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arXiv:1608.04711 · Journal of Physics A Mathematical and Theoretical

Quantum centipedes with strong global constraint

Pascal Grange

Published May 9, 2017• 0 citations

Abstract

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 .

DOI: 10.1088/1751-8121/aa6dcf

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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 .

Abstract We consider the quantum centipede made of N fermionic quantum walkers on the one-dimensional lattice interacting by means of the simplest of all hard-bound constraints: the distance between two consecutive fermions is either one or two lattice spacings. This composite quantum walker spreads ballistically, just as the simple quantum walk. However, because of the interactions between the internal degrees of freedom, the distribution of its center-of-mass velocity displays numerous ballistic fronts in the long-time limit, corresponding to singularities in the empirical velocity distribution. The spectrum of the centipede and the corresponding group velocities are analyzed by direct means for the first few values of N . Some analytical results are obtained for arbitrary N by exploiting an exact mapping of the problem onto a free-fermion system. We thus derive the maximal velocity describing the ballistic spreading of the two extremal fronts of the centipede wavefunction, including its non-trivial value in the large- N limit.

Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first

M

sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit

M\times M

Hermitian matrix, which yields the optimal state by direct diagonalization. For large

M

, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in

M

and a constant prefactor

π^2/4

. Numerical results verify both the finite-

M

spectral formulation and the predicted scaling.

Abstract In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in Wong (2015 J. Phys. A: Math. Theor . 48 435304) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker’s probability distribution.

We treat quantum walk (QW) on the line whose quantum coin at each vertex tends to the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a ``power-law" decay around the origin and a ``strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at

x=0

(the origin) and

x=1

(the bottom) without continuous parts.

In this thesis, we have investigated the spreading of quantum correlations in isolated lattice models with short- or long-range interactions driven far from equilibrium via sudden global quenches. A general theoretical approach relying on a quasiparticle theory is presented. The latter has permitted to unveil a generic expression for the equal-time connected correlation functions valid both for short-range and long-range interacting particle and spin lattice models on a hypercubic lattice. Relying on stationary phase arguments, we have shown that its causality cone displays a universal twofold structure consisting of a correlation edge and a series of local extrema defining the outer and inner structure of the space-time correlations. For short-range interactions, the motion of each structure is ballistic and the associated spreading velocities are related to the group and phase velocities of the quasiparticle dispersion relation of the post-quench Hamiltonian. For long-range interactions of the form

M

, the correlation spreading is substantially different due to a possible divergence of group velocity when tuning the power-law exponent

α

. For a divergent group velocity, i.e. the quasi-local regime, we have presented evidence of a universal algebraic structure for the causality cone. While, the correlation edge motion has been found to be always slower than ballistic, the local extrema propagate faster than ballistically and ballistically for gapless and gapped quantum systems respectively. For the local regime implying a well-defined group velocity, we have recovered similar scaling laws and spreading velocities than the short-range case for the causality cone of correlations.

A continuous-time quantum walk of a phase-modulated wave packet is considered in a one-dimensional equispacing lattice {| k >| k ∈ Z }. The interaction

\hbar c/\Delta x

between adjacent sites drives the packet unidirectionally, where Δ x denotes the lattice spacing. If the initial state is spread over N consecutive sites and phase-modulated by exp (i k θ), the packet travels ballistically with a velocity v = (1 - N -1 ) c cos θ.