Magnetic phase transition in coherently coupled Bose gases in optical lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Magnetic phase transition in coherently coupled Bose gases in optical lattices
L. Barbiero, M. Abad,
2, 3 and A. Recati
2, 4 Dipartimento di Fisica e Astronomia ”Galileo Galiei”, Universit`a di Padova, 35131 Padova, Italy INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, 38123 Povo, Italy Quantum Systems Unit, OIST Graduate University, Onna, Okinawa 904-0495, Japan Technische Universit¨at M¨unchen, James-Franck-Straße 1, 85748 Garching, Germany
We describe the ground state of a gas of bosonic atoms with two coherently coupled internallevels in a deep optical lattice in a one dimensional geometry. In the single-band approximationthis system is described by a Bose-Hubbard Hamiltonian. The system has a superfluid and a Mottinsulating phase which can be either paramagnetic or ferromagnetic. We characterize the quantumphase transitions at unit filling by means of a density-matrix renormalization group technique, andcompare the results with a mean-field approach and an effective spin Hamiltonian. The presenceof the ferromagnetic Ising-like transition modifies the Mott lobes. In the Mott insulating regionthe system maps to the ferromagnetic spin-1/2 XXZ model in a transverse field and the numericalresults compare very well with the analytical results obtained from the spin model. In the superfluidregime quantum fluctuations strongly modify the phase transition with respect to the well establishedmean-field three dimensional classical bifurcation.
PACS numbers: 75.10.Pq, 05.10.Cc, 05.30.Jp, 03.75.Lm
I. INTRODUCTION
Ultra-cold atoms in optical lattices have opened newpossibilities to study quantum phase transitions [1] andto observe the effects of quantum fluctuations [2, 3]. Re-cent experimental advances have also paved the way tothe investigation of quantum magnetism, notable exam-ples being the demonstration of super-exchange inter-actions in bosonic gases [4], the time-evolution of spinimpurities [5, 6], and the engineering of Ising [7] andanisotropic exchange Hamiltonians [8, 9]. On the otherhand cold atoms are also very suitable to study coherencephenomena related to the control of the coupling betweeninternal levels of atomic species. One can obtain coher-ently coupled superfluids, which show many interestingfeatures ranging from a classical bifurcation transitionin internal Josephson effect [10] to dimerization of half-vortices in rotating superfluids [11, 12].In this work we combine the two ingredients by study-ing a coherently coupled Bose gas trapped in a one-dimensional (1D) optical lattice at unit filling, which canbe described by a coupled two-component Bose-Hubbardmodel with on-site interactions (see Eq.(1)). In par-ticular the relative strengths of the coherent coupling(or phase coupling) and the density couplings due tospecies-dependent two-body interactions drive the sys-tem into superfluid (SF) or Mott-insulating (MI), non-polarized/paramagnetic (NP) or polarized/ferromagnetic(FM) phases. We characterize the phase diagram in de-tail by combining mean-field and density matrix renor-malization group (DMRG) approaches [13], and by map-ping to spin chain Hamiltonians. The interest in such asystem is manyfold since it allows for the study of dif-ferent topics such as: the role of quantum fluctuationsdue to confinement and interaction in the NP-FM bifur-cation in the superfluid regime; the change of the lobesin the SF-MI transition, which in 1D (at constant integer density) is of the Berezinskii-Kosterlitz-Thouless (BKT)type [14–16]; the Ising-like ferromagnetic transition in theMI phase; and the possible simulation of a ferromagneticXXZ chain in a transverse field. Moreover, the modelHamiltonian we use is relevant for ladder chain modelsin presence of a density-density interaction between theparticles on different chains (see [17, 18], where the in-commensurate filling case is studied), which has not beenas much studied as the case of non-interacting chains (see,e.g., [19] and references therein).In systems of hard-core bosons or fermions withnearest-neighbor intra-species and on-site intra-speciesinteractions, the NP-FM transition has been studied forthe density (charge) gapless phase [20]. Interestingly ithas been shown that the transition belongs to the Isingin transverse field universality class. We find that thesame holds for our model, but in MI, i.e., density (charge)gapped phase. By means of our accurate numerical toolswe give an explicit expression for the phase transitionpoint. Moreover we characterize completely the variousphases, and find, e.g., as mentioned above, that the NP-FM transition affects the Mott lobes. We also determinethe behavior of the transverse and longitudinal spin cor-relation functions across the phase transition. The latterquantities can be directly measured in cold gases exper-iments [21, 22].
II. COHERENTLY COUPLED BOSE-HUBBARDMODEL
We consider a Bose gas at unit filling confined in a1D geometry with two hyperfine levels that are coher-ently coupled. The atoms feel a deep optical lattice ofnumber of sites L which is the same for the two internallevels. The system can be described by a two-componentsingle-band Bose-Hubbard Hamiltonian with a static lin-ear coupling between the two species: H = X i h X σ U n iσ (ˆ n iσ −
1) + U ab ˆ n ia ˆ n ib i ++ J Ω X i (ˆ a † i ˆ b i + ˆ a i ˆ b † i ) − J X
The Mott-superfluid transition is related to the break-ing of the U (1) symmetry, which leads to the emergenceof a global phase, and thus to quasi-condensation in 1D.In the absence of hopping, J = 0, the ground state ofHamiltonian Eq. (1) is | i = Q i c † i | vac i , where | vac i isthe vacuum of particles and we have introduced the op-erators ˆ c † i = (ˆ a † i − ˆ b † i ) / √ i inthe anti-symmetric state of the internal levels a and b (dressed state). Notice that if J Ω were not real (or posi-tive), a different relative phase would appear between ˆ a † and ˆ b † in the definition of ˆ c † which would not affect theproperties of the system.In the presence of hopping the system undergoes aphase transition between a Mott insulating and a super-fluid phase. It is customary to depict the phase diagramof the system as a function of its chemical potential µ and the tunnelling energy J . This leads to a lobe struc-ture with fixed filling within the Mott lobes. Examples ofphase diagrams of Hamiltonian ( 1) for n = 1 are plottedin Fig. 1 (top panel), both within mean-field approxima-tion and the exact DMRG result (see text below). µ / U U ab /U=0.5U ab /U=1.0U ab /U=1.8U ab /U=6.0U ab /U=20.0 J/U | S z | µ + /U µ - /U FIG. 1: Top panel: MI-SF phase transition predicted by themean-field approach (dashed lines) and DMRG (symbols). Inthe inset we show a typical finite size scaling of µ + and µ − in the superfluid regime for U ab /U = 1 .
8. We characterize acharge gapless phase, i.e. superfluidity, by µ + − µ − = 0 inthe thermodynamic limit. Bottom panel: associated NP-FMtransition calculated with DMRG. All curves correspond to J Ω /U = 0 . A. Mean Field Mott-Superfluid Phase Transition
In order to get an insight into the way the differentparameters of the model enter in the SF-MI phase transi-tion, we apply a mean-field theory [1] to the grandcanon-ical Hamiltonian H − µ P iσ ˆ n iσ . At J = 0, the bor-ders of the Mott lobes are easily determined by requiringunit filling factor. The chemical potential must satisfythe conditions − J Ω < µ and µ < J Ω + ( U + U ab ) / − p J + ( U − U ab ) /
2. For J = 0, second-order pertur-bation theory predicts that the border between the MIand the SF region is given by the condition1 zJ = 1 µ + J Ω + − µ + 6 J Ω + U + U ab (cid:0) − µ + J Ω + U + U ab (cid:1) − J − (cid:0) U − U ab (cid:1) , (2)where the coordination number is z = 2 in 1D. No-tice that in the SU(2) symmetric case for the inter-action, U ab = U , the single component result is re-covered provided the chemical potential is rescaled to˜ µ = µ − J Ω . When the hopping strength J becomeslarger than that given by Eq. (2) the system enters theSF phase and develops a nonzero order parameter givenby ψ − = ( ψ, − ψ ) T / √
2, with ψ = h a i = h b i . Since quan-tum fluctuations are neglected the MI phase is describedby the state | i introduced above. Therefore, the systemcould support a polarized state only in the SF regimeprovided U ab was large enough, in analogy to coupledcondensates (see, e.g., the experiment reported in [10]and references therein).The structure of the mean-field Mott lobes given byEq. (2) is shown as dashed lines in Fig. 1 for differentvalues of U ab . There are a number of features in thestructure of the lobes to be noticed: the lower borderequals − J Ω /U for all values of U ab /U and the upperborder converges at 1 + J Ω /U for U ab > U ; as U ab /U is increased, the lobes saturate at a maximum value of J/U , a feature that also takes place in mixtures. More-over at fixed U one has, as expected by the change in thecompressibility, that for U ab < U the insulating regionis smaller than in the single component case, while for U ab > U the insulating region is enlarged.With respect to quantum systems in higher dimensions[35], in 1D the role of quantum fluctuations can bringrelevant beyond mean-field effects [3]. These are usuallynot properly captured in semi-classical approaches, suchas the mean-field, but can be accounted for in quasi-exactmethods such as DMRG (see next paragraph). B. DMRG Mott-Superfluid Phase Transition
In order to check the previous analysis and to get quan-titative results we use DMRG technique [13] to deter-mine the properties of the ground state of Eq. (1). Thismethod has already proven to give strong beyond mean-field effects in the context of the single-species Bose-Hubbard model [15, 16]. All the numerical results areobtained at unit filling. The Mott lobes calculated withDMRG are shown as symbols in Fig. 1 (top panel).As expected, we find that the Mott-superfluid transi-tion takes place at values of
J/U much higher than pre-dicted by mean field (dashed lines), and that the lobeshave the reentrant shape characteristic of the 1D Bose-Hubbard Hamiltonian [15, 16]. We determine the tran-sition points by the closure of the so-called density (orcharge) gap for different system sizes and then perform-ing finite size scaling as we report in the inset of Fig.1. The density gap, µ = µ + − µ − , for a system with Nparticles with energy E ( N ) is defined by the differencein energy in adding, µ + = E ( N + 1) − E ( N ), or remov-ing, µ − = E ( N ) − E ( N − U = U ab (equivalent to the single component case)are in very good agreement with the ones obtained by calculating the central charge as in [36]. IV. PARA-/FERRO-MAGNETIC PHASETRANSITION
In addition to the Mott-SF transition, Hamiltonian (1)allows for states breaking a Z symmetry, creating a finitepolarization S z = ( N a − N b ) / N , with N σ the numberof atoms in state σ = a, b . A. Global Magnetization
In order to study the breaking of the Z symmetryin Hamiltonian (1) we first of all determine the globalpolarization (or magnetization), S z . In our numericalsimulations this requires special attention, especially inthe superfluid phase, since a sufficiently large size of theHilbert space has to be taken. That is, we need to con-sider an on-site basis containing the states correspondingto a number of bosons up to n max , to allow the fluctu-ations of a and b to explore the relevant configurationsand thus to drive the phase transition. We obtain conver-gence of the results for open boundary conditions using n max = 6, keeping up to 512 DMRG states and 6 sweeps[13], getting a truncation error lower that 10 − . Unlessotherwise stated we show the results for a chain with L = N = 80 [39].The results for the absolute value of the polarization[40] as a function of J/U are reported in the bottom panelof Fig. 1.In the SF phase (corresponding to U ab /U = 1 .
8) thesystem shows strong quantum fluctuations. Indeed, theNP-FM transition has been studied in the continuum andwithin the Gross-Pitaevskii framework (for a recent dis-cussion see, e.g, [37] and references therein), and has beenseen to take place for U ab − U = 2 J Ω /n , with n = 1 the to-tal density of the system. Moreover the critical exponentof the magnetisation is in this case the expected mean-field value β = 1 /
2. In the lattice, instead, the transitionoccurs for an inter-species interaction larger (but still ofthe same order) than the one predicted for a mean-fieldcoherent state, i.e. U ab /U = 1 .
2, and the magnetizationdoes not follow the classical bifurcation law. Notice thatthe magnetization behavior for U ab /U = 1 . | S z | .In the Mott phase, where double occupancy is stronglysuppressed, the inter-species interaction has to be muchstronger, e.g. U ab /U = 6 and U ab /U = 20, to drivethe phase transition. For increasing values of U ab thetransition point is seen to approach a limiting value of J corresponding to the value given by the ITF mappingdiscussed above.Moreover, it can be noticed from Fig. 1 that once themagnetic phase transition has taken place inside the lobe(see for instance the case U ab /U = 20), the latter shrinksslightly, indicating that the SF phase is more favorablethan the MI for the polarized system. Also, in this casethe Mott insulating lobes do no longer strongly dependon the value of U ab , since in the ferromagnetic phase thisinteraction is less effective. This saturation of the Mottlobes for U ab large has a completely different meaningfrom the saturation found in the mean-field analysis. B. Strong Coupling Regime
When the system becomes strongly interacting thefluctuations of the number of atoms in each site areweaker and therefore the effect of the two-body inter-action is reduced, making the polarized state less favor-able. In particular in the deep MI phase ( J ≪ U , U ab )the single particle tunneling is suppressed and exchangeof atoms is the dominant process. In this case the co-herently coupled Bose-Hubbard model Eq. (1) can bemapped into a spin chain model (see, e.g., [27, 31]). Theeffective spin Hamiltonian is the so-called spin-1 / H XXZ = − t X i ( ˆ S xi ˆ S xi +1 + ˆ S yi ˆ S yi +1 +∆ ˆ S zi ˆ S zi +1 )+2 J Ω X i ˆ S xi , (3)where ˆ S zi = (ˆ n ia − ˆ n ib ) /
2, ˆ S xi = (ˆ a † i ˆ b i + ˆ a i ˆ b † i ) /
2, ˆ S yi = − i (ˆ a † i ˆ b i − ˆ a i ˆ b † i ) / t = 4 J /U ab and ∆ = 2 U ab /U − − < ∆ < + ∞ . In suchparameter range the spin model Eq. (3) exhibits only twophases, a paramagnetic phase with magnetization alongthe x -axis and an Ising ferromagnetic phase along the z -axis. For J Ω = 0 the model is exactly solvable and thetransition occurs at ∆ = 1, i.e., U ab = U . For J Ω = 0the transition is shifted to larger values of U ab /U . Onthe other hand for U ab → ∞ the Hamiltonian reduces tothe Ising model in a transverse field (ITF) which is alsoexactly solvable and predicts a transition at t ∆ = 4 J Ω ,i.e., for 2 J = U J Ω , with a critical exponent β = 1 / J Ω , as reported in the top panel ofFig. 2, which shows the DMRG results. As describedabove in the Mott phase for J Ω → U ab /U = 1. For J Ω = 0 the transition is shifted tolarger values of U ab /U . One can obtain an approxima-tion to the critical condition by noticing that Hamilto-nian Eq. (3) can be rewritten as a Heisenberg exchangeterm, P ~S i · ~S i +1 , plus an ITF term. Neglecting the ef- fect of the Heisenberg term (valid for U ab > U ), the phasetransition is driven by the ITF and it takes place at t (∆ −
1) = 8 J (1 /U − /U ab ) = 4 J Ω . (4)The accuracy of this expression with respect to the nu-merical solution of Hamiltonian Eq. (1) is shown in thebottom panel of Fig. 2, where it is seen to be very goodfor a range of values of J Ω . Moreover in the inset of Fig. 2it is possible to notice that the critical exponent β = 1 / J/U . J/U | S z | (J/U-J c /U) J/U | S z | J Ω /U=0.010J Ω /U=0.020J Ω /U=0.050J Ω /U=0.075J Ω /U=0.100J Ω /U=0.125 J Ω /U J / U U ab /U=6.0U ab /U=20.0 FIG. 2: Top panel: NP-FM transition in the MI phase fordifferent values of the linear coupling J Ω /U , for U ab /U =6. Inset: comparison between numerical results and criticalexponent 1 / J Ω = 0 .
1. Bottom panel: NP-FMtransition point calculated with DMRG (symbols) and usingexpression J (1 /U − /U ab ) = J Ω / U ab /U in the MIphase. C. Spin-Spin Correlation Functions
While S z is the global order parameter, we character-ize the NP and FM phases, and in particular the NP-FMtransition, also by determining the behavior of the cor-relation functions around the phase transition point. Westudy the longitudinal and the transverse spin-spin cor-relation functions C s ( i ) = h ˆ S sj ˆ S sj + i i with s = z, x respec-tively. In order to drop boundary effects we exclude themore external sites and evaluate the correlation functionsonly in the central region of the system (in particular wetake j = 15).To have an idea of how the large distance behavior ofthe correlation functions changes along the transition, weplot in the top panel of Fig. 3 the correlation functions for J/U U ab /U=1.0U ab /U=1.8U ab /U=6.0U ab /U=20.0 i -5 -4 -3 -2 -1 C z ( i ) J/U=0.40J/U=0.42J/U=0.44J/U=0.46J/U=0.48J/U=0.50J/U=0.52J/U=0.54 0 10 20 30 40 50 i J/U=0.20J/U=0.21J/U=0.22J/U=0.23J/U=0.24J/U=0.25J/U=0.26 U ab /U=1.8 U ab /U=6.0 FIG. 3: Top panel: Behavior of C x (50) (open symbols)and C z (50) (filled symbols) across the MI-SF transition, for J Ω /U = 0 . C z ( i ) in the SF (left panel) and MI (right panel) phasesclose to the phase transition. a separation i = 50 as a function of J/U . The paramag-netic phase is dominated by transverse spin correlationssince in this regime J Ω is the most important term, whilein the ferromagnetic phase the longitudinal correlationsbecome dominant. Notice that the magnetic transition(see Fig. 1) seems to be well described by the crossingpoint between the long-range values of C x and C z .The longitudinal correlation function across the NP-FM transition is shown in the lower panels of Fig. 3 inthe superfluid ( U ab /U = 1 .
8, left panel) and in the MIphase ( U ab /U = 6, right panel). The behavior of C z changes from an exponential decay in the paramagneticphase to long-range order in the FM phase showing aclear second order phase transition. The critical point isin good agreement with the one obtained with S z (Fig. 1).Notice that in the SF phase the system polarizes more“slowly” than in the insulating case due to strong fluctu-ations, which explains the larger region of intermediatedecays. V. CONCLUSION AND PERSPECTIVES
Let us briefly comment here on the experimental real-ization of the model Hamiltonian Eq. (1) with cold-gases.Even if some relevant ingredients are already availablewithin the present technology some challenging achieve-ments are missing. The two species Bose-Hubbard mod-els have been realized and their mapping to a spin chaintested, see, e.g., [9]. Adding a static Rabi coupling is notan issue. At the same time, in fermionic systems, temper-atures of the same order of the spin exchange have beenreached [8]. In current experiments, where Rb atoms are used, the most difficult and not yet achieved ingredient isto have very different intra- and inter-species interactionto address the ferromagnetic transition in the MI phase.A very helpful tool in this direction is the recent possibil-ity, explored in Esslinger’s group [41], of creating state-dependent lattices for essentially any atomic species. Atthe same time spin-selective microwave fields could allowfor the exploration of resonances in non-standard colli-sion channels [42]. It would open the way towards theachievement of a large range U ab /U values.In conclusion, the system we have studied, describedby Eq. (1), constitutes a quite unexplored system inthe family of Bose-Hubbard Hamiltonians (see, e.g., also[18, 20]). It is fundamentally different from Bose-Bosemixtures and in a way a generalization of two-leg chains.The system shows two quantum phase transitions: su-perfluid to Mott insulator transition – which is of theBerezinskii-Kosterlitz-Thouless kind at fixed integer den-sity – and a paramagnetic/non-polarized to ferromag-netic/polarized transition. We show that the latter tran-sition changes the structure of the Mott lobes. In theMott regime the transition is well described in terms ofa quantum XXZ model in a transverse field. In the SFregime due to quantum fluctuations strong correctionsto the mean-field coherent results are present. Whilewe focused on the unit filling factor case, at low fillingfactor, the system is also interesting, especially consid-ering that its experimental realization should be feasiblewithin current technology as shown in [9]. Indeed in thesmall J/U case both species a and b have a fermionic(Tonks-Girardeau regime) equation of state [43]. There-fore one has the possibility of studying the fate of itin-erant ferromagnetism in one dimension in analogy to therecent analysis in [44] with the inclusion of the linearinterspecies coupling J Ω . Another interesting aspect tostudy is the dynamics of the system. The latter has beenstudied in some detail for the homogeneous weakly in-teracting case. In the presence of a lattice it would beinteresting, e.g., to study the quenching across the ferro-magnetic transition [45–47] or how J Ω would modify thedomain wall dynamics (see, e.g., [48]) or the quenchingacross the ferromagnetic transition. Acknowledgement.
Useful discussions with and G. Fer-rari, Yan-Hua Hou and Tommaso Roscilde are acknowl-edged. This work has been supported by ERC throughthe QGBE grant and by Provincia Autonoma di Trento.L.B. acknowledges support by Cariparo Foundation (Ec-cellenza grant 11/12) and the CNR-INO BEC Center inTrento for CPU time. A.R. acknowledges support fromthe Alexander von Humboldt foundation. M. A. acknowl-edges support from the Okinawa Institute of Science andTechnology Graduate University during the final stagesof the work.During the review process of the current manuscript,the Mott regime has been studied [49] obtaining resultsin agreement with ours. [1]
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