Suppression of polaron self-localization by correlations

We investigate self-localization of a polaron in a homogeneous Bose-Einstein condensate in one dimension. This effect, where an impurity is trapped by the deformation that it causes in the surrounding Bose gas, has been first predicted by mean field calculations, but has not been seen in experiments. We study the system in one dimension, where, according to the mean field approximation, the self-localization effect is particularly robust, and present for arbitrarily weak impurity-boson interactions. We address the question whether self-localization is a real effect by developing a variational method which incorporates impurity-boson correlations non-perturbatively and solving the resulting inhomogeneous correlated polaron equations. We find that correlations inhibit self-localization except for very strongly repulsive or attractive impurity-boson interactions. Our prediction for the critical interaction strength for self-localization agrees with a sharp drop of the inverse effective mass to almost zero found in quantum Monte Carlo simulations of polarons in one dimension.


I. INTRODUCTION
The original Bose polaron problem concerns an electron in a solid which is dressed by small distortions of the crystal lattice and was modelled by Fröhlich 1 .Another type of polaron is formed by an electron or impurity atom in superfluid 4 He.This problem has long been studied 2 and later extended to molecular impurities and impurity aggregates in 4 He, which lead to a new type of low-temperature spectroscopy of molecules 3,4 .More recently, polarons of mobile impurities have been experimentally realized in ultracold Bose gases [5][6][7] .
For electrons in ionic solids 8 and also in superfluid 4 He 9 a mechanism for self-localization, or self-trapping, was proposed 10 .Self-localization implies that, even in the absence of an external trap potential, the impurity probability density ρ(r 0 ) is not uniform but trapped by the distortion of the density of phonons or He atoms created by the impurity itself.In Refs.11 and 12 based on the mean field (MF) approach self-localization has also been predicted for polarons in a Bose-Einstein condensate.According to Cuccietti et al. 11 a polaron in a threedimensional homogeneous Bose gas self-localizes above a critical impurity-boson interaction strength, while below it the polaron ground state is homogeneous.This would imply a phase transition to a translation symmetry breaking ground state.Subsequently, other works have also predicted this effect, e.g. for neutral polarons, again using the MF approximation [13][14][15][16][17] , including finite temperature 18 , and also with other methods such as path integrals 19,20 .Also ionic polarons 21 and angular polarons 22 have been predicted to self-localize.However, other works have not seen evidence of self-localization in three dimensions [23][24][25] , nor has it been observed experimentally.This raises the question whether selflocalization is a methodological artifact or a real effect.
In one dimension the MF approximation predicts a self-trapped polaron regardless of the strength of the impurity-boson interaction 14 .Exact quantum Monte Carlo simulations 26 indeed predict an essentially diver-gent polaron effective mass above a certain impurityboson interaction strength, i.e. the polaron becomes immobile, which would be consistent with self-localization for strong interactions.Conversely, Ref. 27 found a finite effective mass for attractive impurity-boson interaction, using the same Monte Carlo method for similar bosonboson interaction strengths but smaller mass ratio.Indirect measurements of Bose polarons in one dimension gave an even lower effective mass 5 .
The goal of this work is to check if the self-localized ground state predicted by the MF approximation is a real effect or an artifact of the uncorrelated Hartree ansatz of MF.To check this, we take a crucial step beyond the Hartree ansatz by incorporating impurity-boson correlations in a non-perturbative way, while treating the weakly interacting Bose background still in the MF approximation, thus omitting boson-boson correlations.We note that the perturbative treatment of correlations (then usually referred to as quantum fluctuations) has been shown to lead to corrections to the density ρ(x 0 ) of a self-localized impurity in one dimension 13 but still preserves self-localization.In this work we show that with a non-perturbative treatment of impurity-boson correlations impurity self-localization happens only for very strongly attractive or repulsive impurity-boson interactions.

II. THEORY AND METHOD
The Hamiltonian of one impurity and N bosons in one dimension is given by consisting of the kinetic energy of the impurity, the kinetic energy of the bosons, the impurity-boson inter-action and the boson-boson interaction.The bosonboson interaction is modelled by a contact potential with strength λ BB , which is related to the scattering length a BB by λ BB = −2ℏ 2 aBBm 28,29 .The impurity-boson interaction is modelled by a finite range potential, for which we choose a Gaussian, , characterized by the strength and width parameters U 0 and σ U .
The MF approach is usually derived in a variational formulation, with the Hartree ansatz wave function for one impurity in a bath of N bosons: ( This wave function does not account for the correlations caused by the interactions, e.g. the decrease of the probability |Ψ(x 0 , . . ., x i , . . .)| 2 if a boson at x i is close to a repulsive impurity at x 0 .The optimization of Ψ MF leads to one-body equations with effective potentials, the "mean fields".The uncorrelated MF ansatz (2) can be expected to be a poor approximation of the true many-body wave function if impurity-boson interactions are strong (but our results show it is a poor approximation for weak interaction as well).Therefore, we generalize the ansatz by replacing the boson one-body functions ψ(x i ) with impurity-boson pair correlation functions f (x 0 , x i ): where it turns out to be convenient to introduce a prefactor including the normalization volume Ω.This is a Jastrow-Feenberg ansatz wave function 30 but limited to impurity-boson correlations.We refer to it as the inhomogeneous correlated polaron (inh-CP) ansatz.
If the ground state is assumed homogeneous, i.e. translationally invariant like the Hamiltonian, the ansatz (3) simplifies to which was studied already by Gross 31 .Of course, we cannot make this assumption of translational invariance if we want to study the possible symmetry breaking by self-localization of the impurity.But the homogeneous correlated polaron (hom-CP) ansatz (4) will still be useful: if self-localization is indeed energetically favorable, the energy difference between the inh-CP and the hom-CP result is the energy gained by forming a self-localized ground state.Our ansatz (3) includes impurity-boson correlations but still treats the (weakly interacting) Bose background in the MF approximation, as it does not include bosonboson correlations.Since we take only one step beyond the MF approach, this allows for a comprehensible comparison between our method and the MF approach.Impurities immersed in a strongly interacting Bose liquid like 4 He, however, require the inclusion of boson-boson correlations.Optimizing such a full Jastrow-Feenberg ansatz leads to the hypernetted-chain Euler-Lagrange method 32,33 .The method and its time-dependent generalization have been used extensively to study impurities in 4 He [34][35][36] .
Before deriving equations for η(x 0 ) and f (x 0 , x i ) from the Ritz' variational principle, we need an expression for the energy functional E = ⟨Ψ|H|Ψ⟩, where we assume normalization of the wave function, ⟨Ψ|Ψ⟩ = 1.The 4 terms in the Hamiltonian (1) lead to the following 4 terms in E: where dx = dx 0 dx 1 ... dx N , and Ψ is the correlated polaron ansatz (3).Owing to the star-shaped correlation structure, where the impurity is correlated with all bosons but the bosons are not correlated between themselves, most of the N +1 integrals in E factorize and yield dx ′ 1 f (x 0 , x ′ 1 ) 2 .We abbreviate this partially integrated correlation function We obtain the energy functional In a study of self-localization, we are primarily interested in the impurity density ρ I (x 0 ).Without an external trapping potential, the impurity density is constant in the absence of self-localization, ρ I (x 0 ) = 1 Ω , while in the presence of self-localization ρ I (x 0 ) peaks at a random location x0 37 and falls to zero away from x0 .Similarly, the density of the Bose gas ρ B (x 1 ) is constant in the first case, ρ B (x 1 ) = N Ω , while it has a valley/peak for repulsive/attractive impurity-boson interaction in the latter case.For the correlated polaron ansatz (3), the impurity density is given by and the boson density is given by where normalization of the wave function was assumed.
According to the Ritz' variational principle the optimal η(x 0 ) and f (x 0 , x 1 ) are obtained from minimizing the energy (7), i.e. setting its functional derivatives with respect to η(x 0 ) and f (x 0 , x 1 ) to zero.To ensure normalization of the wave function we introduce a Lagrange multiplier λ.Hence, we need to optimize the Lagrangian The inh-CP equations for the general inhomogeneous case are the coupled Euler-Lagrange equations, formally written as Their explicit form is derived in appendix A, where we show that in the thermodynamic limit N → ∞ and Ω → ∞ with ρ = N Ω fixed, we obtain a 1-body equation for the square root of the impurity density g(x 0 ) = ρ I (x 0 ) and a two-body equation for f (x 0 , x 1 ) ≡ g(x 0 )f (x 0 , x 1 ): with the impurity and boson chemical potential µ I and µ B and the effective one-body and two-body potentials We have cast the two coupled inh-CP equations into the form of a one-and a two-body nonlinear Schrödinger equation, respectively, with effective potentials ( 15) and ( 16) that depend on g(x 0 ) and f (x 0 , x 1 ) itself.Similarly to other nonlinear Schrödinger equations 38 , Eqns. ( 13) and ( 14) can be solved self-consistently by imaginary time propagation, where we always start the propagation with self-localized trial states, for example the MF ground state.Details are given in appendix B.

III. RESULTS
We present results for the Bose polaron ground state in one dimension for three levels of approximation: a) solving the full inh-CP equations ( 13) and ( 14 In all three types of calculations, we use the same Gaussian interaction model.Following Bruderer et al. 14 , we measure length in units of the healing length ξ = ℏ/ √ λ BB ρm and energy in units of E 0 = λ BB ρ.This leaves us with three dimensionless essential parameters characterizing the Bose polaron system (1): the mass ratio α = m/M , the relative interaction strength β = λ IB /λ BB and a density parameter γ = 1/(ρξ).λ IB is obtained from the scattering length a IB via λ IB = −ℏ/a IB (1/M + 1/m), and the scattering length a IB is obtained from the parameters U 0 and σ U the Gaussian model interaction using the results of Ref. 39.We have confirmed the universality of the interaction model, i.e. that our results depend only on λ IB and not on the parameters U 0 and σ U if σ U is chosen very small.Too small values for σ U would require a very fine discretization and correspondingly high numerical effort.Therefore, we choose σ U = 0.1, where results differ only insignificantly from the universal limit.
We compare results obtained with the inh-CP and the hom-CP equations to ensure numerical consistency, and also to calculate the formation energy (called binding energy in Ref. 11) gained from self-localization if we do find self-localized polarons.But the main goal of this work is to compare the inh-CP results and MF results, i.e. results with and without including correlations, to see whether self-localization still occurs when impurity-boson correlations are included in the variational ansatz.We note that both solving the hom-CP equation and solving the MF equations is numerically straightforward and fast since all quantities depend on a single coordinate, unlike f (x 0 , x 1 ) in the inh-CP ansatz (3).
In this work we restrict ourselves to equal impurity and boson mass, i.e. α = 1.The parameter γ is related to the gas parameter, ρ|a BB | = 2/γ 2 .A small parameter γ signifies weak boson-boson interactions (i.e.large |a BB |) and/or high density, while γ → ∞ is the strongly correlated Tonks-Girardeau limit 40 .We study two cases, γ = 0.2 and γ = 0.5, which both correspond to a weakly interacting Bose gas, where it may be justified to neglect boson-boson correlations as done in the ansatz (3).We vary the relative impurity-boson interaction strength β over a wide range from strongly attractive to strongly repulsive.

A. Density and localization length
In Fig. 1 we show the impurity density ρ I (x 0 ) (top panels) and the boson density ρ B (x 1 ) (bottom panels) for attractive impurity-boson interactions, −10 ≤ β < 0, (left panels) and repulsive interaction 0 < β ≤ 50 (right panels).We show only half of the densities since they are assumed to be symmetric.The darker lines (positive coordinates) are the solutions of the inh-CP equations, while the lighter lines (negative coordinates) are the solutions of the MF equations, calculated also in Ref. 14.All calculations in Fig. 1 are done for γ = 0.5.
The comparison in Fig. 1 demonstrates that incorporating the impurity-boson correlations strongly reduces the tendency towards self-localization.The MF approximation predicts that the polaron self-localizes for all values of β, where ρ I (x 0 ) and ρ B (x 1 ) becomes narrower for larger |β|. 14Conversely, the ground state of the cor- related polaron is qualitatively and quantitatively quite different: for a wide β-range the polaron does not selflocalize at all, thus ρ I (x 0 ) and ρ B (x 1 ) are simply constant.It may come as a surprise that especially for weak interactions the MF approximation gives a wrong result regarding the question of self-localization, which demonstrates that in one dimension correlations should never be neglected.Only for sufficiently strong attraction or repulsion, the correlated polaron self-localizes, but both ρ I (x 0 ) and ρ B (x 1 ) are significantly broader than in the MF approximation.
A localized polaron can be characterized by a localization length σ, e.g. by fitting a Gaussian exp[−x 2 0 /(2σ 2 )]/(σ √ 2π) to the impurity densities ρ I (x 0 ) shown in Fig. 1. σ → ∞ means the polaron delocalizes.In Fig. 2 we show the localization length σ of the correlated polaron (filled squares) and the corresponding σ mf of the MF polaron (open squares) as functions of the relative interaction strength β for γ = 0.2 (top) and γ = 0.5 (bottom).Since in all our calculations, including the MF calculations, we use a Gaussian interaction of finite width σ U = 0.1 instead of a contact potential, our results for σ mf deviate slightly from Ref. 14, at most by 10%.Since the MF approximation predicts unconditional self-localization in 1D, σ mf is finite for all β ̸ = 0.For the correlated polaron, we get a large range of β where the polaron is delocalized, indicated by the grey area.Therefore, σ is not only significantly larger than σ mf , but it diverges at a critical attractive and repulsive relative interaction strength β cr,1 and β cr,2 , respectively, the value of which depends on γ.Since a large σ γ βcr,1 βcr,2 γP ηcr,2 ηcr,2 0.2 -9.6 16.8 0.04 -0.38 0.67 0.5 -6.2 23.3 0.25 -1.55 5.82 TABLE I.The critical relative interaction strengths β cr,1/2 for self-localization, obtained from solving the inh-CP equations.We also tabulate the results expressed in alternative dimensionless units (see text) for better comparison with Ref. 26.   requires a large computational domain, approaching the critical β becomes numerically expensive and we estimate it by fitting to a 1 |β − β cr,1 | c1 for the attractive side and a 2 |1 − β cr,1 /β| c2 for the repulsive side (where σ seems to saturate at a finite value for large β).The estimates are tabulated in Tab.I.
The Bose polaron in one dimension was studied with diffusion Monte Carlo simulations 26,27 .The trial wave functions used in that work are translationally invariant, which may mask a self-localization effect.Nonetheless, a relatively sharp increase of the polaron effective mass to a very large value was observed on both the attractive and repulsive side.Parisi et al. 26 considered equal masses for impurity and bosons, which allows comparison with the present work.They use the parameters γ P = γ 2 and η = βγ 2 to characterize boson density/interactions and impurity-boson interactions, respectively.For better comparison Tab.I provides the critical interaction strength also in terms of γ P and η.The closest values of γ P compared to our values are γ (MC) P = 0.02 and 0.2.Fig. 4 in Ref. 26 shows that for γ (MC) P = 0.02 the inverse effective mass essentially vanishes for η ≈ −1 and for η ≈ 1 for attractive and repulsive interactions, respectively; for γ (MC) P = 0.2 the corresponding values are η ≈ −2 and η ≈ 10, however the statistical fluctuations and the logarithmic scale makes it hard to give precise numbers.Considering this uncertainty and our slightly different values for γ P , our prediction for the critical interaction strength for a self-localized polaron ground state is consistent with that for an essentially infinite effective mass obtained with diffusion Monte Carlo.

B. Chemical potential
Solving the correlated polaron equations ( 13) and ( 14) yields not only g(x 0 ) and f (x 0 , x 1 ) but also the impurity and boson chemical potentials µ I and µ B .For the latter we obtain the trivial result µ B /E 0 = 1, i.e. the MF approximation of the pure Bose gas, which is not altered by a single impurity in the thermodynamic limit.Slight numerical deviations from unity provide a measure of finite size effects.
The impurity chemical potential µ I provides nontrivial information.According to the Ritz' variational principle, better variational wave functions yield lower ener- gies, closer to the exact ground state energy.This is also true for µ I , because it is obtained by subtracting the constant E 0,N from the ground state energy, see appendix A. Hence, the chemical potential of the correlated impurity must be lower than that of the MF impurity, µ I < µ mf I .In Fig. 3 we show µ I and µ mf I as functions of β for γ = 0.5 (top panels) and 0.2 (bottom panels).For all cases, µ mf I is higher than µ I , as it should be.Furthermore, we expect µ I < 0 for β < 0 and vice versa, which is indeed the case for both µ I and µ mf I .For attractive impurityboson interactions, shown in the left panels, µ I shows no sign of saturating to a finite value when β is decreased to stronger attraction, in fact the slope steepens.For repulsive interactions (right panels), µ I does saturate with increasing β.This is consistent with the behavior of the localization length shown in Fig. 2 for negative and positive β.
The comparison between µ I and µ mf I serves mainly as a check that we did not converge to an unphysical local energy minimum.More interesting is the comparison of the chemical potentials obtained from the inhomogeneous and the homogeneous polaron equations, tion energy E b , which is about two orders of magnitude smaller than µ I , and its determination without numerical bias is challenging.We note that the smallness of E b relative to µ I would render its calculation by Monte Carlo simulation a formidable task. If , thus E b = 0, no energy is gained from self-localization, which therefore does not happen.Indeed, in these cases the inh-CP solver converges to a constant polaron density, ρ I = 1/Ω, with the same correlation function f (x 0 , x 1 ) as that of the hom-CP solution, f hom (x 0 −x 1 ).If µ I < µ hom I , thus E b < 0, self-localization lowers the ground state with respect to a homogenous ground state.The critical relative interaction strength β cr,1 and β cr,2 discussed above is just the point where E b becomes 0.
We illustrate the difference between a homogeneous pair correlation f (x 0 − x 1 ) of a delocalized ground state and the inhomogeneous pair correlation f (x 0 , x 1 ) of a self-localized ground state in Fig. 5 for γ = 0.5.The left panel shows f (x 0 , x 1 ) for β = −10 (localized), which has only inversion symmetry.The right panel shows f (x 0 , x 1 ) = f hom (x 0 − x 1 ) for β = −5 (homogeneous), which has translation symmetry with respect to the center of mass (x 0 + x 1 )/2.

IV. CONCLUSIONS
We revisited the self-localization problem of an impurity in a Bose gas, where the mean field (MF) approximation predicted self-localized polaron ground states in 3D, 11 , and later in 2D and 1D 14 ; in particular in 1D, self-localization was predicted to happen for any strength of the impurity-boson interaction, quantified by the parameter β.Extending the MF method using the Bogoliubov method to account for quantum fluctuations has proven useful in many instances (dipolar interactions 41 , self-bound Bose mixtures 42 ), but is still only a perturbative expansion.In our work, we incorporate optimized, inhomogeneous impurity-boson correlations in a non-perturbative way and derive inhomoge- neous correlated polaron (inh-CP) equations, which we solve numerically for the 1D case.The results of this improved variational ansatz for the ground state wave function shows that the MF approach is not sufficient to study polaron physics in 1D.Impurity-boson correlations suppress the tendency towards self-localization significantly, which happens only for strongly attractive or repulsive impurity-boson interactions.Despite being variational, our results are consistent with the sharp increase of the effective mass of the polaron at a similar critical impurity-boson interaction strength predicted by exact diffusion Monte Carlo simulations 26 .
In case of the MF approximation, it is straightforward to see why it might predict a spurious self-localization even for weak interactions: without correlations, i.e. using a Hartree ansatz (2), a localized impurity density and accordingly an inhomogeneous Bose density "mimics" the effect of a correlations as the most optimal solution of the Ritz' variational problem.For example, for repulsive interactions the Bose density is suppressed around the localized impurity, lowering the total energy of a Hartree ansatz.Instead, in a correlated many-body wave function like (3), repulsion causes a correlation hole in the pair distribution function, which does not require self-localization of the polaron.Our method predicts selflocalization only for strong impurity-boson interactions, but this is not a rigorous proof that such a breaking of the translational invariance of the Hamiltonian (1) is a real effect rather than a variational artifact.Further refinements beyond the variational wave function (3), such as boson-boson correlations or three body impurityboson-boson correlations, may push the transition to selflocalization to even stronger interactions.However, the above-mentioned consistency with exact Monte Carlo results lends credibility to the correlated polaron ansatz (3) in the regime of weak boson-boson interactions that we studied in this work.
Experimental observation of a possibly self-localized polaron is challenging.The smallness of the formation energy E b would require a low temperature, depending on the magnitude of |β|, where strongly attractive interactions, β < 0, are clearly favorable according to our results.In higher dimensions, there is no evidence of a sharp increase of the effective mass of a polaron three dimensions, according to quantum Monte Carlo simulations 24 , but the MF approach 11 does predict self-localization.Correlations tend to be less important in higher dimensions, and the MF approach usually becomes a better approximation.It will be interesting to see if there is a parameter regime where the correlated polaron ansatz (3) is self-localized in more than one dimension.Furthermore, the inh-CP method can be generalized to time-dependent problems, similarly to the time-dependent hypernetted-chain Euler-Lagrange method 43 .This allows to calculate the effective mass but also to study nonequilibrium dynamics of polarons after a quench 44 , such as an interaction quench of β.
Our results pertain only to neutral atomic impurities.For dipolar and especially ionic impurities, which interact via long-ranged attractive potentials with the surrounding Bose gas due to induced dipoles, the situation may be different.Ions in BECs can dress themselves with a substantial cloud of bosons 45 , making ionic polarons a more likely candidate for self-localization.
Note that, when we multiply this equation by η(x 0 ) f (x 0 ) N and integrate over x 0 , we obtain λ = E, i.e. the Lagrange multiplier is indeed the energy.
Using ( 7) and ( 10), the second Euler-Lagrange equation ( 11) becomes, after dividing by 2N We can simplify this lengthy equation by dividing by η(x 0 ) and subtracting eq.(A1) multiplied by f (x 0 , x 1 ), Both sides of this equation scale linearly with N .Therefore, before taking the thermodynamic limit, we subtract the MF energy of N bosons without impurity E 0,N = ρ 2 2 λ BB Ω multiplied by η(x 0 ).With E ≡ E 1,N we can then identify the impurity chemical potential µ I = E 1,N −E 0,N on the right-hand side of the resulting equation.Furthermore, we introduce the square root g(x 0 ) = ρ I (x 0 ) of the impurity density defined in eq.( 8), which in the thermodynamic limit becomes, see eq.(A6), This permits to write the one-body inh-CP equation in the final form given in eq. ( 13).
Note that we still have to divide f (x 0 , x 1 ) by g(x 0 ) for the calculation of V g (x 0 ).This is the price for formulating the two-body equation as nonlinear Schrödinger equation for f (x 0 , x 1 ).This division by g(x 0 ) can be problematic for localized solutions g(x 0 ) if we choose the computation domain too large.
Eqns. ( 13) and (B1) are coupled non-linear one-and two-body Schrödinger equations with effective Hamiltonians H g = T I + V g and H f = T I + T B + Ṽf , containing the potentials ( 15) and (B2), respectively.We obtain the ground state by the imaginary time propagation.We initialize g and f at imaginary time τ = 0 with localized states, e.g. a MF solution, and then use small time steps ∆τ together with the Trotter approximation 46 to calculate an approximation of the ground state by performing a large number M of propagation steps until convergence is reached: g(M ∆τ ) = e −Vg/2 ∆τ e −TI ∆τ e −Vg/2 ∆τ M g(0) (B3) f (M ∆τ ) = e − Ṽf /2 ∆τ e −(TI+TB) ∆τ e − Ṽf /2 ∆τ M f (0).

(B4)
In between time steps we have to normalize g(x 0 ), which is the square root of the impurity density, dx 0 g(x 0 ) 2 = 1.
(B5) Furthermore, in the thermodynamic limit the impurity and bosons should be uncorrelated for large separation, i.e. f (x 0 , x 1 ) → 1 for |x 0 − x 1 | → ∞.In order to ensure this property, we specifically require In summary, we perform the following calculations for each time step ∆τ of the imaginary time propagation: 1. calculate V g (15) and Ṽf (B2) ), derived in this work and based on the ansatz (3); b) solving the special case of the hom-CP equation, derived in Ref.31, based on the ansatz (4), that precludes self-localization; c) solving the MF equations, based on the ansatz (2), which according to Ref. 14 always result in selflocalization in one dimension.

FIG. 1 .
FIG. 1.The impurity density ρI(x0) (top panels) and boson density ρB(x1) (bottom panels) are shown as functions of β.The correlated polaron results are depicted on the positive side of the coordinate axis x0 or x1, while the mean field results are depicted on the negative side.The left and right panels show results for attractive and repulsive impurity-boson interactions, respectively.A constant ρI(x0) and ρB(x1) means there is no self-localization for the corresponding value of β.All results are for γ = 0.5.

FIG. 2 .
FIG. 2. The localization length σ of a polaron is plotted as a function of β for γ = 0.2 (top panel) and 0.5 (bottom panel).The filled and open symbols are the correlated and mean field results, respectively, the latter agreeing with Ref. 14.The shaded area indicates the range of β where no self-localization occurs according to our correlated results.

FIG. 3 .
FIG. 3. The impurity chemical potential µI (filled squares) from the solution of the inhomogeneous correlated polaron equations is plotted as a function of β for γ = 0.2 (top panels) and 0.5 (bottom panels), together with the mean field prediction µ mf I (open squares) and the homogeneous correlated polaron prediction µ hom I (stars).Left and right panels show attractive and repulsive impurity-boson interactions, respectively.

2 FIG. 4 .
FIG. 4. The formation energy E b = µI − µ hom I is plotted as a function of β, split into attractive and repulsive interaction (left and right panel).Self-localization happens only if E b < 0.

2 FIG. 5 .
FIG. 5. Optimal pair correlation f (x0, x1) obtained from solving the inhomogeneous correlated polaron equations.For β = −10 (left panel) the polaron ground state is self-localized and for β = −5 (right panel) the ground state is homogeneous.The values at the upper left and lower right corners of the computational domain are a result of the periodic boundary conditions.