Thermodynamics of Hamiltonian anyons with applications to quantum heat engines

TL;DR

The paper addresses how exchange symmetry affects thermodynamics and how to realize a continuum between bosonic and fermionic statistics via Hamiltonian anyons.It develops a symmetry-driven model in which a latent degree of freedom and a bias parameter $p_F$ produce a temperature- and frequency-dependent overlap between symmetric and antisymmetric sectors, yielding a partition function $Z = \binom{d+N-1}{N} Z_F + \binom{d}{N} e^{-\beta \nu} Z_B$.The work shows phase transitions in the thermodynamics (first-order in trap frequency and second-order in temperature) and demonstrates two quantum engine cycles—the Stirling cycle and a finite-time Otto cycle—that achieve enhanced efficiency and power by exploiting the anyonic crossover.These results point to tunable thermodynamics in low-dimensional quantum gases and practical routes to simulate Hamiltonian anyons on NISQ devices.

Abstract

The behavior of a collection of identical particles is intimately linked to the symmetries of their wavefunction under particle exchange. Topological anyons, arising as quasiparticles in low-dimensional systems, interpolate between bosons and fermions, picking up a complex phase when exchanged. Recent research has demonstrated that similar statistical behavior can arise with mixtures of bosonic and fermionic pairs, offering theoretical and experimental simplicity. We introduce an alternative implementation of such statistical anyons, based on promoting or suppressing the population of symmetric states via a symmetry generating Hamiltonian. The scheme has numerous advantages: anyonic statistics emerge in a single particle pair, extending straightforwardly to larger systems; the statistical properties can be dynamically adjusted; and the setup can be simulated efficiently. We show how exchange symmetry can be exploited to improve the performance of heat engines, and demonstrate a reversible work extraction cycle in which bosonization and fermionization replace compression and expansion strokes. Additionally, we investigate emergent thermal properties, including critical phenomena, in large statistical anyon systems.

Figures (11)

• Figure 1: An illustration of two statistical approaches to anyons. Orange represents fermionic exchange symmetry, while blue indicates bosonic. (a) Hamiltonian anyons as introduced in this paper. A single pair of particles occupies a mixed quantum state which overlaps with both the symmetric and antisymmetric subspaces. The framework can easily be extended to $N>2$ particles. (b) Statistical anyons, introduced in myers_21 as an ensemble of particle pairs, each in a purely symmetric or antisymmetric state. Collectively, the statistical properties of the ensemble interpolate between those of bosons and fermions.
• Figure 2: The fermionic subspace overlap, $p_{\textrm{F}}$, plotted as a function of $\nu/k_B T$ in panel (a) and $\hbar \omega/k_B T$ in panel (b). The system undergoes a transition from bosonic to fermionic statistics, as described by Eq. \ref{['eqn:pf']}. In panel (a) we set $\hbar\omega = k_\mathrm{B} T$, and the characteristic width of the sigmoid curve is $k_\mathrm{B} T$. In panel $(b)$ we set $\nu=0$; and the width of the transition is $\sim 2k_\mathrm{B} T / N^2$, becoming vanishingly narrow for large $N$. In both panels we have taken $N = d = 50$. Note the inflection point at $p_{\textrm{F}} =1/2$.
• Figure 3: Diagrams of the heat capacity per particle, $c_T$ (panels (a)–(c)), and $c_\omega$ (panels (d)–(f)) for $N = d = 50$. The energies are in arbitrary units. Note that for $\omega$, there is an order of magnitude bigger jump in the derivative of $U$ than for $T$. The jump follows the expression given by Eq. (\ref{['eqn:smallepsilon']}) with $\varepsilon=0$, corresponding to equal overlap with the fermionic and bosonic subspaces.
• Figure 4: Internal energy (a) and heat capacity (b) as a function of temperature, for a pair of fermions, bosons, statistical anyons with $k_\textrm{F}=0.5$, and Hamiltonian anyons with $\nu=0$. Statistical anyons are described by a fixed-weighting mean of fermions and bosons (which have identical heat capacities). Note that the pink, dotted line representing statistical anyons overlaps with the lines for bosons and fermions in panel (b). The internal energy of Hamiltonian anyons interpolates according to the temperature-dependent weighting $p_{\textrm{F}}$. This additional temperature dependence means that the heat capacity of Hamiltonian anyons can exceed that of fermions and bosons. These plots are for the case $N=2$, $d=2$.
• Figure 5: Schematic diagram of a modified Stirling cycle based on fermionizing and bosonizing the particle statistics. A pair of Hamiltonian anyons has access to two ladders of energy eigenstates, one with bosonic exchange symmetry (blue) and one fermionic (red), which are offset by a variable energy bias $\nu$. As both particles can occupy the bosonic ground state, the internal energy is lower than for fermions which are subject to Pauli exclusion. By lowering $\nu$, so that the particles have access to bosonic states, energy can be extracted as work. Moreover, by bosonizing at one temperature and fermionizing at another, a net output of work can be achieved in a cycle.

Theorems & Definitions (9)

• Definition 1: Ordered, integer partitions
• Definition 2: Young diagram
• Definition 3: Young Tableau
• Definition 4: Young Symmetrizer
• Theorem 1
• Definition 5: Schur functor
• Theorem 2: Schur-Weyl duality
• Proposition 1
• Proposition 2