Quantum return probability of a system of N non-interacting lattice fermions

Abstract

Abstract. We consider N non-interacting fermions performing continuous-time quantum walks on a one-dimensional lattice. The system is launched from a most compact configuration where the fermions occupy neighboring sites. We calculate exactly the quantum return probability (sometimes referred to as the Loschmidt echo) of observing the very same compact state at a later time t. Remarkably, this probability depends on the parity of the fermion number -it decays as a power of time for even N , while for odd N it exhibits periodic oscillations modulated by a decaying power law. The exponent also slightly depends on the parity of N , and is roughly twice smaller than what it would be in the continuum limit. We also consider the same problem, and obtain similar results, in the presence of an impenetrable wall at the origin constraining the particles to remain on the positive half-line. We derive closed-form expressions for the amplitudes of the power-law decay of the return probability in all cases. The key point in the derivation is the use of Mehta integrals, which are limiting cases of the Selberg integral.

Light-cone and local front dynamics of a single-particle extended quantum walk

“…In this section, we study the localization properties of the time evolved wave packet by studying various measures characterizing the spreading and localization of the wave packet. In particular, we discuss the inverse participation ratio, Shannon entropy and the return probability [21,36,[39][40][41]. The inverse participation ratio is used to characterize topological phases (like topological-band insulator transitions occurring in 2D Dirac materials like silicene), Anderson metal-insulator transition, many-body localization [39,40], etc.…”

Return probability of $N$ fermions released from a 1D confining potential

“…On the one-dimensional lattice, the band structure of a tight-binding particle, E = 2 cos q, possesses two inequivalent stationary points where the group velocity v = −2 sin q vanishes, namely q = 0, as before, and q = π. In the long-time regime of the free expansion phase, both stationary points in momentum space are roughly equally populated [21]. This band-structure effect has two noticeable consequences on the decay exponent: it is roughly twice smaller than its counterpart in the continuum and exhibits a rather unexpected dependence on the parity of N .…”

“…For the Calogero-Sutherland model, an exactly solvable interacting many-particle system on the full line [16], the decay exponent is known to be N (1 + λ(N − 1)), where the coupling constant λ allows to interpolate between free bosons for λ = 0, where the exponent is simply N , and hardcore bosons (equivalent to non-interacting fermions in one dimension) for λ = 1, where the exponent N 2 is recovered. Table 1 presents a comparison of the decay exponent in both geometries with exponents defined similarly in two other situations, namely tight-binding lattice fermions launched from a compact configuration, investigated in our recent work [21], and non-colliding classical random walkers, whose survival and return probabilities are derived in Appendix A. There are both analogies and differences between these three situations.…”

“…where G denotes the Barnes G-function (see Appendix C). The universal decay exponent N 2 can be recovered by the following heuristic argument [21]. The wavefunction of N non-interacting fermions can be estimated by expressing that the particles spread on a ballistic scale L(t) ∼ t and that the wavefunction vanishes when two particles occupy the same position.…”

“…In a recent work [21] we studied the quantum return probability for a system of N free fermions hopping on a discrete infinite or semi-infinite lattice, launched from a compact configuration where the fermions occupy neighboring sites. We showed that this probability decays algebraically in time, with an exponent that exhibits an intriguing dependence on the parity of N , due to the combined effects of quantum interferences and discreteness.…”