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arXiv:1507.05639 · Physical Review B

Fate of classical solitons in one-dimensional quantum systems

Michael Pustilnik, К. А. Матвеев

Published Nov 23, 2015• 12 citations
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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, namely 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.

DOI
10.1103/physrevb.92.195146

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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, namely 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.
A one-dimensional model of bosons with repulsive short-range interactions, solved analytically by Lieb and Liniger many years ago, predicts the existence of two branches of elementary excitations. One of them represents Bogoliubov phonons; the other, as suggested by some authors, might be related to dark solitons. On the other hand, it has been already demonstrated within the framework of the classical field approximation that a quasi-one-dimensional interacting Bose gas at equilibrium exhibits excitations which are phonons and dark solitons. By showing that for lower temperatures statistical distributions of dark solitons obtained within the classical field approximation are well represented by the distributions of quasiparticles of the second kind derived from fully quantum description, we demonstrate that type II excitations in the Lieb-Liniger model are, indeed, quantum solitons.
In one-dimensional quantum systems with strong long-range repulsion particles arrange in a quasiperiodic chain, the Wigner crystal. We demonstrate that besides the familiar phonons, such one-dimensional Wigner crystal supports an additional mode of elementary excitations, which can be identified with solitons in the classical limit. We compute the corresponding excitation spectrum and argue that the solitons have a parametrically small decay rate at low energies. We discuss implications of our results for the behavior of the dynamic structure factor.
Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.
We developed a comprehensive semiclassical theory of solitons in one-dimensional systems at the BCS-BEC crossover to provide a semiclassical explanation of their excitation spectra. Our semiclassical results agree well with the exact solutions on both the deep BCS and the deep BEC side and explain qualitatively the smooth crossover between them. Especially, we showed that the minimum energy of the S=1/2S=1/2S=1/2 excitation is achieved exactly at the Fermi momentum ${k}_{F}=\ensuremath{\pi}n/2$, where nmFn{m}_{F}nmF​ (mF{m}_{F}mF​ is the mass of the fermionic atom) is the total mass density of the system. This momentum remains unchanged along the whole crossover whether the mass is contained in the bosonic molecules as on the deep BEC side or in the fermionic atoms as on the deep BCS side. This phenomenon comes about as a result of a special feature of one-dimensional systems that the conventional quasiparticle is not stable with respect to soliton formation. It is valid not only in exactly solvable models, but also on the level of semiclassical theory. Besides, we also resolved the inconsistency of existing semiclassical theory with the exact solution of solitonlike S=0S=0S=0 excitations on the deep BCS side by a different proposal of soliton configuration.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.
Abstract Solitons are ubiquitous phenomena that appear, among others, in the description of tsunami waves, fiber-optic communication and ultracold atomic gases. The latter systems turned out to be an excellent playground for investigations of matter-wave solitons in a quantum world. This tutorial provides a general overview of the ultracold contact interacting Bose and Fermi systems in a one-dimensional space that can be described by the renowned Lieb–Liniger and Yang–Gaudin models. Both the quantum many-body systems are exactly solvable by means of the Bethe ansatz technique, granting us a possibility for investigations of quantum nature of solitonic excitations. We discuss in details a specific class of quantum many-body excited eigenstates called yrast states and show that they are strictly related to quantum dark solitons in the both considered Bose and Fermi systems.