Renormalization group approach to the Fröhlich polaron model: application to impurity-BEC problem

When a mobile impurity interacts with a many-body system, such as a phonon bath, a polaron is formed. Despite the importance of the polaron problem for a wide range of physical systems, a

unified theoretical description valid for arbitrary coupling strengths is still lacking. Here we develop a renormalization group approach for analyzing a paradigmatic model of polarons, the

so-called Fröhlich model and apply it to a problem of impurity atoms immersed in a Bose-Einstein condensate of ultra cold atoms. Polaron energies obtained by our method are in excellent

agreement with recent diagrammatic Monte Carlo calculations for a wide range of interaction strengths. They are found to be logarithmically divergent with the ultra-violet cut-off, but

physically meaningful regularized polaron energies are also presented. Moreover, we calculate the effective mass of polarons and find a smooth crossover from weak to strong coupling regimes.

Possible experimental tests of our results in current experiments with ultra cold atoms are discussed.

A general class of fundamental problems in physics can be described as an impurity particle interacting with a quantum reservoir. This includes Anderson’s orthogonality catastrophe1, the

Kondo effect2, lattice polarons in semiconductors, magnetic polarons in strongly correlated electron systems and the spin-boson model3. The most interesting systems in this category can not

be understood using a simple perturbative analysis or even self-consistent mean-field (MF) approximations. For example, formation of a Kondo singlet between a spinful impurity and a Fermi

sea is a result of multiple scattering processes4 and its description requires either a renormalization group (RG) approach5 or an exact solution6,7, or introduction of slave-particles8.

Another important example is a localization delocalization transition in a spin bath model, arising due to “interactions” between spin flip events mediated by the bath3.

While the list of theoretically understood non-perturbative phenomena in quantum impurity problems is impressive, it is essentially limited to one dimensional models and localized

impurities. Problems that involve mobile impurities in higher dimensions are mostly considered using quantum Monte Carlo (MC) methods9,10,11. Much less progress has been achieved in the

development of efficient approximate schemes. For example a question of orthogonality catastrophe for a mobile impurity interacting with a quantum degenerate gas of fermions remains a

subject of active research12,13.

Recent experimental progress in the field of ultracold atoms brought new interest in the study of impurity problems. Feshbach resonances made it possible to realize both

Fermi14,15,16,17,18,19 and Bose polarons20,21 with tunable interactions between the impurity and host atoms. Detailed information about Fermi polarons was obtained using a rich toolbox

available in these experiments. Radio frequency (rf) spectroscopy was used to measure the polaron binding energy and to observe the transition between the polaronic and molecular states14.

The effective mass of Fermi polarons was studied using measurements of collective oscillations in a parabolic confining potential15. Polarons in a Bose-Einstein condensate (BEC) received

less experimental attention so far although polaronic effects have been observed in nonequilibrium dynamics of impurities in 1d systems20,21,22.

The goal of this paper is two-fold. Our first goal is to introduce a theoretical technique for analyzing a common class of polaron problems, the so-called Fröhlich polarons. We develop a

unified approach that can describe polarons all the way from weak to strong couplings. Our second goal is to apply this method to the problem of impurity atoms immersed in a BEC. We focus on

calculating the polaron binding energy and effective mass, both of which can be measured experimentally. For this particular polaron model in a BEC we address the long-standing question how

the polaron properties depend on the polaronic coupling strength and whether a true phase transition exists to a self-trapped regime. Our results suggest a smooth cross-over and do not show

any non-analyticity in the accessible parameter range. Moreover we investigate the dependence of the groundstate energy on the ultra-violet (UV) cut-off and point out a logarithmic UV

divergence. Considering a wide range of atomic mixtures with tunable interactions23 and very different mass ratios available in current

experiments24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44 we expect that many of our predictions can be tested in the near future. In particular we discuss that currently

available technology should make it possible to realize intermediate coupling polarons.

Previously the problem of an impurity atom in a superfluid Bose gas has been studied theoretically using self-consistent T-matrix calculations45 and variational methods46 and within the

Fröhlich model in the weak coupling regime47,48,49, the strong coupling approximation50,51,52,53,54, the variational Feynman path integral approach55,56,57 and the numerical diagrammatic MC

simulations58. These four methods predicted sufficiently different polaron binding energies in the regimes of intermediate and strong interactions, see Fig. 1. While the MC result can be

considered as the most reliable of them, the physical insight gained from this approach is limited. The method developed in this paper builds upon earlier analytical approaches by

considering fluctuations on top of the MF state and including correlations between different modes using the RG approach. We verify the accuracy of this method by demonstrating excellent

agreement with the MC results58 at zero momentum and for intermediate interaction strengths.

By applying a rf-pulse to flip a non-interacting (left inset) into an interacting impurity state (right inset) a Bose polaron can be created in a BEC.

From the corresponding rf-spectrum the polaron groundstate energy can be obtained. In the main plot we compare polaronic contributions to the energy Ep (as defined in Eq. (25)) predicted by

different models, as a function of the coupling strength α. Our results (RG) are compared to calculations with correlated Gaussian wavefunctions (CGWs)59, MC calculations by Vlietinck et

al.58, Feynman variational calculations by Tempere et al.56 and MF theory. We used the standard regularization scheme to cancel the leading power-law divergence of Ep. However, to enable

comparison with the MC data, we did not regularize the UV log-divergence reported in this paper. Hence the result is sensitive to the UV cutoff chosen for the numerics and we used the same

value Λ0 = 2000/ξ as in58. Other parameters are M/m = 0.263158 and P = 0.

Our method provides new insight into polaron states at intermediate and strong coupling by showing the importance of entanglement between phonon modes at different energies. A related

perspective on this entanglement was presented in Ref. 59, which developed a variational approach using correlated Gaussian wavefunctions (CGWs) for Fröhlich polarons. Throughout the paper

we will compare our RG results to the results computed with CGWs. In particular, we use our method to calculate the effective mass of polarons, which is a subject of special interest for

many physical applications and remains an area of much controversy.

The Fröhlich Hamiltonian represents a generic class of models in which a single quantum mechanical particle interacts with the phonon reservoir of the host system. In particular it can

describe the interaction of an impurity atom with the Bogoliubov modes of a BEC50,52,56. In this case it reads (ħ = 1)

Here M denotes the impurity mass and m will be the mass of the host bosons, is the annihilation operator of the Bogoliubov phonon excitation in a BEC with momentum k, P and R are momentum

and position operators of the impurity atom, d is the dimensionality of the system and Λ0 is a high momentum cutoff needed for regularization. The dispersion of phonon modes of the BEC and

their interaction with the impurity atom are given by the standard Bogoliubov expressions56

with n0 being the BEC density and ξ = (2mgBBn0)−1/2 the healing (or coherence) length and c = (gBBn0/m)1/2 the speed of sound of the condensate. Here gIB denotes the interaction strength

between the impurity atom with the bosons, which in the lowest order Born approximation is given by gIB = 2πaIB/mred, where aIB is the scattering length and is the reduced mass of a pair

consisting of impurity and bosonic host atoms. Similarly, gBB is the boson-boson interaction strength. The analysis of the UV divergent terms in the polaron energy will require us to

consider a more accurate cutoff dependent relation between gIB and the scattering length aIB (see methods).

The Fröhlich Hamiltonian (1) for an impurity atom in a BEC is characterized by only two dimensionless coupling constants when expressing lengths in units of ξ and energies in units of c/ξ.

Firstly the mass ration M/m enters the kinetic energy of the impurity and determines the strength of long-range phonon-phonon interactions mediated by the impurity atom. Secondly,

impurity-phonon interactions are determined by the scattering length aIB and the BEC density n0 and can be parametrized by the dimensionless coupling strength56

In order to calculate the energy of the impurity atom in the BEC one needs to consider the full expression , where is the MF interaction energy of the impurity with bosons from the

condensate and is the groundstate energy of the Fröhlich Hamiltonian. From now on we will call the impurity-condensate interaction energy and EB the polaron binding energy. Only EIMP is

physically meaningful and can be expressed in a universal cutoff independent way using the scattering length aIB. Precise conditions under which one can use the Fröhlich model to describe

the impurity BEC interaction and parameters of the model for specific cold atoms mixtures are discussed in the discussion section. We point out that the Fröhlich type Hamiltonians (1) are

relevant for many systems besides BEC-impurity polarons. Its original and most common use is in the context of electrons coupled to crystal lattice fluctuations in solid state systems60.

Another important application area is for studying doped quantum magnets, in which electrons and holes are strongly coupled to magnetic fluctuations. Motivated by this generality of the

model (1) we will analyze it for a broader range of parameters than may be relevant for the current experiments with ultra cold atoms.

As a first step we apply the standard Lee-Low-Pines (LLP)61 unitary transformation to the Fröhlich Hamiltonian, which separates the total polaron momentum P as a conserved quantity. Next, we

apply a second exact unitary transformation, which displaces the phonons by the MF polaron solution 61. This brings the Hamiltonian into the form (see also method section for a

self-contained derivation)

Here is the polaron binding energy obtained from MF theory and the MF phonon dispersion is denoted by . MF polaron theory was formulated for this problem in48 and we give a self-contained

summary in the methods section. Moreover we defined , : ... : stands for normal-ordering and we introduced the short-hand notation .

In this section we provide the RG solution of the Hamiltonian (4), describing quantum fluctuations on top of the MF polaron state. We begin with a dimensional analysis of different terms in

(4) in the long wavelength limit, which establishes that only one of the interaction terms is marginal and all others are irrelevant. Then we present the RG flow equations for parameters of

the model, including the expression for the polaron binding energy. We note that electron-phonon interactions of the Fröhlich type, see Eq. (1), have been treated before using a different RG

formalism62,63. There, phonons were integrated out exactly and in contrast to the method introduced below all information about phonon correlations in the polaron cloud was lost.

Our approach to the RG treatment of the model (4) is similar to the “poor man’s RG” in the context of the Kondo problem. We use Schrieffer-Wolff type transformation to integrate out high

energy phonons in a thin shell in momentum space near the cutoff, Λ − δΛ