Spin-charge separation in two-component Bose-gases
A. Kleine, C. Kollath, I. McCulloch, T. Giamarchi, U. Schollwoeck
aa r X i v : . [ c ond - m a t . o t h e r] J un Spin-charge separation in two-component Bose-gases
A. Kleine, C. Kollath, I.P. McCulloch, T. Giamarchi, and U. Schollw¨ock Institute for Theoretical Physics C, RWTH Aachen, D-52056 Aachen, Germany DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland (Dated: October 24, 2018)We show that one of the key characteristics of interacting one-dimensional electronic quantumsystems, the separation of spin and charge, can be observed in a two-component system of bosonicultracold atoms even close to a competing phase separation regime. To this purpose we determinethe real-time evolution of a single particle excitation and the single-particle spectral function us-ing density-matrix renormalization group techniques. Due to efficient bosonic cooling and goodtunability this setup exhibits very good conditions for observing this strong correlation effect. Inanticipation of experimental realizations we calculate the velocities for spin and charge perturbationsfor a wide range of parameters.
One of the most exciting recent events is the ever-growing interplay between previously disconnected fieldsof physics, such as quantum optics and condensed matterphysics. In particular, cold atomic systems have openedthe way to engineer strongly interacting quantum many-body systems of unique purity. The unprecedented con-trol over interaction strength and dimensionality allowsthe realization of “quantum simulators” where funda-mental but hard to analyze phenomena in strongly corre-lated systems could be observed and controlled. Exam-ples are the observation of superfluid to Mott insulatortransition for Bose gases [1] and the fermionization ofstrongly interacting one dimensional bosons [2, 3].Among interacting systems, the physics depends verystrongly on the dimensionality. In one-dimensional sys-tems the interactions play a major role and lead to dras-tically different physics than for their higher dimensionalcounterpart. Typically, interactions in one-dimensionalsystems lead to a Luttinger liquid state where the exci-tations of the system are collective excitations [4]. Theimportance of such a state for a large variety of exper-imental devices in condensed matter has led to a huntto observe its properties. A remarkable consequence ofsuch a state is the absence of single particle excitations.This means that a quantum particle, that would normallycarry both charge and spin degrees of freedom, fraction-alizes into two different collective excitations, a spin anda charge excitation. Such a fractionalization of a sin-gle particle excitation is the hallmark of collective effectscaused by interactions. However, just as detecting frac-tional excitations in the case of the quantum hall effect isdifficult [5], observing spin-charge separation has provenelusive despite several experimental attempts [6, 7, 8].So far, the best experimental evidence is provided bytunneling between quantum wires where interferences ef-fects are due to the existence of two different velocities[9]. However, in these systems it is hard to quantify or totune the interaction between the particles which causesthe collective effects. Since control of the interaction isa routing procedure in ultracold gases, the possible real-ization of the phenomenon of spin charge separation has also been discussed in the context of cold fermionic gases[10, 11, 12, 13] and strongly interacting bosonic gases[14].However, proposals to observe spin-charge separationin ultracold fermionic gases are still plagued by the cur-rently quite high temperatures in such systems. A muchbetter setup to test for spin-charge separation wouldbe two-component Bose gases, for example using the | F = 2 , m F = − i and the | F = 1 , m F = 1 i hyper-fine states of Rb [15, 16]. Experimentally, this sys-tem retains the advantages of the fermionic ultracoldatom setup while allowing for much lower temperaturesdue to the more efficient cooling techniques available forbosons. However, theoretical studies [17, 18] for one-dimensional systems predict that close to the experimen-tally accessible parameter regime of almost equal inter-and intra-species interaction strength phase separationoccurs. This is the remaining potential experimentalcomplication in the setup.In this work we demonstrate the phenomenon of spin-charge separation in the experimentally relevant parame-ter regime, allowing to use this system to unambiguouslytest for spin-charge separation. We calculate both thereal-time evolution of a single particle excitation and thedynamical single particle spectral function of the two-component bosonic systems. We show that both thesequantities demonstrate the separation of a single-particleexcitation into spin and charge. We further determinethe velocity of spin and charge and the Luttinger param-eters for experimentally relevant parameter regimes. Toperform the calculations we use variants of the densitymatrix renormalilzation group method (DMRG) [19, 20].The numerical treatment is necessary to obtain reliablepredictions for experimental realizations, due to the closeproximity of this regime to phase separation.A one-dimensional two-component Bose gas in an op-tical lattice [21] can be described by the two-componentBose-Hubbard model H = − J P j,ν (cid:16) b † j +1 ,ν b j,ν + h.c. (cid:17) + P j,ν U ν ˆ n j,ν (ˆ n j,ν − + U P j ˆ n j, ˆ n j, + P j,ν ε j,ν ˆ n j,ν (1)Here j is the site index and ν = 1 , b and b † are theannihilation and creation operators and ˆ n is the num-ber operator. The first term models the kinetic energyof the atoms. The intra-species interaction is describedby the U ν term. We use U := U = U as it is ap-proximately realized for commonly used hyperfine statesof Rb [22]. The inter-species interaction is given bythe U term and the last term describes external po-tentials. In the following we use the dimensionless pa-rameters u = U/J and u = U /J . We define the‘charge’ density n j,c = n j, + n j, and the ‘spin’ density n j,s = n j, − n j, . We focus on systems with average fill-ing n = P j n j, /L = P j n j, /L smaller than one particleper site and parameter regimes up to close to the tran-sition to the phase separation (approximately u ≈ u [18]). Here L is the number of sites in the system.In a superfluid phase away from the transition to phaseseparation the low energy physics can be approximatedby a density-phase representation of the bosons as usedin the bosonization method [4]. In this representationthe Hamiltonian is totally separated into one part forthe charge and one for the spin degrees of freedom. Thephysics is fully determined by the velocities v c,s andthe so-called Luttinger parameters K c,s for spin (s) andcharge (c). Therefore the separation of a single particleexcitation into spin and charge excitations is expected.The parameters of two interacting species of bosons canbe related to the parameters K and v for the singlespecies case [23] by v c,s = v p ± ( g K ) / ( πv ) (2)and K c,s = K/ p ± ( g K ) / ( πv ) . Here g is the interspecies interaction strength in thecontinous model. In the limit of small interactions, thesingle species parameters K and v can be directly re-lated to the Bose-Hubbard Hamiltonian Eq. (1) [4]. Forhigher values of the interaction strength the relation evenfor the single species situation is unkown, and has tobe determined numerically [24]. For large values of theinter-species interaction of the order of the intra-speciesinteraction the system approaches the transition to phaseseparation and the bosonization approach becomes a pri-ori inaccurate.Snapshots of the real time-evolution of a single par-ticle excitation in a two-component bosonic system areshown in Fig. 1. The single particle excitation at time t = 0 is prepared by the application of the creation op-erator of one species, say 1, on site L/ | ψ i , i.e. b † L/ , | ψ i . The resulting sharp peaks in the density distributions are shown in Fig. 1 (a). For t > k| Ψ( t + ∆ t ) i − exp[ − iH ∆ t ] | Ψ( t ) ik < − . Thestepsize ∆ t = 0 . A ν ( q, ω ) = π ℑh b † q,ν H + ω +ı η − E b q,ν i as shownin Fig. 2. For fermions this function is known to ex-hibit two peaks at the spin and charge excitation en-ergies [28, 29], showing a direct signature of the spin-charge separation. For the bosons computing this spec-tral function is more involved and up to very recentlyit was only derived for a single component bosonic sys-tem [30, 31, 32]. An expression for the correlation func-tions which allow to obtain the single-particle correlationfunction within the bosonization treatment for a two-component bosonic system was derived in [33]. Powerlaw singularities at qv c,s are obtained with respective ex-ponents 1 / K c,s + 1 / K s,c −
1. For the values of theLuttinger parameters (as shown in Fig. 2) one thus ex-pects two divergent peaks. We show in Fig. 2 the fullspectral function for our microscopic model, as calculatednumerically using a matrix product state generalizationof the correction vector method[31, 34]. Our results showclearly the appearance of the two separated peaks, thelower representing the spin and the upper the charge ex-citation branch [42]. Thus both the real time evolution ofa single particle function and the single particle spectralfunction show clear signatures of separation of spin andcharge.To observe the separation of spin and charge exci-tations experimentally in a system of ultracold bosons,knowledge of the spin and charge velocities is indispens-able. We therefore determined the velocities for a widerange of parameters. This was done calculating the time-evolution of a small spin and charge density perturbationrespectively. The density perturbation was created attime t = 0 applying an external potential of Gaussianform ε ν,j = ε exp (cid:16) − ( j − j ) σ j (cid:17) where ν = c, s . At time t = 0 the potential is switched off and the time-evolutionof the density perturbation is calculated. The errors inthe obtained velocities are of the order of 0 . aJ/ ¯ h forsmall u and increase with larger u .In Fig. 3 we show the dependence of the velocities c - 1.25n s -0.2 0 0.2 0.4 0.6 0 8 16 24 32site(b) n c - 1.25n s -0.2 0 0.2 0.4 0.6 0 8 16 24 32site(c) n c - 1.25n s FIG. 1: (Color online) Snapshots of the time-evolution of the charge and spin density distribution of a single particle excitationcreated at time t = 0¯ h/J ; (a) at time t = 0¯ h/J , (b) at time t = 1 . h/J and (c) at time t = 2 . h/J . The system parameterswere n , = 0 . u = 3, u = 2 .
1. The charge density is shifted by 1 .
25 for better visibility. The arrows in (c) mark the clearseparation of the charge and the spin density waves. A ( q , ω ) ω FIG. 2: One-particle spectral function at momentum q =20 / π/a . Two peaks corresponding to the spin and thecharge excitation can be distinguished. The following param-eters were used n = 0 . u = 3, u = 2 . L = 64 sites and a broadening η = 0 .
1. Here a is the latticespacing. on the inter-species interaction for two different valuesof the parameter γ = u/ (2 n c ). For both parameterregimes, the charge velocity increases with increasing in-teraction whereas the spin velocity descreases. For avanishing inter-species interaction it was shown in [24]that for γ < γ ≈ . u close to phase separation. For γ ≈ . v and K . Thisholds even up to close to the regime of phase separa-tion, i.e. u ≈ u where the difference in the velocities ismaximal. However, for u ≈ u the results for the spinvelocity start to deviate for both values of γ . In Fig. 4 we show for two different fixed inter-speciesinteraction strengths the dependence of the velocities onthe density. The charge and the spin velocities rise withincreasing background charge density. (Note, even at n c = 1 the system is in the superfluid regime.) Theincrease of the velocities is described to good accuracyusing the analytical form Eq. (2), provided we use nu-merically obtained values of K and v . For large u and small n c the results for the velocities from DRMG,in particular the spin velocity, deviate considerably fromEq. (2), showing that the approximate relation cannotbe used in this regime. Note that in this regime theextraction of the spin velocity from the real-time evolu-tion becomes also more involved since the spin pertur-bation shows a strong spreading in space (cf. [35]). Atthe timescales over which we calculated the velocity, theleft- and the right-moving spin perturbations are not yetfully separated and show strong amplitude damping. Ourfinding of the dependencies can be used to predict the ve-locities for experimentally interesting parameter regimes.In recent experiments for the preparation of a mixtureof two bosonic components in optical lattices mostly twohyperfine states of Rb are used, e.g. the | F = 2 , m F = − i and the | F = 1 , m F = 1 i hyperfine states. Theintra-species scattering lengths are a = 91 . a B and a = 100 . a B [22], respectively where a B is the Bohr ra-dius. For these states the inter-species scattering lengthis of the same order of magnitude as the intra-speciesscattering length and can be tuned about 20% using aFeshbach resonance [15, 16]. Thereby the experimen-tal parameters are close to the competing phase sepa-ration regime. These mixtures can be confined to one-dimensional structures using strongly anisotropic lattices[2, 3, 36, 37]. The most intuitive observation of the phe-nomenon of spin-charge separation in these systems isto generate a single particle excitation and then followthe evolution of the excitation in real time. This can bedone measuring the spin-resolved density over a certainregion. The creation of a single particle excitation can v u (a)DMRG: chargeDMRG: spinBosonization 0 1 2 3 0 1 2 3 v u (b)DMRG: chargeDMRG: spinBosonization FIG. 3: (Color online) Dependence of the charge and spinvelocity on the interparticle interaction strength for (a) u =2 and n ≈ .
88 and (b) u = 3 and n = 0 .
63. A comparison ofanalytical results (line, see text) and numerical DMRG results(symbol) is shown. The velocities are measured in units aJ/ ¯ h .Note that the errors of the DMRG results increase close to u ≈ u . v nu = 2.7, chargeu = 2.7, spinu = 1.2, chargeu = 1.2, spinBosonization FIG. 4: (Color online) Dependence of the charge and spinvelocity on the charge background density., comparison ana-lytical results of bosonization and numerical DMRG results.The parameters used are (a) u = 3, u = 1 . u = 3, u = 2 . be done e.g. using outcoupling of single particles by theapplication of a magnetic field gradient for addressabilityand a microwave field [38, 39, 40][43]. The efficiency ofsuch a technique for generating single particle exciationswas demonstrated [39] using a cavity. The microwavefield could be chosen to couple the | F = 1 , m F = 1 i hy- perfine state to e.g. the | F = 2 , m F = 2 i . This has theadvantage that scattering with the | F = 1 , m F = 1 i stateare supressed. The measurement of the density resolvedover a region of approximately 10 lattice sites can then beperformed using again the magnetic field gradient to getan unambigious signal. In an array of one-dimensionaltubes the broadening of the signal caused by the trap-ping potential could be suppressed by preparing most ofthe tubes in a Mott-insulating state as shown in [12].We would like to thank J.S. Caux, S. F¨olling, M. K¨ohl,and B. Paredes for fruitful discussions. AK and US ac-knowledge support by the DFG and CK and TG by theSwiss National Fund under MaNEP and Division II andthe CNRS. CK thanks the Institut Henri Poincare for itshospitality during the final part of the work. [1] M. Greiner et al. , Nature , 39 (2002).[2] B. Paredes et al. , Nature , 277 (2004).[3] T. Kinoshita, T. Wenger, and D. S. Weiss, Science ,1125 (2004).[4] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2004).[5] C. Glattli, in
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