Renormalization group approach to the fröhlich polaron model: application to impurity-bec problem

ABSTRACT 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. SIMILAR CONTENT BEING VIEWED BY OTHERS MEDIATED INTERACTIONS BETWEEN FERMI

POLARONS AND THE ROLE OF IMPURITY QUANTUM STATISTICS Article Open access 26 October 2023 IONIC POLARON IN A BOSE-EINSTEIN CONDENSATE Article Open access 11 May 2021 MANY-BODY BOUND STATES

AND INDUCED INTERACTIONS OF CHARGED IMPURITIES IN A BOSONIC BATH Article Open access 24 March 2023 INTRODUCTION 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. 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 _n_0 being the BEC density and ξ = (2_mg_BB_n_0)−1/2 the healing

(or coherence) length and _c_ = (_g_BB_n_0/_m_)1/2 the speed of sound of the condensate. Here _g_IB denotes the interaction strength between the impurity atom with the bosons, which in the

lowest order Born approximation is given by _g_IB = 2_πa_IB/_m_red, where _a_IB is the scattering length and is the reduced mass of a pair consisting of impurity and bosonic host atoms.

Similarly, _g_BB 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 _g_IB and the scattering length _a_IB (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 _a_IB and the BEC density _n_0 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

_E_B the polaron binding energy. Only _E_IMP is physically meaningful and can be expressed in a universal cutoff independent way using the scattering length _a_IB. 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.

RESULTS 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 . RG ANALYSIS 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 < _k_ < Λ, using 1/Ω_K_ as a small parameter (Ω_K_ being the frequency of phonons in the thin momentum-shell, where

initially ). This transformation renormalizes the effective Hamiltonian for the low energy phonons. Iterating this procedure we get a flow of the effective Hamiltonian with the cutoff

parameter Λ. To analyze whether the system flows to strong or weak coupling in the long wavelength limit |_K_|ξ ≪ 1 we consider scaling dimensions of different operators in (4). We fix the

dimension of using the condition that is scale invariant. In the long wavelength limit phonons have linear dispersion Ω_K_ ∝ |_K_|, which requires scaling of as . Scaling dimensions of

different contributions to the interaction part of the Hamiltonian (26) is shown in Table 1. We observe that, as the cutoff scale tends to zero, most terms are irrelevant and only the

quartic term is marginal. As we demonstrate below, this term is marginally irrelevant, i.e. in the process of RG flow the impurity mass _M_ flows to large values. This feature provides

justification for doing the RG perturbation expansion with 1/_M_ as the interaction parameter. Also, an irrelevance of the interaction under the RG flow physically means that at least slow

phonons in the system are Gaussian. This provides an insight why the variational correlated Gaussian wavefunctions59 are applicable for the Fröhlich Hamiltonian under consideration. By

starting from the Hamiltonian (4) and calculating its RG flow we find that the general RG Hamiltonian can be written in the following way, Note that the interaction is now characterized by a

general tensor [the indices μ = _x_, _y_, _z_, ... label cartesian coordinates and they are summed over when occurring twice], where the anisotropy originates from the conserved total

momentum of the polaron _P_ = _P__E__x_, breaking the rotational symmetry of the system. Due to the cylindrical symmetry of the problem, the mass tensor has the form and we find different

flows for the longitudinal and the transverse components of the mass tensor. While can be interpreted as the (tensor-valued) renormalized mass of the impurity, it should not be confused with

the mass of the polaron. The first line of Eq. (5) describes the diagonal quadratic part of the renormalized phonon Hamiltonian. It is also renormalized compared to the MF expression , The

momentum carried by the phonon-cloud, _P_ph, acquires an RG flow, describing corrections to the MF result . In addition there is a term linear in the phonon operators, weighted by By

comparing Eq. (5) to Eq. (4) we obtain the initial conditions for the RG, starting at the original UV cutoff Λ0 where , We derive the following flow equations for the parameters in (see

methods for details), Here we use the notation for the integral over the _d_ − 1 dimensional surface defined by momenta of length |_P_| = Λ. The energy correction to the binding energy of

the polaron beyond MF theory, , is given by Note that, in this expression, we evaluated the renormalized impurity mass at a value of the running cutoff Λ = _k_ given by the integration

variable _k_ = |_K_|. Similarly, it is implicitly assumed that _P_ph(_K_) and appearing in the expressions for _W__K_ and Ω_K_, see Eqs (7) and (6), are evaluated at Λ = _k_. Figure 2 shows

typical RG flows of and . For we observe quick convergence of these coupling constants. One can see comparison of the MC and RG calculations58 for the polaron binding energy at momentum _P_ 

= 0 in Fig. 1. The agreement is excellent for a broad range of interaction strengths. We will further discuss these results below. CUTOFF DEPENDENCE In a three dimensional system (_d_ = 3),

examination of the binding energy of the polaron _E_B shows that it is UV divergent. To leading order it scales linearly with the UV cut-off, _E_B ~ Λ0. This divergence is known from MF

polaron theory and it can be regularized by using the second order Lippman-Schwinger equation to relate the scattering length _a_IB to the interaction strength _g_IB which determines the

impurity-condensate interaction (see methods for a brief review). From our RG protocol, in addition, we obtain a sub-leading logarithmic UV divergence, _E_B ~ −log(Λ0ξ). To obtain cut-off

independent polaron energies _E_IMP we developed a regularization scheme which cancels the log-divergence, by including beyond-MF effects in the Lippman-Schwinger equation. The details of

this scheme will be presented in a separate publication64. All other observables discussed in this paper, such as the effective polaron mass, are UV convergent and thus no regularization

will be required. POLARON ENERGY The impurity energy (calculated from the RG and using our regularization scheme) is shown in Fig. 3 as a function of the polaronic coupling strength α. The

resulting energy is close to, but slightly above, the MF energy. [This result might be surprising at first glance, because MF polaron theory relies on a variational principle and thus yields

an upper bound for the groundstate energy. However, this bound holds only for the binding energy _E_B, defined as the groundstate energy of the Fröhlich Hamiltonian and not for the entire

impurity Hamiltonian including the condensate-impurity interaction.] We thus conclude that the large deviations observed in Fig. 1 of beyond MF theories (MC, RG, variational) from MF are

merely an artifact of the logarithmic UV divergence which was not properly regularized. We note that this also explains — at least partly — the unexpectedly large deviations of Feynman’s

variational approach from the numerically exact MC results, reported in58, since Feynman’s model does not capture the log-divergence58. Our results have important implications for

experiments. The relatively small difference in energy between MF and (properly regularized) RG in Fig. 3 demonstrates that a measurement of the impurity energy alone does not allow to

discriminate between uncorrelated MF theories and extensions thereof (like RG or MC). Yet such a measurement would still be significant as a consistency check of our regularization scheme.

To find smoking gun signatures for beyond MF behavior, i.e. for a regime dominated by quantum fluctuations, other observables are required such as the effective polaron mass, which we

discuss next. EFFECTIVE POLARON MASS In Fig. 4 we show the polaron mass calculated using several different approaches. In the weak coupling limit α → 0 the polaron mass can be calculated

perturbatively in α and the lowest-order result is shown in Fig. 4. We observe that in this limit, all approaches follow the same line which asymptotically approaches the perturbative result

(as α → 0). The only exception is the strong coupling Landau-Pekkar approach, which yields a self-trapped polaron solution only beyond a critical value of α54. For larger values of α, MF

theory sets a lower bound for the polaron mass. Naively this is expected, because MF theory does not account for quantum fluctuations due to couplings between phonons of different momenta.

These fluctuations require additional correlations to be present in beyond MF wavefunctions, like e.g. in our RG approach, which should lead to an increased polaron mass. Indeed, for

intermediate couplings the RG, as well as the variational approach using CGWs, predict a polaron mass which is considerably different from the MF result48. In Fig. 4 we present the most

interesting aspect of our analysis, which is related to the nature of the cross-over from weak to strong coupling polaron regime. While Feynman’s variational approach predicts a sharp

transition, the RG and CGWs results show no sign of any discontinuity in the accessible parameter range. Instead they suggest a smooth cross-over from one into the other regime. This is also

expected on general grounds and rigorous proofs were given for generic polaron models in Refs 65, 66. The proofs do not apply to the Fröhlich Hamiltonian in a BEC, see Eq. (1), however.

Interestingly, for closely related Fröhlich polarons with acoustic phonons, indications for a true phase transition were found in the solid-state context67. It is possible that the sharp

crossover obtained using Feynman’s variational approach is an artifact of the limited number of parameters used in the variational action. It would be interesting to consider a more general

class of variational actions20,68. In Fig. 4 we calculated the polaron mass in the strongly coupled regime, where α ≫ 1 and the impurity-boson mass ratio _M_/_m_ = 0.26 is small. It is also

instructive to see how the system approaches the integrable limit _M_ → ∞ when it becomes exactly solvable48. Figure 5 shows the (inverse) polaron mass as a function of α for different mass

ratios _M_/_m_. For _M_ ≫ _m_, as expected, the corrections from the RG are negligible and MF theory is accurate. When the mass ratio _M_/_m_ approaches unity, we observe deviations from the

MF behavior for couplings above a critical value of α which depends on the mass ratio. Remarkably, for very large values of α the mass predicted by the RG follows the same power-law as the

MF solution, with a different prefactor. This can be seen more clearly in Fig. 6, where the case _M_/_m_ = 1 is presented. This behavior can be explained from strong coupling theory. As

shown in54 the polaron mass in this regime is proportional to α, as is the case for the MF solution. However prefactors entering the expressions for the weak coupling MF and the strong

coupling masses are different. To make this more precise, we compare the MF, RG and strong coupling polaron masses for _M_/_m_ = 1 in Fig. 6. We observe that the RG smoothly interpolates

between the weak coupling MF and the strong coupling regime. While the MF solution is asymptotically recovered for small α → 0 (by construction), this is not strictly true on the strong

coupling side. Nevertheless, the observed value of the RG polaron mass in Fig. 6 at large α is closer to the strong coupling result than to the MF theory. Now we return to the discussion of

the polaron mass for systems with a small mass ratio _M_/_m_ < 1. In this case Fig. 5 suggests that there exists a large regime of intermediate coupling, where neither strong coupling nor

MF theory can describe the qualitative behavior of the polaron mass. This is demonstrated in Fig. 4, where our RG approach predicts values for the polaron mass midway between MF and strong

coupling, for a wide range of couplings. In this intermediate-coupling regime, the impurity is constantly scattered on phonons, leading to strong correlations between them. Thus measurements

of the polaron mass rather than the binding energy should be a good way to discriminate between different theories describing the Fröhlich polaron at intermediate couplings. Quantum

fluctuations manifest themselves in a large increase of the effective mass of polarons, in strong contrast to the predictions of the MF approach based on the wavefunction with uncorrelated

phonons. Experimentally, both the quantitative value of the polaron mass, as well as its qualitative dependence on the coupling strength can provide tests of our theory. The mass of the

Fermi polaron has successfully been measured using collective oscillations of the atomic cloud15 and we are optimistic that similar experiments can be carried out with Bose polarons in the

near future. Alternatively, momentum resolved radio-frequency spectroscopy can be used to measure the mass of the polaron, see e.g.48. If imbalanced atomic mixtures are used, the

polaron-polaron interactions need to be sufficiently weak to prevent the system from phase-separation, as discussed in Ref. 69using the strong-coupling approximation. DISCUSSION Now we

discuss conditions under which the Fröhlich Hamiltonian can be used to describe impurities in ultra cold quantum gases. We also present typical experimental parameters and show that the

intermediate coupling regime α ~ 1 can be reached with current technology. Possible experiments in which the effects predicted in this paper could be observed are also discussed. To derive

the Fröhlich Hamiltonian Eq. (1) for an impurity atom immersed in a BEC52,56, the Bose gas is described in Bogoliubov approximation, valid for weakly interacting BECs. Then the impurity

interacts with the elementary excitations of the condensate, which are Bogoliubov phonons. In writing the Fröhlich Hamiltonian to describe these interactions, we included only terms that are

linear in the Bogoliubov operators. This implicitly assumes that the condensate depletion Δ_n_ caused by the impurity is much smaller than the original BEC density, Δ_n_/_n_0 = 1, giving

rise to the condition52 When this condition is not fulfilled, other interesting phenomena like the formation of a bubble polaron70 can be expected which go beyond the physics described by

the Fröhlich model. To reach the intermediate coupling regime of the Fröhlich model, coupling constants α larger than one are required (for mass ratios _M_/_m_ ≃ 1 of the order of one). This

can be achieved by a sufficiently large impurity-boson interaction strength _g_IB, which however means that condition (12) becomes more stringent. Now we discuss under which conditions both

and Eq. (12) can simultaneously be fulfilled. To this end we express both equations in terms of experimentally relevant parameters _a_BB (boson-boson scattering length), _m_ and _M_ which

are assumed to be fixed and we treat the BEC density _n_0 and the impurity-boson scattering length _a_IB as experimentally tunable parameters. Using the first-order Born approximation result

_g_IB = 2_πa_IB/_m_red Eq. (12) reads and similarly the polaronic coupling constant can be expressed as Both α and ε are proportional to the BEC density _n_0, but while α scales with , ε is

only proportional to _a_IB. Thus to approach the strong coupling regime _a_IB has to be chosen sufficiently large, while the BEC density has to be small enough in order to satisfy Eq. (13).

When setting ε = 0.3 ≪ 1 and assuming a fixed impurity-boson scattering length _a_IB, we find an upper bound for the BEC density, where _a_0 denotes the Bohr radius. For the same fixed

value of _a_IB the coupling constant α takes a maximal value compatible with condition (12). Before discussing how Feshbach resonances allow to reach the intermediate coupling regime, we

estimate values for αmax and for typical background scattering lengths _a_IB. Despite the fact that these _a_IB are still rather small, we find that keeping track of condition (13) is

important. To this end we consider two experimentally relevant mixtures, (i) 87Rb (majority) -41K20,29 and (ii) 87Rb (majority) -133Cs25,28. For both cases the boson-boson scattering length

is _a_BB = 100_a_023,24 and typical BEC peak densities realized experimentally are _n_0 = 1.4 × 1014cm−3 29. In the first case (i) the background impurity-boson scattering length is _a_Rb−K 

= 284_a_023, yielding αRb−K = 0.18 and ε = 0.21 ≪ 1. By setting ε = 0.3 for the same _a_Rb−K, Eq. (15) yields an upper bound for the BEC density above the value of _n_0 and a maximum

coupling constant . For the second mixture (ii) the background impurity-boson scattering length _a_Rb−Cs = 650_a_028 leads to αRb−Cs = 0.96 but ε = 0.83 < 1. Setting ε = 0.3 for the same

value of _a_Rb−Cs yields and . We thus note that already for small values of , Eq. (13) is _not_ automatically fulfilled and has to be kept in mind. The impurity-boson interactions, i.e.

_a_IB, can be tuned by the use of an inter-species Feshbach resonance23, available in a number of experimentally relevant mixtures26,31,37,38,39,42,43. In this way, an increase of the

impurity-boson scattering length by more than one order of magnitude is realistic. In Table 2 we show the maximally achievable coupling constants αmax for several impurity-boson scattering

lengths and imposing the condition ε < 0.3. We consider the two mixtures from above (87Rb − 41K and 87Rb − 133Cs), where broad Feshbach resonances are available20,26,37,38. We find that

coupling constants α ~ 1 in the intermediate coupling regime can be realized, which are compatible with the Fröhlich model and respect condition (12). The required BEC densities are of the

order _n_0 ~ 1013 cm−3, which should be achievable with current technology. Note that when Eq. (12) would not be taken into account, couplings as large as α ~ 100 would be possible, but then

ε ~ 8 ≫ 1 indicates the importance of the phonon-phonon scatterings neglected in the Fröhlich model. Bose polarons in such close vicinity to a Feshbach resonance have also been discussed in

Refs 45,46. METHODS FRÖHLICH HAMILTONIAN IN THE IMPURITY FRAME The Hamiltonian (1) describes a translationally invariant system. It is convenient to perform the LLP transformation61 that

separates the system into decoupled sectors of conserved total momentum, The transformed Hamiltonian (18) does no longer contain the impurity position operator _R_. Thus _P_ in equation (18)

is a conserved net momentum of the system and can be treated as a -number (rather than a hermitian operator). Alternatively, the transformation (17) is commonly described as going into the

impurity frame, since the term describing boson scattering on the impurity in (18) is obtained from the corresponding term in (1) by setting _R_ = 0. The Hamiltonian (18) has only phonon

degrees of freedom but they now interact with each other. This can be understood physically as a phonon-phonon interaction, mediated by an exchange of momentum with the impurity atom. This

impurity-induced interaction between phonons in Eq. (1) is proportional to 1/_M_. Thus in our analysis of the polaron properties, which is based on the LLP transformed Fröhlich Hamiltonian,

we will consider 1/_M_ as controlling the interaction strength. REVIEW OF THE MEAN FIELD APPROXIMATION In this section we briefly review the MF approach to the polaron problem, which

provides an accurate description of the system when quantum fluctuations do not play an important role, e.g. for weak coupling α  1 or large impurity mass. We discuss how one should

regularize the MF interaction energy, which is UV divergent for _d_ ≥ 2. To set the stage for subsequent beyond MF analysis of the polaron problem, we derive the Hamiltonian that describes

fluctuations around the MF state. The MF approach to calculating the ground state properties of (18) is to consider a variational wavefunction in which all phonons are taken to be in a

coherent state61. The MF variational wavefunction reads It becomes exact in the limit of an infinitely heavy (i.e. localized) impurity. Energy minimization with respect to the variational

parameters α_K_ gives where is the momentum of the system carried by the phonons. It has to be determined self-consistently from the solution (20), The MF character of the wave function (19)

is apparent from the fact that it is a product of wave functions for individual phonon modes. Hence it contains neither entanglement nor correlations between different modes. The only

interaction between modes is through the selfconsistency equation (21). Properties of the MF solution have been discussed extensively in Refs 48,61,71. Here we reiterate only one important

issue related to the high energy regularization of the MF energy45,48,56. In _d_ ≥ 2 dimensions the expression for the MF energy, is UV divergent as the high momentum cutoff Λ0 is sent to

infinity. In order to regularize this expression we recall that the physical energy of the impurity is a sum of and the polaron binding energy _E_B. If we use the leading order Born

approximation to express _g_IB = 2_πa_IB/_m_red, we observe that the MF polaron energy has contributions starting with the second order in _a_IB. Consistency requires that the

impurity-condensate interaction energy is computed to order . The Lippman-Schwinger equation provides the relation between the microscopic interaction _g_IB, the cutoff Λ0 and the physical

scattering length _a_IB72 To second order in _a_IB one has Now we recognize that in the physically meaningful impurity energy the UV divergence cancels between and . Thus separation of the

impurity energy into the impurity-condensate interaction and the binding energy _E_B is not physically meaningful. A decomposition of the impurity energy in orders of the scattering length

_a_IB, on the other hand, is well defined, where the second-order term is referred to as the polaronic energy of the impurity56. The main shortcoming of the MF ansatz (19) is that it

discards correlations between phonons with different energies and at different momenta. In this paper we develop a method that allows us to go beyond the MF solution (19) and include

correlations between modes. In the following we demonstrate how this can be accomplished and discuss physical consequences of phonon correlations. To simplify the subsequent discussion we

perform a unitary transformation that shifts the phonon variables in Eq. (18) by the amount corresponding to the MF solution, see Eq. (4) in the main text. Note that the absence of terms

linear in in the last equation reflects the fact that correspond to the MF (saddle point) solution. We emphasize that (26) is an exact representation of the original Fröhlich Hamiltonian,

where the operators describe quantum fluctuations around the MF polaron. DERIVATION OF RG FLOW EQUATIONS We now derive the RG flow equations of the general Hamiltonian (5), which should be

supplemented by the initial conditions in Eq. (8). To this end we separate phonons into “fast” ones with momenta _P_ and “slow” ones with momenta _K_, according to Λ − δΛ < |_P_| < Λ

and |_K_| ≤ Λ − δΛ. Then the Hamiltonian (5) can be split into where we use the short-hand notations and . The slow-phonon Hamiltonian is given by Eq. (5) except that all integrals only go

over slow phonons, . In we do not have a contribution due to the interaction term since it would be proportional to δΛ2 and we will consider the limit δΛ → 0. We can obtain intuition into

the nature of the transformation needed to decouple fast from slow phonons, by observing that for the fast phonons the Hamiltonian (27) is similar to a harmonic oscillator in the presence of

an external force (recall that contains only linear and quadratic terms in ). This external force is determined by the state of slow phonons. Thus it is natural to look for the

transformation as a shift operator for the fast phonons, with coefficients depending on the slow phonons only, i.e. . One can check that taking eliminates non-diagonal terms in up to second

order in 1/Ω_P_. After the transformation we find which is valid up to corrections of order or δΛ2. The last equation describes a change of the zero-point energy _δE_0 of the impurity in the

potential created by the phonons and it is caused by the RG flow of the impurity mass. To obtain this term we have to carefully treat the normal-ordered term in Eq. (5). [The following

relation is helpful to perform normal-ordering, .] We will show later that this contribution to the polaron binding energy is crucial because it leads to a UV divergence in _d_ ≥ 3

dimensions. From the last term in Eq. (30) we observe that the ground state |gs〉 of the Hamiltonian is obtained by setting the occupation number of high energy phonons to zero, . Then from

Eq. (32) we read off the change in the Hamiltonian for the low energy phonons. From the form of the operator in Eq. (31) one easily shows that the new Hamiltonian is of the universal form ,

but with renormalized couplings. This gives rise to the RG flow equations for the parameters in presented in Eqs (9, 10, 11). CALCULATION OF THE POLARON MASS In this section we provide a few

specifics on how we calculate the polaron mass. We relate the average impurity velocity to the polaron mass _M_p and obtain where _M_ is the bare impurity mass. The argument goes as

follows. The average polaron velocity is given by _v_p = _P_/_M_p. The average impurity velocity _v_I, which by definition coincides with the average polaron velocity _v_I = _v_p, can be

related to the average impurity momentum _P_I by _v_I = _P_I/_M_. Because the total momentum is conserved, _P_ = _P_ph + _P_I, we thus have _P_/_M_p = _v_p = _v_I = (_P_ − _P_ph)/_M_.

Because the total phonon momentum _P_ph in the polaron groundstate is obtained from the RG by solving the RG flow equation in the limit Λ → 0, we have _P_ph = _P_ph(0) as defined above and

Eq. (34) follows. We note that in the MF case this result is exact and can be proven rigorously, see48. This is also true for the variational approach based on CGWs59. ADDITIONAL INFORMATION

HOW TO CITE THIS ARTICLE: Grusdt, F. _et al._ Renormalization group approach to the FrÖhlich polaron model: application to impurity-BEC problem. _Sci. Rep._ 5, 12124; doi: 10.1038/srep12124

(2015). REFERENCES * Anderson, P. W. “Infrared catastrophe in fermi gases with local scattering potentials,” Phys. Rev. Lett. 18, 1049 (1967). Article  CAS  ADS  Google Scholar  * Kondo, J.

“Resistance minimum in dilute magnetic alloys,” Progr. Theoret. Phys. 32, 37–49 (1964). Article  CAS  ADS  Google Scholar  * Leggett, A. J. et al. “Dynamics of the dissipative two-state

system,” Rev. Mod. Phys. 59, 1–85 (1987). Article  CAS  ADS  Google Scholar  * Anderson, P. W. “Localized magnetic states in metals,” Phys. Rev. 124, 41 (1961). Article  CAS  ADS  MathSciNet

  Google Scholar  * Wilson, K. G. “The renormalization group: Critical phenomena and the kondo problem,” Rev. Mod. Phys. 47, 773–840 (1975). Article  ADS  MathSciNet  Google Scholar  *

Andrei, N. “Diagonalization of the kondo hamiltonian,” Phys. Rev. Lett. 45, 379–382 (1980). Article  CAS  ADS  Google Scholar  * Wiegmann, P. B. & Tsvelik, A. M. “Solution of the kondo

problem for an orbital singlet,” JETP Lett. 38, 591–596 (1983). ADS  Google Scholar  * Read, N. & Newns, D. M. “A new functional integral formalism for the degenerate anderson model,” J.

Phys. C 16, 1055–1060 (1983). Article  ADS  Google Scholar  * Prokof,ev, N. V. & Svistunov, B. V. “Polaron problem by diagrammatic quantum monte carlo,” Phys. Rev. Lett. 81, 2514–2517

(1998). Article  CAS  ADS  Google Scholar  * Gull, E. et al. “Continuous-time monte carlo methods for quantum impurity models,” Rev. Mod. Phys. 83, 349–404 (2011). Article  CAS  ADS  Google

Scholar  * Anders, P., Gull, E., Pollet, L., Troyer, M. & Werner, P. “Dynamical mean-field theory for bosons,” New J. Phys. 13, 075013 (2011). Article  ADS  Google Scholar  * Rosch, A.

“Quantum-coherent transport of a heavy particle in a fermionic bath,” Adv. Phys. 48, 295–394 (1999). Article  CAS  ADS  Google Scholar  * Knap, M. et al. “Time-dependent impurity in

ultracold fermions: Orthogonality catastrophe and beyond,” Phys. Rev. X 2, 041020 (2012). Google Scholar  * Schirotzek, A., Wu, C., Sommer, A. & Zwierlein, M. W. “Observation of fermi

polarons in a tunable fermi liquid of ultracold atoms,” Phys. Rev. Lett. 102, 230402 (2009). Article  ADS  Google Scholar  * Nascimbene, S. et al. “Collective oscillations of an imbalanced

fermi gas: Axial compression modes and polaron effective mass,” Phys. Rev. Lett. 103, 170402 (2009). Article  CAS  ADS  Google Scholar  * Koschorreck, M. et al. “Attractive and repulsive

fermi polarons in two dimensions,” Nature 485, 619 (2012). Article  CAS  ADS  Google Scholar  * Kohstall, C. et al. “Metastability and coherence of repulsive polarons in a strongly

interacting fermi mixture,” Nature 485, 615 (2012). Article  CAS  ADS  Google Scholar  * Zhang, Y., Ong, W., Arakelyan, I. & Thomas, J. E. “Polaron-to-polaron transitions in the

radio-frequency spectrum of a quasi-two-dimensional fermi gas,” Phys. Rev. Lett. 108, 235302 (2012). Article  CAS  ADS  Google Scholar  * Massignan, P., Zaccanti, M. & Bruun, G. M.

“Polarons, dressed molecules and itinerant ferromagnetism in ultracold fermi gases,” Rep. Prog. Phys. 77, 034401 (2014). Article  ADS  Google Scholar  * Catani, J. et al. “Quantum dynamics

of impurities in a one-dimensional bose gas,” Phys. Rev. A 85, 023623 (2012). Article  ADS  Google Scholar  * Fukuhara, T. et al. “Quantum dynamics of a mobile spin impurity,” Nature Phys.

9, 235–241 (2013). Article  CAS  ADS  Google Scholar  * Palzer, S., Zipkes, C., Sias, C. & KÖhl, M. “Quantum transport through a tonks-girardeau gas,” Phys. Rev. Lett. 103, 150601

(2009). Article  ADS  Google Scholar  * Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010). Article  CAS  ADS

  Google Scholar  * Egorov, M. et al. “Measurement of s-wave scattering lengths in a two-component bose-einstein condensate,” Phys. Rev. A 87, 053614 (2013). Article  ADS  Google Scholar  *

Spethmann, N. et al. “Dynamics of single neutral impurity atoms immersed in an ultracold gas,” Phys. Rev. Lett. 109, 235301 (2012). Article  ADS  Google Scholar  * Pilch, K. et al.

“Observation of interspecies feshbach resonances in an ultracold rb-cs mixture,” Phys. Rev. A 79, 042718 (2009). Article  ADS  Google Scholar  * Lercher, A. D. et al. “Production of a

dual-species bose-einstein condensate of rb and cs atoms,” Eur. Phys. J. D 65, 3–9 (2011). Article  CAS  ADS  Google Scholar  * McCarron, D. J., Cho, H. W., Jenkin, D. L., KÖppinger, M. P.

& Cornish, S. L. “Dual-species bose-einstein condensate of 87 rb and 133 cs,” Phys. Rev. A 84, 011603 (2011). Article  ADS  Google Scholar  * Catani, J., De Sarlo, L., Barontini, G.,

Minardi, F. & Inguscio, M. “Degenerate bose-bose mixture in a three-dimensional optical lattice,” Phys. Rev. A 77, 011603 (2008). Article  ADS  Google Scholar  * Wu, C., Park, J. W.,

Ahmadi, P., Will, S. & Zwierlein, M. W. “Ultracold fermionic feshbach molecules of 23na 40k,” Phys. Rev. Lett. 109, 085301 (2012). Article  ADS  Google Scholar  * Park, J. W. et al.

“Quantum degenerate bose-fermi mixture of chemically di_erent atomic species with widely tunable interactions,” Phys. Rev. A 85, 051602 (2012). Article  ADS  Google Scholar  * Schreck, F. et

al. “Quasipure bose-einstein condensate immersed in a fermi sea,” Phys. Rev. Lett. 87, 080403 (2001). Article  CAS  ADS  Google Scholar  * Truscott, A. G., Strecker, K. E., McAlexander, W.

I., Partridge, G. B. & Hulet, R. G. “Observation of fermi pressure in a gas of trapped atoms,” Science 291, 2570–2572 (2001). Article  CAS  ADS  Google Scholar  * Shin, Y., Schirotzek,

A., Schunck, C. H. & Ketterle, W. “Realization of a strongly interacting bose-fermi mixture from a two-component fermi gas,” Phys. Rev. Lett. 101, 070404 (2008). Article  ADS  Google

Scholar  * Bartenstein, M. et al. “Precise determination of li-6 cold collision parameters by radio-frequency spectroscopy on weakly bound molecules,” Phys. Rev. Lett. 94, 103201 (2005).

Article  CAS  ADS  Google Scholar  * Roati, G., Riboli, F., Modugno, G. & Inguscio, M. “Fermi-bose quantum degenerate k-40-rb-87 mixture with attractive interaction,” Phys. Rev. Lett.

89, 150403 (2002). Article  CAS  ADS  Google Scholar  * Ferlaino, F. et al. “Feshbach spectroscopy of a k-rb atomic mixture,” Phys. Rev. A 73, 040702 (2006). Article  ADS  Google Scholar  *

F. Ferlaino, F. et al. “Feshbach spectroscopy of a k-rb atomic mixture - Erratum,” Phys. Rev. A 74, 039903 (2006). Article  ADS  Google Scholar  * Inouye, S. et al. “Observation of

heteronuclear feshbach resonances in a mixture of bosons and fermions,” Phys. Rev. Lett. 93, 183201 (2004). Article  CAS  ADS  Google Scholar  * Scelle, R., Rentrop, T., Trautmann, A.,

Schuster, T. & Oberthaler, M. K. “Motional coherence of fermions immersed in a bose gas,” Phys. Rev. Lett. 111, 070401 (2013). Article  CAS  ADS  Google Scholar  * Hadzibabic, Z. et al.

“Two-species mixture of quantum degenerate bose and fermi gases,” Phys. Rev. Lett. 88, 160401 (2002). Article  CAS  ADS  Google Scholar  * Stan, C. A., Zwierlein, M. W., Schunck, C. H.,

Raupach, S. M. F. & Ketterle, W. “Observation of feshbach resonances between two different atomic species,” Phys. Rev. Lett. 93, 143001 (2004). Article  CAS  ADS  Google Scholar  * T.

Schuster, T. et al. “Feshbach spectroscopy and scattering properties of ultracold li+na mixtures,” Phys. Rev. A 85, 042721 (2012). Article  ADS  Google Scholar  * Schmid, S., Härter, A.

& Denschlag, J. H. “Dynamics of a cold trapped ion in a bose-einstein condensate,” Phys. Rev. Lett. 105, 133202 (2010). Article  ADS  Google Scholar  * Rath, S. P. & Schmidt, R.

“Field-theoretical study of the bose polaron,” Phys. Rev. A 88, 053632 (2013). Article  ADS  Google Scholar  * Li, W. & Das Sarma, S. “Variational study of polarons in bose-einstein

condensates,” Phys. Rev. A 90, 013618 (2014). Article  ADS  Google Scholar  * Bei-Bing, H. & Shao-Long, W. “Polaron in bose-einstein-condensation system,” Chin. Phys. Lett. 26, 080302

(2009). Article  Google Scholar  * Shashi, A., Grusdt, F., Abanin, D. A. & Demler, E. “Radio frequency spectroscopy of polarons in ultracold bose gases,” Phys. Rev. A 89, 053617 (2014).

Article  ADS  Google Scholar  * Kain, B. & Ling, H. Y. “Polarons in a dipolar condensate,” Phys. Rev. A 89, 023612 (2014). Article  ADS  Google Scholar  * Cucchietti, F. M. &

Timmermans, E. “Strong-coupling polarons in dilute gas bose-einstein condensates,” Phys. Rev. Lett. 96, 210401 (2006). Article  CAS  ADS  Google Scholar  * Sacha, K. & Timmermans, E.

“Self-localized impurities embedded in a one-dimensional bose-einstein condensate and their quantum uctuations,” Phys. Rev. A 73, 063604 (2006). Article  ADS  Google Scholar  * Bruderer, M.,

Klein, A., Clark, S. R. & Jaksch, D. “Polaron physics in optical lattices,” Phys. Rev. A 76, 011605 (2007). Article  ADS  Google Scholar  * Bruderer, M., Klein, A., Clark, S. R. &

Jaksch, D. “Transport of strong-coupling polarons in optical lattices,” New J. Phys. 10, 033015 (2008). Article  ADS  Google Scholar  * Casteels, W., Van Cauteren, T., Tempere, J. &

Devreese, J. T., “Strong coupling treatment of the polaronic system consisting of an impurity in a condensate,” Laser Phys. 21, 1480–1485 (2011). Article  CAS  ADS  Google Scholar  *

Feynman, R. P. “Slow electrons in a polar crystal,” Phys. Rev. 97, 660–665 (1955). Article  CAS  ADS  Google Scholar  * Tempere, J. et al. “Feynman path-integral treatment of the

bec-impurity polaron,” Phys. Rev. B 80, 184504 (2009). Article  ADS  Google Scholar  * Casteels, W., Tempere, J. & Devreese, J. T. “Polaronic properties of an impurity in a bose-einstein

condensate in reduced dimensions,” Phys. Rev. A, 86, 043614 (2012). Article  ADS  Google Scholar  * Vlietinck, J. et al. “Diagrammatic monte carlo study of the acoustic and the bec

polaron,” New J. Phys. 17, 033023 (2014). Article  Google Scholar  * Shchadilova, Y. E., Grusdt, F., Rubtsov, A. N. & Demler, E. “Polaronic mass renormalization of impurities in bec:

correlated gaussian wavefunction approach,” arXiv:1410.5691v1 (2014). * FrÖhlich, H. “Electrons in lattice fields,” Adv. Phys. 3, 325 (1954). Article  ADS  Google Scholar  * Lee, T. D., Low,

F. E. & Pines, D. “The motion of slow electrons in a polar crystal,” Phys. Rev. 90, 297–302 (1953). Article  ADS  MathSciNet  Google Scholar  * Tsai, S.-W., Castro Neto, A. H., Shankar,

R. & Campbell, D. K. “Renormalization-group approach to strong-coupled superconductors,” Phys. Rev. B 72, 054531 (2005). Article  ADS  Google Scholar  * Klironomos, F. D. & Tsai,

S.-W. “Phonon-mediated tuning of instabilities in the hubbard model at half-filling,” Phys. Rev. B 74, 205109 (2006). Article  ADS  Google Scholar  * Grusdt, F. & Demler, E. A., “New

theoretical approaches to bose polarons,” Proceedings of the International School of Physics Enrico Fermi (In preparation). * Gerlach, B. & LÖwen, H. “Proof of the nonexistence of

(formal) phase-transitions in polaron systems,” Phys. Rev. B 35, 4297–4303 (1987). Article  CAS  ADS  Google Scholar  * Gerlach, B. & LÖwen, H. “Analytical properties of polaron systems

or - do polaronic phase-transitions exist or not,” Rev. Mod. Phys. 63, 63–90 (1991). Article  ADS  MathSciNet  Google Scholar  * Fantoni, R. “Localization of acoustic polarons at low

temperatures: A path-integral monte carlo approach,” Phys. Rev. B 86, 144304 (2012). Article  ADS  Google Scholar  * Giamarchi, T. & Le Doussal, P. “Variational theory of elastic

manifolds with correlated disorder and localization of interacting quantum particles,” Phys. Rev. B 53, 15206–15225 (1996). Article  CAS  ADS  Google Scholar  * Santamore, D. H. &

Timmermans, E. “Multi-impurity polarons in a dilute bose-einstein condensate,” New J. Phys. 13, 103029 (2011). Article  ADS  Google Scholar  * Blinova, A. A., Boshier, M. G. &

Timmermans, E. “Two polaron avors of the bose-einstein condensate impurity,” Phys. Rev. A 88, 053610 (2013). Article  ADS  Google Scholar  * Devreese, J. T. “Lectures on frÖhlich polarons

from 3d to 0d - including detailed theoretical derivations,” arXiv:1012.4576v4 (2013). * Pethick, C. J. & Smith, H. Bose-Einstein Condensation in Dilute Gases, 2nd Edition (Cambridge

University Press, 2008). Download references ACKNOWLEDGEMENTS We acknowledge useful discussions with I. Bloch, S. Das Sarma, M. Fleischhauer, T. Giamarchi, S. Gopalakrishnan, W. Hofstetter,

M. Oberthaler, D. Pekker, A. Polkovnikov, L. Pollet, N. Prokof’ev, R. Schmidt, V. Stojanovic, L. Tarruell, N. Trivedi, A. Widera and M. Zwierlein. We are indebted to Aditya Shashi and Dmitry

Abanin for invaluable input in the initial phase of the project. We are grateful to Wim Casteels for providing his calculations for the effective polaron mass using Feynman’s path-integral

formalism. F.G. is a recipient of a fellowship through the Excellence Initiative (DFG/GSC 266) and is grateful for financial support from the “Marion Köser Stiftung”. Y.E.S. and A.N.R. thank

the Dynasty foundation for financial support. The authors acknowledge support from the NSF grant DMR-1308435, Harvard-MIT CUA, AFOSR New Quantum Phases of Matter MURI, the ARO-MURI on

Atomtronics, ARO MURI Quism program. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany F. Grusdt *

Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663, Kaiserslautern, Germany F. Grusdt * Department of Physics, Harvard University, 02138, Cambridge,

Massachusetts, USA F. Grusdt, Y. E. Shchadilova & E. Demler * Russian Quantum Center, 143025, Skolkovo, Russia Y. E. Shchadilova & A. N. Rubtsov * Department of Physics, Moscow State

University, 119991, Moscow, Russia A. N. Rubtsov Authors * F. Grusdt View author publications You can also search for this author inPubMed Google Scholar * Y. E. Shchadilova View author

publications You can also search for this author inPubMed Google Scholar * A. N. Rubtsov View author publications You can also search for this author inPubMed Google Scholar * E. Demler View

author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS All authors contributed substantially to the writing of the manuscript. F.G., Y.S., A.R. and

E.D. contributed to the theoretical analysis of the data. F.G. and Y.S. performed the numerical calculations. F.G. and E.D. conceived the RG method. ETHICS DECLARATIONS COMPETING INTERESTS

The authors declare no competing financial interests. RIGHTS AND PERMISSIONS This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third

party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative

Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Grusdt, F., Shchadilova, Y., Rubtsov, A. _et al._ Renormalization group approach to the Fröhlich polaron model: application to

impurity-BEC problem. _Sci Rep_ 5, 12124 (2015). https://doi.org/10.1038/srep12124 Download citation * Received: 03 March 2015 * Accepted: 08 June 2015 * Published: 17 July 2015 * DOI:

https://doi.org/10.1038/srep12124 SHARE THIS ARTICLE Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently

available for this article. Copy to clipboard Provided by the Springer Nature SharedIt content-sharing initiative