Thermodynamics of Permutation-Invariant Quantum Many-Body Systems: A Group-Theoretical Framework

Quantum systems of indistinguishable particles are commonly described using the formalism of second quantisation, which relies on the assumption that any admissible quantum state must be either symmetric or anti-symmetric under particle permutations. Coherence-induced many-body effects such as superradiance, however, can arise even in systems whose constituents are not fundamentally indistinguishable as long as all relevant dynamical observables are permutation-invariant. Such systems are not confined to symmetric or anti-symmetric states and therefore require a different theoretical approach. Focusing on non-interacting systems, here we combine tools from representation theory and thermodynamically consistent master equations to develop such a framework. We characterise the structure and properties of the steady states emerging in permutation-invariant ensembles of arbitrary multi-level systems that are collectively weakly coupled to a thermal environment. As an application of our general theory, we further explore how these states can in principle be used to enhance the performance of quantum thermal machines. Our group-theoretical framework thereby makes it possible to analyse various limiting cases that would not be accessible otherwise. In addition, it allows us to show that the properties of multi-level ensembles differ qualitatively from those of spin ensembles, which have been investigated earlier using the standard Clebsch-Gordan theory. Our results have a large scope for future generalisations and pave the way for systematic investigations of collective effects arising from permutation-invariance in quantum thermodynamics.