Enhancement of boson superfluidity in a one-dimensional Bose-Fermi mixture
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Enhancement of boson superfluidity in a one-dimensional Bose-Fermi mixture
Chenrong Liu, Yongzheng Wu, Jie Lou, ∗ and Yan Chen † Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China The 32nd Research Institute of China Electronics Technology Group Corporation, Shanghai 200433, China (Dated: February 16, 2021)We examine the effect of boson-fermion interaction in a one-dimensional Bose-Fermi mixture byusing the density matrix renormalization group method. We show that the boson superfluidity isenhanced by fermions for a weak boson-fermion coupling at an approximate integer boson fillingfactor (e.g., 0 . ≤ ρ b ≤ . I. INTRODUCTION
Bose-Fermi mixture, which has been recently realizedin ultracold atoms experiments [1–3], represents a newsystem for studying strongly correlated many-bodyphysics. The physics of the one-dimensional Bose-Fermimixture was extensively studied based on theories be-yond the mean-field approximation or the perturbationtheory [4–13]. In such a mixture, the effective interac-tion induced by the boson-fermion interaction plays anessential role in the emergence of a new phase and newphysical effect. For the effect of bosons on fermions,Kinnunen et al. [14] presented a strong coupling theoryfor the critical temperature of p -wave pairing betweenspin-polarized (spinless) fermions immersed in a Bose-Einstein condensate. Quite recently, a quantum MonteCarlo study on a two-dimensional Bose-Fermi mixturerevealed that an effective p -wave interaction betweenfermions will be induced as far as the bosons are ina superfluid state, and the composite fermion pairsmay appear at low temperatures [15]. For the effectof fermions on bosons, an investigation of the effectiveinteraction between bosons was previously carried outin a two-dimensional Bose-Fermi mixture by using thelinear response theory [16]. Their results show that thesecond-order term of the effective boson interaction canbe attractive. This attractive boson effective interactionwas also investigated using phenomenological bosoniza-tion and Green’s function techniques in one dimension[17]. Numerically, a quantum Monte Carlo study of thisinduced boson-boson interaction was proposed in thenormal fermi state for a narrow boson-fermion couplingregion [18], and they show that the boson superfluidphase region expands. Moreover, the fermion-mediatedinteractions between bosonic atoms have been observedin experiments. Their results indicate that when aBose-Einstein condensate of Cesium atoms is embedded ∗ [email protected] † [email protected] in a degenerate Fermi gas of lithium atoms, interspeciesinteractions can give rise to an attractive boson-bosoninteraction [19]. However, few theoretical works havebeen done on how the particle filling factor and fermionstate influence the effective boson-boson interaction.It is well known that in the one-dimensional Bose-Hubbard model, there exists a quantum phase transitionfrom superfluid phase to the Mott insulator phase whenthe onsite boson repulsive interaction U bb switches on[20, 21]. On the other hand, there is a metal-insulatorphase transition in the one-dimensional spinless Fermi-Hubbard model with a repulsive nearest-neighborinteraction V ff at half fermion filling [22]. In theweak-coupling limit U bf →
0, one can expect thatbosons and fermions are decoupled, e.g., the groundstate of the mixture is a combination of boson super-fluid and fermi metal. Otherwise, the mixture is ina phase separation state for the strong coupling limit U bf → ∞ (see Appendix). The effective boson-boson(fermion-fermion) interaction, induced by the exchangeof fermions (bosons), leads to new phases and new effectsbetween these two limits.In this paper, we mainly study the physical effect in-duced by the effective boson-boson interaction in a one-dimensional Bose-Fermi mixture. We find that the bo-son superfluidity is enhanced at a weak repulsive onsiteboson-fermion coupling (e.g. U bf /t < .
0) when theboson filling factor ρ b is 1.0 or extremely close to thisinteger (0 . ≤ ρ b ≤ . t is the boson-bosonand fermion-fermion nearest-neighbor hopping ampli-tude. The correlation functions and fermi momentumdistribution are calculated at different filling factors us-ing the density matrix renormalization group (DMRG)method. To analyze how different fermion states affectthe enhancement of boson superfluidity, the interactingand non-interacting fermions are both considered in ourmodel. In the strong boson-fermion coupling limit, theboson density distribution in real space is also addressed. II. MODEL AND METHOD
We consider a mixture of spinless fermions and bosonsin one dimension. The canonical ensemble Hamiltonianreads H = H t + H int ,H t = − X h ij i (cid:16) t b b † i b j + t f c † i c j + H. c. (cid:17) , (1) H int = U bb X i n bi (cid:0) n bi − (cid:1) + U bf X i n bi n fi + V ff X h ij i n fi n fj . where t b ( t f ) is the boson-boson (fermion-fermion)nearest-neighbor hoping amplitude, b i ( c i ) is the boson(fermion) annihilation operator, U bb ( U bf ) is the onsiteboson-boson (boson-fermion) repulsive interaction, V ff is Coulomb repulsive interaction between the nearest-neighbor fermions, and n bi ( n fi ) is the onsite boson(fermion) number operator.In our calculations, t b = t f = t is set to 0.1. To studythe effect of fermions on bosons, we fixed U bb /t = 3 . ρ b = 1 because U bb /t = 3 . U c bb /t ≈ . V ff /t < . V ff /t > . V ff /t = 0) and interactingfermions ( V ff /t > ρ b ( f ) = N b ( f ) /N s . Besides, the latticesize N s is set as 62 for fermion half filling ρ f = 1 / N s = 60 for fermion quarter filling ρ f = 1 / n cut b = 2per site on account of the boson repulsive interaction U bb .We use the DMRG method [25] to perform the numer-ical calculations. The maximum number of kept states is4500, and we use up to 40 sweeps. Besides, we add somenoises in the first nine sweeps to avoid a metastable state.Under the setup, the largest truncation error during thesweep is ∼ − , and the energy convergence precision is ∼ − . All the calculations were conducted using theITensor library [26] with periodic boundary condition. III. NUMERICAL RESULTS
In this part, we will discuss our numerical resultsin two subsections. One is for free fermions with V ff = 0, and the other is for interacting fermions with V ff >
0. In the free fermion case, we mainly studythe fermion-mediated effective boson-boson interactionand investigate the effect of boson-fermion coupling U bf at different particle filling factors. We focus on therelations between the boson-boson correlation functionand the fermion state in the interacting fermion section. A. Free fermions: V ff /t = 0 In order to study the influence of the onsite boson-fermion repulsive interaction. One of the primary effortshere is to obtain the boson-boson correlation functionthat corresponds to the boson superfluid order. Thecorrelations h b † i b j i as a function of lattice distance r ij were calculated for several different values of U bf /t atboson filling factor ρ b = 1 and fermion filling factor ρ f = 1 / U bf /t .
0, a power-law decay behavior of h b † i b j i is observed, which indicates that the boson is ina superfluid state. The power-law relation also revealsthe so called quasi-long-range order which means thereis no truly boson superfluidity in one dimension. When U bf ≫ t , we expect that the superfluid should be brokenby the fermions so that h b † i b j i decay exponentially. Inour simulations, a density-wave state is established at U bf /t = 7 . U bf /t is increased from 0 to 3.0. It ap-pears that the boson-boson correlation h b † i b j i at distance r = N s / U bf /t is increased from 0 to3 .
0, and then this value decreases when U bf /t continueto grow up. In other words, the boson-boson correlationfunction decays more slowly if U bf /t goes from 0 to 3 . h b † i b j i with a power-law relation, h b † i b j i = Ar αij , (2)where α is the decay exponent, and A denotes a con-stant. Thus, α can be used to describe the strength ofthe boson superfluid order. These exponents are shownin Fig. 1(b). As one can see, α indeed increases in region0 < U bf /t < . U bf /t = 3 . U bb andfavor superfluidity. A simple picture enables this to beunderstood as follows: for the repulsive boson-fermion . . . .
81 0 5 10 15 20 25 30 35(a) − . − . − . − . . . . . . . . .
81 0 10 20 30 40 50 60 70(c) 00 . . .
18 0 0 . − . − . − . − . − . h b † i b j i r ij U bf /t = 0 . . . . . . . . α U bf /t h b † i b j i r = N s / U bf /t h b † i b j i r ij N s = 6290122 ∆ α /N s α U bf /tρ b = 1 . , ρ f = 1 / ρ b = 1 . , ρ f = 1 /
41 bh , ρ f = 1 /
24 bh , ρ f = 1 / ρ b = 0 . , ρ f = 1 / FIG. 1. (Color online) Boson-boson correlation functions and its decay exponents. (a)-(c) are calculated with fixed particlefilling factors, e.g. ρ b = 1 . ρ f = 1 /
2. (a) The boson-boson correlation functions h b † i b j i vs lattice distance r ij for differentvalues of U bf /t , the insert figure is plotted by using a logarithmic scale for both the x -axis and the y -axis. (b) The power lawdecay exponents α of (a), the insert figure is the value of h b † i b j i at the distance r = N s /
2. (c) Finite size effect of boson-bosoncorrelation function. The full line and the line-symbol indicates the boson-fermion coupling is 2.8 and 0 respectively. The insertfigure shows the difference of α between U bf /t = 2 . U bf /t = 0, the dash line is a fitting function. (d) α as a function of U bf /t at different boson and fermion filling factors. Here, 1bh represents ρ b = N b /N s = ( N s − /N s = 0 .
984 and 4bh labels ρ b = N b /N s = ( N s − /N s = 0 . coupling, fermions are repelled from the bosons, andthus the bosons feel an attractive interaction where thefermion density is quite low [19]. Therefore, this inducedattractive boson-boson interaction would be determinedby the boson-fermion interaction.To check whether the enhancement is due to the finitesize effect, we calculate the boson-boson correlationfunction for three different lattice sizes at a fixed bosonand fermion filling factor. The results are shown inFig. 1(c). Here, the separation of the two correlationfunctions is expanded for a larger lattice size, and theinsert figure also indicates that the difference ∆ α is increased with a growing lattice size. The finite-sizescaling of ∆ α which is presented in the insert figure ofFig. 1(c) predicts ∆ α = 0 .
12 when
N s → ∞ . This resultclearly shows that the observed enhancement of the bo-son superfluidity does not come from the finite size effect.If there are only a few bosons in the mixture, thenthe boson can not drive the fermionic system. The totalnumber of bosons would have an important role in theformation of the enhancement. We examine α against U bf /t at different boson and fermion filling factors. Westart with an integer boson filling factor ρ b = 1 and thengradually reduce the total boson number. Numericalresults are presented in Fig. 1(d). One can observe that α is increased in a weak bose-fermi coupling region whenthe boson filling factor is an integer( ρ b = 1) or extremelyclose to this integer. For boson filling factor ρ b = 1, thisincrement behavior exists at both a half and a quarterfermion filling factor. But for a non-integer boson fillingfactor, e.g. ρ b = 0 . α decreases monotonically inthe whole range of U bf /t . These features show thatthe boson superfluidity enhancement can only exist at ρ b = 1 . U bf is added, the bosons would feel a repulsiveinteraction on the fermion occupied site, and thus theboson number fluctuation among the sites is induced.In other words, the Mott phase region is shrunk, andthus the enhancement is observed. However, this is onlytrue for a boson filling factor close to an integer. At afractional boson filling factor( ρ b = 0 . . . . .
81 0 0 . . . . h n f k i k/π U bf /t = 0 . . . . . . . . . FIG. 2. (Color online) The fermion particle distribution inmomentum space at a fixed boson filling factor ρ b = 1 . ρ f = 1 /
2. The blue dashed vertical lineis the position of Fermi momentum k F /π = 1 / The effect of bosons on fermions is also addressed.We calculate the fermion momentum distribution h n fk i inFig. 2 at a fixed boson filling factor ρ b = 1 . ρ f = 1 /
2. The h n fk i is defined as, h n fk i = 1 N s X i,j e − i~k · ( ~r i − ~r j ) h c † i c j i , (3) which is the Fourier transformation of the equal timeGreen’s function [29], and gives the occupation offermions in momentum space.As the violet line-cross of Fig. 2 shows, all the fermionsare free when U bf /t is zero and this leads to the factthat h n fk i has a discontinuity ( Z = 1) at the correspond-ing Fermi momentum k F /π = 1 /
2. When U bf /t turnson and rises up, instead of a jump, one could finds apower-law singularity at k = k F . This is the feature of aTomonaga-Luttinger liquid (TLL) [30], and it means thatthere exists an effective interaction between fermions,which is induced by the exchange of bosons. If the boson-fermion coupling is too strong, e.g., U bf /t > .
0, thenthe fermions and bosons can not occupy the same site,and the fermions exhibit a charge density wave (CDW)pattern in real space (See Fig. 5(b) in Appendix). Conse-quently, the singularity does not exist as the empty blacktriangle-line of Fig. 2 shown. This evidence shows thatalthough there is no fermion-fermion interaction in theHamiltonian, there is still an induced phase transitionfrom a TLL phase to a CDW phase when we switch on U bf . B. Interacting fermions: V ff /t > In this section, we discuss the effect of interactingfermions on bosons by setting the fermion interaction V ff to be non-zero. The particle filling factors here are fixedat ρ b = 1 and ρ f = 1 /
2. Under these conditions, wecan tune the fermion state from a metal to an insulatorand examine how different fermion states affect the bo-son superfluidity. The result is presented in Fig. 3(a).Interestingly, the enhancement of boson superfluidity isobserved in both situations of the fermion state. Forthe fermion insulator case, the scattering of fermions stillcan expand the boson superfluid phase region, and thusthe boson superfluidity is enhanced. However, for strongboson-fermion coupling, bosons are localized.If fermions are in a deep insulating region, fermionsare pinned on the lattice site. The bosons should havea density wave feature in real space at a finite U bf .Therefore, the boson superfluid order and density waveorder can coexist. We examine this by calculating theboson correlation function and boson density distribu-tion. Here, we set fermion-fermion hopping t f = 0 andfermion-fermion nearest-neighbor repulsive interaction V ff /t b = 100. The results are shown in Fig. 3(b)and (c). For a weak boson-fermion coupling U bf (e.g. U bf /t b = 1 .
0) a boson-boson correlation function withpower-law decay is observed in Fig. 3(b) indicatingthat the bosons are in a superfluid state. On the otherhand, its density distribution in real space (blue curvein Fig. 3(c)) clearly shows a density wave feature. Theseresults indicate that the boson superfluid order coexistswith the density wave order in the deep fermion insulator − . − . − . . . . . .
21 1 10(b)0 . . . . α U bf /tV ff /t = 1 . . h b † i b j i r ij U bf /t b = 0 . . . . h n b i i r i FIG. 3. (Color online) The exponents, correlation functionsand particle density distribution are calculated at finite V ff .(a) α as a function of U bf /t for fermions in a metallic state(black line) and an insulator state (red line). (b) Boson-bosoncorrelation functions and (c) boson density distribution in realspace are evaluated in a deep fermion insulator region wherethe fermion hopping t f is set to be zero and V ff /t b = 100.The same line color and symbol in (b)-(c) represents the samevalue of U bf . region. IV. CONCLUSIONS
We have studied the enhancement of boson superflu-idity in a one-dimensional Bose-Fermi mixture for freeand interacting fermions at different particle filling fac-tors. Close to an integer boson filling factor (e.g., 0 . ≤ ρ b ≤ . U bf although V ff is zero and thus lead-ing to a phase transition from a fermion metallic stateto a fermion CDW state. The boson-boson correlationfunction and its density distribution are also calculatedfor interacting fermions in a very deep fermion insulatorregion with t f = 0 and V ff /t b = 100, and we find thatthe boson superfluid order and density wave order coex- ist there. Recently, the boson-fermion interactions canbe tuned in Rb - K mixtures via Feshbach resonancetechnology [31]. Our predictions could be tested in futureexperiments.
V. ACKNOWLEDGMENTS
We are thankful for the useful discussions with T. K.Lee, X. W. Guan, and C. S. Ting. This work is sup-ported by the National Key Research and DevelopmentProgram of China (Grants Nos. 2017YFA0304204 and2016YFA0300504), the National Natural Science Foun-dation of China Grant No. 11625416, and the ShanghaiMunicipal Government (Grants Nos. 19XD1400700 and19JC1412702).
Appendix A: Comparing with quantum Monte Carlomethod(QMC). . . . . . h b † i b j i , | h c † i c j i | | r i − r j | QMC-bcorrDMRG-bcorrQMC-fcorrDMRG-fcorr
FIG. 4. (Color online) Comparing with quantum MonteCarlo method(QMC). The bcorr (fcorr) represents boson-boson (fermion-fermion) correlation function. QMC data isfrom Ref.[9].
In order to compare with QMC, the parameters inHamiltonian Eq.(1) are set as t b = t f = 1 . N s = 70, U bb = 5 . V ff = 0 and U bf = 1 .
0. The particle fillingfactors are fixed at ρ f = ρ b = N s /
2. These results areshown in Fig. 4. Because of the fractional 1 / Appendix B: Phase separation and induced fermiondensity wave state in the free fermion case. . . . . . . . h n i i r i bosonfermion h n f i i r i FIG. 5. (Color online) (a) Boson density distribution in realspace with U bf /t = 10, ρ b = 1 . ρ f = 1 / U bf /t = 7 . ρ b = 1 . ρ f = 1 / In this part, we calculate the particle distribution inreal space at a strong boson-fermion coupling with fixed V ff /t = 0. The results are presented in Fig. 5. Whenparticle filling factors are set to be ρ b = 1 . ρ f = 1 / U bf /t = 10 as shown in Fig. 5(a). As onecan see, the particle density in real space shows a cluster-like distribution, and the boson clusters are incompatiblewith fermion clusters. If we set ρ b = 1 . ρ f = 1 / U bf /t = 7 asproduced in Fig. 5(b). All these features indicate thatfermions can still obtain an effective interaction via theexchange of bosons. [1] G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J.Brecha, and M. Inguscio, Science , 2240 (2002).[2] F. Ferlaino, E. de Mirandes, G. Roati, G. Modugno, andM. Inguscio, Phys. Rev. Lett. , 140405 (2004).[3] I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier,M. Pierce, B. S. Rem, F. Chevy, and C. Salomon, Science , 1035 (2014).[4] A. Imambekov and E. Demler, Phys. Rev. A , 021602(2006).[5] X. Barillier-Pertuisel, S. Pittel, L. Pollet, and P. Schuck,Phys. Rev. A , 012115 (2008).[6] A. Mering and M. Fleischhauer, Phys. Rev. A , 023601(2008).[7] C. N. Varney, V. G. Rousseau, and R. T. Scalettar, Phys.Rev. A , 041608 (2008).[8] X. Yin, S. Chen, and Y. Zhang, Phys. Rev. A , 053604(2009).[9] L. Pollet, M. Troyer, K. Van Houcke, and S. M. A. Rom-bouts, Phys. Rev. Lett. , 190402 (2006).[10] M. T. Batchelor, M. Bortz, X.W. Guan, and N. Oelkers,Phys. Rev. A , 061603 (2005).[11] L. Mathey, D.-W. Wang, W. Hofstetter, M. D. Lukin,and E. Demler, Phys. Rev. Lett. , 120404 (2004).[12] L. Mathey and D.-W. Wang, Phys. Rev. A , 013612(2007).[13] M. A. Cazalilla and A. F. Ho, Phys. Rev. Lett. , 150403(2003).[14] J. J. Kinnunen, Z. Wu, and G. M. Bruun, Phys. Rev. Lett. , 253402 (2018).[15] Y. Z. Wu, Z. Yan, Z. Lin, J. Lou and Y. Chen, Sci. Rep. , 10822 (2020).[16] H. P. B¨uchler and G. Blatter, Phys. Rev. Lett. , 130404(2003).[17] E. Orignac, M. Tsuchiizu, and Y. Suzumura, Phys. Rev.A , 053626 (2010).[18] L. Pollet, C. Kollath, U. Schollw¨ock, and M. Troyer,Phys. Rev. A , 023608 (2008).[19] B. J. DeSalvo, K. Patel, G. Cai, and C. Chin, Nature , 61 (2019).[20] S. Ejima, H. Fehske, F. Gebhard, K. zu M¨unster, M.Knap, E. Arrigoni, and W. von der Linden, Phys. Rev.A , 053644 (2012).[21] T. D. K¨uhner, S. R. White, and H. Monien, Phys. Rev.B , 12474 (2000).[22] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz,Phys. Rev. Lett. , 1691 (1982).[23] R. V. Pai and R. Pandit, Phys. Rev. B , 104508 (2005).[24] J. Des Cloizeaux and M. Gaudin, J. Math. Phys. , 1384(1966).[25] S. R. White, Phys. Rev. B , 10345 (1993).[26] ITensor Library (version 2.0.11), http://itensor.org .[27] H. P. B¨uchler and G. Blatter, Phys. Rev. A , 063603(2004).[28] S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev.B , 054511 (2009).[29] T. Giamarchi, Quantum Physics in One Dimension (2010).[30] J. M. Luttinger, J. Math. Phys. , 1154 (1963).[31] T. Best, S. Will, U. Schneider, L. Hackerm¨uller, D. van Oosten, I. Bloch, and D.-S. L¨uhmann, Phys. Rev. Lett.102