling. (c) Dynamical quantum protocol that uses reset together with a coherent rotation $R_Y(\theta) = e^{-i \frac{\theta}{2} Y}$ to inject energy, creating highly excited states for the PXP Hamiltonian. After relaxation, the subsystem of the steady state possesses extensive energy extractable by unitary operation.
• Figure 2: Ergotropy and entanglement for eigenstates of PXP model by interpolating between scar and thermal states within the zero energy shell i.e. in the middle of the energy spectrum. (a)Ergotropy density$W/L$ exhibits a crossover from extensive to sub-extensive scaling. Data for $L=\infty$ is obtained by extrapolating finite size data to thermodynamic limit. The non-vanishing ergotropy at $\lambda=1$ reflects imperfect thermal state separation and finite size effects. Inset: Bipartite half chain entanglement entropy density $S_\mathrm{vN}/L$ showing transition from area law to volume law. We adopt phenomenological fitting forms: $S_{\mathrm{vN}}=a+ v L + c \ln L/3$, yielding thermodynamic limit ergotropy density $\lim_{L\to\infty} W/L$. Both observables show clear crossover around $\lambda_c \approx 0.6$. (b)Relation between entanglement entropy and bound energy. Entanglement entropy squared $S_{\mathrm{vN}}^2$ versus bound energy $Q$ for system sizes $L=10$--$22$ shows a linear relation $S_{\mathrm{vN}}^2/Q = n+m L$, as an interacting many-body case generalizing the free fermion results in Refs. Mula2023Mitra_2025. Each $\lambda$ is annotated with their slope $m$, and larger $\lambda$ (more thermal) yields larger slope $m$, indicating stronger suppression of ergotropy by entanglement. $m$ saturates quickly after $\lambda_c$ and thus only $\lambda = 0, 0.2, 0.3, 1$ are shown here. (c)Multipartite entanglement witnessed by quantum Fisher information (QFI) density $f_Q = \expval{( O-\langle O\rangle)^2}/L$ with respect to the antiferromagnetic operator $O = \sum_i (-1)^{i+1} Z_i$, showing a consistent crossover. Scar states, despite obeying area law entanglement entropy, possess extensive QFI density scaling $\propto L$, indicating genuine multipartite entanglement, in contrast to thermal states with sub-extensive QFI density ExtensiveQFI_pappalardi.
• Figure 3: Quantum quench dynamics and steady state properties.(a)Time evolution of ergotropy$W(t)$ for $L=28$ reveals distinct relaxation patterns across rotation angles ($\theta = 0, \pi/4, \pi/2$). Scarred dynamics ($\theta=0$) exhibit persistent oscillations with revivals, while thermal dynamics ($\theta=\pi/2$) show rapid decay. The intermediate case ($\theta=\pi/4$) displays scar like behavior with reduced initial ergotropy due to lower injected energy (cf. Appendix \ref{['eq:energy_density']}). (b)Steady state ergotropy$\bar{W}$ exhibits non-monotonic $\theta$ dependence with extensive scaling for scar and sub-extensive scaling for thermal limit across system sizes $L=12$--$28$. The minimum approaches thermal regime in thermodynamic limit, suggesting the thermodynamic advantage of scarred states. (c)Steady state entanglement entropy$\bar{S}_{\mathrm{vN}}$ shows behavior inversely related to ergotropy, transiting from low entanglement scar regime to volume law thermal regime, demonstrating fundamental entanglement-ergotropy anti-correlation. The detailed dynamics of bound energy $Q(t)$ and entanglement entropy $S_{\mathrm{vN}}(t)$ is presented in Appendix \ref{['fig:additional_dynamics']}.
• Figure 4: (a) Schematic illustration of ergotropy definition and its relationship to bound energy. The density matrix eigenvalue population of subsystem state $\rho_A$ on energy spectrum $H_A$ (deep blue solid curve) is optimally redistributed by $U_{{\rm opt}}$ to anti-align with the subsystem energy spectrum (blue axis), maximizing work extraction $W$ while minimizing bound energy $Q$, with average energy drop (dash line). The gradient blues represents the magnitude of population. Here total energy satisfies $E = W + Q$. (b) System size scaling of ergotropy $W$ and total energy $E$ that is only supported on subsystem $A$: while total subsystem energy $E$ scales linearly with system size $L$, ergotropy exhibits extensive scaling $W \sim L-\ln^2(L)/L$ for scar states versus sub-extensive scaling $W \sim L^{-1}$ for thermal states across $L=10$--$22$.
• Figure 5: Many-body entanglement.(a) Tripartite mutual information (TMI) $I(A:B:C)$ between intervals $A,B,C$ by partitioning system into four equal intervals as shown in (b), serves as a diagnostic for the entanglement crossover from area law to volume law phases. The transition exhibits scaling from constant to extensive behavior $I(A:B:C) \propto -L$, with pronounced even odd effects reflecting distinct entanglement structures for $L=4N$ versus $L=4N+2$ systems. (b) Bipartite mutual information (MI) $I(A:C)$ scaling shows the transition from extensive to sub-extensive behavior, with similar even odd effects, so we only show $L=4N+2$ data. (c) System size dependence of QFI density across all superpositions: extensive scaling characterizes the scar regime ($\lambda=0$), while sub-extensive, approximately constant behavior characterizes the thermal regime ($\lambda=1$).