Impurities in a one-dimensional Bose gas: the flow equation approach
SSciPost Physics Submission
Impurities in a one-dimensional Bose gas:the flow equation approach
F. Brauneis
1, 2 , H.-W. Hammer
1, 3 , M. Lemeshko , A. G. Volosniev , Technische Universit¨at Darmstadt, Department of Physics, Institut f¨ur Kernphysik, 64289Darmstadt, Germany Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur SchwerionenforschungGmbH, 64291 Darmstadt, Germany*[email protected] 3, 2021
Abstract
A few years ago, flow equations were introduced as a technique for calculating theenergies of cold Bose gases with and without impurities. In this paper, we extendthis approach to compute observables other than the energy. As an example, wecalculate the energies, densities, and phase fluctuations of one-dimensional Bosegases with one and two impurities.For a single mobile impurity, we show that the flow equation results agree wellwith the mean-field results obtained upon the Lee-Low-Pines transformation ifthe phase coherence length is larger than the healing length of the condensate.This agreement occurs for all values of the boson-impurity interaction strengthas long as the boson-boson interaction is weak.For two static impurities, we calculate repulsive and attractive impurity-impuri-ty correlations mediated by the Bose gas. We find that leading order perturba-tion theory fails when the boson-impurity interaction strength is larger than theboson-boson interaction strength. We note that the mean-field approximationreproduces the flow equation results for weak boson-boson interactions.
Contents a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b ciPost Physics Submission c > c < A.1 Normal ordering 25A.2 Flow equation 26A.3 One-body basis 27A.4 Truncation of the one-body basis 28
References 29
The flow equation approach provides an ab-initio method for solving many-body problems [1].A related method, the In-medium similarity renormalization group (IM-SRG), was recentlydeveloped and successfully applied to nuclei (see, e.g., Refs. [2,3]) . The main concept behindboth methods is the same. Therefore, we use ‘IM-SRG’ and ‘flow equations’ interchangeablyin this paper. IM-SRG was recently extended to cold Bose gases [7, 8], where it was testedby calculating energies of the one-dimensional Bose gas, and the self-energy of an impurityembedded in this gas (the ‘Bose-polaron problem’). Our present work explores the possibilityto employ flow equations to calculate further observables for Bose gases, in particular, thedensity and phase fluctuations. Motivated by recent cold-atom experiments [9,10], we focus onthe one-dimensional Bose-polaron problem. This problem is of current theoretical interest, seeRefs. [8,11–25], which provide us with data for benchmarking and interpreting some of our IM-SRG results. In particular, it is interesting to compare to the (Wilson-type) renormalizationgroup technique developed to study large systems with impurities in momentum space [14,26, 27]. Note that the word ‘in-medium’ in the name of the method is used to separate the IM-SRG from thestandard SRG approaches, which are used in nuclear physics to ‘soften’ nuclear forces before using them in abinitio methods (see, e.g., Refs. [4–6]), such as a no-core shell model. ciPost Physics Submission It has been noticed that the mean-field approximation (MFA) in a frame co-moving with theimpurity can accurately describe the self-energy of the impurity in a Bose gas [8, 17, 19, 22].This observation is somewhat counter-intuitive, since strong phase fluctuations in one spatialdimension require a beyond-mean-field approach. A counterargument to this point can bebased on the observation that (when solving a Bose-polaron problem) one is interested only inlocal properties of a Bose gas, and therefore, the absence of long-range order is not necessarilyrelevant. The mean-field approach can be useful (cf. [28]) as long as the phase coherencelength, ξe √ π /γ , is larger than the length scale associated with the polaron, which is of theorder of ξ [8]. Here ξ is the healing length of the condensate, and γ is the dimensionless Lieb-Liniger parameter which characterizes the boson-boson interaction strength, see Sec. 3. Thisargument implies that as long as e √ π /γ (cid:29) The IM-SRG is an extension of the SRG [1, 30, 31] based upon the flow equation d H ds = [ η , H ] , (1)3 ciPost Physics Submission which transforms the Hamiltonian matrix into a block-diagonal form. The flow equation isdefined once the initial condition, H ( s = 0), and the generator of the transformation, η , arespecified. It is worthwhile noting that Eq. (1) is equivalent to the unitary transformation H ( s ) = U ( s ) H ( s = 0) U † ( s ), assuming that the antihermitian operator η and the unitaryoperator U are connected as η ( s ) = d U ( s )d s U † ( s ) . (2)We prefer to write the unitary transformation in the form of Eq. (1) because it allows us tochoose the operator η ( s ) during the flow, i.e., for every parameter s , and, hence, to steer theflow in the desired direction.The IM-SRG was previously applied to study Bose gases with a large condensate fraction [7].In this work, we use and extend that formalism to calculate the density and phase fluctua-tions of the Bose gas. In addition, we discuss the role of a reference state in more detail incomparison to previous works. For convenience of the reader, we shall present in this chapterthe main ingredients of the IM-SRG method for bosons, see also Ref. [7], Appendix A, andFig. 1.In general, it is impossible to solve Eq. (1) for a many-particle system, unless certain ap-proximations are implemented. The complexity is due to the commutator [ η , H ]: It leadsto many-body terms, which are not present in the initial Hamiltonian H ( s = 0). To solveEq. (1), these forces must be truncated at some order. To define a truncation hierarchy,we write the Hamiltonian H in second quantization using normal ordering with respect toa reference state. As a reference state, we use a product state, which can approximate theground state of a mesoscopic ensemble of bosons well, at least for weak interactions. In ourwork, we truncate three-body excitations and beyond, see Fig. 1. To estimate the introducedtruncation error, we use the three-body elements and second order perturbation theory formatrices, see Fig. 1. The result of the IM-SRG in our implementation is an approximatemapping of the reference state to the ground state of the Hamiltonian H .In practice, the following steps constitute the IM-SRG method: (i) find a one-body basis towrite a Hamiltonian matrix in second quantization, (ii) find a reference state to normal orderthe operators, (iii) solve the flow equation (1). We discuss (i) in Appendix A.3. The item(ii) is discussed in the next subsection. We use Python’s standard routines [in particular, theexplicit Runge-Kutta method 5(4)] to perform (iii). One of the key ingredients of the IM-SRG is the reference state, Ψ ref , which is used to normalorder operators, see Appendix A.1. The reference state should well approximate an eigenstate(here the ground state) of the Hamiltonian, otherwise the IM-SRG transformation cannot mapΨ ref onto the exact state. In previous works on flow equations for bosons [7, 8], the groundstate of the non-interacting Hamiltonian was used as a reference state. This is the simplestchoice; one of the aims of the present paper is to explore other options.This work focuses on ground-state properties of a bosonic system. Therefore, it is logical to4 ciPost Physics Submission
Figure 1: Illustration of the action of the flow equation (1) on the Hamiltonian. The Hamil-tonian matrix is unitarily transformed to a block-diagonal form, such that the ground state(green) becomes decoupled. The upper row illustrates the exact transformation without atruncation. Note that many-body excitations appear during the flow. The bottom row il-lustrates a truncation scheme adopted to circumvent this problem. The induced three-bodyterms (blue) can be estimated and used to evaluate the accuracy of the truncation scheme.The excitations are defined with respect to the adopted normal ordering, see Appendix A.5 ciPost Physics Submission use a product state as a reference state, i.e.,Ψ ref ( x , ..., x N ) = N (cid:89) α =1 f ( x α ) , (3)where x α is the coordinate of the α th boson, and f is some function, which we discussbelow. The choice of a product-state ansatz is natural for cold Bose gases with macroscopicpopulation of a single mode, i.e., with a large condensate fraction. Note that the referencestate of Eq. (3) is in general not an accurate approximation to one-dimensional systems inthe thermodynamic limit, where the correlations decay algebraically [32–36]. However, as ouranalysis below shows, the product state is a useful starting point for analyzing mesoscopicsystems with impurities.This work mainly focuses on two (single-body) reference states f and f . The function f is the ground-state wave function of the one-body Hamiltonian as in Refs. [7, 8]. The secondfunction is obtained within a mean-field approximation, i.e., f is the solution of the Gross-Pitaevskii equation. To distinguish the IM-SRG method with f from IM-SRG with f , weintroduce the notation IM-SRG( f ) and IM-SRG( f ), respectively . It is worthwhile notingthat one can rely on an iterative procedure to find a good reference state, starting from anyreasonable initial guess f (0) a . Indeed, IM-SRG( f (0) a ) may provide a new reference state as f (1) a = √ ρ , where ρ is the density obtained from IM-SRG( f (0) a ). The iterative procedure iscontinued until f ( i +1) a → f ( i ) a , which signals that the results are converged. In addition, thisprocedure can be used to validate the convergence of our results. We have checked that theresults within the zeroth-order iteration (i.e., of IM-SRG( f (0) a ) with f (0) a = f , f ) are alreadyaccurate for the systems discussed here. In nuclear physics, the IM-SRG method was used not only to calculate the energy, but alsoto estimate other observables [3, 37]. One of the goals of the present paper is to develop (andtest the accuracy of) the IM-SRG method for calculating the density and phase fluctuationsof cold Bose gases.In order to calculate observables other than the energy, the corresponding Hermitian operator O should be transformed together with the Hamiltonian. To this end, we write O in secondquantization, normal order it with respect to the reference state Ψ ref , and solve the flowequation d O d s = [ η , O ] . (4)Equations (1) and (4) are solved simultaneously since the generator η depends on H .In this work, we focus on calculating one-body observables. The commutator in Eq. (4) leadsto two- and higher-order terms for such observables at s >
0, which should be truncatedaccording to our scheme. We cannot estimate the associated error using the strategy adopted Note that this differs from the convention in nuclear physics, where the notation IM-SRG( n ) is used tospecify the order n of the truncation scheme for many-body forces. ciPost Physics Submission for the energy (see Appendix A), as in general the operators O and H do not commute.Instead we define the “relative truncation error” as∆ = δee , (5)where e is the energy calculated using flow equations and δe is our estimation of the truncationerror for the energy. We estimate the error due to truncation for O as δO ≈ ∆ · (cid:104) O (cid:105) . (6)By comparing to the exact density, we will show below that δO can estimate accurately theerror of the IM-SRG. However, we do not expect this always to be the case. In general, onecannot infer the accuracy of an observable from the accuracy of the energy using a linearapproximation , which means that δO is no more than a useful phenomenological estimate. To illustrate calculations of observables based upon flow equations, we investigate a one-dimensional system of N bosons and a single impurity atom. This system, which is oftenreferred to as the one-dimensional Bose-polaron problem, is one of the simplest models whereour approach is useful. The corresponding Hamiltonian in first quantization reads as ( (cid:126) = 1) H = − m ∂ ∂y − M N (cid:88) i =1 ∂ ∂x i + V ib ( { x i } ) + V bb ( { x i } ) , (7)where y is the position of the impurity, x i is the position of the i th boson, m is the mass ofthe impurity atom, and M is the mass of a boson. For simplicity, we shall use the system ofunits in which M = 1. To model atom-atom interactions, we use zero-range potentials [38]: V ib ( { x i } ) = c N (cid:88) i =1 δ ( x i − y ) , V bb ( { x i } ) = g (cid:88) i,j δ ( x i − x j ) , (8)where c defines the strength of the boson-impurity interactions, and g determines the boson-boson interactions. We consider periodic boundary conditions, i.e., particles are confined toa ring of length L , see Fig. 2. The density of the Bose gas is ρ = N/L . The importantquantities that characterize the gas are the Lieb-Liniger parameter γ = g/ρ , and the healinglength ξ = 1 / ( √ γρ ). We focus on weakly-interacting Bose gases with small values of γ , forwhich the phase coherence length is larger or comparable to L , so that the system is in a(quasi)-condensed state and our flow equation approach is useful. To illustrate this statement, let us assume that a numerical method produces the following approximationto the ground state: ψ + αf , where ψ is the exact ground-state wave function, and f is an element of theHilbert space, which is orthogonal to ψ , i.e., (cid:104) ψ | f (cid:105) = 0. If the numerical method is accurate, then α → α for theexpectation value of the energy. However, the error for a general observable can be much larger, as it scales as α . ciPost Physics Submission Figure 2: Illustration of the one-dimensional Bose-polaron problem: N bosons and a singleimpurity on a ring of length L . The mass of the impurity atom is m while the mass of thebosons is M . The coordinates of the impurity and the bosons are y and { x i } , respectively.For convenience, the problem is solved using the set of coordinates { z i } , which describe therelative distances between the impurity and the bosons.In this work, we consider the case of repulsive boson-impurity interactions ( c > z i = Lθ ( y − x i ) + x i − y where θ ( x ) is the Heaviside stepfunction. The coordinates z i allow one to calculate mean-field properties of the Bose-polaronproblem in a simple manner, see Refs. [8, 17]. The transformation { y, x i } → { z i } is related tothe unitary Lee-Low-Pines transformation [40] performed in coordinate space [19, 22]. In thenew coordinates, the Hamiltonian (7) is written as H P = − N (cid:88) i ∂ ∂z i − m (cid:32) N (cid:88) i ∂∂z i (cid:33) + iPm N (cid:88) i ∂∂z i + g (cid:88) i 2) =2 cκ . The parameter N is determined from the normalization condition (cid:82) | f ( z ) | d z = 1: N = (cid:112) k/ ( kL + sin( kL )).The second reference state, f , solves the Gross-Pitaevskii equation that corresponds to H : − κ d f dz + g ( N − f ( z ) + cδ ( z ) = µf ( z ) , (11)where µ is the chemical potential. The solution to this equation is given by f ( z ) = (cid:115) K ( p ) pκgL δ ( N − 1) sn (cid:18) K ( p ) (cid:20) zδL + 12 − δ (cid:21) , p (cid:19) , (12)where sn is the sn − Jacobi elliptic function, and K ( p ) is the complete elliptic integral of thefirst kind [41]. The parameters p ∈ [0 , 1) and δ are fixed by the boundary conditions dueto the delta-function potential cδ ( z ) [ f (cid:48) | +0 − f (cid:48) | − = 2 κcf (0)], and by the normalizationcondition [ (cid:82) | f ( z ) | d z = 1]. The corresponding chemical potential µ reads as: µ = 2 p + 1 κδ L K ( p ) . (13)The mean-field solution f is discussed in more detail in Ref. [8], see also Refs. [19, 22] forthe discussion of the thermodynamic limit, and Refs. [42, 43] for the discussion of the limit c → ∞ . In this section, we calculate the density, ρ ( z ) = (cid:104) Φ gr | ρ ( z ) | Φ gr (cid:105) , of the Bose gas in the frameco-moving with the impurity: ρ ( z ) = (cid:104) Φ gr | N (cid:88) i =1 δ ( z − z i ) | Φ gr (cid:105) , (14)where Φ gr is the ground state of H . ρ ( z ) should not be confused with the density of the Bosegas without the impurity, ρ = N/L . To use the density operator ρ ( z ) in the flow equationapproach (see Sec. 2.3), we write it in second quantization as ρ ( z ) = (cid:88) i,j φ i ( z ) φ j ( z ) a † i a j , (15)where φ i ( z ) is the i th element of the one-body basis employed for writing the Hamiltonian insecond quantization. Note that in our implementation the basis { φ i ( z ) } depends on the usedreference state, see Appendix A.3. 9 ciPost Physics Submission . . . . . . . zρ . . . . . . ρ ( z ) / ρ ExactRef. StateIM-SRG ( f ) 0 . . . . . . . zρ . . . . . . ρ ( z ) / ρ ExactRef. StateIM-SRG ( f ) Figure 3: Density of the Bose gas in the frame co-moving with the impurity. The dots arecalculated with the flow equations. Left [right] panel shows the results of the IM-SRG( f )[IM-SRG( f )] method. The densities obtained directly from the reference state f [ f ] aregiven by the solid green curves. The exact pair-correlation function from Ref. [44] is shownas a dashed orange curve. Our results are presented for N = 6 bosons plus a single impurityatom. The interaction strengths are c/ρ = g/ρ = 1, and the masses of impurity and bosonsare equal, m = M = 1.To test flow equations, we first calculate ρ ( z ) assuming equal masses ( m = M ) and equalinteractions ( c = g ). These assumptions turn our system into the exactly solvable Lieb-Linigermodel for N + 1 particles [45], since we can no longer distinguish between the impurity anda boson. One can show that the pair correlation function of the Lieb-Liniger model, g (2) LL , isidentical to the density of the bosons defined in Eq. (14), i.e., ρ ( z ) = g (2) LL (0 , z ) (cid:18) N + 1 L (cid:19) . (16)The function g (2) LL is known for mesoscopic systems for certain parameter regimes (see, e.g.,Ref. [44]), and we use those results to benchmark our findings, see Fig. 3 for N = 6. Thedensity calculated using flow equations agrees well with the exact values for all values of z .Near the impurity, the density of the bosons is suppressed, since the boson-impurity interactionis repulsive. Note that the suggested error estimate in Eq. (6) is accurate.Figure 3 shows that the results of the IM-SRG( f ) method agree well with the results of theIM-SRG( f ) method, which means that both f and f are suitable reference states for theparameters considered in Fig. 3. In our studies, we have noticed that the reference state f allows us to investigate a larger range of parameters (i.e., more particles, larger boson-bosoninteractions) than f . Figure 3 explains this observation: The more complicated mean-fieldfunction f provides a better approximation of the exact density, and, hence, it is easier forthe flow equation method to map this reference state onto the real ground state of the system.In what follows, we present our results only for IM-SRG( f ).Finally, we calculate the density for parameters for which the system is no longer exactlysolvable. Our goal here is to test the mean-field treatment of H . It is already known that10 ciPost Physics Submission zρ − . . . . δ Φ z c) c/ρ = 0 . c/ρ = 0 . zρ . . . . . δ Φ z d) c/ρ = 0 . c/ρ = 0 . Figure 4: (Upper row): The density of the Bose gas in the frame co-moving with the impurity.Dots are calculated using the IM-SRG( f ) method. Mean-field densities are shown usingsolid, dashed and dot-dashed curves. The dotted vertical line indicates the relevant healinglength ξ/ √ κ . Results are presented for N = 50, m = M = 1 and different impurity-bosoninteractions c/ρ listed in the legend. (Bottom row): Phase fluctuations for the Bose gas in theframe co-moving with the impurity. Dots with error bars are calculated using the IM-SRG( f )method. The dashed curves are added to guide the eye. The parameters N , m and M areas in the upper row. The boson-boson interaction strength in panels a) and c) is g/ρ = 0 . g/ρ = 0 . ξ/ √ κ [8, 22]. We estimate that ξρ (cid:39) . g/ρ = 0 . 02 and ξρ (cid:39) . g/ρ = 0 . 1. All in all, we observe that the mean-field approximationdescribes ρ ( z ) accurately for all values of the boson-impurity interaction strength c , as longas the parameter g is small . We have checked that the mean-field approximation is accuratefor up to γ (cid:39) . c → ∞ by comparing to the Monte-Carlo results presented in Ref. [14]. We expect the mean-field approximation to break down when the boson-boson interaction is increased.With the IM-SRG method we were not able to investigate this, since the truncation error also increases withthe boson-boson interaction, e.g., for N = 15, c/ρ = 1 and g/ρ = 1 the truncation error is ≈ 5% but for g/ρ = 2 it is already above 10%. These large error bars make it impossible to pinpoint parameters for whichthe mean-field description starts to fail. ciPost Physics Submission The comparison of our IM-SRG results to the RG results of Ref. [14] suggests that it ismore advantageous to work with a real-space formulation of the problem. Large values of c require beyond-Fr¨ohlich-polaron treatment of the problem in momentum space, in particular,one should include phonon-phonon interactions. In contrast, in our implementation, alreadymean-field results are accurate. The accuracy of MFA is probably not surprising after wenotice that the (phase) coherence length is larger than the length scale we are interestedin, for instance, the phase coherence length for γ = 0 . ξ . We illustrate thisstatement further by calculating phase fluctuations in the next subsection. Phase fluctuations are strong in the one-dimensional world [35, 36, 46, 47], incapacitating themean-field treatment. However, as long as one is interested in the physics on the length scalessmaller than the coherence length, the mean-field approach can give accurate results. Wewill now calculate phase fluctuations using flow equations, and explicitly justify the use ofthe MFA. Another way to validate the mean-field ansatz could be based on calculating thecondensate fraction in the considered mesoscopic ensembles. For example, the flow equationapproach predicts that the condensate fraction for systems with N = 50 and g ≤ . ∼ ρ (0 , z ) ≡ (cid:104) Φ gr | ρ (0 , z ) | Φ gr (cid:105) = (cid:104) Φ gr | (cid:88) i,j φ ∗ i (0) φ j ( z ) a † i a j | Φ gr (cid:105) , (17)and then extract the phase fluctuations δ Φ z using the expression [46]: ρ (0 , z ) = (cid:112) ρ (0) ρ ( z ) exp (cid:26) − δ Φ z (cid:27) , (18)The result is shown in Fig. 4. For g/ρ = 0 . 02, phase fluctuations are negligibly small. For g/ρ = 0 . 1, phase fluctuations play a more important role, however even then they can beneglected so that ρ (0 , z ) (cid:39) (cid:112) ρ (0) ρ ( z ). This implies that for these parameters the Bose gascan be described using a mean-field ansatz. One could anticipate that phase fluctuationsdepend noticeably on the value c , which determines the density of bosons, and, hence, theeffective Lieb-Liniger parameter in the vicinity of the impurity. However, we observe thatphase fluctuations depend only weakly on the boson-impurity interaction strength, c . Notethat in practice it is difficult to extract phase fluctuations according to the definition (18) insystems with c/g (cid:29) ρ (0) → 0. Therefore we do not present phase fluctuations inthis limit. By condensate fraction, we mean here the expectation value of the operator a † a /N . ciPost Physics Submission − − − c/ρ . . . . . C g/ρ = 0 . g/ρ = 2 10 − − c/ρ . . . . . . C g/ρ = 0 . g/ρ = 2 Figure 5: Comparison of the mean-field contact parameter from Eq. (19) to the QuantumMonte-Carlo result [12] for two different boson-boson interaction strengths g/ρ = 0 . 02 (bluecircles) and g/ρ = 2 (orange squares). The results are shown as a function of the impurity-boson interaction strength, c/ρ with c < c > 0] displayed in the left [right] panels, re-spectively. The solid curves give the mean-field results, while the circles/squares are fromRef. [12]. The masses of the impurity and the bosons are equal ( m = M = 1). At the end of our discussion of the one-dimensional Bose-polaron problem, we mention onceagain that this paper focuses on repulsive boson-impurity interactions, and leaves a rigorousIM-SRG study of the attractive case for the future [39]. However, we must note that ourconclusion that the mean-field approximation describes accurately an impurity in a Bose gascannot be directly extended to a Bose gas with an attractive impurity. There is an importantdifference between the cases with c > c < 0: The latter supports a formation of tightlybound states at large negative values of c . Many-body bound states are highly-correlatedand cannot be accurately described by the mean-field ansatz, unlike the opposite case with c → ∞ . This implies that the region of applicability for c > C = ρ (0) /ρ , in the mean-field approximation in the thermodynamic limit: C MF = tanh ± ( D ) , D = 12 asinh (cid:18) ρc (cid:114) gρκ (cid:19) , (19)where the positive sign is for c > c < 0. We compare the parameter C MF to the contact parameter calculated using QuantumMonte-Carlo [12] in Fig. 5. The mean-field values agree well with the data for c > 0. Forattractive interactions, the agreement is only qualitative for large values of | c | ( | c | > g ),indicating the breakdown of the mean-field approximation. In the previous section, we considered a single impurity in a Bose gas, which is the standardstarting point in the analysis of systems with impurities. However, the physics of systems13 ciPost Physics Submission Figure 6: Illustration of a system of N bosons and two static impurities confined to a ringof length L . The i th boson has the coordinate x i . One impenetrable impurity ( c → ∞ ) isplaced at y = 0, and another one with the interaction strength c is at y = d .with many impurity atoms can be drastically different from that of a system with a singleimpurity atom, in particular, because the Bose gas mediates interactions between impuri-ties. Therefore, the next step for a reliable description of a (quasispin-)polarized systemsmust be an assessment of the strength of the induced impurity-impurity interaction poten-tial. One possible way to do this is to consider a Bose gas with two mobile impurities, see,e.g., Refs. [20, 25, 48, 49]. We choose another approach. We estimate the induced impurity-impurity interaction using the Born-Oppenheimer approximation [50–52] (see Refs. [53–57]for related studies in three spatial dimensions), i.e., for m → ∞ . It is known that theBorn-Oppenheimer approximation describes well short-range correlations, which define over-all properties of impurity-impurity interactions [50, 58]. It is worthwhile noting that thelong-range impurity-impurity correlations induced by quantum fluctuations are beyond therange of applicability of the Born-Oppenheimer approximation, in particular, since they candepend on the mass of the impurity [21, 29, 59]. These long-range correlations are not rele-vant for our discussion – they are weak for the considered parameters regimes, and can beneglected.The Hamiltonian for two static impurities is written in first quantization as: H = − M N (cid:88) i =1 ∂ ∂x i + g (cid:88) i,j δ ( x i − x j ) + c N (cid:88) i =1 δ ( x i − y ) + c N (cid:88) i =1 δ ( x i − y ) , (20)where c and c describe the strength of the impurity-boson interactions, and y and y arethe positions of the impurities. Without loss of generality, we place one impurity at y = 0and the other at y = d , see Fig. 6. As before, we shall use a system of units in which M = 1. For the sake of discussion, we assume that the impurity at y is impenetrable, i.e.,1 /c = 0. In other words we consider an impurity in a semi-infinite Bose gas [23, 24]. This14 ciPost Physics Submission assumption allows us to simplify the presentation. We will show that for c > c < H . We also show that theinduced interactions can be accurately calculated using the mean-field approximation, at leastfor weak boson-boson interactions. Finally, we estimate when first-order perturbation theory,which is commonly used to estimate impurity-impurity interactions [24, 51], fails. c > To use the IM-SRG( f ) scheme for a numerical study of this system, we first find the referencestate, f , by solving the Gross-Pitaevskii equation: − ∂ f ∂x + g ( N − f ( x ) + ( c δ ( x ) + c δ ( x − d )) f ( x ) = µf ( x ) . (21)The solution to this equations reads f ( x ) = (cid:115) K ( p ) p gL δ ( N − 1) sn (cid:18) K ( p ) (cid:20) xδ L (cid:21) + a, p (cid:19) x ∈ [0 , d ] (cid:115) K ( p ) p gL δ ( N − 1) sn (cid:18) K ( p ) (cid:20) L − xδ L (cid:21) + a, p (cid:19) x ∈ [ d, L ] , (22)where the parameters p , p , δ , δ , a are determined by the conditions: L (cid:90) f dx = 1 , (23) f ( d + ) = f ( d − ) , (24) p + 1 δ L K ( p ) = p + 1 δ L K ( p ) , (25) ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) d + d − = 2 c f ( d ) , (26) ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) + L − = 2 c f (0) . (27)The chemical potential is µ = 2 p +1 δ L K ( p ). The mean-field solution presented here is validfor any positive values of c . For the special case 1 /c = 0 that we consider in this section,15 ciPost Physics Submission dρ . . . . . . (cid:15) / ρ a) Pertubation TheoryMean-fieldIM-SRG( f ) dρ . . . . . . (cid:15) / ρ b) Pertubation TheoryMean-fieldIM-SRG( f ) dρ . . . . (cid:15) / ρ c) Pertubation TheoryMean-fieldIM-SRG( f ) dρ . . . . . . (cid:15) / ρ d) Pertubation TheoryMean-fieldIM-SRG( f ) Figure 7: The impurity-impurity interaction induced by the Bose gas for repulsive boson-impurity interactions, c > 0. Dots with error-bars represent the IM-SRG( f ) results. Thesolid (blue) curves are calculated using perturbation theory, and the dashed (green) curvesshow the mean-field energy. The data are for N = 60, g/ρ = 0 . 02 and four different valuesof the boson-impurity interaction: a) c /ρ = 0 . 02, b) c /ρ = 0 . 04, c) c /ρ = 0 . 06, andd) c /ρ = 0 . 1. For comparison, the dotted lines give the self-energies of a single impurity, E ( c = 0 , c ) − E ( c = 0 , c = 0), for those parameters.one should set the parameter a to 0, and solve only the boundary conditions (23)-(26). Oncethe function f is obtained, the mean-field energy of the system is calculated as E MF = µN − gN ( N − (cid:90) L f ( x ) dx . (28) We first calculate the induced interaction potential, (cid:15) = E ( c , d ) − E ( c = 0 , d ), where E isthe ground-state energy of H . Our results for this quantity for N = 60 particles are presentedin Fig. 7. Note that E ( c = 0 , d ) = E ( c , d = 0), hence (cid:15) ( d = 0) = 0. The potential (cid:15) isattractive, because the boson-impurity repulsion is minimized when the two impurities are ontop of each other. The considered system is not yet in the thermodynamic limit (see the nextsubsection), but it is large enough to see the saturation of the impurity-impurity interactionat large values of d , i.e., far from the impentrable impurity.16 ciPost Physics Submission Figure 7 compares the results obtained via IM-SRG( f ) and perturbation theory. The latterassumes that the second impurity does not affect the density of the Bose gas, and therefore (cid:15) = c n ( d ), where n ( d ) is the density of the Bose gas for c = 0. This is the usual startingpoint for estimating the induced interaction [23, 24, 51]. The perturbation theory leads to theYukawa-type potential when both impurities are weakly interacting [25,50,51,58]. Hence, ourresults indicate the limits of applicability of that standard potential. The figure also presentsthe mean-field approximation, which uses Eq. (28) to estimate (cid:15) . Our conclusion is that theperturbation theory can be used to describe the system if c < g . However, it fails alreadyfor c (cid:38) g , and more involved calculations are required to find the induced potential in thisregime. Perturbation theory implies much stronger impurity-impurity interactions. Its usewill lead to wrong predictions for a number of experimentally relevant observables, such asthe limits of stability of the polaronic gas [60]. For all considered parameters, the mean-fieldapproximation agrees with the IM-SRG( f ) method, and can be used to calculate the inducedpotential. Finally, we note that far from the impenetrable impurity, the energy (cid:15) does notapproach the self-energy of a single impurity, E ( c = 0 , c ) − E ( c = 0 , c = 0), (dottedlines in Fig. 7). This is a finite-size effect. Far from the impenetrable impurity, the densityof the Bose gas in a finite system is larger than ρ , see Fig. 4, which leads to the difference at dρ = 20 between the dotted lines and dots in Fig. 7.We also employ the IM-SRG( f ) to calculate the density and phase fluctuations of the Bosegas for N = 60, g = 0 . c = 0 . ρd = 11 , 20. The results are presented in Fig. 8.The density vanishes at xρ = 0 and xρ = 60 due to the impenetrable impurity and periodicboundary conditions. The Bose gas is strongly affected by the second impurity, which islocated at xρ = d . This explains the discrepancy between perturbation theory and the flowequation approach in Fig. 7. All in all, the IM-SRG( f ) results for ρ ( x ) agree well with themean-field prediction.Figures 8 c) and d) show phase fluctuations. Here, we choose the position of the secondimpurity, d , as the reference point. As for a single impurity in a ring, phase fluctuationsincrease far from d . The maximum value of δ Φ dx is small (in agreement with the resultsof the previous section), and does not depend strongly on d . A condensate fraction for theconsidered system is about ∼ In this subsection, we study the approach to the thermodynamic limit, i.e., we increase N and L , while keeping the density constant, ρ = N/L . The key question here is how many bosonsare needed to simulate the infinite system. For a single impurity in a weakly-interacting Bosegas, this was briefly considered in Ref. [8]. Here, we discuss what happens for two impurities.The IM-SRG results for c = 0 . c = 0 . g = 0 . 02 and different numbers of particles areshown in Fig. 9 a) and b). For the considered values of N , the induced interaction is far fromthe thermodynamic limit, i.e., it changes with the numbers of particles. The thermodynamiclimit is reached for system sizes which are beyond the flow equation approach. Still, we canuse IM-SRG to predict the induced potential in the thermodynamic limit by fitting the IM-17 ciPost Physics Submission xρ . . . . . . ρ ( x ) / ρ a) Reference stateIM-SRG( f ) xρ . . . . . . ρ ( x ) / ρ b) Reference stateIM-SRG( f ) xρ . . . . δ Φ d x c) xρ . . . . δ Φ d x d) Figure 8: (Upper row): The density of the Bose gas for N = 60, g/ρ = 0 . c /ρ = 0 . ρd = 11 and b) ρd = 20. Dots showthe results of the IM-SRG( f ). For comparison, the solid curve gives the mean-field density.(Bottom row): Phase fluctuations of the Bose gas for N = 60, g/ρ = 0 . c /ρ = 0 . ρd = 11 and d) ρd = 20. The dots arecalculated using the IM-SRG( f ).SRG energies to the function C + C N C , where C , C and C are fitting parameters. Theparameter C defines the potential in the thermodynamic limit. The fitting parameters C and C have no direct physical interpretation. In Figs. 9 a) and b), we present the value of C .The truncation error of the IM-SRG grows with the number of particles, which complicatesthe calculation of errorbars. Therefore, we give no estimate for the accuracy of C , whichexplains an apparently non-smooth character of the potential.In Figs. 9 c) and d), we compare the potential in the thermodynamic limit with calculationsbased upon perturbation theory, which includes quantum fluctuations [23, 24]. We see, thatfor a small boson-impurity interaction, c = g , both curves agree well. For larger interactions,however, the curves deviate. In this case, the density of bosons is strongly influenced by thesecond impurity (see Fig. 8), and, therefore, perturbation theory is not a valid approximation.We draw two conclusions from Fig. 9. First, high compressibility of a weakly-interacting Bosegas leads to a large number of particles needed to reach the thermodynamic limit. This shouldbe contrasted with systems of strongly interacting bosons or fermions, for which a handful ofparticles can screen the impurity [61], for more detail, see Refs. [62,63] and references therein.18 ciPost Physics Submission dρ . . . (cid:15) / ρ a) Thermod. LimitN=40N=50N=60N=70 dρ . . . . . . (cid:15) / ρ b) Thermod. LimitN=40N=50N=60N=70 dρ . . . . . (cid:15) / ρ c) Pert. + QF., Ref. [23]Thermod. Limit (IM-SRG)Thermod. Limit (MF) dρ . . . . . . (cid:15) / ρ d) Pert. + QF., Ref. [23]Thermod. Limit (IM-SRG)Thermod. Limit (MF) Figure 9: Interaction potential for two repulsive impurities induced by the Bose gas for g/ρ =0 . 02: panel a) shows c /ρ = 0 . 02 and panel b) shows c /ρ = 0 . 1. The curves display differentvalues of N listed in the insets. We also show our prediction for the induced interactionpotential in the thermodynamic limit. In the lower panels, we compare our prediction forthe thermodynamic limit (blue dots) with the corresponding result based upon perturbationtheory, which includes quantum fluctuations [23] (solid curve), and the mean-field result (greendots). Panels c) and d) are for c /ρ = 0 . 02 and c /ρ = 0 . 1, respectively. The dotted lines inc) and d) show the self-energies of a single impurity, E ( c = 0 , c ) − E ( c = 0 , c = 0) in thethermodynamic limit while the dashed curves are added to guide the eye.Here we find that for g/ρ = 0 . 02, one needs more than 100 particles to accurately capture theeffective short-range interaction between two static impurities. This implies that any studythat aims to relate measurements in current cold-atom set-ups with small number of particlesto the thermodynamic limit must provide an estimation of finite-size effects. This is especiallyimportant for any prospective experimental study of induced long-range interactions. Second,calculations beyond first-order perturbation theory are required to estimate induced impurity-impurity interactions also in the thermodynamic limit. As we show here, these calculationscan be based upon the mean-field approximation. c < In this section, we consider attractive boson-impurity interactions, c < 0. This case ismore complicated because each boson is attached to the impurity if gN (cid:46) | c | , see [39, 64].Therefore, to understand the induced interactions, we must consider N (cid:29) | c | /g .19 ciPost Physics Submission4.3.1 Reference state For attractive interactions, we do not obtain the reference state f from the Gross-Pitaevskiiequation. The full mean-field solution seems to be cumbersome. Instead, we choose thereference state as f ( x ) = N f one − rep ( x ) f one − attr ( x ) , (29)where f one − rep is the mean-field solution for a Bose gas with one repulsive impurity, and f one − attr is the mean-field solution for a Bose gas with a single attractive impurity. Therefore,the function f in Eq. (29) is the full mean-field solution for two impurities in the limitof large separation between the impurities, i.e., dρ (cid:29) 1. Otherwise, the function f is anapproximation to the solution of the Gross-Pitaevski equation. We show below that f isan accurate approximation, see, in particular, Fig. 11, where f is plotted together with thedensity of the bosons calculated using flow equations.The function f one − rep is given by (see Sec. 3.1) f one − rep ( x ) = (cid:115) K ( p ) p gL δ ( N − 1) sn (cid:18) K ( p ) (cid:20) xδ L + 12 − δ (cid:21) , p (cid:19) , (30)where δ = 1, since we work with an impenetrable impurity, c → ∞ . The function f one − attr reads as [39] f one − attr ( x ) = (cid:115) K ( p ) gL δ ( N − 1) ns (cid:18) K ( p ) (cid:20) xδ L + 12 − δ (cid:21) , p (cid:19) . (31)The parameter N is given by the normalization condition: (cid:90) f ( x ) dx = 1 . (32) We compare the induced interaction potential obtained with the IM-SRG( f ) to the oneobtained using first-order perturbation theory, see Fig. 10. In contrast to the case of c > c > 0, perturbation theory fails to describe impurity-impurityinteractions when | c | (cid:38) g . The important difference is that now perturbation theory leadsto a weaker induced interaction in comparison to the exact result.As in the previous case, perturbation theory fails because the density of the Bose gas is stronglymodified in the vicinity of the impurity for | c | (cid:38) g . To illustrate a strong modification ofthe density, we calculate ρ ( x ), and phase fluctuations of the Bose gas for N = 60, g = 0 . c = − . ρd = 11 , 20 within the IM-SRG( f ), see Fig. 11.20 ciPost Physics Submission dρ − . − . − . . (cid:15) / ρ a) Pertubation TheoryIM-SRG( f ) c > , IM-SRG( f ) dρ − . − . − . − . − . − . . (cid:15) / ρ b) Pertubation TheoryIM-SRG( f ) c > , IM-SRG( f ) dρ − . − . − . − . . (cid:15) / ρ c) Pertubation TheoryIM-SRG( f ) c > , IM-SRG( f ) dρ − . − . − . . (cid:15) / ρ d) Pertubation TheoryIM-SRG( f ) c > , IM-SRG( f ) Figure 10: Induced impurity-impurity interaction for a repulsive and an attractive impurity, c < N = 60, and g/ρ = 0 . 02. The four panels are for different values of c : a) c /ρ = − . c /ρ = − . 04, c) c /ρ = − . 06, and d) c /ρ = − . 1. The circles with error bars showthe result of the IM-SRG( f ) calculation. The solid curve gives the result of first-orderperturbation theory. For comparison, we also show the IM-SRG result for c > E ( c = 0 , c ) − E ( c = 0 , c = 0).Figure 11 demonstrates that the reference state f gives an accurate approximation to theexact density of the Bose gas. Phase fluctuations are small, and we have checked that thecondensate fraction for these parameters is above 95%. Therefore, our conclusion for theconsidered parameters is similar to the case with c > 0: The mean-field approach can beused to describe the impurity-impurity mediated interactions as long as the boson-bosoninteraction is weak. Although, we do not demonstrate this here, we expect that f and themean-field approach in general to be less accurate for one attractive impurity in comparisonto one repulsive impurity when | c | (cid:29) g . This expectation is based on our analysis of thecontact parameter, see Fig. 5, as well as on Ref. [39]. Finally, we calculate the induced interactions for different values of N , in order to understandthe few- to many-body transition in this system. We illustrate our findings in Fig. 12 for g/ρ = 0 . 02 and c /ρ = − . c /ρ = − . 1. Like in the case with c > 0, we fit theenergy to estimate the induced impurity-impurity potential in the thermodynamic limit, and21 ciPost Physics Submission xρ . . . . . ρ ( x ) / ρ a) Reference stateIM-SRG( f ) x ( x ) / b) Reference stateIMSRG( f ) xρ − . . . . . . δ Φ d x c) xρ . . . . . δ Φ d x d) Figure 11: (Upper Row): The density of the Bose gas for N = 60, g/ρ = 0 . c /ρ = − . ρd = 11 and b) ρd = 20. We show the densitycalculated using flow equations (dots) together with the reference state, f (solid curve).(Bottom Row): Phase fluctuations of the Bose gas for N = 60, g/ρ = 0 . c /ρ = − . ρd = 11 and d) ρd = 20. The dots show the IM-SRG( f ) results. Thedashed curves are added to guide the eye.compare the estimate to the result of Ref. [23]. Just like in the case with c > 0, we see thatperturbation theory is accurate for weak boson-impurity interactions but fails to describestronger interactions. The paper explores the possibility to calculate observables for degenerate Bose gases using theflow equation approach. For illustration purposes, the focus is on a one-dimensional Bose gaswith one and two impurity atoms. The considered system allows us to benchmark the IM-SRGresults against the existing exact data based upon the Bethe ansatz, and to study in detail theBose-polaron problem and polaron-polaron interactions, topics of current theoretical interest.In the single impurity case, we consider repulsive boson-impurity interactions ( c > c < ciPost Physics Submission dρ − . − . − . . (cid:15) / ρ a) Thermod. LimitN=40N=50N=60N=70 dρ − . − . − . . (cid:15) / ρ b) Thermod. LimitN=40N=50N=60N=70 dρ − . − . − . − . . (cid:15) / ρ c) Pert. + QF., Ref. [23]Thermod. Limit c > , Thermod. Limit dρ − . − . − . − . − . − . . (cid:15) / ρ d) Pert. + QF., Ref. [23]Thermod. Limit c > , Thermod. Limit Figure 12: Interaction potential between two impurities induced by the Bose gas for g/ρ = 0 . c < 0, with two different strengths a) c /ρ = − . 02 and b) c /ρ = − . N listed in the insets. We also give our prediction forthe induced interaction potential in the thermodynamic limit. In the lower panels, we displayour prediction for the values in the thermodynamic limit (blue dots) and the correspondingresult based upon perturbation theory, which includes quantum fluctuations [23] (solid curve).Panels c) and d) are for c /ρ = − . 02 and c /ρ = − . 1, respectively. For comparison, wealso show the negative IM-SRG result for c > E ( c = 0 , c ) − E ( c = 0 , c = 0), in the thermodynamic limit.density of the Bose gas for the repulsive Bose-polaron problem can be calculated accuratelyusing the mean-field approximation in the coordinate frame, which is ‘co-moving’ with theimpurity. To explain the validity of the mean-field approach, we show that the condensatefraction is large and phase fluctuations are small for the considered mesoscopic ensembles( N < ciPost Physics Submission teraction is smaller or equal to the boson-boson interaction, however, fails otherwise. Finally,we discuss the few- to many-body transition, and show the importance of finite-size effectsfor impurities in weakly-interacting Bose gases.Our results show that the mean-field approach is a robust tool to study weakly-interactingBose gases with impurities. In the other limit of strongly interacting Bose gases, one canstudy the self-energy and the density of an impurity using tools developed for Fermi gases,e.g., based upon variational wave functions [65] or the Bethe ansatz [66]. In the future, itwill be interesting to investigate the transition between these two limits, and the evolutionof the Bose polaron into the impurity in a Tonks-Girardeau gas. A modification of the flowequation approach with two reference states might be useful to study this transition. It will beinteresting to compare the obtained IM-SRG results to those calculated using beyond-Gross-Pitaevski effective theories (as, e.g., introduced in Ref. [67]).The present work paves the way for IM-SRG studies of Bose gases in higher spatial dimensions.A starting point for such an extension might be a study of dilute bosonic droplets, for which anumber of interesting analytical predictions exist [68, 69]. It should also be possible to inves-tigate two- and three-dimensional systems with impurities. The relevant mean-field solutionscan be found in the literature [70–73], giving reference states for flow equations. A modifica-tion of the IM-SRG, which takes into account Hilbert space associated with an impurity canallow one to study composite impurities in Bose gases, and corresponding quasiparticles, inparticular, angulons [74, 75]. Note added after ArXiv submission: After submission of this manuscript, we learned of arecent work [76], where the accuracy of the mean-field approximation to polaron-polaroninteractions is also discussed. Funding information We thank Fabian Grusdt for sharing with us the data for the den-sities presented in Ref. [14]. This work has received funding from the DFG Project No.413495248 [VO 2437/1-1] (F. B., H.-W. H., A. G. V.) and European Union’s Horizon 2020research and innovation programme under the Marie Sk(cid:32)lodowska-Curie Grant Agreement No.754411 (A. G. V.). M. L. acknowledges support by the European Research Council (ERC)Starting Grant No. 801770 (ANGULON). H.-W.H. thanks the ECT* for hospitality duringthe workshop ”Universal physics in Many-Body Quantum Systems – From Atoms to Quarks”.This infrastructure is part of a project that has received funding from the European Union’sHorizon 2020 research and innovation programme under grant agreement No 824093.24 ciPost Physics Submission A Details on Method Below, we explain the IM-SRG method in more detail. We start by discussing normal ordering.Then, we present the form of the flow equation (1) after our truncation of many-body terms,and discuss our estimate of the truncation error. Finally, we present further technical detailsof our implementation of the IM-SRG method.In the Appendix, the Einstein summation rule is implied, when Arabic indices are used. Thereis no summation for Greek indices. A.1 Normal ordering We study a system of N bosons whose ground state can be approximated well by the conden-sate wave function Φ ref = N (cid:89) α =1 f ( x a ) . (33)Following Ref. [7], we shall normal order the Hamiltonian with respect to Φ ref . To normalorder a Hamiltonian with one- and two-body operators, we define the contractions : a † α a α : = a † α a α − I C α α , (34): a † α a † α a α a α : = a † α a † α a α a α − I C α α α α − N − N (1 + P α α )(1 + P α α ) C α α : a † α a α : , (35)where a † α is a bosonic creation operator, C α α = (cid:104) Φ ref | a † α a α | Φ ref (cid:105) and C α α α α = (cid:104) Φ ref | a † α a † α a α a α | Φ ref (cid:105) . The parameter N is the number of bosons. The operator P α α swaps the indices α and α .A generic Hamiltonian with one- and two-body operators, H = A ij a † i a j + 12 B ijkl a † i a † j a k a l , (36)reads in the normal ordered prescription as H = (cid:15)N I + f ij : a † i a j : + 12 Γ ijkl : a † i a † j a k a l : , (37)with (cid:15)N = A ij C ij + 12 B ijkl C ijkl , (38) f α α = A α α + N − N B α ijα C ij , (39)Γ α α α α = B α α α α . (40) (cid:15)N = E is the energy of the ground state, f α α describes one-particle excitations, andΓ α α α α shows two-particles excitations. We use the notation : O : for the normal-ordered form of the operator O . ciPost Physics Submission A.2 Flow equation Our flow equation reads as d H ds = [ η , H ] , (41)where the generator η is written as η ( s ) = ξ ij ( s ) : a † i a j : + 12 η ijkl ( s ) : a † i a † j a k a l : . (42)The matrices ξ ij and η ijkl must be chosen such that the couplings to the ground state -the matrix elements f i and Γ ij - vanish (see Fig. 1). Note that the commutator in theright-hand-side of Eq. (1) induces many-body terms:[: a † i a † j a k a l : , : a † a a † b a c a d :] = a † i a † j a † b a k a c a d + ... . (43)This three-body operator induces a three-body operator in the Hamiltonian at s > 0. Sinceall couplings from the ground state need to vanish, we would now need a three-body operatorin the generator as well, which would in turn generate a four-body operator and so on. Itis therefore impossible to treat the flow equation exactly and we need to truncate many-body terms. We choose to truncate at the two-body level, keeping only the terms from thethree-body operator that contain at least one a † a operator (see Ref. [7] for a more detaileddiscussion). We neglect the remaining pieces (called W ), see also Fig. 1, which illustrates theused truncation scheme.Upon truncation, we derive a closed system of coupled differential equations (note that Ref. [7]has typos in this system of equation, which we correct here):d (cid:15) d s = S + ( N − (cid:18) S ii − S (cid:19) , (44)d f α α ( s )d s = − ( N − ( S α α + S α α + S α α ) + ( N − S α ii α + ( N − S α α + ( N − N − S α α + S α α D α D α )+ ( N − N − S α α D α + S α α D α ) + S α α , (45)dΓ α α α α ( s )d s = (1 + P α α )(1 + P α α )2 (cid:18) S α α α α − ( N − S α α α α + S α α α α ) + 12 S α α iiα α + ( N − D α D α S α α α α + ( N − I α α D α S α α α α + ( N − D α I α α S α α α α + ( N − I α α I α α S α α α α (cid:19) , (46)with D α = 2 − δ α , I α α = 1 + δ α δ α − δ α , and S (1) α α = ξ α i f iα − f α i ξ iα ,S (2) α α α α = ξ α i Γ iα α α − Γ α α α i ξ iα + η α α α i f iα − f α i η iα α α ,S α α α α α α = η α α α i Γ iα α α − Γ α α α i η iα α α . (47)26 ciPost Physics Submission A b i t r a r y un i t s EnergyCouplings Figure 13: Illustration of a solution of the flow equations (44), (45), (46) as a functionof s . The ground-state energy (cid:15)N is shown using (blue) dots. The sum of the couplings, (cid:80) i | f i | + (cid:80) ij | Γ ij | , is presented as (orange) squares. The dashed curves are added to guidethe eye.To estimate the truncation error for the ground-state energy we use second-order perturbationtheory [7] δe (cid:39) N (cid:88) p (cid:0) (cid:104) Φ p | (cid:82) ∞ W ( s ) ds | Φ ref (cid:105) (cid:1) (cid:104) Φ p | H | Φ p (cid:105) − (cid:104) Φ ref | H | Φ ref (cid:105) , (48)where Φ p is a state that contains three-body excitations.The explicit choice of the generator can be justified a posteriori if the couplings to the excitedstates vanish [1]. Our choice of the generator is η = f i : a † i a : + 12 Γ ij : a † i a † j a † a : − H.c. . (49)For this generator, the couplings (the matrix elements f i and Γ ij ) vanish, and the energy ofthe ground state, (cid:15)N , converges, see Fig. 13 for an illustration of a typical convergence patternin our study. We expect that other standard choices of the generator, see, e.g., Ref. [3], leadto similar results, see also Ref. [7]. A.3 One-body basis To write the Hamiltonian in second quantization, H = A ij a † i a j + B ijkl a † i a † j a k a l , we needto calculate the matrix elements A ij and B ijkl using some one-body basis. In this section, wediscuss the one-body basis used in the present study.First, we solve the single-boson problem whose eigenstates produce the basis set { φ i } . Thisset is used as a basis set when we work with the (single-body) reference state f = φ , where27 ciPost Physics Submission 14 16 18 20 22 n E / a) 14 16 18 20 22 n ( ) / b) 14 16 18 20 22 n ( L / ) / c) 14 16 18 20 22 n ( , L / ) / d) Figure 14: Convergence of observables for a system with one repulsive impurity as a functionof the truncation parameter, n . The parameters of the system are N = 50, g/ρ = 0 . 02 and c/ρ = 0 . 5. Panel a) shows the energy of the system, panel b) presents the density at z = 0,panel c) shows the density at z = L/ 2, and panel d) gives the one-body density matrix at ρ (0 , L/ φ is the ground-state of the single-boson problem. The corresponding contractions enjoythe simple form C α α = δ α δ α N and C α α α α = δ α δ α δ α δ α N ( N − f , we construct anotherone-body basis set: We take f as the zeroth element of our basis, and use the Gram-Schmidtprocess to build an orthogonal basis set from f , φ , φ , ... . A.4 Truncation of the one-body basis We truncate the one-body basis to solve the flow equations numerically. Let us denote thesize of the truncated basis by n . As argued in Ref. [7], to calculate the energy of the systemin the limit n → ∞ , one should compute (cid:15) for a few values of n , and then fit the obtainedsequence using the function (cid:15) ( n ) = (cid:15) ( n → ∞ ) + An δ , (50)where (cid:15) ( n → ∞ ) , A, δ are fit parameters. (cid:15) ( n → ∞ ) is the value of the energy in the limit n → ∞ . This value is presented in the figures of the main text. In our calculations, we fitresults for n = 13 − 23. We estimate the uncertainty by the standard deviation error of thefit. We show convergence of other observables in Figs. 14 b), c), and d).28 ciPost Physics Submission Observables, which depend on the density of the bosons in the vicinity of the impurity [seeFigs. 14 b) and d)] approach the limit n → ∞ in a similar to the energy fashion. Such behavioris the result of a slow convergence of the wave function due to the delta-function potential. Tocalculate such observables in the limit n → ∞ , we use the fitting procedure described above.Observables that do not depend on the density of bosons in the vicinity of the impurity, e.g.,the density of the Bose gas far away from the impurity [see Fig. 14 c)], are virtually convergedfor n = 23. 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