Far-from-equilibrium thermodynamics of non-Abelian thermal states

Abstract

Noncommutativity of observables is a central feature of quantum physics. It plays a fundamental role in the formulation of the uncertainty principle for complementary variables and strongly affects the laws of thermodynamics for systems with noncommuting, that is, non-Abelian, conserved quantities. We here derive nonequilibrium generalizations of the second law of thermodynamics in the form of fluctuation relations, both for mechanically and thermally driven quantum systems. We identify a non-Abelian contribution to the energy and entropy balances, without which these relations would be violated. The latter term can be controlled to enhance both work extraction and nonequilibrium currents compared to what is obtained in commuting thermodynamics. These findings demonstrate that noncommutativity maybe a useful thermodynamic resource.

Figures (2)

• Figure 1: Non-Abelian exchange fluctuation relation. a) Non-Abelian contribution ${\mathcal{E}}_z$, current ${Q}_z$ and entropy production $\left \langle \Sigma \right \rangle_z$, for a two-qubit Heisenberg chain, as a function of the angle $\theta$ between the spin states. In the commuting case, $\theta=(0, \pi)$, ${\mathcal{E}}_z$ vanishes, implying that entropy production and nonequilibrium current are equal. For $\pi/2<\theta<\pi$, the non-Abelian term is negative. b) In this noncommuting regime, the ratio of current and dissipation is enhanced compared to the Abelian case, showing that noncommuting charges are a thermodynamic resource. c) The exchange fluctuation relation (11) is only satisfied, when the non-Abelian contribution ${\boldsymbol{\mathcal{E}}}$, Eq. (3), is included. Parameters are $J = \omega = 1$, $\beta = 1$, $\beta^R = 0.5$ and $\tau = \pi$.
• Figure 2: Non-Abelian work fluctuation relation. a) Work $\boldsymbol{W}+ \boldsymbol{\mathcal{E}}$, heat $\boldsymbol{Q}$ and entropy production $\boldsymbol{\Sigma}$ for a linearly driven system spin of the two-qubit Heisenberg chain, as a function of the angle $\theta$ between the spin states. Work is done on the system in the Abelian case $\theta=0$. As coherences are increased, work is reduced before becoming negative around $\theta=\pi/2$. b) In this non-Abelian regime, work is extracted out of the system, and the ratio of extracted work and dissipation is maximum. c) The work fluctuation relation (7) is obeyed when the non-Abelian term $\boldsymbol{\mathcal{E}}$, Eq. (3), is included (here $g_0 = 10$ and $g_\tau = 0.1$).