Entanglement entropies of an interval in the free Schrödinger field theory at finite density

Abstract

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

Entanglement entropies of an interval in the free Schrödinger field theory on the half line

“…We find that also these entanglement entropies are finite functions of the dimensionless parameter η ≡ R k F 0. In these models the entanglement entropy displays an oscillatory behaviour, differently from the entanglement entropy of the interval on the line considered in [41]. We remark that in our analyses the dispersion relation ω(k) ∝ k 2 is not approximated through a linear dispersion relation at the Fermi points (Tomonaga's approximation) [46,47].…”

“…The finiteness of the entanglement entropies in these models on the half line is a consequence of the analogous property which holds for the entanglement entropies of an interval on the line [41]. The latter follows from the properties of the solution of the sine kernel spectral problem in the interval on the line, which has been found in a series of seminal papers by Slepian, Pollak and Landau [48][49][50][51] and it is written in terms of the prolate spheroidal wave functions (PSWF) of order zero (see also the overview [52] and the recent book [53]).…”

“…Since for the Schrödinger problem on the whole line [41] the mean particle number in the interval [−R, R] is N 2A ⊂ R = 2η/π, one can rewrite (2.23) as…”

“…This is a free non-relativistic quantum field theory with z = 2 which describes the dilute spinless Fermi gas in d = 1 [39,40]. When this model is defined on the line, the entanglement entropies of an interval [−R, R] ⊂ R have been studied in [41], finding that they are finite functions of the dimensionless parameter η ≡ R k F 0, where k F is the Fermi momentum, and that the entanglement entropy S A is a monotonically increasing function of η . The µ = 0 case has been considered earlier in [42][43][44].…”

“…The procedure described in [41] for the entanglement entropies of the interval on the line, which follows the one discussed in [55][56][57][58] for some lattice models, can be adapted to the entanglement entropies of an interval adjacent to the boundary of the half line in a straightforward way and this leads us to write analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of η. These results are based on the expansions of the Bessel kernel tau function reported in [59][60][61][62], specialised to two specific values of the parameter in the Bessel kernel.…”