Quantum centipedes with strong global constraint

TL;DR

This paper analyzes a quantum walk model of a centipede with a strong global constraint, deriving exact probability laws, dispersion relations, and group velocities, revealing that the maximal group velocity diminishes as the number of legs increases.

Contribution

It introduces a quantum centipede model with a maximal global constraint and derives exact analytical expressions for its dynamics, contrasting with unconstrained models.

Findings

Maximal group velocity approaches zero as N increases.

Probability law of the first leg expressed in closed form with Bessel functions.

Relaxing the constraint linearly increases the maximal group velocity.

Abstract

A centipede made of $N$ quantum walkers on a one-dimensional lattice is considered. The distance between two consecutive legs is either one or two lattice spacings, and a global constraint is imposed: the maximal distance between the first and last leg is $N+1$. This is the strongest global constraint compatible with walking. For an initial value of the wave function corresponding to a localized configuration at the origin, the probability law of the first leg of the centipede can be expressed in closed form in terms of Bessel functions. The dispersion relation and the group velocities are worked out exactly. Their maximal group velocity goes to zero when $N$ goes to infinity, which is in contrast with the behaviour of group velocities of quantum centipedes without global constraint, which were recently shown by Krapivsky, Luck and Mallick to give rise to ballistic spreading of extremal wave-front at non-zero velocity in the large-$N$ limit. The corresponding Hamiltonians are implemented numerically, based on a block structure of the space of configurations corresponding to compositions of the integer $N$. The growth of the maximal group velocity when the strong constraint is gradually relaxed is explored, and observed to be linear in the density of gaps allowed in the configurations. Heuristic arguments are presented to infer that the large-$N$ limit of the globally constrained model can yield finite group velocities provided the allowed number of gaps is a finite fraction of $N$.