We use a mobile impurity or depleton model to study elementary excitations in one-dimensional integrable systems. For Lieb-Liniger and bosonic Yang-Gaudin models we express two phenomenological parameters characterising renormalised inter- actions of mobile impurities with superfluid background: the number of depleted particles, $N$ and the superfluid phase drop $\pi J$ in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impuritiesWe use a mobile impurity or depleton model to study elementary excitations in one-dimensional integrable systems. For Lieb-Liniger and bosonic Yang-Gaudin models we express two phenomenological parameters characterising renormalised inter- actions of mobile impurities with superfluid background: the number of depleted particles, $N$ and the superfluid phase drop $\pi J$ in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impurities

Jan 5, 20170 citations

Quantum impurities: from mobile Josephson junctions to depletons

Michael Schecter, Dimitri M. Gangardt, Alex Kamenev

We overview the main features of mobile impurities moving in one-dimensional superfluid backgrounds by modeling it as a mobile Josephson junction, which leads naturally to the periodic dispersion of the impurity. The dissipation processes, such as radiative friction and quantum viscosity, are shown to result from the interaction of the collective phase difference with the background phonons. We develop a more realistic depleton model of an impurity-hole bound state that provides a number of exact results interpolating between the semiclassical weakly interacting picture and the strongly interacting Tonks–Girardeau regime. We also discuss the physics of a trapped impurity, relevant to current experiments with ultra cold atoms.We overview the main features of mobile impurities moving in one-dimensional superfluid backgrounds by modeling it as a mobile Josephson junction, which leads naturally to the periodic dispersion of the impurity. The dissipation processes, such as radiative friction and quantum viscosity, are shown to result from the interaction of the collective phase difference with the background phonons. We develop a more realistic depleton model of an impurity-hole bound state that provides a number of exact results interpolating between the semiclassical weakly interacting picture and the strongly interacting Tonks–Girardeau regime. We also discuss the physics of a trapped impurity, relevant to current experiments with ultra cold atoms.

arXiv:1601.00628New Journal of PhysicsMay 31, 201629 citations

Mobile impurities in one-dimensional quantum liquids

A. S. Campbell

For a long time the Luttinger liquid has served as the standard low energy description of gapless one-dimensional quantum systems. More recently, using an effective mobile impurity model has allowed progress to be made beyond the linearity of the spectrum. In this thesis we show that the depleton model of mobile impurities can be extended to become equivalent to the effective impurity model calculate the dynamic response functions. We also use exact Bethe ansatz solutions to conorm predictions of the effective model to demonstrate the reliability of both the depleton model and the response functions.For a long time the Luttinger liquid has served as the standard low energy description of gapless one-dimensional quantum systems. More recently, using an effective mobile impurity model has allowed progress to be made beyond the linearity of the spectrum. In this thesis we show that the depleton model of mobile impurities can be extended to become equivalent to the effective impurity model calculate the dynamic response functions. We also use exact Bethe ansatz solutions to conorm predictions of the effective model to demonstrate the reliability of both the depleton model and the response functions.

Jul 1, 20131 citation

Finite size effects and dynamics in one-dimensional integrable systems

Maria Colomé‐Tatché

Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.

Dec 17, 20080 citations

Mobile impurity in a one-dimensional quantum gas: Exact diagonalization in the Bethe Ansatz basis

Evgeni Burovski, Oleksandr Gamayun, Oleg Valentinovich Lychkovskiy

We consider a mobile impurity particle injected into a one-dimensional quantum gas. The time evolution of the system strongly depends on whether the mass of the impurity and the masses of the host particles are equal or not. For equal masses, the model is Bethe Ansatz solvable, but for unequal masses, the model is no longer integrable and the Bethe Ansatz technique breaks down. We construct a controllable numerical method of computing the spectrum of the model with a finite number of host particles, based on exact diagonalization of the Hamiltonian in the truncated basis of the Bethe Ansatz states. We illustrate our approach on a few-body system of 5+1 particles, and trace the evolution of the spectrum depending on the mass ratio of the impurity and the host particles.We consider a mobile impurity particle injected into a one-dimensional quantum gas. The time evolution of the system strongly depends on whether the mass of the impurity and the masses of the host particles are equal or not. For equal masses, the model is Bethe Ansatz solvable, but for unequal masses, the model is no longer integrable and the Bethe Ansatz technique breaks down. We construct a controllable numerical method of computing the spectrum of the model with a finite number of host particles, based on exact diagonalization of the Hamiltonian in the truncated basis of the Bethe Ansatz states. We illustrate our approach on a few-body system of 5+1 particles, and trace the evolution of the spectrum depending on the mass ratio of the impurity and the host particles.

arXiv:2112.06627Dec 13, 20211 citation

Exponents of Spectral Functions in the One-Dimensional Bose Gas

P. Schlottmann

The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge $c=1$ . For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge $c=1$. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.

Condensed MatterOct 23, 20183 citations

N

and the superfluid phase drop

\pi J

in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impurities

c=1

. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.