Phase transitions at high energy vindicate negative microcanonical temperature

The notion of negative absolute temperature emerges naturally from Boltzmann's definition of “surface” microcanonical entropy in isolated systems with a bounded energy density. Recently, the well-posedness of such construct has been challenged, on account that only the Gibbs “volume” entropy—and the strictly positive temperature thereof—would give rise to a consistent thermodynamics. Here we present analytical and numerical evidence that Boltzmann microcanonical entropy provides a consistent thermometry for both signs of the temperature. In particular, we show that Boltzmann (negative) temperature allows the description of phase transitions occurring at high energy densities, at variance with Gibbs temperature. Our results apply to nonlinear lattice models standardly employed to describe the propagation of light in arrays of coupled wave guides and the dynamics of ultracold gases trapped in optical lattices. Optically induced photonic lattices, characterized by saturable nonlinearity, are particularly appealing because they offer the possibility of observing states and phase transitions at both signs of the temperature.

Figure 1

Relation between the grand-canonical $\beta$ and the kinetic energy per particle in a 1D lattice, as provided by the transfer-matrix approach (symbols). Panels (a) and (b) refer to saturable and standard nonlinearities, respectively. Despite the non-negligible interaction strengths, the data points closely follow the corresponding noninteracting result, Eq. (9).

Figure 2

Phase transition in a 2D lattice model with saturable self-focusing nonlinearity ($a=1,U=-0.75$); (a) BKT transition at negative Boltzmann temperature. The horizontal gray line signals the expected critical point; (b) condensation into the localized ground state at positive temperature. The dashed lines are guides to the eye.

Figure 3

Condensation transition at negative Boltzmann temperature in a 3D lattice model with standard self-focusing nonlinearity ($U=-0.75,a=0.5$). The solid line is the prediction from the Bogoliubov approximation.

Figure 4

Boltzmann entropy density for a 1D noninteracting lattice model comprising $L=20$ sites. Main plot: semiclassical case, Eq. (2). The inset shows the situation for the corresponding quantum (Bose-Hubbard) model. In all cases $a=1$.

Figure 5

Occupation of the ground and highest-energy state for a $64\times 64\times 64$ noninteracting lattice model with $a=1$, as provided by a grand-canonical calculation.

Figure 6

Instantaneous value of the microcanonical inverse temperature and kinetic energy per particle. Left: $128\times 128$ lattice with saturable nonlinearity at a negative temperature. Figure 7 has been obtained by time averaging the same data in the time window $4\times {10}^4<tJ\hslash <8\times {10}^4$. Right: $64\times 64\times 64$ lattice with standard nonlinearity at a positive temperature. The rightmost green circle in Fig. 3 has been obtained by time averaging the same data in the time window $4\times {10}^4<tJ\hslash <8\times {10}^4$. In both cases $U=0.75,a=1$.

Figure 7

Time-averaged distribution for the occupation of the single-particle modes in equilibrium states at small $\left|\beta \right|$. The green dots refer to the whole lattice, and the yellow dots refer to a sublattice whose volume is $1/16$ of the whole lattice; the slope of the dashed black (straight) line is the time-averaged value of the instantaneous microcanonical inverse temperature. The density plots in the upper insets show the average distribution according to quasimomentum. The lower insets contain histograms of the values assumed by the instantaneous inverse temperature in the considered time window of $\mathrm{\Delta }t=4\times {10}^4\hslash J^{-1}$. In both cases $a=1,U=0.75$, and the nonlinearity is of the saturable kind. Panels (a) and (b) refer to 2D $128\times 128$ and $256\times 256$ lattices, respectively.

Figure 8

Time-averaged distribution for the mode occupation in a $128\times 128$ lattice with standard nonlinearity, $U=10.0,a=1.0$. Green dots refer to the single-particle modes: ${\xi }_{\mathbf{q}}={\tilde{z}}_{\mathbf{q}}$ and ${\epsilon }_{\mathbf{q}}={\varepsilon }_{\mathbf{q}}-{\varepsilon }_{\mathbf{0}}$. We translated the energy spectrum for better comparison with the orange dots, which refer to the Bogoliubov quasiparticle modes: ${\xi }_{\mathbf{q}}=b_{\mathbf{q}}$ and ${\epsilon }_{\mathbf{q}}={\varpi }_{\mathbf{q}}$. The slope of the black straight line corresponds to the average microcanonical temperature. The insets are obtained as in Fig. 7, except that we used a logarithmic scale in the density plot of the quasimomentum distribution.