Bosonization and the many-body problem

We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.

arXiv:1011.2453Journal of Physics B Atomic Molecular and Optical PhysicsFeb 18, 201148 citations

Many-Body Boson Systems

André F. Verbeure

Theoretical and mathematical physicsNov 24, 201061 citations

Some Problems in Many-Body Field Theory

G.M. Vagradov

Springer eBooksJan 1, 19670 citations

Few-body states of bosons interacting with a heavy quantum impurity

Shuhei M. Yoshida, Zhe-Yu Shi, Jesper Levinsen +1 more

We consider the problem of a fixed impurity coupled to a small number $N$ of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body $(N=2)$ problem analytically and determine the trimer energy for impurity-boson scattering lengths $a>0$ . By analyzing the atom-dimer scattering amplitude, we find a critical scattering length ${a}^{*}$ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$ and show that the tetramer also merges with the continuum at ${a}^{*}$ . Indeed, since the critical point ${a}^{*}$ formally resembles the unitary point $1/a=0$ , we find that all higher-body bound states (involving the impurity and $N>1$ bosons) emerge and disappear at both of these points. We show that the behavior at these ``multibody resonances'' is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.We consider the problem of a fixed impurity coupled to a small number $N$ of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body $(N=2)$ problem analytically and determine the trimer energy for impurity-boson scattering lengths $a>0$. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length ${a}^{*}$ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$ and show that the tetramer also merges with the continuum at ${a}^{*}$. Indeed, since the critical point ${a}^{*}$ formally resembles the unitary point $1/a=0$, we find that all higher-body bound states (involving the impurity and $N>1$ bosons) emerge and disappear at both of these points. We show that the behavior at these ``multibody resonances'' is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.

arXiv:1807.09992Physical Review ADec 11, 201814 citations

We consider the problem of a fixed impurity coupled to a small number

N

of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body

(N=2)

problem analytically and determine the trimer energy for impurity-boson scattering lengths

a>0

. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length

{a}^{*}

at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for

a>0

and show that the tetramer also merges with the continuum at

{a}^{*}

. Indeed, since the critical point

{a}^{*}

formally resembles the unitary point

1/a=0

, we find that all higher-body bound states (involving the impurity and

N>1

bosons) emerge and disappear at both of these points. We show that the behavior at these ``multibody resonances'' is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.