Fate of classical solitons in one-dimensional quantum systems

TL;DR

This paper analyzes how classical solitons in one-dimensional systems transition into quantum particles, showing they become fermionic excitations, with detailed studies on the Lieb-Liniger and quantum Toda models.

Contribution

It provides an analytical description of the classical-to-quantum crossover of solitons in 1D systems and demonstrates the universality of these results across models.

Findings

Classical solitons evolve into fermionic particles and holes in the quantum regime.

Exact solutions for the Lieb-Liniger and quantum Toda models illustrate the crossover.

Results are universally applicable to 1D quantum systems with a classical KdV limit.

Abstract

We study one-dimensional quantum systems near the classical limit described by the Korteweg-de Vries (KdV) equation. The excitations near this limit are the well-known solitons and phonons. The classical description breaks down at long wavelengths, where quantum effects become dominant. Focusing on the spectra of the elementary excitations, we describe analytically the entire classical-to-quantum crossover. We show that the ultimate quantum fate of the classical KdV excitations is to become fermionic particles and holes. We discuss in detail two exactly solvable models exhibiting such crossover, the Lieb-Liniger model of bosons with weak contact repulsion and the quantum Toda model. We argue that the results obtained for these models are universally applicable to all quantum one-dimensional systems with a well-defined classical limit described by the KdV equation.