Nature of polaron-molecule transition in Fermi polarons
NNature of polaron-molecule transition in Fermi polarons
Cheng Peng,
1, 2, ∗ Ruijin Liu, ∗ Wei Zhang,
3, 4 and Xiaoling Cui
1, 5, † Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, Renmin University of China, Beijing 100872, China Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,Renmin University of China, Beijing 100872, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Dated: March 1, 2021)It has been commonly believed that a polaron to molecule transition occurs in three-dimensional(3D) and two-dimensional(2D) Fermi polaron systems as the attraction between thesingle impurity and majority fermions gets stronger. The conclusion has been drawn from the sep-arate treatment of polaron and molecule states and thus deserves a close reexamination. In thiswork, we explore the polaron and molecule physics by utilizing a unified variational ansatz with upto two particle-hole(p-h) excitations(V-2ph). We confirm the existence of a first-order transition in3D and 2D Fermi polarons, and show that the nature of such transition lies in an energy competitionbetween systems with different total momenta Q = 0 and | Q | = k F , where k F is the Fermi mo-mentum of majority fermions. The literally proposed molecule ansatz is identified as an asymptoticlimit of | Q | = k F state in strong coupling regime, which also implies a huge SO (3)(for 3D) or SO (2)(for 2D) ground state degeneracy in this regime. The recognization of such degeneracy is cruciallyimportant for evaluating the molecule occupation in realistic experiment with finite impurity den-sity and at finite temperature. To compare with recent experiment of 3D Fermi polarons, we havecalculated various physical quantities under the V-2ph framework and obtained results that are ingood agreements with experimental data, especially in the unitary regime. Further, to check thevalidity of our conclusion in 2D, we have adopted a different variational method based on the Gaus-sian sample of high-order p-h excitations(V-Gph), and found the same conclusion on the nature ofpolaron-molecule transition therein. For 1D system, the V-2ph method predicts no sharp transitionand the ground state is always at Q = 0 sector, consistent with exact Bethe ansatz solution. Thepresence/absence of polaron-molecule transition is analyzed to be closely related to the interplayeffect of Pauli-blocking and p-h excitations in different dimensions. I. INTRODUCTION
Fermi polaron is a typical quasi-particle describing animpurity immersed in and dressed by a Fermi-sea envi-ronment. The attractive and repulsive branches of Fermipolarons have attracted great attention in recent yearsin the field of ultracold atoms both experimentally[1–8]and theoretically[9–35], thanks to the high controllabilityof species, number and interaction in this ideal platform.For the attractive Fermi polaron, depending on the in-teraction strength between the impurity and fermions, itcould end up with two distinct destinies as revealed byprevious studies[13–19, 22–25]: one destiny is that theimpurity is dressed with the surrounding cloud of major-ity fermions and forms a fermionic polaron ; the otheris that the impurity essentially binds with one singlefermion on top of the Fermi surface to form a bosonic molecule . To characterize these distinct pictures, the fol-lowing variational ansatz for polaron and molecule stateswith truncated n particle-hole(p-h) excitations have been ∗ These authors contributed equally to this work. † [email protected] proposed[9, 11, 12, 14, 15, 19–23, 30, 33]: P n +1 (0) = ψ c † ↓ + n (cid:88) l =1 (cid:88) k i q j ψ k i q j c † P ↓ l (cid:89) i =1 c † k i ↑ l (cid:89) j =1 c q j ↑ | FS (cid:105) N ;(1) M n +2 (0) = (cid:34)(cid:88) k φ k c † − k , ↓ c † k , ↑ + n (cid:88) l =1 (cid:88) k i q j φ k i q j c † P ↓ l +1 (cid:89) i =1 c † k i ↑ l (cid:89) i =1 c q j ↑ | FS (cid:105) N − .. (2)Here c † k ,σ is the creation operator of spin- σ fermions atmomentum k and the ↓ -spin is the impurity, | FS (cid:105) N isthe Fermi sea of ↑ -spin with number N ; all q ( k ) arebelow (above) the Fermi surface of ↑ -atoms and P = (cid:80) j q j − (cid:80) i k i . The two ansatz above have been shownto lead to a first-order transition between polaron andmolecule for both 3D[14, 15, 17–19] and 2D[22, 23] Fermipolaron systems. The same conclusion was also drawnfrom Monte-Carlo methods[13, 24, 25], where the polaronand molecule were treated separately with their energiesextracted from different physical quantities. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b The separate treatment of polaron and molecule,though physically inspiring, has its own drawback asthe transition appears to be artificially designed at thevery beginning. As a result, the conclusion of polaron-molecule transition can easily get questioned. For in-stance, a previous theory[35] claimed the absence of suchtransition by showing that the two types of variationalansatz are mutually contained in a generalized momen-tum space if more p-h excitations are included. There-fore, the relation and competition between polaron andmolecule deserve a close re-examination under a unifiedframework.On the experimental side, the polaron-molecule transi-tion has been identified by a continuous zero-crossing ofquasi-particle residue, instead of a sudden jump as in thefirst-order transition, in both 3D and 2D Fermi gases[1,5]. In particular, a recent experiment on 3D Fermi po-larons by Isreal group has observed a smooth evolu-tion of various physical quantities across the polaron-molecule transition, as well as a coexistence of polaronand molecule near their transition[8]. All these obser-vations need to be reconsidered carefully following theunified treatment of polaron and molecule states.With above motivations, in a recent work[36] wehave adopted a unified variational method with one p-hexcitations(V-1ph) to study the Fermi polaron problemin 3D. Specifically, the unified ansatz we used is P ( Q ),i.e., the extension of P (0) in Eq. (1) to finite momen-tum. By this, we found that the bare molecule state M (0) actually constitutes part of P ( Q ) with | Q | = k F (denoted as P ( k F ) for short), here k F the Fermi momen-tum of majority fermions. Due to the incomplete vari-ational space of M (0) even within the lowest-order p-hexcitations, it always has a higher energy than P ( k F ).The significance of introducing M (0) is found to lie inthe strong coupling regime, where it can serve as a goodapproximation for P ( k F ). Within V-1ph method, weconcluded that the nature of “polaron-molecule transi-tion” is given by an energy competition between P (0)and P ( k F ). This naturally resolves the theoretical de-bate in Ref.[35] because the transition is between differ-ent Q -states rather than between different forms of varia-tional ansatz. Furthermore, near the transition point, wefound the double-minima (at | Q | = 0 and k F ) structureof the impurity dispersion curve, providing the underly-ing mechanism for polaron-molecule coexistence in real-istic systems. Based on this, we qualitatively explainedthe smooth polaron-molecule transition as observed in Is-real experiment[8] with a finite impurity density and atfinite temperature.In the present work, we extend the study of Fermi po-laron problem to various dimensions using the unifiedvariational method with up to two p-h excitations (V-2ph), namely, under variational ansatz P ( Q ). WithV-2ph method, we confirm the existence of polaron-molecule transition in 3D and 2D, and re-enforce the con-clusion made in Ref.[36] that the nature of such transitionlies in an energy competition between different momenta Q = 0 and | Q | = k F . Same as Ref.[36], near the tran-sitions the two momenta states appear as double min-ima in the dispersion curve, signifying the coexistence oftwo states under realistic situation. Here, we find themain effect of including two p-h excitations is to shiftthe transition point and the coexistence region to weakercoupling regime, from which we obtain a reasonably bet-ter prediction to various physical quantities as measuredin the unitary regime of Fermi polaron experiment[8].Moreover, we emphasize in this work that the resem-blance between molecule ansatz and the | Q | = k F statein strong coupling regime tells us a fact that has beenoverlooked for long time, i.e., the ground state in thisregime has a huge degeneracy ( SO (3) for 3D and SO (2)for 2D). The recognization of such degeneracy is cruciallyimportant for correctly evaluating the individual occupa-tion of polaron and molecule in their coexistence regionfor realistic Fermi polaron systems.To further check the validity of our results in 2D, weadopt a different variational method based on the Gaus-sian sample of high order p-h excitations(V-Gph)[37],which gives the same conclusion for the nature of polaron-molecule transition therein. For 1D system, the V-2phmethod predicts no sharp transition and the ground stateis always the Q = 0 state for any coupling strength, con-sistent with the Bethe ansatz solutions. These compar-isons further justify the validity of V-2ph method and thereliability of our results in various dimensions. We an-alyze that the presence or absence of polaron-moleculetransition is closely related to the interplay effect ofPauli-blocking and p-h excitations in different dimen-sions.The rest of the paper is organized as follows. In Sec.II,we present the algorithm from two variational ansatz totreat the Fermi polaron problem: one is the variationalansatz with up to two p-h excitations(V-2ph), and theother is the Gaussian variational ansatz with high orderp-h excitations(V-Gph). In Sec.III, we present the resultsof polaron-molecule transition for single impurity systemin various dimensions from the two methods, and ana-lyze the intrinsic reason for the presence/absence of suchtransition in different dimensions. In Sec. IV, we usethe single-impurity results to investigate the coexistenceand smooth crossover between polaron and molecule in3D Fermi polaron systems, in comparison with the ex-perimental data from Isreal group[8]. Finally the resultsare summarized in Sec. V. II. METHODS
We consider the following Hamiltonian describinga spin- ↓ impurity interacting with spin- ↑ majorityfermions: H = (cid:88) k σ (cid:15) k ,σ c † k σ c k σ + g/L d (cid:88) Q , k , k (cid:48) c † Q − k , ↑ c † k , ↓ c k (cid:48) , ↓ c Q − k (cid:48) , ↑ (3)where (cid:15) k = k / (2 m ); d is the dimension of the system; g is the bare coupling constant which needs to be renor-malized in 2D and 3D due to the induced ultraviolet di-vergence in two-body scattering process. Specifically, for3D g is related to the s-wave scattering length a s via1 /g = m/ (4 πa s ) − /V (cid:80) k / (2 (cid:15) k ) with V = L the vol-ume of the system; for 2D, the scattering length a d de-fines the two-body binding energy E b = − /ma d and g is related to E b via 1 /g = − /S (cid:80) k / (2 (cid:15) k − E b ) where S = L is the area of the system. In this work we take (cid:126) as unity for brevity.In this section, we present the algorithm of two varia-tional methods used to treat Fermi polaron problems.One is the the unified variational ansatz P ( Q ) withup to two p-h excitations(V-2ph), in comparison withthe molecule ansatz M ( Q M ). The other is the Gaus-sian variational ansatz with high order p-h excitations(V-Gph). A. Unified variational approach with up to two p-hexciations (V-2ph)
In the following, we will present the algorithm of P ( Q ), the polaron ansatz with arbitrary momentum andwith up to two p-h excitations, as well as the algorithmof M ( Q M ), the molecule ansatz with arbitrary momen-tum and with one p-h excitations. It is noted that the Q = 0 case of P ( Q ) have been studied previously in3D[12, 14], 2D[23], and 1D[28] Fermi polaron systems;the Q M = 0 case of M ( Q M ) have also been studied pre-viously in 3D[14–16] and 2D[22, 23] systems. Here wegeneralize the study to arbitrarily finite momenta, whichevolves more numerical work than the zero-momentumcase. The intrinsic relation between the two ansatz willalso be discussed. P ( Q ) The generalized polaron ansatz with up to two p-h ex-citations is: P ( Q ) = ψ c † Q ↓ + (cid:88) kq ψ kq c † Q + q − k ↓ c † k ↑ c q ↑ +14 (cid:88) kk (cid:48) qq (cid:48) ψ kk (cid:48) qq (cid:48) c † Q + q + q (cid:48) − k − k (cid:48) ↓ c † k ↑ c † k (cid:48) ↑ c q ↑ c q (cid:48) ↑ | FS (cid:105) N . (4)By imposing the Schr¨odinger equation, we can obtain thecoupled integral equations for all variational coefficients,from which the ground state energy can be obtained.This is equivalent to minimizing the energy functional E tot = (cid:104) H (cid:105) for a normalized ansatz. In this paper, wetake the unperturbed Fermi sea | FS (cid:105) N as the reference system and define the impurity energy as E = E tot − E FS ,with E FS is the energy of | FS (cid:105) N .The equations for the variational coefficients are: − g ( E − E (0) Q ) ψ = (cid:88) kq ψ kq ; − g ( E − E (1) kq ) ψ kq = ψ + (cid:88) K ψ Kq − (cid:88) q (cid:48) ψ kq (cid:48) − (cid:88) Kq (cid:48) ψ kKqq (cid:48) ; − g ( E − E (2) kk (cid:48) qq (cid:48) ) ψ kk (cid:48) qq (cid:48) = − ψ kq − ψ k (cid:48) q (cid:48) + ψ kq (cid:48) + ψ k (cid:48) q + (cid:88) K ψ Kk (cid:48) qq (cid:48) + (cid:88) K ψ kKqq (cid:48) − (cid:88) Q (cid:48) ψ kk (cid:48) Q (cid:48) q (cid:48) − (cid:88) Q (cid:48) ψ kk (cid:48) qQ (cid:48) , where E (0) Q = (cid:15) Q , E (1) kq = (cid:15) Q + q − k + (cid:15) k − (cid:15) q , E (2) kk (cid:48) qq (cid:48) = (cid:15) Q + q + q (cid:48) − k − k (cid:48) + (cid:15) k + (cid:15) k (cid:48) − (cid:15) q − (cid:15) q (cid:48) . As before, all q ( k ) inthese equations are by default below (above) the Fermisurface of | FS (cid:105) N .Above equations can be solved in 1D using iterativemethod. For 2D and 3D, due to the renormalizationscheme of bare coupling g , the equations can be sim-plified by using g (cid:88) k (cid:48) α k (cid:48) q E − E (2) kk (cid:48) qq (cid:48) ∼ g (cid:88) k (cid:48) E − E (2) kk (cid:48) qq (cid:48) ∼ . The final equations for numerical simulation are E = (cid:15) Q + (cid:88) q A q ; (5) α kq = A q − (cid:80) q (cid:48) G ( k , q , q (cid:48) ) E − E (1) kq ; (6) G ( k , q , q (cid:48) ) = α kq (cid:48) − α kq − (cid:80) k (cid:48) G ( k (cid:48) , q , q (cid:48) ) E − E (2) kk (cid:48) qq (cid:48) h ( k , q , q (cid:48) ) , (7)with A q = 1 − (cid:80) kq (cid:48) G ( k , q , q (cid:48) ) E − E (1) kq h ( q ) ; (8) h ( q ) = 1 g − (cid:88) k E − E (1) kq ; (9) h ( k , q , q (cid:48) ) = 1 g − (cid:48) (cid:88) k (cid:48) E − E (2) kk (cid:48) qq (cid:48) , (10)where we have defined α kq = ψ kq /ψ , A q = g (1 + (cid:80) k α kq ), G ( k , q , q (cid:48) ) = g (cid:80) k (cid:48) ψ ( k , k (cid:48) , q , q (cid:48) ) /ψ .Due to the rotational invariance of momentum Q , inthis work we have taken it along z axis for simplicity.Compared to the zero momentum case, here the finite Q in 3D and 2D introduces more momentum variablesin the simulation and thus requires a heavier numericalwork. In practice, we have used iterative scheme to solveEqs. (5) to (7). In updating E in Eq. (5) and updating G ( k , q , q (cid:48) ) in Eq. (7), we have used the successive over-relaxation method to reduce the fluctuation and ensurethe convergency of the results. M ( Q M ) The generalized molecule ansatz with one p-h excita-tions is written as: M ( Q M ) = (cid:34)(cid:88) k φ k c † Q M − k , ↓ c † k , ↑ +12 · (cid:88) kk (cid:48) q φ kk (cid:48) q c † Q M + q − k − k (cid:48) ↓ c † k ↑ c † k (cid:48) ↑ c q ↑ | FS (cid:105) N − .. (11)By imposing the Schr¨odinger equation, one can obtainthe equations for all variables φ k , φ kk (cid:48) q . Again for 2Dand 3D cases, the equations can be simplified. Namely,by introducing two auxiliary functions γ = g (cid:80) k φ k and η kq = g (cid:80) k (cid:48) φ kk (cid:48) q , we can arrive at the following integralequations for ˜ η kq = η kq /γ (see the Q M = 0 case in [14,15, 22, 23]):1 g − (cid:88) k E + E F − E (1) k = (cid:88) kq ˜ η kq E + E F − E (1) k ; (12) (cid:34) g − (cid:88) k (cid:48) E + E F − E (2) kk (cid:48) q (cid:35) ˜ η kq = − (cid:80) q (cid:48) ˜ η kq (cid:48) E − E (1) k − (cid:88) k (cid:48) ˜ η kq E + E F − E (2) kk (cid:48) , q , (13)with E (1) k = (cid:15) Q M − k + (cid:15) k and E (2) kk (cid:48) , q = (cid:15) Q M − k − k (cid:48) + q + (cid:15) k + (cid:15) k (cid:48) − (cid:15) q .Again in the calculation we take Q M along z axis dueto its rotational invariance. Compared to P ( Q ), thesimulation of M ( Q M ) is easier due to the smaller vari-ational space. One can obtain the molecule energy E either by using iterative method or by solving large ma-trix equations with respect to ˜ η kq . We have confirmedthat these two methods produce consistent results.
3. Relation between P ( Q ) and M ( Q M ) In our previous work[36], we have discussed the inti-mate relation between M (0) and P ( Q ) with | Q | = k F .The discussion can be straightforwardly extended toother momentum sectors and to arbitrary levels of p-h excitations. Here we consider the case of P ( Q ) and M ( Q M ) and discuss their relation as below. We startwith the following equality between two Fermi sea states | FS (cid:105) N − = c k F ↑ | FS (cid:105) N . (14)Here k F is the Fermi momentum that can point to anydirection on the Fermi surface. Given (14), one can seethat if we further take ψ = 0 , ψ kq = φ k δ q , k F , ψ kk (cid:48) qq (cid:48) = φ kk (cid:48) q δ q (cid:48) , k F , (15)then P ( Q ) in (4) exactly reproduces M ( Q M ) in (11)under the relation Q M = Q + k F . (16)Eqs. (15,16), which can be directly generalized to arbi-trary order of p-h excitations, immediately tell us twoimportant facts:(i) M ( Q M ) has a smaller variational space than P ( Q = Q M − k F ). Specifically, the former correspondsto only considering a particular configuration of p-h ex-citations in the latter, i.e., with one hole pinning at theFermi surface [see Eq. (15)]. In principle, such config-uration is not isolated and can be coupled to other p-hexcitations via interactions, which will further reduce thevariational energy. Due to such incomplete variationalspace, M ( Q M ) always has a higher variational energythan P ( Q = Q M − k F ) for the ground state of the sys-tem. When reduced to the special case Q M = 0 and | Q | = k F , we arrive at the conclusion that M (0) alwaysproduces a higher energy than P ( Q ) with | Q | = k F .This is a direct extension of the conclusion in our previ-ous work with one p-h excitations[36].(ii) The correspondence (16) tells that, the previouslystudied zero-momentum molecule M (0) actually stays ina different momentum sector from the zero-momentum polaron P (0). Such momentum difference, k F , originatesfrom the relation (14) between two Fermi seas with differ-ent numbers. When choose the reference state as | FS (cid:105) N ,the total momentum of P ( Q ) is Q , while the total mo-mentum of M ( Q M ) is Q M − k F . This is to say, M (0)and P (0) should have zero overlap since they belong todifferent total momentum space (note that the Hamilto-nian (3) preserves the total momentum). This momen-tum difference is robust against the choice of referencestate. Recognizing such difference is crucially importantfor understanding the nature of polaron-molecule transi-tion, as addressed in section III.Based on (i,ii), we can conclude that up to two p-h ex-citations, the generalized polaron ansatz P ( Q ) can serveas the unified variational wave function for both polaronand molecule states. The ground state of the system canthen be obtained by searching for the energy minimumin the total momentum ( Q ) space. B. Gaussian variational method with high-orderparticle-hole excitations (V-Gph)
For 2D system, besides the V-2ph method we adoptthe Gaussian variational method with high-order p-h ex-citations (V-Gph)[37]. The essence of this method isthe combination of fermionic Gaussian state[38, 39] andthe Lee-Low-Pines (LLP) transformation[40]. To be self-contained, in the following we give a brief introductionto this method.Applying the LLP transformation U LLP = e − i ˆKˆr ,where ˆK = (cid:80) k k c † k ↑ c k ↑ is the total momentum of thebackground spin-up atoms and ˆr is the coordinate of theimpurity, the Hamiltonian (3) can be transformed as H LLP = U † LLP HU LLP = (cid:88) k ( (cid:15) k − µ ) c † k ↑ c k ↑ + ˆp m − (cid:88) k ˆp · k m c † k ↑ c k ↑ + (cid:88) k , k (cid:48) k · k (cid:48) m c † k ↑ c k ↑ c † k (cid:48) ↑ c k (cid:48) ↑ + gL (cid:88) k , k (cid:48) c † k ↑ c k (cid:48) ↑ . (17)Here ˆp is the momentum operator of the impurity.Note that here we have introduced an additional term“ − µ (cid:80) k c † k ↑ c k ↑ ” into the original Hamiltonian Eq. (3) totune the particle number of the background Fermi sea.After the LLP transformation, the conserved total mo-mentum of the system transforms into the momentum ofthe impurity, i.e., U † LLP ( ˆp + ˆK ) U LLP = ˆp . (18)Thus we can replace ˆp in H LLP with its eigenvalue Q ,which eliminates the degree of the impurity.We further use fermionic Gaussian state to approx-imate the ground state with total momentum Q of thetransformed Hamiltonian, Eq. (17). The fermionic Gaus-sian state is defined as | Ψ GS (cid:105) = c † Q ↓ U GS | (cid:105) , (19) where | (cid:105) is chosen to be the vacuum state and U GS = e i A T ξA (20)is called the Gaussian unitary operator, A =( a , k , . . . , a , k Nk , a , k , . . . , a , k Nk ) T , N k is the numberof k modes satisfying | k | ≤ k c with cutoff k c , the Ma-jorana operators are defined as a , k j = c † k j , ↑ + c k j , ↑ , a , k j = i ( c † k j , ↑ − c k j , ↑ ), and the variational parameter ξ isan antisymmetric Hermitian matrix which has 2 N k − N k free matrix elements. We point out that the use of Majo-rana operators is just for computational convenience andthe operators can be re-expressed in terms of c † k j , ↑ and c k j , ↑ as in Ref[41].To eliminate the gauge degree of freedom in ξ , it isconvenient to introduce a covariance matrix [37](Γ) s , k ; s , k = i (cid:104) Ψ GS | [ a s , k , a s , k ] | Ψ GS (cid:105) , (21)with s ( s ) = 1 ,
2. The covariance matrix is related to ξ as Γ = − U m (cid:18) − N k N k (cid:19) U Tm , (22)where U m = e iξ and N k is the identity matrix of dimen-sion N k .By reversing the LLP transformation, the eigenstateof the original Hamiltonian (3) with a total conservedmomentum Q can be expressed as a non-Gaussian state | Ψ (cid:105) = U LLP c † Q ↓ U GS | (cid:105) . (23)The imaginary-time evolution equation for the non-Gaussian state Eq. (23) can be written as d τ | Ψ (cid:105) = −P ( H − E tot ) | Ψ (cid:105) , (24)where P is the projection operator onto the subspacespanned by tangent vectors of the variational manifold, E tot = (cid:104) Ψ | H | Ψ (cid:105) can be calculated using Wick’s theorem.Finally we obtain E tot = 12 (cid:88) k ε k − µN k (cid:88) k ( ε k − µ − Q · k m )(Γ , k ;2 , k − Γ , k ;1 , k ) + Q m + g L N k + g L (cid:88) k , k (cid:48) (Γ , k ;2 , k (cid:48) − Γ , k ;1 , k (cid:48) ) + 18 m (cid:88) k k + 132 m [ (cid:88) k k (Γ , k ;2 , k − Γ , k ;1 , k )] − m (cid:88) k , k (cid:48) k · k (cid:48) Γ , k ;1 , k (cid:48) Γ , k ;2 , k (cid:48) + 18 m (cid:88) k , k (cid:48) k · k (cid:48) Γ , k ;2 , k (cid:48) Γ , k ;1 , k (cid:48) . (25)To be consistent with the variational approach withtruncated p-h exciations, we calculate the energy E = E tot + µN ↑ − E FS . The imaginary time equation of mo- tion (EOM) for the covariance matrix Γ is ∂ τ Γ = − h − Γ h Γ , (26)with h = 4 δE GS δ Γ . (27)Evolving Γ according to Eq. (26) until the variationalenergy converges, we can finally obtain the approximatedground state.Now we discuss the level of p-h excitations in V-Gph.Since the Fermi sea | FS (cid:105) N is also a Gaussian state, wecan replace | (cid:105) as | FS (cid:105) N in Eq. (19) and immediately onecan see that it can include multiple p-h excitations. Byexpanding U GS in terms of ξ : U GS = 1 + i A T ξA + ... ,the wave function Ψ can also be expanded in terms of ξ . We note that the first two terms in the expansionhave included all the bare and one p-h excitation termsin P ( Q ), while the coefficients of two and higher p-hexcitation terms in Ψ are strongly correlated with thoseof one p-h terms and thus are not free variables. Thismeans that V-Gph can be a better variational approachthan V-1ph, but not necessarily better than V-2ph. Inthis work, we use it as a complementary method to testthe reliability of V-2ph. III. POLARON-MOLECULETRANSITION/CROSSOVER FOR SINGLEIMPURITY SYSTEMS
In this section, we study the polaron to molecule tran-sition or crossover for single impurity systems in vari-ous dimensions. We will apply the V-2ph method forall dimensions, in combination with V-Gph method for2D and Bethe-ansatz method for 1D. The conclusion forthe presence/absence of polaron-molecule transition fromthese methods are consistent.
A. 3D
In our previous work[36], we have used the V-1phmethod based on ansatz P ( Q ) to unveil the natureof polaron-molecule transition in 3D. Here by using V-2ph method with up to two p-h excitations, we will re-examine the polaron and molecule physics in this system.In our numerical simulations, we have taken the momen-tum cutoff as k c = 30 k F .First, we investigate the relation between M (0) and P ( Q ) with Q = k F e z , and we will denote the latterstate as P ( k F ) for short. As discussed in above sec-tion, due to the incomplete variational space of M (0),it should be energetically unfavorable as compared to P ( k F ). In Fig.1, we show their energies, in compar-ison with P ( k F ) and M (0), as functions of couplingstrength. It is found that the molecule state M (0)(or M (0)) always has a higher energy than P ( k F ) (or P ( k F )), as expected. Only in the strong coupling side,the energy difference between M (0) and P ( k F ) (or be-tween M (0) and P ( k F )) becomes invisible. For in- k F a S ( EE b ) / E F M (0) P ( k F ) M (0) P ( k F ) P (0) P (0) FIG. 1. (Color online). Energy comparison between variousansatz for 3D single impurity system. All energies are shiftedby E b = − / ( ma s ) in a s > cos q || k F a S = 0.2 : kk qq k F a S = 0.2 : kq k F a S = 0.9 : kk qq k F a S = 0.9 : kq FIG. 2. (Color online). Hole angular distribution of varia-tional coefficients in P ( k F ) at different coupling strengths.Here we use the polar coordinate ( | k | , θ k , φ k ) to charac-terize momentum k , with θ k ∈ [0 , π ) and φ k ∈ [0 , π ).In the figure we choose k = (1 . k F , . , . , k (cid:48) =(2 . k F , . , . , q (cid:48) = ( k F , , . q = ( k F , θ q , . α kq and α kk (cid:48) qq (cid:48) . stance, M (0) energetically approaches P ( k F ) at cou-plings 1 / ( k F a s ) (cid:38) .
3, and M (0) energetically ap-proaches P ( k F ) at 1 / ( k F a s ) (cid:38) .
6. Moreover, we cansee that the V-2ph method produces a lower energy forboth polaron and molecule states, as compared to thosefrom V-1ph method.To explain why the energies of M (0) and P ( k F ) be-come so close in the strong coupling limit, we examinethe wave-function of P ( k F ) in Fig.2. Specifically, weshow the hole angular distribution of variational coeffi-cients at two different coupling strengths. It is foundthat at intermediate coupling 1 /k F a s = 0 .
2, the angulardistribution of the hole ( q ) spreads in a broad region,while at stronger coupling 1 /k F a s = 0 . θ q = π , i.e., along the oppo-site direction of Q (= k F e z ). Recalling Eqs. (15,16), thiscorresponds to locking the hole at − Q so as to produce amolecule state with Q M = 0. We have checked that suchpronounced hole distribution at − Q applies for generalexcited momenta k and k (cid:48) . Together with the energyresemblance as shown in Fig.1, this serves as a strong ev-idence that M (0) indeed can well approximate P ( k F )in the strong coupling limit. Q / k F ( EE ( )) / E F FIG. 3. (Color online). (a) Energy dispersion of P ( Q )(solidlines) in 3D at various couplings (from top to bottom)1 / ( k F a s ) = 0 . , . , . , . , . , . , . , .
2, shifted bythe value at Q = 0. The rectangular point mark the positionof maximum energy, and the small black dots show the ener-gies of M ( Q M ), with | Q M | shifted by k F in order to comparewith the energies of P ( Q ). Here Q = | Q | . Given the fact that the molecule M (0) is nothing butjust a good approximation for the finite-momentum state P ( k F ), now we are ready to investigate the polaron-molecule competition by examining the energy disper-sion E ( Q ) from P ( Q = Q e z ). In Fig.3, we show E ( Q )for various coupling strengths. We can see that for weakcoupling 1 / ( k F a s ) (cid:46) .
5, there is only one minimum inthe dispersion and Q = 0 polaron is the only groundstate. As increasing 1 / ( k F a s ) to ∼ . Q = k F as a metastable state.At 1 /k F a s = 0 .
91, the two minima has the same en-ergy, signifying a first-order transition between Q = 0and Q = k F states, or between polaron and molecule states given that M (0) can well approximate P ( k F )near the transition (see black dots). At even strongerattractions, the local minimum at Q = 0 is bendeddownwards and the only stable state is at Q = k F , themolecule state. Here with V-2ph method, the doubleminima structure of the dispersion appears in the cou-pling window 1 /k F a s ∈ (0 . , . k F from both V-2ph and V-1ph methods. One cansee that under V-2ph, the critical point for the transitionis at (1 /k F a s ) c = 0 .
91, very close to the critical pointobtained from Monte-Carlo[13] and diagrammatic[14]methods. Clearly, this critical point shifts to weaker cou-pling side as compared to the value (1 /k F a s ) c = 1 . M (0) under V-2ph and M (0) under V-1ph) can well approximate the Q = k F states, and thusthe transition between Q = 0 and Q = k F states canindeed be interpreted as the polaron-molecule transition.This sets the nature of such first-order transition betweenpolaron and molecule. k F a S Z P (0) P (0) P ( k F ) P (0.9 k F ) P (0.95 k F ) P ( k F ) FIG. 4. (Color online). Residue Z as a function of couplingstrength 1 / ( k F a s ) for different momentum states using V-1phor V-2ph methods. In Fig.4, we further show the residue Z = | ψ | as afunction of 1 / ( k F a s ) for different momentum ( Q = | Q | )states. For zero-momentum Q = 0, we can see that Z is insensitive to the variational approach used (V-1phor V-2ph). However, for momentum Q = k F , Z canchange a lot between V-1ph and V-2ph methods, or be-tween P ( k F ) and P ( k F ). Moreover, for a given cou-pling strength, Z can be greatly reduced by increasingthe momentum Q . In particular, as Q approaches k F ,the reduction of Z is quite substantial in the weak cou-pling limit, implying the failure of quasi-particle picturefor Q ∼ k F state in this regime.Finally, we comment on the huge ground state degener-acy in the molecule limit. As shown above, in the strongcoupling limit, the ground state of the system has a fi-nite momentum with amplitude | Q | = k F . Since thedirection of Q can point to any direction in 3D space,this state has a huge SO (3) degeneracy. Physically, thisis because the impurity can pair with any fermion atthe Fermi surface to form a zero-momentum molecule, orequivalently, the vector k F in Eq. (16) can point to anydirection on the Fermi surface. Such huge degeneracy re-sembles the single-particle ground state degeneracy underan isotropic spin-orbit coupling[42, 43]. Similarly, thereis an important consequence of such degeneracy, namely,it greatly enhances the density of state (DOS) at low-energy space near the ground state (or near | Q | ∼ k F ).One can expect that such degeneracy can significantlyenhance the molecule occupation in realistic system withfinite impurity density and at finite temperature, as wewill discuss later. B. 2D
For 2D Fermi polaron system, we have carried out nu-merical simulations using both the V-2ph and V-Gphmethods and found consistent results. We use the dimen-sionless coupling strength ln( k F a d ) to characterize theinteraction effect. In our numerical calculations, we setthe momentum cutoff as k c = 30 k F in V-2ph method. InV-Gph method, we discretize the whole space to 40 × N = 49and the momentum cutoff as k c = 8 k F . ln ( k F a d )0.80.40.00.40.8 ( EE b ) / E F M (0) P ( k F ) M (0) P ( k F ) P (0) P (0) FIG. 5. Energy comparison in 2D. All energies are shifted by E b = − / ( ma d ) in order to highlight the difference. In Fig.5, we show the energies of P ( k F ), P (0) and M (0) as functions of ln( k F a d ), in comparison with the energies of P ( k F ), P (0) and M (0). One can see thatsimilar to the 3D case, the molecule state M (0) alwayshas a higher energy than P ( k F ); however, in the strongcoupling regime ln( k F a d ) < − .
7, the two states are in-distinguishable in energy, indicating that the former canserve as a good approximation for the latter. Moreover,we note from Fig.5 that the V-2ph method can producevisibly lower energy for both polaron and molecule statesthan V-1ph. For instance, within one p-h framework, P (0) always has a lower energy than P ( k F ) and M (0).However, by adding two p-h excitations, the molecule en-ergy can be significantly reduced. In the strong couplinglimit ln( k F a d ) → −∞ , the energies of P ( k F ) and M (0)(from V-2ph) both approach E b − E F , much lower thanthe asymptotic energy E b + E F of P ( k F ) and M (0)states (from V-1ph) in this limit. This shows a significantrole played by p-h excitations in 2D, as will be discussedlater.In Fig.6(a,b), we plot out the energy dispersions atvarious couplings from both V-2ph and V-Gph methods,from which we see that the results from the two methodsare qualitatively consistent. Namely, as increasing the at-traction between impurity and fermions, there is a first-order transition at certain coupling strength where theground state of the system switches from total momen-tum Q = 0 to Q = k F . Near the transition and beyond,the dispersion near Q ∼ k F can indeed be well approxi-mated by the molecule state M ( Q M ) near Q M ∼
0, seetriangular points in Fig.6(a). To see more clearly thetransition point, we show the energies at these two mo-menta as functions of coupling strengths in Fig.7. Thecritical coupling at which the ground state switches from Q = 0 to Q = k F is ln( k F a d ) c ≈ − .
97 from V-2phmethod, and − .
81 from V-Gph. In comparison, thecritical coupling obtained from the comparison between P (0) and M (0) is ln( k F a d ) c ≈ − . Q = 0 and | Q | = k F . Since Q can pointto any direction in the 2D plane, there will be a SO (2)ground state degeneracy in the molecule regime with afixed | Q | = k F .We note that the polaron-molecule competition in 2Dhas also been investigated by Monte-Carlo methods[24–26]. Among these studies, Refs.[24, 25] have claimed atransition while Ref.[26] has claimed a smooth crossoverbetween polaron and molecule. However, we note thatin Ref.[26] the number of majority fermions used in theweak coupling regime is different (by one) from that inthe strong coupling regime. This automatically changethe total momentum of the system by k F and thus theconclusion of smooth crossover is not for the same sys-tem with a fixed total momentum. Moreover, Fig.7shows that the shifted energy E ( k F ) − E b evolves non-monotonically with ln( k F a d ), different from the 3D case(see Fig.1). In particular, in weak coupling regime it FIG. 6. (a)Energy dispersion of P ( Q )(solid line) in 2Dat various couplings (from top to bottom) ln( k F a d ) = − . , − . , − . , − . , − .
2, shifted by the values at Q = 0.The small black dots show the energies from M ( Q M ), with | Q M | shifted by k F in order to compare with the ener-gies of P ( Q ). (b) Energy dispersion from V-Gph methodat various couplings (from top to bottom) ln( k F a d ) = − . , − . , − . , − . , − . , − .
0, again shifted by the valuesat Q = 0. Here Q = | Q | . shares similar functional lineshape as E (0) − E b , whichmay also cause the confusion that the polaron-moleculeconversion in 2D is a smooth crossover. C. 1D
We will briefly go through the 1D case, where the cou-pling strength is governed by a dimensionless parameter k F a d , with a d = − / ( mg ) the 1D scattering length.In our numerical calculations, we are able to computewith different momentum cutoff k c and finally obtain theresults for k c → ∞ by extrapolation.In Fig.8, we show the energy dispersion at weak andstrong couplings from V-2ph method (solid lines), in ln ( k F a d )0.70.60.50.40.30.20.1 ( EE b ) / E F Q = k F , V GphQ = k F , V phQ = 0, V GphQ = 0, V ph FIG. 7. Energies of Q = 0 and Q = k F states as functions ofcoupling strengths in 2D, obtained from both the V-2ph andV-Gph methods. All energies are shifted by E b = − / ( ma d )in order to highlight the difference. comparison with those from the exact Bethe ansatzsolutions[45–47] (dashed lines). It is found that the twomethods give consistent conclusion that there is no tran-sition in the system and the ground state is always atzero momentum Q = 0, on the contrary to 2D and 3D.Remarkably, the energy from V-2ph method fits the ex-act solution remarkably well in the weak coupling limit,see Fig.8(a). For strong coupling (see Fig.8(b)), the devi-ation between the two energies is attributed to the insuf-ficiency of V-2ph method and thus more p-h excitationsare required. In the strong coupling limit, the groundstate energy(at Q = 0) is given by E → E b − E F , signi-fying a smooth crossover to molecule regime for the 1Dsingle-impurity system. D. Discussion
In above we have shown that the presence of polaron-molecule transition sensitively depends on the dimensionof the system, namely, there is such a transition in 3Dand 2D but not in 1D. In the following we point outsome intrinsic reasons for this sensitive dependence ondimensionality.Let us start from the weak coupling regime that canbe smoothly connected to the non-interacting limit. Inthis regime one can easily anticipate that the groundstate should be the Q = 0 polaron, describing a zero-momentum impurity dressed with a limited number ofp-h excitations of background fermions. Therefore, thekey question is to find out the ground state in the strongcoupling regime, which determines whether there is atransition (switch of ground state) as the attraction is0 Q / k F ( EE b ) / E F (a) V phBA Q / k F ( EE b ) / E F (b) V phBA FIG. 8. Dispersion for 1D system at different couplings k F a d = 1(a) and 0 . E b = − / ( ma d ). increased from weak to strong. Since the molecule state(belong to Q = k F sector) is an important candidate forthe ground state in strong coupling regime, in the fol-lowing we will analyze how its energy depends on the di-mension. In particular, we will highlight the roles playedby the Pauli-blocking effect and the p-h excitations ofbackground fermions in different dimensions.Let us consider the bare molecule M (0) and analyzethe Pauli-blocking effect to the molecule energy. For d -dimensional system, it has been shown that in the strongcoupling or deep molecule regime (when | E b | → ∞ ), themolecule energy (with respect to the energy of | FS (cid:105) N )is[21] E M = E b − E F + c d E F (2 E F / | E b | ) ( d − / , (28)with c d is a positive constant. One can see that in deepmolecule regime, the shift of E M from E b − E F is negligi-ble for 3D, a constant ( ∝ E F ) for 2D and an exceedinglylarge number for 1D. It means that the effect of Pauliblocking by the underlying Fermi sea is very little for3D molecule, but gets more and more significant if go tolower dimensions. This is because in 3D, the phase spaceblocked by the Fermi sea is negligible as compared to thefull phase space, while in lower dimensions the difference between the two phase spaces is not that substantial. Asa result, the molecule becomes energetically less favoredas going to lower- d systems, which may serve as a crucialreason for the absence of polaron-molecule transition in1D.Moreover, we note that the p-h excitations also becomemore and more important to affect the molecule energyas going to lower- d systems. As one can see from the en-ergy comparison between M (0) and M (0) in Fig.1 andFig.5, adding one more p-h excitations will reduce themolecule energy by a small proportion of E F in 3D, butby a visible constant (as large as ∼ E F ) in 2D. WithinV-2ph, the molecule energy in 3D and 2D in strong cou-pling regime all approaches to E b − E F (with respect tothe energy of | FS (cid:105) N ), which is the lowest energy one canimagine for the system. Therefore, there must be a tran-sition between polaron ( Q = 0) and molecule ( Q = k F ) atcertain intermediate coupling strength for 3D and 2D. Onthe contrary, for 1D system, the ground state is alwaysat Q = 0(see Fig.8), and in strong coupling regime theenergy at Q = 0 approaches E b − E F while at Q = k F approaches E b − E F /
2. It means that in 1D, the po-laron to molecule conversion is completed entirely withinzero momentum sector, and thus the process is a smoothcrossover rather than a transition.Above analysis show that it is important to considerthe effects of Pauli-blocking and p-h excitations in lowerdimensional Fermi polaron systems. The interplay ofthese effects significantly influence the presence or ab-sence of polaron-molecule transitions in different dimen-sions.
IV. POLARON-MOLECULE COEXISTENCEAND SMOOTH CROSSOVER IN REALISTICFERMI POLARON SYSTEMS
In our previous work[36], we have used the single-impurity result from V-1ph method to qualitativelyexplain the polaron-molecule coexistence and smoothcrossover as observed in recent 3D Fermi polaronsystems with a finite impurity density and at finitetemperature[8]. Recently, a theoretical study[48] ex-tended the finite-momentum V-1ph method to finite-temperature and explained the smooth crossover betweenpolaron and moleclue. Here we will refine the explanationby utilizing the results from V-2ph and incorporating thetrap effect through local density approximation(LDA).In our calculation, we will take the same temperature( T = 0 . T F ) and the same impurity concentration asused in the experiment[8].As seen from Fig.3, the double minima structure ofthe single-impurity dispersion provides a clear pictureof polaron-molecule coexistence under a finite impuritydensity and at finite temperature. Same as Ref.[36], wewill neglect the thermal distortion of majority Fermi seaand mediated interactions between the same spins, andonly focus on two possible configurations for the dressed1impurities: one is “polaron” nearby zero-momentumand obeying fermionic statistics; the other is “molecule”nearby | Q | = k F and obeying bosonic statistics.Now we discuss how to separate polaron and moleculein the dispersion curve. In the polaron-molecule coex-istence regime 1 / ( k F a s ) ∈ (0 . , . Q c , that can be chosen as the location of energy max-imum between Q = 0 and Q = k F , as marked bysquares in Fig.3. After defining Q c , the energy cutofffor the thermal excitation of impurities is also fixed as E c = E ( Q c ). More specifically, the polaron occupies at | Q | < Q c and the molecule occupies at | Q | > Q c withenergy cutoff E c . The value of Q c outside the coexistenceregime is defined as follows. In the weak coupling regime1 / ( k F a s ) < .
5, the impurities occupy as polarons andthere is no molecule distribution; in this case, we define Q c as the polaron momentum when its residue reduces to0 .
01. In the strong coupling regime, 1 / ( k F a s ) > .
2, thepolaron vanishes and all impurities occupy as molecules;in this case we simply take Q c = 0.Next we incorporate the trap effect. For the majorityfermions, we use the zero-temperature density distribu-tion as the approximation: n ↑ ( r ) = 16 π (2 m [ µ ↑ − V ( r )]) , (29)where V ( r ) = mω r / µ ↑ = k F ↑ (0) / (2 m ) = (6 N ↑ ) / ω is chemical potential of ma-jority fermions at the center of trap. Under LDA,one can define the local Fermi momentum as k F ↑ ( r ) =(6 π n ↑ ( r )) / , which determines the local occupation of polaron and molecule states.Under above assumptions, the local impurity densitycan be written as (with θ ( x ) step function) n ↓ ( r ) = (cid:90) d Q (2 π ) [ n F ( E ( Q , r ) , µ ↓ , V ↓ ( r )) θ ( Q c − | Q | )+ n B ( E ( Q , r ) , µ ↓ , V ↓ ( r )) θ ( | Q | − Q c ) θ ( E c − E ( Q , r ))](30)where n F/B ( E, µ ↓ , V ↓ ( r )) = [1 ± exp( E − µ ↓ + V ↓ ( r ) k B T )] − and V ↓ ( r ) = V ↑ ( r )(1 − EE F ) is the renormalized trap potentialfelt by impurity atoms [8, 10]. Note that because of the r -dependence of local k F ↑ , the quantities E, Q c , E c inthe above equation all depend locally on r .Following the definition of averaged density ratio in [8]: (cid:104) n ↓ n ↑ (cid:105) = (cid:82) d r n ↓ ( r ) · n ↓ ( r ) n ↑ ( r ) (cid:82) d r n ↓ ( r ) , (31)in our calculation we will fix (cid:104) n ↓ n ↑ (cid:105) = 0 .
15 as in Ref.[8],which is used to determine µ ↓ in Eq. (30). Then we gofurther to calculate trap averaged residue, contact, andthe polaron energy by¯ Z = (cid:82) d r n ↓ ( r ) Z ( r ) (cid:82) d rn ↓ ( r ) , ¯ C = (cid:82) d r n ↓ ( r ) C ( r ) (cid:82) d rn ↓ ( r ) , ¯ E pol = (cid:82) d r n ↓ ( r ) E pol ( r ) (cid:82) d r n ↓ ( r ) , (32)where Z ( r ) , C ( r ) , E pol ( r ) are : Z ( r ) = 1 n ↓ ( r ) (cid:90) d k (2 π ) Z ( k ) n F ( E ( k , r ) , µ ↓ , V ↓ ( r )) · θ ( Q c − | k | ) ,C ( r ) = 4 πm n ↓ ( r ) k F (cid:90) d k (2 π ) dE ( k , r ) d (1 /k F a s ) n F ( E ( k , r ) , µ ↓ , V ↓ ( r )) θ ( Q c − | k | )+ 4 πm n ↓ ( r ) k F (cid:90) d k (2 π ) dE ( k , r ) d (1 /k F a s ) n B ( E ( k , r ) , µ ↓ , V ↓ ( r )) θ ( | k | − Q c ) θ ( E c − E ( k , r )) ,E pol ( r ) = E ( k = 0 r ) . (33)In Fig.9(a,b,c), we show the calculated ¯ Z , ¯ C , and ¯ E p (see orange circles and lines) as functions of couplingstrength. One can see that they fit reasonably well tothe experimental data in Ref.[8](shown as blue circleswith error bars). In particular, compared to the theoryprediction based on V-1ph plus LDA[8] (see black dashedlines), our prediction of ¯ Z is visibly lower and the pre-diction of ¯ C is visibly higher, which give a better fit tothe experimental data near the unitary regime. Thesevisible improvements brought by the V-2ph method can be attributed to the following two reasons:First, compared to the V-1ph method, the inclusion oftwo p-h excitations in V-2ph does not change too muchthe polaron energy but reduces the molecule energy con-siderably, see Fig.1. A direct consequence of this changeis to move the polaron-molecule transition point and theircoexistence region to weaker coupling side. The otherconsequence is to enhance the molecule occupation in theco-existence regime. These two effects both contribute toreducing the residue ¯ Z and increasing the contact ¯ C for2 FIG. 9. (Color online). Residue ¯ Z (a), contact ¯ C (b) andthe polaron energy ¯ E p (c) as functions of coupling strengthgiven the realistic experimental condition in Ref.[8]. The bluecircles with error bar shows the experimental results in Ref.[8].Black dashed-dot lines show the theoretical prediction basedon V-1ph plus LDA (see theory in Ref.[8]). Orange circlesand lines show our results based on V-2ph plus LDA. Here k F is the Fermi momentum of majority fermions at the trapcenter. a given coupling strength.Secondly, we have a different classification and sam-pling scheme for polaron and molecule as compared toRef.[8]. In particular, we note that the momentum ofmolecule state in Ref.[8] is near zero rather than k F .This significantly underestimates the molecule occupa-tion number due to the small density of state(DoS) near Q ∼
0. In comparison, in this work we point out thatthe molecule actually stays around | Q | ∼ k F with a huge SO (3) degeneracy and thus a much larger DoS at low en-ergy. This will help to enhance the molecule occupationfurther and lead to a smaller ¯ Z and a larger ¯ C than thetheory prediction in Ref.[8]. Again, we emphasize that the recognization of the molecule momentum is cruciallyimportant for correctly evaluating its occupation in thepolaron-molecule coexistence regime.Finally, we note that our results do not fit well to theexperimental data of ¯ Z near its zero crossing. As shownin Fig.9(a), ¯ Z from our prediction continuously dropsto nearly zero around 1 / ( k F a s ) ∼ .
9, very close to thepolaron-molecule transition point ( ∼ .
91) for the single-impurity system. Nevertheless, the experimental data of¯ Z near its zero-crossing seem to be better fit by the V-1phprediction. The underlying reason is unclear and awaitsmore exploration. V. SUMMARY
In this work we have investigated the polaron andmolecule physics in 3D, 2D and 1D Fermi polaron sys-tems by utilizing a unified variational ansatz with up totwo p-h excitations(V-2ph). Moreover, we have checkedthe reliability of our results by comparing with the resultfrom the variational method in 2D based on the Gaus-sian sample of high-order p-h excitations(V-Gph), andwith the result of Bethe-ansatz solutions in 1D. Thesemethods produce consistent conclusions, which are sum-marized as follows:(I) There exists a first-order transition for single-impurity system in 3D and 2D as the attraction betweenthe impurity and fermions increases. The nature of suchtransition lies in an energy competition between differenttotal momenta Q = 0 and | Q | = k F , with k F the Fermimomentum of majority fermions. From V-2ph method,the transition point is at 1 / ( k F a s ) = 0 .
91 for 3D and atln( k F a d ) = − .
97 for 2D. In 1D, there is no transitionand the ground state is always at Q = 0 for all couplings.The underlying reason for the presence/absence of suchtransition is analyzed to be closely related to interplayeffect of Pauli-blocking and p-h excitations in differentdimensions.(II) The literally proposed molecule state has an in-complete variational space in terms of p-h excitations,but can serve as a good approximation for the Q = k F state in strong coupling regime. Due to the finite momen-tum, the ground state in the molecule regime has a hugedegeneracy ( SO (3) for 3D and SO (2) for 2D), which cangreatly enhance the low-energy density of state for themolecule occupation in realistic Fermi polaron systemswith a finite impurity density. Our theory well explainsthe coexistence and smooth crossover between polaronand molecule as observed in recent 3D Fermi polaronexperiment[8], and also produces quantitatively good fitsto various physical quantities measured in the unitaryregime of the system.In the future, it would be interesting to extend ourtheory to various other impurity systems, such as withdifferent mass ratios between the impurity and the back-ground, as well as the regime with strong three-body cor-relations where the trimer physics can dominate.3 Acknowledgements.
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