Prethermalization with negative specific heat

We study noncanonical relaxation in an aggregate of subsystems with negative specific heat. The Thirring instability drives the constituent subsystems towards the edges of their energy spectrum, so that the existence of a single adiabatic invariant results in structured noncanonical steady states that are spectacularly different from the grand-canonical prediction. For parameter regimes where this adiabatic invariance breaks down, the system exhibits prethermalization far away from integrability, with an unprecedented contrast between the prethermal- and thermal states.

Figure 1

The dimensionless action variable $m\left(\varepsilon \right)$ (solid) and microcanonical entropy $S\left(\varepsilon \right)/k_B=ln[m^{\prime }\left(\varepsilon \right)/m^{\prime }\left({\varepsilon }_{-}\right)]$ (dashed) of the Bose-Hubbard dimer as a function of dimer energy $\varepsilon$ for $n=2$ and $U=0.15$ (a) or $U=0.35$ (b). The dotted straight line in (a) is to help show the upward curvature of $m$. The curves are representative of all $0<Un<1$; upward curvatures increase with $Un$. The slope of $S$ is everywhere positive, meaning positive microcanonical temperature $T=1/S^{\prime }\left(\varepsilon \right)$. As implied by the monotonically increasing slope of the $S\left(\varepsilon \right)$ curve, the microcanonical specific heat $C=-S^{\prime \prime }/{\left(S^{\prime }\right)}^2$ of a dimer is negative for all $\varepsilon$ for $0<Un<1$.

Figure 2

Long-time numerical propagation: The Bose-Hubbard dimer populations $n_{ij}$ (a),(b) and energies ${\varepsilon }_{ij}$ (c),(d) at $Jt={10}^3$ are plotted throughout the $60\times 60$ dimers array. Initial conditions correspond to a uniform population and energy distribution (see black dot in Fig. 3); panels (a),(c) and (b),(d) show the steady state outcome at $u=0.1$ and $u=0.5$, respectively. In all simulations $J=0.05\mathrm{\Omega }$.

Figure 3

Comparison of the one-dimer number and energy distribution in the results of Fig. 2 (gray dots, each corresponding to a single dimer) with the GCE distribution of Eq. (2), (a),(c) for $u=0.1$ and $u=0.5$, respectively) and with the generalized Gibbs distribution of Eq. (4), (b),(d) for the same parameters). The black dot corresponds to the initial uniform distribution. While the GCE fails to describe the observed distribution, the GGE captures the numerical outcome, thus demonstrating restricted ergodicity within the $N,E,M$ shell.

Figure 4

Prethermalization: (a) Time evolution of the total joson number $M$ for $u=0.5$ (solid), $0.7$ (dashed) and $0.9$ (dotted) and $J=0.05\mathrm{\Omega }$; (b) Joson number and total entropy dynamics for $\overline{u}=0.84$, just above the critical value $u_c=0.8$. Arrows mark the times at which panels (c),(e),(g) and (d),(f),(h) are taken; (c) The ${\varepsilon }_r$ distribution at the end of the M-conserving prethermalization stage; (d) same after complete thermalization; (e) Prethermal state matches the GGE with the appropriate $N,M,E$; (f) Thermal state matches GCE with the same $N,E$. Panels (g),(h) compare crosscuts through the GGE and GCE distributions at the most probable value of $n$ [marked by the dashed line in (e),(f)] showing good agreement with the numerical result at the corresponding stage.

Figure 5

(a) Dynamics of total dimer action variable $M$ for two representative values of $u$ below and above the critical $u_c$ for $J=0.05$. The corresponding $u_c$ is $0.8$ as marked by the vertical dotted line in (b), where the long time value $M_{\infty }$ is plotted against $u$. $M_0$ denotes the the total $M$ at $t=0$. (c), (d) Final distribution of initially prepared maximally excited array of dimers in the $\varepsilon ,n$ plane for $J=0.05$. Below $u_c$ the nearly uniform state remains stable, as seen in (c), while above $u_c$ it evolves into a dispersed ensemble of many different single-dimer configurations, scattered over the array.