Maximum Quantum Work at Criticality: Stirling Engines and Fibonacci-Lucas Degeneracies

TL;DR

The paper investigates how quantum criticality can be harnessed to achieve Carnot-limited performance in quantum Stirling engines with finite work output. It develops a general analytical framework showing that, in the low-temperature and large-gap regime, the cycle between a QCP with degeneracy $g_{\mathrm{crit}}$ and a high-field point with degeneracy $g_0$ yields $W = k_B\,\delta\ln(g_{\mathrm{crit}}/g_0)$ and $\eta = \eta_C = 1 - T_L/T_H$, without the need for a classical regenerator. The authors validate this universal result across a variety of spin models, including two-spin-1/2 systems, open chains and rings, and the Lipkin–Meshkov–Glick model, and reveal Fibonacci- and Lucas-type degeneracies in the 1D Ising model that lead to non-extensive finite-size scaling $W \sim \ln(F_N)$ (open chains) or $W \sim \ln(L_N)$ (rings), with a return to classical extensivity as $N$ grows. The work provides a regulator-free pathway to Carnot efficiency in quantum thermal machines and offers design principles for exploiting critical degeneracies in engineered quantum devices.

Abstract

Many-body effects and quantum criticality play a central role in determining the performance of quantum thermal machines. Although operating near a quantum critical point (QCP) is known to enhance engine performance, the precise thermodynamic conditions required to attain the Carnot efficiency limit remain unsettled. Here, we derive the exact conditions for a quantum Stirling engine to achieve Carnot efficiency when a QCP drives its working medium. In the low-temperature regime, where only the ground-state manifold is populated, the net work output is given by $ W = k_B δ\ln (g_{\text{crit}}/g_0) $ with $ δ= T_H - T_L $, which directly yields the Carnot efficiency $ η_C = 1 - T_L/T_H $, independent of microscopic details. Notably, whereas ideal Stirling cycles attain Carnot efficiency only with a perfect regenerator, here no regenerator is required because, at low temperatures, the thermal population remains confined to the degenerate ground state; this represents a clear quantum advantage over engines with classical working substances. We validate this universal result by recovering known behaviors in various quantum systems, including spin chains with Dzyaloshinskii-Moriya interactions and magnetic anisotropies. Applying the framework to the one-dimensional antiferromagnetic Ising model, we predict non-extensive scaling of the work output governed by Fibonacci and Lucas numbers for open chains and closed rings, respectively, which converges to classical extensivity in the thermodynamic limit. This analysis establishes a general and robust foundation for designing quantum thermal machines that reach the Carnot bound while delivering finite work.

Figures (4)

• Figure 1: Schematic diagram of energy $E$ versus the tuning parameter $\lambda$. At the QCP located at $\lambda_{\mathrm{crit}}$, the $g_{\mathrm{crit}}$-fold degenerate ground state energy levels separate. The new ground state $\epsilon_0$ with degeneracy $g_0$ and the first excited state $\epsilon_1$ with degeneracy $g_1$ are separated by an energy gap $\Delta$ at high tuning parameter values $\lambda_H$. The thermal populations of higher excited states (with energies $\epsilon_{n>1}$) are negligible near absolute zero; only their degeneracies at the QCP influence the thermodynamic properties of the system.
• Figure 2: Schematic representation of the Stirling cycle in terms of entropy and the control parameter $\lambda$. Segments $AB$ and $CD$ correspond to the isothermal processes at $T=T_H$ and $T=T_L$, respectively. The entropy at point $A$ ($D$) is lower than at point $B$ ($C$), and the cycle proceeds in a counterclockwise direction.
• Figure 3: Stirling cycle diagram for a system with a QCP, represented in terms of entropy and the control parameter $\lambda$. The critical value $\lambda_{\text{crit}}$ denotes the QCP. Segments $AB$ and $CD$ correspond to the isothermal processes at $T_H = T_L + \delta$ and $T_L$, respectively. The entropy satisfies $S_B = S_C = g_{\text{crit}}$, while $S_D = g_0$ and $S_A$ may include contributions from $g_1$. The heat exchange along $BC$ vanishes ($Q_{BC}=0$), whereas $Q_{DA}$ is proportional to the temperature difference between reservoirs.
• Figure 4: Quantum Stirling work $W$ [meV] as a function of the number of sites $N$ for the antiferromagnetic Ising chain and ring ($J=1$), evaluated at the QCP $\lambda_L=B/J=1$ and a high-field parameter $\lambda_H=B/J=3$, with reservoir temperatures $T_L=0.1$ and $T_H=0.2$. In all cases the system operates at Carnot efficiency $\eta_C=0.5$. Red (black) dots represent simulated cycles for the chain (ring) system. Yellow and green markers show the predictions of Eqs. (\ref{['trabajo_fibo_Formula']}-\ref{['trabajo_lucas_Formula']}) based on Fibonacci and Lucas degeneracies at the QCP, respectively, while blue (orange) lines correspond to the large-$N$ asymptotic expressions in Eqs. (\ref{['trabajo_fibo_Formula_Lineal1']}-\ref{['trabajo_fibo_Formula_Lineal2']}).