Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A compellingly simple proof of the speed of sound for interacting bosons

J. Eisert

Abstract

On physical grounds, one expects locally interacting quantum many-body systems to feature a finite group velocity. This intuition is rigorously underpinned by Lieb-Robinson bounds that state that locally interacting Hamiltonians with finite-dimensional constituents on suitably regular lattices always exhibit such a finite group velocity. This also implies that causality is always respected by the dynamics of quantum lattice models. It had been a long-standing open question whether interacting bosonic systems also feature finite speeds of sound in information and particle propagation, which was only recently resolved. This work proves a strikingly simple such bound for particle propagation - shown in literally a few elementary, yet not straightforward, lines - for generalized Bose-Hubbard models defined on general lattices, proving that appropriately locally perturbed stationary states feature a finite speed of sound in particle numbers.

A compellingly simple proof of the speed of sound for interacting bosons

Abstract

On physical grounds, one expects locally interacting quantum many-body systems to feature a finite group velocity. This intuition is rigorously underpinned by Lieb-Robinson bounds that state that locally interacting Hamiltonians with finite-dimensional constituents on suitably regular lattices always exhibit such a finite group velocity. This also implies that causality is always respected by the dynamics of quantum lattice models. It had been a long-standing open question whether interacting bosonic systems also feature finite speeds of sound in information and particle propagation, which was only recently resolved. This work proves a strikingly simple such bound for particle propagation - shown in literally a few elementary, yet not straightforward, lines - for generalized Bose-Hubbard models defined on general lattices, proving that appropriately locally perturbed stationary states feature a finite speed of sound in particle numbers.
Paper Structure (2 theorems, 13 equations, 1 figure)

This paper contains 2 theorems, 13 equations, 1 figure.

Key Result

Theorem 1

For any interacting bosonic Hamiltonian (Hamiltonian) defined on a lattice $G$ and any initial state $\rho(0)$ that locally perturbs a stationary state $\omega$ as in Definition 1, adding particles locally on a region ${\cal R}\subset {\cal L}$, it holds true that for all times $t\geq 0$, for all $j$ with ${\rm dist} (j,{\cal R})\geq l$, where $c>0$ is a universal constant and $v: = v_0 + D$ is a

Figures (1)

  • Figure 1: A schematic picture of the Bose-Hubbard model in one spatial dimension, with a hopping at rate $\tau>0$ to neighbouring ones and on-site interactions with strength $U>0$, which is a paradigmatic example of the general interacting bosonic particle Hamiltonians considered here. This work shows a finite speed of sound for particle densities for general such models. Local excitations initially confined to a region ${\cal R}$ will at most ballistically propagate through the lattice ${\cal L}$ with a speed of sound, up to exponentially small corrections.

Theorems & Definitions (5)

  • Definition 1: Initial excitations
  • Theorem 1: Finite speed of sound of particle propagation for interacting bosons
  • Lemma 1: Positivity
  • proof
  • proof