Entanglement entropy, single-particle occupation probabilities, and short-range correlations

For quantum many-body systems with short-range correlations (SRCs), the intimate relationship between their magnitude, the behavior of the single-particle occupation probabilities at momenta larger than the Fermi momentum, and the entanglement entropy is a new qualitative aspect not studied and exploited yet. A large body of recent condensed matter studies indicates that the time evolution of the entanglement entropy describes the nonequilibrium dynamics of isolated and strongly interacting many-body systems, in a manner similar to the Boltzmann entropy, which is strictly defined for dilute and weakly interacting many-body systems. Both theoretical and experimental studies in nuclei and cold atomic gases have shown that the fermion momentum distribution has a generic behavior $n\left(k\right)=C/k^4$ at momenta larger than the Fermi momentum, due to the presence of SRCs, with approximately 20% of the particles having momenta larger than the Fermi momentum. The presence of the long momentum tails in the presence of SRCs changes the textbook relation between the single-particle kinetic energy and occupation probabilities, $n_{\text{mf}}\left(k\right)=1/\{1+exp\beta [\epsilon \left(k\right)-\mu ]\}$ for momenta very different form the Fermi momentum, particularly for dynamics processes. SRCs induced high-momentum tails of the single-particle occupation probabilities increase the entanglement entropy of fermionic systems, which in its turn affects the dynamics of many nuclear reactions, such as heavy-ion collisions and nuclear fission.

Figure 1

Typical behavior of the $s$-wave canonical occupation probabilities in the case of pairing in a finite nuclear system. Three different amplitudes of the pairing field were considered here ${\mathrm{\Delta }}_0=$ 1, 5, and 15 MeV [4], the last one comparable to the strength of the pairing field in the unitary limit [56]. For $\epsilon \left(k\right)>75$ MeV the momentum occupation probabilities have the behavior $n\left(k\right)=C/k^4\propto 1/{\left[\epsilon \left(k\right)\right]}^2$. The UV cutoff is determined by $\epsilon \left(k\right)$ of the highest canonical state with a wave function located inside the system. The states with $\epsilon \left(k\right)$ beyond the UV cutoff are not expected to be physically relevant [4], as their corresponding canonical occupation probabilities vanish in the continuum limit. In the inset, the occupation probabilities $n\left(k\right)$, shown in a linear scale, have an expected Bardeen-Cooper-Schrieffer [76].

Figure 2

The dependence of the dimensionless “contact” $C\left(k_0\right)/k_F^7$ on the choice of momentum scale $k_0$ extracted by imposing the normalization condition Eq. (12) on the occupation probability $n\left(k\right)$, for a thermal mean field distribution, see Eq. (10) and where the temperature $\beta =1/T$, increases from the lowest to the highest curve.

Figure 3

The dependence of $\eta \left(k_0\right)$ on the choice of momentum scale $k_0$ extracted from Eq. (13), for different temperatures when $n_{\text{mf}}\left(k\right)$ is a thermal distribution for free fermion gas. For $k_0>1.05k_f$ the temperature decreases from the lowest to the highest curve.

Figure 4

The fraction of the particles $n(k>k_0)/n_0$ with momenta $k\ge k_0$, where $n\left(k\right)\propto 1/k^4$. The curves rank by temperature.

Figure 5

The final proton and neutron canonical occupation probabilities $n_{\alpha }=n\left(k\right)$ extracted from the TDDFT extended to superfluid systems treatment of induced fission reaction ${}^{235}\mathrm{U}(n,f$) as a function of ${\epsilon }_{\alpha }=\epsilon \left(k\right)$. The upper inset shows a small energy interval near the Fermi level. Above ${\epsilon }_{\alpha }\approx 50$ MeV one can see a clear power law behavior compatible with theory prediction $n\left(k\right)\propto 1/{\left[\epsilon \left(k\right)\right]}^2$. The initial state was the compound nucleus close to the top of the outer fission barrier at $t=0$ fm/$c$ and in the final state the fission fragments are spatially separated by $\approx 30$ fm at $t=1700$ fm/$c$ [4]. The nonequilibrium time-evolution of the orbital entanglement entropy $S\left(t\right)$ is shown in the lower inset, with solid and dashed lines corresponding to no nucleon number projections and with nucleon number projections, respectively [4].