The impurity problem in a bilayer system of dipoles
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov The impurity problem in a bilayer system of dipoles
N. Matveeva and S. Giorgini
Dipartimento di Fisica, Universit`a di Trento and CNR-INO BEC Center, I-38050 Povo, Trento, Italy
We consider a bilayer geometry where a single impurity moves in a two-dimensional plane andis coupled, via dipolar interactions, to a two-dimensional system of fermions residing in the secondlayer. Dipoles in both layers point in the same direction oriented by an external field perpendicularto the plane of motion. We use quantum Monte Carlo methods to calculate the binding energy andthe effective mass of the impurity at zero temperature as a function of the distance between layersas well as of the in-plane interaction strength. In the regime where the fermionic dipoles form aWigner crystal, the physics of the impurity can be described in terms of a polaron coupled to thebath of lattice phonons. By reducing the distance between layers this polaron exhibits a crossoverfrom a free-moving to a tightly-bound regime where its effective mass is orders of magnitude largerthan the bare mass.
PACS numbers: 05.30.Fk, 03.75.Hh, 03.75.Ss
The polaron problem, in the broad sense of an impuritycoupled to a bath of elementary excitations, is of generaland fundamental interest in condensed matter physics.The original formulation of the polaron model addressedthe motion of electrons coupled to the lattice vibrationsof a crystal [1]. In this context phonon excitations arefound to dress the impurity, thereby increasing its effec-tive mass. For strong interactions, self-trapping of po-larons was predicted as a result of the dragged phononcloud which creates a confining potential where the im-purity is finally trapped [2]. Since the variational calcu-lation by Landau and Pekar, the polaron problem in thestrong-coupling regime has been investigated using manydifferent theoretical tools [3], including exact quantumMonte Carlo (QMC) methods [4–8]. On the experimen-tal side, clear evidence of the self-trapping of polarons isstill lacking, also due to the difficulty of accessing largeinteraction strengths in solid-state devices [1].An important recent extension of the polaron conceptconcerns the field of ultracold atoms. A polaron-like be-havior is indeed expected from an impurity immersed ina Bose-Einstein condensate, which provides the phononmodes of the bath [11–13], as well as in a Fermi sea,where the excitations of the medium have fermionic na-ture. This latter case is particularly interesting sinceboth attractive [14–16] and repulsive [17–19] polaronshave been considered theoretically and characterized inexperiments [20–22]. The quantum dynamics of an impu-rity in a Bose gas has also been recently observed [23, 24].In this Letter we propose a realization of the polaronmodel by using an impurity coupled via dipolar interac-tions to a two-dimensional (2D) system of fermions ina bilayer geometry [25]. By tuning the in-plane dipo-lar interaction strength the state of the fermions can beturned from the Fermi liquid (FL) to the Wigner crystal(WC) phase, thereby changing the nature of the elemen-tary excitations of the bath from fermionic to bosonic.QMC simulations are performed to calculate the bind-ing energy and the effective mass of the impurity as a function of the distance between layers assuming thatthe interlayer potential barrier is high enough to sup-press tunneling. In the WC phase the impurity exhibitsa crossover from a free-moving to a tightly-bound regimesimilar to the self-trapping transition. However, in con-trast to the paradigmatic case of electrons in a crystal,the coupling to phonons is found to decrease the effec-tive mass of the impurity with respect to the band massdetermined by the static periodic potential and to favorhopping processes of the impurity between lattice sites.We consider a system of N + 1 identical dipolarfermions of mass m and dipole moment d . N fermions oc-cupy the first layer (bottom layer) and the extra fermionoccupies alone the second layer (top layer). The layersare 2D parallel planes separated by a distance λ and mo-tion in the transverse direction is completely frozen by astrong confining potential provided, for example, by anintense optical lattice. Since interlayer tunneling is as-sumed to be completely suppressed, the particle in thetop layer can be considered as a distinguishable impuritycoupled via dipolar interactions to the particles of thebottom layer. An external field aligns the dipoles in thedirection perpendicular to the layers, so that in-planeinteractions are purely repulsive and scale with the in-terparticle distance as 1 /r while the interlayer particle-impurity potential is given by V ( r ai ) = d ( r ai − λ )( r ai + λ ) / , (1)where r ai = | r i − r a | is the in-plane distance betweenthe i -th particle and the projection of the impurity posi-tion onto the bottom layer. The full Hamiltonian of thesystem is written as H = − ¯ h m N X i =1 ∇ i + X i 3. Here, we use the same numerical tech-nique to calculate the binding energy of the impurity, de-fined as the energy difference between the ground statewith and without the impurity, µ = E N +1 − E N , and itseffective mass. The latter is obtained from the diffusioncoefficient of the impurity in imaginary time τ [30, 31] mm ∗ = lim τ →∞ h| ∆ r a ( τ ) | i Dτ , (3)where D = ¯ h / m is the diffusion constant of a free par-ticle and h| ∆ r a ( τ ) | i = h| r a ( τ ) − r a (0) | i is the meansquare displacement of the impurity. Simulations are car-ried out both in the WC phase [ k F r > ( k F r ) c ] and inthe FL phase [ k F r < ( k F r ) c ].The technical details of the simulations are similar tothe ones reported in Ref. [27]. To simulate the bottom m / m * k F λ FL, k F r =0.5FL, k F r =2.5FL, k F r =5FL, k F r =10FL, k F r =20WC, k F r =28WC, k F r =35WC, k F r =40static, k F r =35 FIG. 2: (color online). Inverse effective mass of the impurityas a function of the distance between layers for different val-ues of the in-plane interaction strength k F r . Circles refer tothe FL and squares to the WC phase. The stars correspond to m/m ∗ when only the static periodic potential U ( r a ) is consid-ered and phonons are frozen. Lines connecting the symbolsare a guide to the eye. The solid black line corresponds toEq. (7) in the WC phase. layer we use a box of volume Ω = L x L y , with L x ≤ L y ,and periodic boundary conditions (PBC) in both spatialdirections. The fermionic density in the bottom layer is n = N/ Ω. The calculation of the in-plane and interlayerdipolar interaction energy are performed by consideringreplicas of the simulation box and by carrying out thesummation over pairs of particles with separation up tothe cut-off distances R c = 0 . L x and R c = 2 L x , re-spectively [28]. The contribution from distances largerthan R c ( R c ) is accounted for by assuming a uni-form distribution of particles which yields the tail energy E tail = πnd /R c ( E tail = 2 πnR c / ( λ + R c ) / ). Wechecked that larger values of R c and R c give the same µ and m ∗ within statistical uncertainty. We notice that R c is significantly smaller than R c because in-plane in-teractions largely cancel when the difference E N +1 − E N is considered. Calculations were performed with differentnumbers of particles, ranging from N = 30 to N = 90 inthe WC and from N = 29 to N = 61 in the FL phase,and no appreciable finite-size dependence is found forthe binding energy and the effective mass. All resultsreported in this Letter are obtained using N = 30 in theWC phase and N = 29 in the FL phase.The FNDMC method is based on the choice of atrial wavefunction giving the many-body nodal surfacewhich is kept fixed during the simulation [29]. We use aJastrow-Slater function of the form ψ T ( r , ..., r N , r a ) = N Y i =1 h ( r ai ) Y i 1, to values m/m ∗ ≪ 1. For example, at k F λ = 1, we find m/m ∗ = 0 . k F r = 35. Theincrease of the mean square displacement of the impu-rity, | ∆ r a ( τ ) | , as a function of imaginary time is shownin Fig. 3 in the case k F r = 35. The results for m/m ∗ areobtained, following Eq. (3), by fitting a line to the long-time behavior of these curves. The dramatic increase ofthe effective mass at small values of k F λ is clearly shownby the tendency of the long-time tail to approach an hor-izontal line.Direct contact with the acoustic polaron model can bemade when the value of k F r is deep enough in the WCphase that the harmonic approximation for the single-layer Hamiltonian is valid [27, 32]. By expanding theparticle-impurity interaction term in a sum over the ex-citations of the lattice, similarly to the derivation of theelectron-phonon interaction in crystals [1], one can writethe Hamiltonian of the bilayer system as: H = U + X q ,s ¯ hω q ,s (cid:18) a † q ,s a q ,s + 12 (cid:19) − ¯ h m ∇ a + U ( r a )+ i Ω X q ,s V q e i q · r a q · e ∗ q ,s s ¯ hN mω q ,s ( a q ,s + a †− q ,s ) . (5)Here U = 1 . k F r N ǫ F is the energy of the lattice in < | ∆ r a | > / a τ k F λ = 1k F λ = 1.2k F λ = 1.4k F λ = 1.75k F λ = 2 FIG. 3: (color online). Mean square displacement, in units ofthe lattice spacing a , of the impurity coupled to the WC phaseat k F r = 35. The imaginary time τ is in units of mr / ¯ h . Thedashed line corresponds to the free diffusion m/m ∗ = 1. the classical limit [32]. U ( r a ) = P Nm =1 V ( | R m − r a | ) isthe static periodic potential when the atoms occupy thelattice sites R m , whose spatial average over the primitivecell is vanishing. Sums run over the wavevectors q of thefirst Brillouin zone and over the two branches s , corre-sponding to phonons with energy ¯ hω q ,s whose creationand annihilation operators are denoted by a † q ,s and a q ,s respectively. The interlayer potential (1) enters the aboveequation with its Fourier transform V q = − πd qe − qλ and e q ,s denotes the polarization unit vector obeying to e ∗ q ,s · e q ,s ′ = δ s,s ′ . We notice that higher-order phononterms as well as umklapp processes are neglected in theHamiltonian (5).Perturbation theory can be applied to the Hamiltonian(5) in the limit of a weak interlayer coupling potential V q .The increase in energy with respect to the unperturbedground-state E k = U + P q ,s ¯ hω q ,s + ¯ h k m for an im-purity moving with a small momentum ¯ h k is given by δE k = µ + ¯ h k ( m ∗ − m ) and allows one to determineboth the binding energy and the effective mass. The con-tribution from the static potential U ( r a ) is exponentiallysuppressed at large k F λ and can be neglected. The cou-pling to phonons gives instead the result µ = − ǫ F . k F λ ) k F r , (6)for the binding energy and m ∗ m = 1 + 0 . k F λ ) , (7)for the effective mass. The derivation of results (6)-(7) makes use of the excitation energy ¯ hω q ,ℓ = c ℓ q of longitudinal long-wavelength phonons, where c ℓ =0 . √ k F r hk F m is the corresponding speed of sound ob-tained numerically using the approach of Ref. [32].The above predictions of perturbation theory are com-pared with QMC results in Fig. 1 and in Fig. 2, respec-tively for µ and m/m ∗ . In the case of the binding en-ergy, a good agreement is found when k F λ > ∼ k F λ the increase of the effec-tive mass is such a tiny effect that the limited accuracyof the QMC results does not allow for a useful compari-son with Eq. (7). Instead, as Fig. 2 clearly shows, m/m ∗ is found to be appreciably smaller than unity for valuesof the coupling where Eq. (7) is still very close to theunperturbed value.In order to understand better the role of phonons, weperformed calculations of m/m ∗ using the single-particleHamiltonian − ¯ h m ∇ a + U ( r a ) and we thereby determinedthe inverse band mass in the static potential U ( r ). Inthese simulations the particles in the bottom layer areconsidered to be fixed in the positions R m of the Bra-vais lattice and phonon excitations are thus completelyfrozen. The results are reported in Fig. 2 for the value k F r = 35 of the in-plane coupling strength. We noticethat the suppression of m/m ∗ with decreasing distance k F λ is much more pronounced in the static case thanwhen quantum fluctuations are included. For k F λ < ∼ m/m ∗ , in con-trast to what is typically expected from the coupling tophonon excitations. A possible physical explanation ofthis effect is phonon-assisted hopping of the impurity be-tween lattice sites, which results in a reduction of theimpurity effective mass. This mechanism competes withthe increase of the effective mass arising from the phonondrag and becomes dominant for small enough values of k F λ .Finally in Fig. 4(a) we show the results for the particle-impurity correlation function, related to the probabilityof finding the impurity and a particle at a distance r apart. When k F λ is large, the impurity is highly mo-bile and at distances k F r > k F r < ∼ Impurity coupled to a Fermi liquid. The FNDMC results for | µ | and m/m ∗ when k F r < ( k F r ) c , corresponding to the dipolar fermions in the bot-tom layer in the FL phase, are reported in Figs. 1-2. Forthe smallest value of the interaction strength, k F r = 0 . g (r) k F rk F λ = 1.2k F λ = 1.75k F λ = 2.5 (a) g (r) k F rk F λ = 0.5k F λ = 0.83k F λ = 1.25 (b) FIG. 4: (color online). Particle-impurity distribution functionin terms of the in-plane distance. WC phase at k F r = 35[panel (a)] and FL phase at k F r = 2 . but, in contrast to the WC phase, it approaches a finiteasymptotic value for small k F λ . If k F r ≪ m ∗ ≃ m , the mass of a dimer. Interactioneffects in the bottom layer make the value of the effectivemass as large as m ∗ ≃ m at k F r = 20. The particle-impurity correlation function is reported in Fig. 4(b). Itbehaves qualitatively similar to the WC case with a peakcaused by the particle-impurity attraction and a hole dueto the in-plane repulsion. The main difference is the morepronounced deformation of the medium around the im-purity in the FL phase.An important issue concerns the experimental real-izability of the dipolar bilayer system in the properregime of parameters where strongly-coupled polaronscan be investigated. 2D Fermi gases have been real-ized [38, 39] with inverse Fermi wavevectors in the range1 /k F ∼ − 500 nm. Optical lattices with high barri-ers and lattice spacing on the order of 500 nm are alsoavailable. By using polar molecules produced with mix-tures of Na- K [40] or of Cs- Li [41] the dipolarlength can be as large as r ∼ . − µ m and largevalues of k F r can be achieved. If such molecules can beproduced in their ground state and brought to quantumdegeneracy, ultracold dipolar systems can become a use-ful tool to investigate the rich physics of polarons in thestrong-coupling regime.Useful discussions with G. Astrakharchik and A.Pikovski are gratefully aknowledged. This work has beensupported by ERC through the QGBE grant and byProvincia Autonoma di Trento. We thank the Aurora-Science project (funded by PAT and INFN) for allocat-ing part of the computing resources for this work and fortechnical support. [1] See e.g. G.D. Mahan, in Many-Particle Physics (Plenum,New York, 1990), Chap. 6, 2nd ed. [2] L.D. Landau, and S.I. Pekar, Zh. Eksp. Teor. Fiz. e.g. J.T. 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TREATMENT OF THE DIPOLE-DIPOLE INTERACTION ENERGY Since the dipole-dipole force is long range, the potential energy contributions arising fromin-plane ( V dd ) and interlayer ( V ad ) interactions require a careful treatment. The in-planecontribution is given by V dd = X i The trial wavefunction used in the QMC simulations has the general Jastrow-Slater form ψ T ( r , ..., r N , r a ) = N Y i =1 h ( r ai ) Y i 2) =0, being L the side of the square simulation box. In the WC phase, when a rectangularsimulation box of volume Ω = L x L y is used in order to accommodate an integer number ofprimitive cells of the Bravais lattice, the length L in Eq. (6) is replaced by the smallest ofthe two sides L x ≤ L y . An identical choice for the particle-particle Jastrow correlation wasused in the study of Ref. . 3 N. Matveeva and S. Giorgini, Phys. Rev. Lett. , 200401 (2012)., 200401 (2012).