Quantum Otto heat-engine with Kitaev-Heisenberg cluster: Possible roles of frustration, magnons, and duality

TL;DR

This study investigates a quantum Otto engine using Kitaev-Heisenberg clusters as the working medium, driven by a linearly time-dependent Zeeman field $h(t)$. By exact diagonalization and Suzuki–Trotter methods, it reveals that maximum efficiency arises when the KH spectrum forms narrow bands with quantized total spin $S^z_{\mathrm{tot}}$ and emergent magnons, particularly when Kitaev and Heisenberg couplings have opposite signs. Finite $J$ can further enhance efficiency, linked to spectral banding and duality of the spectrum under sign reversal, with entanglement patterns accompanying the efficiency gains. Extending to large spin $S$, the paper shows a quantum advantage persisting up to $S=5/2$, with $\eta_{\max}$ scaling approximately as $S^{-1}$ in both AFM and FM Kitaev models and distinct scaling of the maximum work; these findings suggest KH materials as practical platforms for QOE and provide insight into the roles of frustration, magnons, and spectral dualities in quantum thermodynamics.

Abstract

We study the performance of Kitaev-Heisenberg (KH) clusters as working media realizing a quantum Otto engine (QOE). An external Zeeman field with linear time dependency is used as the driving mechanism. The efficiency strongly depends on Kitaev ($κ$) and Heisenberg ($J$) exchange interaction. Interestingly, efficiency is comparable when the relative magnitude of $κ$ and $J$ is the same but of opposite signs. The above results are explained due to a subtle interplay of frustration, quantum fluctuation, and duality of eigen-spectra for the KH system when both the signs of $κ$ and $J$ are reversed. The maximum efficiency is shown to be dynamically related to eigen-spectra forming discrete narrow bands, where total spin angular momentum becomes a good quantum number. We relate this optimum efficiency to the realization of weakly interacting magnons, where the system reduces to an approximate eigen-system of the external drive. Finally, we extend our study to the large spin Kitaev model and find a quantum advantage in efficiency for $S=1/2$. The results could be of practical interest for materials with KH interactions as a platform for QOE.

Figures (11)

• Figure 1: (a) The red, blue, and green bonds indicate $x$-$x$, $y$-$y$, $z$-$z$-type interactions in Kitaev model. Panels (b), (c), and (d) depict the four, six, and eight-site clusters considered, respectively.
• Figure 2: $|{\langle} B_p{\rangle}|$ in magnetic field ($h_z$) - temperature ($T$) plane for 8-site cluster is plotted. The upper and lower panels are for the AFM and FM Kitaev model, respectively. Value of $J$ is taken to be -0.3, 0, and 0.3 in panels (a,d), (b,e), and (c,f), respectively. For better visualization, panels (a), (c), and (f) have been scaled up; the actual data correspond to the colored data multiplied by the factor indicated in the respective panels.
• Figure 3: Total magnetization $\langle M^z \rangle$ for 8-site cluster is plotted in $h_z-T$ plane. The upper and lower panels correspond to $\kappa=1$ and $\kappa=-1$, respectively. Panels (a,d), (b,e), and (c,f) correspond to $J=-0.3, 0, 0.3$, respectively.
• Figure 4: Otto cycle is shown schematically. At (1), the system is described by the initial magnetic field $h_2$ and density matrix $\rho^{(1)}$, in equilibrium with cold-bath. The system is then detached from the cold bath and unitarily evolved to (2) with a magnetic field linearly increased up to $h_1$, and characterized by a density matrix $\rho^{(2)}$. Now the system is brought in contact with a hot bath and described by a density matrix $\rho^{(3)}$. Finally, the system is detached from the hot bath and undergoes a unitary evolution to reach the state with $h_2$ with $\rho^{(4)}$ at (4).
• Figure 5: Operational regions for 8-site cluster in absence of Heisenberg interaction with (a) $\kappa=+1$, and (b) $\kappa=-1$, at $T_h=10^{2}$ and $h_1=1.0$ in both panels. Green, blue, yellow, and red region describes a heat engine, a refrigerator, a heat-accelerator, and a heater, respectively. For the heat-engine, efficiency is mentioned in contours (for better visibility, it is multiplied by 100). In panels (c-f), the colored plot shows the energy of the working medium along the path $1 \rightarrow 2 \rightarrow 3 \rightarrow 4$, the smallest circle being the starting point. The black contour shows the estimated effective temperature. (c-f) panels correspond to the specific $(T_c,h_2)$ values as indicated with magenta diamonds $\blacklozenge$ in (a). For visual clarity, the vertical axes of (c-f) have been rescaled. Ticks of the energy axis correspond to $[\epsilon_{_1},\epsilon_{_2}]$ = (c) $[0,-0.11]$, (d) $[-0.008,-0.016]$, (e) $[-0.008,-0.014]$, (f) $[-0.007,-0.014]$. Ticks of the effective temperature axis correspond to $[T'_{_1},T'_{_2}]$ = (c) $[105,5]$, (d) $[101,57]$, (e) $[108,66]$, (f) $[140,69]$. On the $T_{\rm eff}$ axis, the hot and cold bath temperatures are indicated with fire and ice icon respectively.