Temperature of a single chaotic eigenstate

The onset of thermalization in a closed system of randomly interacting bosons at the level of a single eigenstate is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both interparticle interaction and energy is complemented by numerical data.

Figure 1

Pictorial description of the increase of temperature for the eigenstate in Fig. 2.

Figure 2

Average occupation numbers $\langle n_s^{\alpha }\rangle$ for weak ($V=0.1$, top panels) and strong ($V=0.4$, bottom panels) perturbation and different energies in the middle of the energy spectrum, (a) $E^{\alpha }=28.51$ and (c) $E^{\alpha }=25.93$, and on the edges, (b) $E^{\alpha }=11.27$ and (d) $E^{\alpha }=6.05$. Error bars indicate one standard deviation and are obtained by averaging over 20 close eigenstates. Dashed curves are obtained from the BED with $E=E^{\alpha }$ in Eq. (10). Solid curves correspond to the BED with the energy $E^{\mathrm{dres}}$ in Eq. (12).

Figure 3

Average energy shift $\langle {\mathrm{\Delta }}_{\alpha }\rangle$ as a function of the energy $E^{\alpha }$ for two different values of the interaction $V$. Symbols stand for numerical results, while solid lines represent the Gaussian approximation (error bars indicate one standard deviation). The average has been done over 20 close eigenstates. Due to the symmetry only half of the energy spectrum (where the density of states is an increasing function of the energy) is shown.

Figure 4

Distributions of relative fluctuations $\mathrm{\Delta }{n}_s/n_s$, $s=1,\cdots ,M=11$: (a) weak perturbation $V=0.1$, high energy, (b) weak perturbation $V=0.1$, low energy, (c) strong perturbation $V=0.4$, high energy, and (d) strong perturbation $V=0.4$, low energy. Statistics have been obtained with ${10}^3$ different realizations of the random potential and by choosing different eigenstates in a small energy window in order to have approximately 20 eigenstates for each realization of the random potential. In all of them, where it is significant (all but the yellow distributions) we superimposed a Gaussian fit (black curve).

Figure 5

(a) $d_{\mathrm{loc}}$ as a function of the number of principal components $N_{pc}$ for each eigenstate $\left|\alpha \right.〉$ and two different values of $V$: 0.1 (lower blue points) and 0.4 (upper red points). Dashed horizontal lines represent $d_{\mathrm{loc}}=V$. Arrows define the critical values $N_{cr}$. (b) Average relative fluctuations in OND $\langle \mathrm{\Delta }{n}_s/n_s\rangle$ vs $N_{pc}/N_{cr}$ for two different $V$ values as indicated in the legend. The dashed line is drawn to guide the eye and stands for $2/\sqrt{N_{pc}}$. Average fluctuations in OND have been obtained by averaging over 20 close eigenenergies.

Figure 6

Occupation numbers for a single eigenstate $n_s^{\alpha }$ for weak ($V=0.1$, top panels) and strong ($V=0.4$, bottom panels) perturbation and different energies in the middle of the energy spectrum, (a) $E^{\alpha }=28.51$ and (c) $E^{\alpha }=25.93$, and on the edges, (b) $E^{\alpha }=11.27$ and (d) $E^{\alpha }=6.05$. Dashed curves are obtained from the BED with $E=E^{\alpha }$ in Eq. (10). Solid curves correspond to the BED with the energy $E^{\mathrm{dres}}$ in Eq. (12).