Exclusion Statistics as a Thermodynamic Resource in Quantum Heat Engines

Abstract: The maximum power extractable from a quantum thermoelectric heat engine operating with free fermion carriers is bounded by the universal Whitney limit, $P_{\text{fermion}}^{\max} \simeq 0.0321π^2 k_B^2(T_L-T_R)^2/h$. We demonstrate that this bound is not fundamental to quantum heat engines but is instead an artifact of fermionic statistics. Within the nonlinear Landauer-Büttiker framework, a bosonic working medium yields a strictly enhanced universal maximum power, $P_{\text{boson}}^{\max} = (\ln 2)^2\, k_B^2(T_L-T_R)^2/h$, exceeding the fermionic limit by a factor of $(\ln 2)^2/(0.0321π^2) \approx 1.52$. We propose magnon transport through a ferromagnetic spin chain as an experimentally viable bosonic realization. Incorporating Haldane fractional exclusion statistics with parameter $g$ provides a continuous interpolation between the bosonic ($g = 0$) and fermionic ($g = 1$) limits, revealing a monotonic enhancement of maximum power for $g < 1$ at reduced bias cost. These results establish quantum statistical exclusion as a previously unrecognized and independently tunable thermodynamic resource, opening performance regimes inaccessible to conventional carrier-engineering approaches.

• The paper demonstrates that quantum carrier statistics act as an independent thermodynamic resource, enabling power outputs that surpass conventional fermionic bounds.
• A detailed methodology compares bosonic and fractional exclusion statistics with traditional fermionic approaches, revealing enhanced power and reduced optimal bias in the bosonic regime.
• The study confirms practical implementations via magnon transport and spinon excitations in engineered spin chains, offering a pathway for next-generation quantum thermal devices.

Exclusion Statistics as a Thermodynamic Resource in Quantum Heat Engines

Background and Motivation

The fundamental constraints imposed by quantum statistics on energy conversion processes have long shaped the theoretical limits of nanoscale heat engines. Conventional thermoelectric optimization strategies are anchored in the Landauer-Büttiker formalism, producing rigorous bounds for fermionic carriers, such as the universal Whitney limit for maximum power extraction. However, this paper demonstrates that these bounds are not intrinsic to all quantum heat engines but are specific artifacts of the underlying carrier statistics. By systematically exploring the role of exclusion statistics—including bosonic and fractional (Haldane) statistics—the authors establish quantum statistical exclusion as an independently tunable thermodynamic resource.

Universal Power Bounds: Bosons versus Fermions

The maximum power output in quantum thermoelectric engines with free fermions is governed by the Whitney bound, $$P_\text{fen} \approx 0.032172 k_B (T_L - T_R)^2 / h$$. This bound is derived under the assumption of particle-hole symmetry breaking and optimization over energy-dependent transmission functions. Extending the analysis to bosonic carriers reveals a substantially elevated universal power bound: $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$, which exceeds the fermionic limit by a factor of approximately $1.52$. This enhancement arises from the fundamental difference in quantum state occupation and exclusion principles governing bosons.

The required bias to achieve maximum power in the bosonic case is also reduced relative to the fermionic regime, with the optimal bias given by $\mu_{\text{opt}} = \ln 2 k_B (T_L - T_R)$ versus $1.146 k_B (T_L - T_R)$ for fermions. These results underscore the role of exclusion statistics as a lever for performance enhancement in quantum thermal machines.

Physical Realization via Magnon Transport

The paper proposes magnon transport across ferromagnetic spin chains as a realistic implementation of a bosonic working medium. Using a linearized Holstein-Primakoff transformation, the magnetic spin system is mapped to an effective quadratic bosonic tight-binding model. Experimental feasibility is established by analyzing magnon transmission characteristics with optimized magnetic fields in the reservoirs and central chain. With parameters such as $T_L = 300$ mK, $T_R = 10$ mK, and $B_L \approx 2$ T, the magnon system achieves over $91\%$ of the ideal bosonic bound, surpassing the universal fermionic limit by a substantial margin. The deviation from the absolute maximum is attributed to the non-ideal sharpness of the magnon transmission step.

Fractional Exclusion Statistics and Performance Interpolation

Introducing Haldane exclusion statistics provides a continuous interpolation between bosonic ($g = 0$) and fermionic ($$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$0) limits. The statistical parameter $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$1 controls particle occupancy and modifies the power bounds: for $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$2, both maximum power and optimal bias are monotonically enhanced. The intersection energy for optimal transmission remains fixed across statistical species, $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$3, ensuring that the optimal transmission window is statistics-invariant.

The distribution function, $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$4, with $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$5 defined by $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$6, introduces intrinsic particle-hole asymmetry, yielding greater thermoelectric optimization potential for fractional statistics. Practical realization of $$P_\text{max}^{\text{boson}} = (\ln 2)^2 k_B (T_L - T_R)^2 / h$$7 emerges in antiferromagnetic Heisenberg spin chains via spinon excitations.

Efficiency-Power Trade-off and Thermodynamic Implications

Analysis of efficiency at finite power reveals a statistics-dependent power-efficiency trade-off. At maximum power, bosonic carriers consistently exhibit higher efficiency relative to fermions, with efficiency converging to the Carnot limit only as power approaches zero. The optimal transmission profiles adjust dynamically with both bias and statistical parameter, further optimizing performance by narrowing or expanding the transport window.

Efficient filtering of high-energy particles leads to significant gains in practical efficiency, and the framework generalizes to all exclusion statistics, forming a benchmark for nanoscopic thermoelectric media.

Theoretical and Practical Implications

The results demonstrate that quantum heat engine performance is not strictly set by structural or transmission engineering but is critically dependent on carrier statistics. Exclusion statistics can be exploited for performance enhancement, establishing new regimes of operation unattainable by conventional approaches. The statistics-driven paradigm suggests several key implications:

• Carrier statistics is a fundamental resource for thermodynamic optimization, orthogonal to traditional mechanisms.
• Physical systems with tunable statistics (e.g., engineered spin chains or fractional quantum Hall states) can realize enhanced power and efficiency.
• Future quantum device architectures can leverage statistical interpolation for adaptive thermal transport control.

These findings also motivate further theoretical work on quantum transport in fractionalized systems and experimental investigation of magnonic and spinonic heat engines.

Conclusion

This study rigorously establishes that the traditional upper bounds for quantum heat engines are statistical artifacts, not fundamental limits. Bosonic and fractional exclusion statistics yield consistently higher power and efficiency outcomes than the fermionic regime, with physical implementations via magnon transport proving feasible and robust. Exclusion statistics emerge as a new axis of thermodynamic optimization, expanding the theoretical landscape and practical toolkit for quantum thermal machines (2606.19310).