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Topological Transport in Spin-Orbit Coupled Bosonic Mott Insulators C. H. Wong and R. A. Duine Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Received 11 December 2012; published 12 March 2013) We investigate topological transport in spin-orbit coupled bosonic Mott insulators. We show that interactions can lead to anomalous quasiparticle dynamics even when the spin-orbit coupling is Abelian. To illustrate the latter, we consider the spin-orbit coupling realized in the experiment of Lin et al. [Nature (London) 471, 83 (2011)]. For this spin-orbit coupling, we compute the quasiparticle dispersions and spectral weights, the interaction-induced momentum-space Berry curvature, and the momentum-space distribution of spin density, and propose experimental signatures. Furthermore, we find that, for a generic spin orbit coupling, in our approximation for the single-particle propagator, the ground state can in principle support an integer Hall conductivity if the sum of the Chern numbers of the hole bands is nonzero. DOI: 10.1103/PhysRevLett.110.115301
PACS numbers: 67.85.d, 03.75.b, 37.10.Jk
Introduction.—Spin-orbit coupled electronic systems that break time-reversal symmetry can exhibit the quantum anomalous Hall (QAH) phase, which supports an integer Hall conductivity in the absence of a magnetic field and was first proposed in a model for graphene [1]. At the single-particle level, this is due to the nontrivial momentum-space topology of the Hamiltonian, which is characterized by a momentum-space Berry curvature that causes in the semiclassical equations of motion an anomalous velocity transverse to external forces, and its integral over the Brillouin zone is a topological invariant known as the Chern number [2]. Since it is now possible to engineer spin-orbit couplings in cold atomic systems [3], there have been several recent proposals of cold atomic optical lattice realizations of the QAH phase [4], the related atom topological insulators (TI) [5], and topological superfluids [6]. All these proposals concern fermions. While TIs and QAH phases have been extensively studied at the noninteracting and weakly interacting limit [7,8], the strongly interacting regime is much less understood. This regime is relevant for some strongly correlated materials, such as the transition metal oxides [9], and can be accessed in cold atomic optical lattices [10], where interactions strengths can be controlled. Furthermore, ultracold atoms are a natural environment to realize topological states for bosons, which have so far been studied mainly at a theoretical level [11]. In this Letter, we consider whether these states can be realized in a spin-orbit coupled bosonic Mott insulator. So far, only Abelian spin-orbit couplings have been achieved in cold atoms, as in the experiments of Ref. [3]. This spin-orbit coupling, equivalent to an equal amount of Rashba and Dresselhaus coupling, has zero Berry curvature in the absence of interactions. In this Letter, we consider the effect of an optical lattice and interactions in the Mott-insulating regime for this system. We find that the interactions renormalize the momentum-space spin texture 0031-9007=13=110(11)=115301(5)
and generate Berry curvature, resulting in an anomalous velocity for the quasiparticles. In two-dimensional topological insulators such as HgTe quantum wells, one can change the topological phase by tuning a time-reversal-breaking gap in the single-particle Hamiltonian [12]. In a spinful Mott insulator, one may consider whether such a gap can arise due to interactions. We find that this is indeed possible and that the Chern number can change in the (ferromagnetic) Mott-insulating phase. If the sum of the hole band Chern numbers is nonzero, we find, in the approximation that the propagator contains only quasiparticle poles, that the ground state can support an integer Hall conductivity. Interacting Berry curvature and Hall conductivity.—For an interacting system, the Hall conductivity H can be expressed in terms of the momentum-space single-particle propagator G ð!; kÞ, which reads in two spatial dimensions at zero temperature [2,13], ij Z þ ^ i G^ 1 G@ ^ j G^ 1 ; (1) d!dkei!0 tr½@! G@ H ¼ 2 8 h where, here and below, i ¼ ðkx ; ky Þ, 0þ denotes a positive infinitesimal, and h is the Planck constant (this conductivity gives a mass current, as the atoms are neutral). This expression is a topological invariant in frequency and momentum space [14] called the Chern number for an interacting system and is related to the number of edge states, and a nonzero integer value defines the QAH phase [15,16]. Next, we show that this phase can exist for a bosonic Mott insulator that supports quasihole excitations with nontrivial Berry curvature, as is typically the case in our approximation. Deep in the Mott-insulating phase, the relevant excitations are long-lived quasiparticle or quasihole states whose spin-orbit coupling is encoded in the Hermitian matrix ^ which can be expressed the local spin basis structure of G, ^ ^ as Gð!; kÞ ¼ U^ G^ d U^ y , where Uð!; kÞ is a unitary matrix
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Ó 2013 American Physical Society
and G^ d is a diagonal matrix. Below, we will compute ^ Gð!; kÞ in the Mott insulator. Using this basis, picking up the contributions from the quasiparticle poles in G^ d in the ! integration in Eq. (1), and making the finite temperature generalization of P Eq. (1), the Hall conductivity can be expressed as H ¼ sn sn =h, where sn ¼
~z Z dk B ksn ; 2 expðksn =kB TÞ 1
(2)
where T is the temperature; kB is the Boltzmann constant; s, n ¼ are indices for the spin-orbit and the quasiparticle or hole bands, respectively; and the interacting Berry curvature is given by [17] ~ z ¼ Bz þ vksn E ksn ; B (3) ksn
ksn
where vksn ¼ @k ksn are the band velocities and ksn are the quasiparticle energies relative to the chemical potential. The Berry electric field E and magnetic field Bz , which in 2D point in plane and out of plane, respectively, can be expressed in terms of a field strength tensor ðBzs ; E is Þ ¼ ðF sxy ; F s!i Þ, where F s @ As @ As , with , ¼ ð!; kx ; ky Þ, is defined in terms of the diagonal ^ ss . components of the matrix gauge field As ½iU^ y @ U It is convenient to define on-shell matrix rotations and ^ ~ ksn ¼ Uð! ¼ ksn ; kÞ and gauge fields in k space by U y s ~ ksn i½U ~ @k U ~ ss ¼ ½A! vksn þ As j!¼ ; A k
ksn
~z ¼ and then with analogous definitions, as above, B ksn s ~ xy . In the zero temperature limit, only the quasihole F contributions in Eq. (2) with ksn < 0 remain, 1X 1 Z ~z ; dkB (4) H ¼ ks h s 2 BZ where the integral over the Brillouin zone (BZ) is the Chern number, which is a topological property of each band. If the sum of these numbers is nonzero, this results in a ground state, integer Hall conductivity. Thus, although the QAH phase cannot occur for noninteracting bosons, it can occur in the bosonic Mott insulator or, more generally, for any bosonic system with hole bands of which the sum of Chern numbers is nonzero. Spin-orbit coupled Bose Hubbard model.—In the remainder of this Letter, we compute the Green’s function in the Mott insulator and we specifically consider a 2D square optical lattice of bosons with two spin components deep in the Mott-insulating phase at a commensurate filling, with spin-dependent hopping amplitudes. We will describe the system by a pseudospin 1=2 Bose Hubbard model with on-site repulsive interactions and write the Hamiltonian as H ¼ H0 þ V, where 1X U ay ay a a ; 2 i; i i i i X X V ¼ a~ yi t^ij a~ j ¼ a~ yk h^k a~ k ;
H0 ¼
hi;ji
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where i denotes lattices sites; the angled brackets denote nearest neighbors; , ¼ are spin indices; hats denote matrices in spin space; a~ k and a~ i denote two-component spinor field operators in momentum and real space, respectively; and t^ij are hopping matrices. It is convenient to ^ where ^ is the vector of Pauli write h^k ¼ hk þ hk , matrices and hk is the noninteracting spin-orbit field. The dispersions of the noninteracting particles are given by 1k ¼ hk þ jhk j. We will first derive results for a general hk and then apply them to the spin-orbit coupling of Ref. [3]. Strong-coupling perturbation theory.—We will compute the single-particle Green function in the strong-coupling limit V=U 1, using a perturbation theory in which the hopping Hamiltonian V is the perturbation to the interaction Hamiltonian H0 [18,19]. To this end, consider the grand canonical partition function with external sources ~ which has the path integral representation (here and J, below, we set @ ¼ 1) Z ~ J~ ¼ DaD ~ a~ exp S0 ½a ~ Z½J; Z X X a~ i t^ a~ j þ J~i a~ i þ a~ i J~i ; þ d hiji
i
(6) R P ~ and where S0 ¼ d i ½ai ð@ Þai þ H0 ðaÞ denotes imaginary time. We will use it as a generating functional for the correlation functions given by hai ayj i ¼ ð1=ZÞð 2 Z= Ji Jj ÞjJ¼0 . We will specify the unperturbed eigenstates in the occupation number basis by the number of spin-up and spin-down particles per site (Nþ , N ) with energies per site given by XU ENþ N ¼ N ðN 1Þ N þ Uþ Nþ N : 2 (7) The unperturbed imaginary-time ordered correlation function is defined by ij g ð; 0 Þ ¼ 0 y hNþ N jT ai ð Þaj ðÞjNþ N i, which is local in space and diagonal in spin, and its Fourier transform reads g ði!n Þ ¼
1 þ N N ; i!n ðNþ ; N Þ i!n ðN 1; N Þ (8)
where ðNþ ; N Þ EN þ1;N EN ;N are the excitation energies and !n are bosonic Matsubara frequencies. We will only be interested in the zero temperature limit, and the dependence on N above has been calculated in this limit. Carrying out this perturbation theory, we find
(5)
k
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^ G^ 1 ði!n ; k; N Þ ¼ g^ 1 ði!n ; N Þ þ hðkÞ:
(9)
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Although this expression is first order in hopping, its inverse, the propagator, contains all orders in hopping. In the following, we will denote by the replacement i!n ! ! the analytic continuation of the Green functions, which will be necessary to compute the spectral functions and quasiparticle dispersions and to define the spin texture in (!, k) space. Defining components by G^ 1 ¼ ^ the spin textures are given by d þ d , dð!; k; N Þ ¼ ½g1 ð!; N Þz z^ þ hðkÞ
(10)
and dð!; k; N Þ ¼ hðkÞ ½g1 ð!; N Þ0 , where g1 0 ¼ 1 Þ=2 and g1 ¼ ðg1 g1 Þ=2. The in-plane þ g ðg1 z þ þ texture is unchanged, while the z component is shifted by the z component of the inverse propagator. The quasiparticle dispersions are computed by solving ðG1 d Þss ð! ¼ ksn Þ ¼ d þ sjdj ¼ 0. The Berry electromagnetic fields are given by s ^ @! d^ @i dÞj ^ !¼ ; (11) ðBzksn ; E iksn Þ ¼ d^ ð@x d^ @y d; ksn 2 where d^ d=jdj and the expressions above are evaluated at ! ¼ ksn . Defining the quasiparticle spin-orbit field in k space by fksn ¼ dð! ¼ ksn ; kÞ, it is readily verified that ~ z =2 ¼ ðs=4Þf^ ksn @x f^ ksn @y f^ ksn , and thus the B ksn Chern number is the integer winding number of fksn . 2 d2 Þ, ^ From the simple poles of G^ ¼ ðd d Þ=ðd we find that quasiparticle peaks in the spectral function ^ ^ þ i0þ Þ Gð! ^ i0þ Þ are given by Að!Þ ¼ i½Gð! X 1 þ sf^ ksn ^ ^ ð! !ksn Þ; Að!; kÞ ¼ 2 Zksn 2 sn
(12)
where Zksn ¼ ½@! ðd þ sjdjÞj!¼ksn 1 are the quasiparticle weights. The numerator above is a spin projection operator that can be written as ~ ksn ~ yksn , where ksn is the normalized spinor part of the quasiparticle wave functions, which points in the direction f^ ksn . We have checked that the trace of the spectral function contains only these quasiparticle peaks. The momentum distributions of particle nk and spin sk density are given by fnk ; sk g ¼ R ^ ^ Að!; kÞ=ðe!=kB T 1Þ, which ð1=Nl Þ ðd!=2Þtr½f1; g in the ground state contains only quasihole contributions, P so that sk ¼ s snks f^ ks , where nks ¼ Zks =Nl is the band s density and Nl is the number of lattice sites. Next, we consider when the interaction-induced spinorbit texture f^ ksn can have a nonzero winding number. For this to be true, hk must have a nonzero in-plane winding number and hence contains vortices. We denote the position of these vortices by ki , and the gaps at the vortex cores which determine the Chern number are given by z fsn ðki Þ ¼ hz ðki Þ ðg1 Þz ½sn ðki Þ:
(13)
To have a nonzero winding number, the gaps need to have different signs at different ki ’s. Naively, since ðg1 Þz is of order U , one might expect it to be the dominant term and set a large uniform gap. However, in our perturbation
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theory, the quasiparticle energies are always close to the on-site particle or hole spectra, which are the zeros of g^ 1 ð!Þ, so that, as long as the the on-site spectrum is nearly spin symmetric [20], ðg1 Þz ½sn ðki Þ Oðh^1 Þ. Therefore, the winding number is determined by the competition between the terms in Eq. (13) and may differ from the noninteracting one determined by hz ðki Þ. A simple case.—To illustrate the relevant physics in the simplest setting, we turn off the interspin interaction, setting Uþ ¼ 0 and Uþþ ¼ U ¼ U. The excitation energies are ðN Þ ¼ UN , and the zeroth order ground state has filling factors N when N 1 < =U < N . Setting ¼ UðN 1=2Þ [21], the inverse total particle and spin propagator is fðg1 Þ0 ; ðg1 Þz g ¼
Uð4!2 1Þ 2ð1 þ 2Nþ þ 2!Þð1 þ 2N þ 2!Þ f1 þ Nþ þ N þ 2!; N Nþ Þg: (14)
In the cases of equal filling fractions ðNþ ; N Þ ¼ ðN; NÞ, ðg1 Þz ¼ 0, so that there are no corrections to the spin texture. The simplest filling fraction which gives an out-of-plane spin texture is (1, 0). This filling represents a ferromagnetic ground state, which is generally present in a bosonic Mott insulator. The quasiparticle dispersions can be expressed implicitly as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ksn 1 ksn þ n 9 þ 20~ (15) ¼ ð1 þ 2~ ksn þ 4~ 2ksn Þ; 4 U where ~ksn ¼ ðhk þ sjfksn jÞ=U and s, n ¼ 1. There is z no s ¼ þ (which has fksn < 0) quasihole band because there are no spin-down atoms in the unperturbed (1, 0) ground state. Thus, the ground state momentum distribution of spin density is given by sk ¼ nk f^ k . To give an example of hole bands with nonzero Chern number, we consider one component of the BernevigHughes-Zhang Hamiltonian which describes HgTe quantum wells [12] and we take parameters such that h ¼ 2tðcoskx þ cosky Þ, h ¼ 2tso ðsinkx ; sinky ; 0Þ. In the absence of interactions, these parameters give zero Chern number. However, by computing the gaps in Eq. (13), we find for the filling factor (1, 2) with t=U 0:01 and an additional on-site spin splitting of =U 0:01 that point to the existence of a quantum spin Hall effect, whereas the sum of the Chern numbers is zero for this model. An Abelian spin-orbit coupling.—We now consider the spin-orbit coupling of Ref. [3] by taking a Hamiltonian that at low energy is specified by hk ¼ k2 =2m and hk ¼ ðx ; kx ; 0Þ, where m is the effective mass [22], is the spin-orbit coupling, and x is an applied detuning. We plot the quasiparticle dispersions in Figs. 1(a) and 1(b) and sk in Fig. 1(c). The in-plane components of sk are qualitatively the same as hk , but interactions generate an out-of-plane component. Although the noninteracting Berry curvature vanishes, h^ @x h^ @y h^ ¼ 0, the interacting Berry electric ~ z ¼ vy E x , respectively, are field and curvature E xk and B k k k
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pffiffiffiffiffiffiffiffiffiffiffi FIG. 1 (color online). For the filling fraction (1, 0): Dispersions with energy in units of U with m=U ¼ x =U ¼ 0:01 for (a) band sp ¼ffiffiffiffiffiffiffiffiffiffi quasiparticles and (b) band s ¼ quasiholes. (c) Ground state momentum distribution of spin density (in units of @) sk for ffi m=U ¼ x =U ¼ 0:04. The vector field represents the in-plane components, and the colored density plot represents the z x and interacting Berry curvature B ~ z of the spin-up quasiparticle band, plotted component. (d) The ground state Berry electric field Ep ffiffiffiffiffiffiffiffiffiffiffi as a function of dimensionless wave vectors k0 ¼ k= 2mU, in the region jk0x;y j < 0:2.
nonzero, which we plot in Fig. 1(d). The Chern number is still zero because h has zero in-plane winding, and this is ~ z is antisymmetric verified in our computation because B k under inversion and hence vanishes upon integration. There are still nontrivial physical consequences due to the ~ z ðr z^ Þ, quasiparticle anomalous velocity r_ ksn ¼ B ksn which gives an edge current in the presence of a trapping potential . Furthermore, it was shown in Ref. [17] for the Fermi liquid that the quasiparticle weight is renormalized by a factor Z0 E r . Since the only key assumption was a Hermitian self-energy matrix, which is true for our case, we expect this to be valid. Experimental signatures.—There are many experimental signatures of the interaction-induced Berry curvature and topological transport that we studied in this Letter. The ground state particle and spin density can be measured using rf spectroscopy [23] and time of flight measurements [24]. The quasiparticle dispersions and spectral weights enter into the density-density response function, which can be measured with Bragg spectroscopy [25]. A nonzero
anomalous Hall conductivity causes transverse oscillations in collective modes which can be seen by absorption imaging of the density [26]. Furthermore, bulk and edge quasiparticle transport can be probed directly by exciting the particles and holes with lasers. Conclusion and outlook.—We have shown that, in a spinful bosonic Mott insulator, interactions significantly modify the quasiparticle’s spin-orbit momentum-space texture. In our strong-coupling perturbative approximation, we have computed the interacting Green function for a general spin-orbit coupled hopping Hamiltonian and show that the ground state can in principle support an integer Hall conductivity given by the sum of the hole Chern numbers. For the optical lattice version of the experiment in Ref. [3], we computed quasiparticle dispersions, Berry curvature, and momentum distributions of spin density. We hope this will motivate experiments to measure these quantities and to create the spin-orbit couplings with in-plane vortex textures which we have predicted will exhibit topological transport.
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It seems to be a generic property of two-band models, which have opposite signs of the Berry charge, that their Chern numbers tend to cancel. For future work, we will investigate whether it is possible to find a model with a nonzero sum of hole Chern numbers, for example, in a three-band model. According to the arguments presented in Ref. [11], the Hall conductivity for bosons should be quantized in even integers. Further research is needed to investigate the mutual consistency of these arguments with our approximation and the zero temperature Hall conductivity that follows from it. By choosing the (1, 0) filling fraction, we have in a sense broken time-reversal symmetry ‘‘by hand.’’ In the presence of interspin interactions Uþ , time-reversal symmetry will be broken spontaneously because the ground state will have magnetic order [27]. This can readily be included in our theory, where Eq. (9) still holds but g^ 1 acquires offdiagonal components [28]. A uniform ferromagnetic state gives qualitatively similar results to the case studied in this Letter. Furthermore, even in the presence of magnetization textures, our momentum-dependent results will be approximately valid for momenta larger than the inverse of the typical length scale of the texture. In general, textures will result in interplay between momentum-space and realspace Berry curvature, which motivates further study. Last, we leave the analysis of the Mott insulatorsuperfluid transition to future work. Note that, in the latter regime, the Green function in the Bogoliubov approximation has a trivial frequency dependence and does not contain the physics discussed in this Letter, which stems from the nontrivial frequency dependence in the Green function due to the on-site propagator in Eq. (8). We also intend to investigate the bulk and edge state transport properties of these systems in more detail. We thank Henk Stoof, Allan MacDonald, Yaroslav Tserkovnyak, and Ari Turner for valuable discussions. This work was supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM), by the Netherlands Organization for Scientific Research (NWO), and by the European Research Council (ERC) under the Seventh Framework Program (FP7).
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[5] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, Phys. Rev. Lett. 105, 255302 (2010). [6] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, and S. Das Sarma, Phys. Rev. B 83, 140510 (2011); L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [8] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [9] A. Shitade, H. Katsura, J. Kunesˇ, X.-L. Qi, S.-C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009); D. Pesin and L. Balents, Nat. Phys. 6, 376 (2010). [10] M. Greiner, O. Mandel, T. Esslinger, T. W. Ha¨nsch, and I. Bloch, Nature (London) 415, 39 (2002). [11] T. Senthil and M. Levin, arXiv:1206.1604; X. Chen, Z. X. Liu, and X. G. Wen, Phys. Rev. B 84, 235141 (2011); Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119 (2012). [12] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006). [13] K. Ishikawa and T. Matsuyama, Nucl. Phys. B280, 523 (1987). [14] G. Volovik, The Universe in a Helium Droplet (Oxford University Press, New York, 2003). [15] Z. Wang, X. L. Qi, and S. C. Zhang, Phys. Rev. Lett. 105, 256803 (2010). [16] X.-L. Qi, Y. S. Wu, and S. C. Zhang, Phys. Rev. B 74, 045125 (2006). [17] R. Shindou and L. Balents, Phys. Rev. Lett. 97, 216601 (2006); Phys. Rev. B 77, 035110 (2008). [18] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989). [19] D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A 63, 053601 (2001); D. B. M. Dickerscheid, D. van Oosten, P. J. H. Denteneer, and H. T. C. Stoof, ibid. 68, 043623 (2003). [20] This requires that N Þ 0, so that there are quasiholes for both spins. [21] The chemical potentials will generally acquire perturbative OðV=UÞ corrections. [22] It has an order of magnitude of m 1=jtja20 , where a0 is the lattice spacing and jtj is the order of magnitude of the hopping matrix. [23] M. Feld, B. Fro¨hlich, E. Vogt, M. Koschorreck, and M. Ko¨hl, Nature (London) 480, 75 (2011). [24] E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos, and J. J. Garcia-Ripoll, Phys. Rev. Lett. 107, 235301 (2011). [25] D. van oosten, Ph.D. thesis, Utrecht University, 2004. [26] E. van der Bijl and R. A. Duine, Phys. Rev. Lett. 107, 195302 (2011). [27] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi, Phys. Rev. Lett. 109, 085302 (2012); J. Radic´, A. Di Ciolo, K. Sun, and V. Galitski, ibid. 109, 085303 (2012); L. He, Y. Li, E. Altman, and W. Hofstetter, Phys. Rev. A 86, 043620 (2012); Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A 85, 061605 (2012). [28] T. Graß, K. Saha, K. Sengupta, and M. Lewenstein, Phys. Rev. A 84, 053632 (2011).
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Topological Transport in Spin-Orbit Coupled Bosonic Mott Insulators C. H. Wong and R. A. Duine Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Received 11 December 2012; published 12 March 2013) We investigate topological transport in spin-orbit coupled bosonic Mott insulators. We show that interactions can lead to anomalous quasiparticle dynamics even when the spin-orbit coupling is Abelian. To illustrate the latter, we consider the spin-orbit coupling realized in the experiment of Lin et al. [Nature (London) 471, 83 (2011)]. For this spin-orbit coupling, we compute the quasiparticle dispersions and spectral weights, the interaction-induced momentum-space Berry curvature, and the momentum-space distribution of spin density, and propose experimental signatures. Furthermore, we find that, for a generic spin orbit coupling, in our approximation for the single-particle propagator, the ground state can in principle support an integer Hall conductivity if the sum of the Chern numbers of the hole bands is nonzero. DOI: 10.1103/PhysRevLett.110.115301
PACS numbers: 67.85.d, 03.75.b, 37.10.Jk
Introduction.—Spin-orbit coupled electronic systems that break time-reversal symmetry can exhibit the quantum anomalous Hall (QAH) phase, which supports an integer Hall conductivity in the absence of a magnetic field and was first proposed in a model for graphene [1]. At the single-particle level, this is due to the nontrivial momentum-space topology of the Hamiltonian, which is characterized by a momentum-space Berry curvature that causes in the semiclassical equations of motion an anomalous velocity transverse to external forces, and its integral over the Brillouin zone is a topological invariant known as the Chern number [2]. Since it is now possible to engineer spin-orbit couplings in cold atomic systems [3], there have been several recent proposals of cold atomic optical lattice realizations of the QAH phase [4], the related atom topological insulators (TI) [5], and topological superfluids [6]. All these proposals concern fermions. While TIs and QAH phases have been extensively studied at the noninteracting and weakly interacting limit [7,8], the strongly interacting regime is much less understood. This regime is relevant for some strongly correlated materials, such as the transition metal oxides [9], and can be accessed in cold atomic optical lattices [10], where interactions strengths can be controlled. Furthermore, ultracold atoms are a natural environment to realize topological states for bosons, which have so far been studied mainly at a theoretical level [11]. In this Letter, we consider whether these states can be realized in a spin-orbit coupled bosonic Mott insulator. So far, only Abelian spin-orbit couplings have been achieved in cold atoms, as in the experiments of Ref. [3]. This spin-orbit coupling, equivalent to an equal amount of Rashba and Dresselhaus coupling, has zero Berry curvature in the absence of interactions. In this Letter, we consider the effect of an optical lattice and interactions in the Mott-insulating regime for this system. We find that the interactions renormalize the momentum-space spin texture 0031-9007=13=110(11)=115301(5)
and generate Berry curvature, resulting in an anomalous velocity for the quasiparticles. In two-dimensional topological insulators such as HgTe quantum wells, one can change the topological phase by tuning a time-reversal-breaking gap in the single-particle Hamiltonian [12]. In a spinful Mott insulator, one may consider whether such a gap can arise due to interactions. We find that this is indeed possible and that the Chern number can change in the (ferromagnetic) Mott-insulating phase. If the sum of the hole band Chern numbers is nonzero, we find, in the approximation that the propagator contains only quasiparticle poles, that the ground state can support an integer Hall conductivity. Interacting Berry curvature and Hall conductivity.—For an interacting system, the Hall conductivity H can be expressed in terms of the momentum-space single-particle propagator G ð!; kÞ, which reads in two spatial dimensions at zero temperature [2,13], ij Z þ ^ i G^ 1 G@ ^ j G^ 1 ; (1) d!dkei!0 tr½@! G@ H ¼ 2 8 h where, here and below, i ¼ ðkx ; ky Þ, 0þ denotes a positive infinitesimal, and h is the Planck constant (this conductivity gives a mass current, as the atoms are neutral). This expression is a topological invariant in frequency and momentum space [14] called the Chern number for an interacting system and is related to the number of edge states, and a nonzero integer value defines the QAH phase [15,16]. Next, we show that this phase can exist for a bosonic Mott insulator that supports quasihole excitations with nontrivial Berry curvature, as is typically the case in our approximation. Deep in the Mott-insulating phase, the relevant excitations are long-lived quasiparticle or quasihole states whose spin-orbit coupling is encoded in the Hermitian matrix ^ which can be expressed the local spin basis structure of G, ^ ^ as Gð!; kÞ ¼ U^ G^ d U^ y , where Uð!; kÞ is a unitary matrix
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and G^ d is a diagonal matrix. Below, we will compute ^ Gð!; kÞ in the Mott insulator. Using this basis, picking up the contributions from the quasiparticle poles in G^ d in the ! integration in Eq. (1), and making the finite temperature generalization of P Eq. (1), the Hall conductivity can be expressed as H ¼ sn sn =h, where sn ¼
~z Z dk B ksn ; 2 expðksn =kB TÞ 1
(2)
where T is the temperature; kB is the Boltzmann constant; s, n ¼ are indices for the spin-orbit and the quasiparticle or hole bands, respectively; and the interacting Berry curvature is given by [17] ~ z ¼ Bz þ vksn E ksn ; B (3) ksn
ksn
where vksn ¼ @k ksn are the band velocities and ksn are the quasiparticle energies relative to the chemical potential. The Berry electric field E and magnetic field Bz , which in 2D point in plane and out of plane, respectively, can be expressed in terms of a field strength tensor ðBzs ; E is Þ ¼ ðF sxy ; F s!i Þ, where F s @ As @ As , with , ¼ ð!; kx ; ky Þ, is defined in terms of the diagonal ^ ss . components of the matrix gauge field As ½iU^ y @ U It is convenient to define on-shell matrix rotations and ^ ~ ksn ¼ Uð! ¼ ksn ; kÞ and gauge fields in k space by U y s ~ ksn i½U ~ @k U ~ ss ¼ ½A! vksn þ As j!¼ ; A k
ksn
~z ¼ and then with analogous definitions, as above, B ksn s ~ xy . In the zero temperature limit, only the quasihole F contributions in Eq. (2) with ksn < 0 remain, 1X 1 Z ~z ; dkB (4) H ¼ ks h s 2 BZ where the integral over the Brillouin zone (BZ) is the Chern number, which is a topological property of each band. If the sum of these numbers is nonzero, this results in a ground state, integer Hall conductivity. Thus, although the QAH phase cannot occur for noninteracting bosons, it can occur in the bosonic Mott insulator or, more generally, for any bosonic system with hole bands of which the sum of Chern numbers is nonzero. Spin-orbit coupled Bose Hubbard model.—In the remainder of this Letter, we compute the Green’s function in the Mott insulator and we specifically consider a 2D square optical lattice of bosons with two spin components deep in the Mott-insulating phase at a commensurate filling, with spin-dependent hopping amplitudes. We will describe the system by a pseudospin 1=2 Bose Hubbard model with on-site repulsive interactions and write the Hamiltonian as H ¼ H0 þ V, where 1X U ay ay a a ; 2 i; i i i i X X V ¼ a~ yi t^ij a~ j ¼ a~ yk h^k a~ k ;
H0 ¼
hi;ji
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where i denotes lattices sites; the angled brackets denote nearest neighbors; , ¼ are spin indices; hats denote matrices in spin space; a~ k and a~ i denote two-component spinor field operators in momentum and real space, respectively; and t^ij are hopping matrices. It is convenient to ^ where ^ is the vector of Pauli write h^k ¼ hk þ hk , matrices and hk is the noninteracting spin-orbit field. The dispersions of the noninteracting particles are given by 1k ¼ hk þ jhk j. We will first derive results for a general hk and then apply them to the spin-orbit coupling of Ref. [3]. Strong-coupling perturbation theory.—We will compute the single-particle Green function in the strong-coupling limit V=U 1, using a perturbation theory in which the hopping Hamiltonian V is the perturbation to the interaction Hamiltonian H0 [18,19]. To this end, consider the grand canonical partition function with external sources ~ which has the path integral representation (here and J, below, we set @ ¼ 1) Z ~ J~ ¼ DaD ~ a~ exp S0 ½a ~ Z½J; Z X X a~ i t^ a~ j þ J~i a~ i þ a~ i J~i ; þ d hiji
i
(6) R P ~ and where S0 ¼ d i ½ai ð@ Þai þ H0 ðaÞ denotes imaginary time. We will use it as a generating functional for the correlation functions given by hai ayj i ¼ ð1=ZÞð 2 Z= Ji Jj ÞjJ¼0 . We will specify the unperturbed eigenstates in the occupation number basis by the number of spin-up and spin-down particles per site (Nþ , N ) with energies per site given by XU ENþ N ¼ N ðN 1Þ N þ Uþ Nþ N : 2 (7) The unperturbed imaginary-time ordered correlation function is defined by ij g ð; 0 Þ ¼ 0 y hNþ N jT ai ð Þaj ðÞjNþ N i, which is local in space and diagonal in spin, and its Fourier transform reads g ði!n Þ ¼
1 þ N N ; i!n ðNþ ; N Þ i!n ðN 1; N Þ (8)
where ðNþ ; N Þ EN þ1;N EN ;N are the excitation energies and !n are bosonic Matsubara frequencies. We will only be interested in the zero temperature limit, and the dependence on N above has been calculated in this limit. Carrying out this perturbation theory, we find
(5)
k
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^ G^ 1 ði!n ; k; N Þ ¼ g^ 1 ði!n ; N Þ þ hðkÞ:
(9)
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Although this expression is first order in hopping, its inverse, the propagator, contains all orders in hopping. In the following, we will denote by the replacement i!n ! ! the analytic continuation of the Green functions, which will be necessary to compute the spectral functions and quasiparticle dispersions and to define the spin texture in (!, k) space. Defining components by G^ 1 ¼ ^ the spin textures are given by d þ d , dð!; k; N Þ ¼ ½g1 ð!; N Þz z^ þ hðkÞ
(10)
and dð!; k; N Þ ¼ hðkÞ ½g1 ð!; N Þ0 , where g1 0 ¼ 1 Þ=2 and g1 ¼ ðg1 g1 Þ=2. The in-plane þ g ðg1 z þ þ texture is unchanged, while the z component is shifted by the z component of the inverse propagator. The quasiparticle dispersions are computed by solving ðG1 d Þss ð! ¼ ksn Þ ¼ d þ sjdj ¼ 0. The Berry electromagnetic fields are given by s ^ @! d^ @i dÞj ^ !¼ ; (11) ðBzksn ; E iksn Þ ¼ d^ ð@x d^ @y d; ksn 2 where d^ d=jdj and the expressions above are evaluated at ! ¼ ksn . Defining the quasiparticle spin-orbit field in k space by fksn ¼ dð! ¼ ksn ; kÞ, it is readily verified that ~ z =2 ¼ ðs=4Þf^ ksn @x f^ ksn @y f^ ksn , and thus the B ksn Chern number is the integer winding number of fksn . 2 d2 Þ, ^ From the simple poles of G^ ¼ ðd d Þ=ðd we find that quasiparticle peaks in the spectral function ^ ^ þ i0þ Þ Gð! ^ i0þ Þ are given by Að!Þ ¼ i½Gð! X 1 þ sf^ ksn ^ ^ ð! !ksn Þ; Að!; kÞ ¼ 2 Zksn 2 sn
(12)
where Zksn ¼ ½@! ðd þ sjdjÞj!¼ksn 1 are the quasiparticle weights. The numerator above is a spin projection operator that can be written as ~ ksn ~ yksn , where ksn is the normalized spinor part of the quasiparticle wave functions, which points in the direction f^ ksn . We have checked that the trace of the spectral function contains only these quasiparticle peaks. The momentum distributions of particle nk and spin sk density are given by fnk ; sk g ¼ R ^ ^ Að!; kÞ=ðe!=kB T 1Þ, which ð1=Nl Þ ðd!=2Þtr½f1; g in the ground state contains only quasihole contributions, P so that sk ¼ s snks f^ ks , where nks ¼ Zks =Nl is the band s density and Nl is the number of lattice sites. Next, we consider when the interaction-induced spinorbit texture f^ ksn can have a nonzero winding number. For this to be true, hk must have a nonzero in-plane winding number and hence contains vortices. We denote the position of these vortices by ki , and the gaps at the vortex cores which determine the Chern number are given by z fsn ðki Þ ¼ hz ðki Þ ðg1 Þz ½sn ðki Þ:
(13)
To have a nonzero winding number, the gaps need to have different signs at different ki ’s. Naively, since ðg1 Þz is of order U , one might expect it to be the dominant term and set a large uniform gap. However, in our perturbation
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theory, the quasiparticle energies are always close to the on-site particle or hole spectra, which are the zeros of g^ 1 ð!Þ, so that, as long as the the on-site spectrum is nearly spin symmetric [20], ðg1 Þz ½sn ðki Þ Oðh^1 Þ. Therefore, the winding number is determined by the competition between the terms in Eq. (13) and may differ from the noninteracting one determined by hz ðki Þ. A simple case.—To illustrate the relevant physics in the simplest setting, we turn off the interspin interaction, setting Uþ ¼ 0 and Uþþ ¼ U ¼ U. The excitation energies are ðN Þ ¼ UN , and the zeroth order ground state has filling factors N when N 1 < =U < N . Setting ¼ UðN 1=2Þ [21], the inverse total particle and spin propagator is fðg1 Þ0 ; ðg1 Þz g ¼
Uð4!2 1Þ 2ð1 þ 2Nþ þ 2!Þð1 þ 2N þ 2!Þ f1 þ Nþ þ N þ 2!; N Nþ Þg: (14)
In the cases of equal filling fractions ðNþ ; N Þ ¼ ðN; NÞ, ðg1 Þz ¼ 0, so that there are no corrections to the spin texture. The simplest filling fraction which gives an out-of-plane spin texture is (1, 0). This filling represents a ferromagnetic ground state, which is generally present in a bosonic Mott insulator. The quasiparticle dispersions can be expressed implicitly as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ksn 1 ksn þ n 9 þ 20~ (15) ¼ ð1 þ 2~ ksn þ 4~ 2ksn Þ; 4 U where ~ksn ¼ ðhk þ sjfksn jÞ=U and s, n ¼ 1. There is z no s ¼ þ (which has fksn < 0) quasihole band because there are no spin-down atoms in the unperturbed (1, 0) ground state. Thus, the ground state momentum distribution of spin density is given by sk ¼ nk f^ k . To give an example of hole bands with nonzero Chern number, we consider one component of the BernevigHughes-Zhang Hamiltonian which describes HgTe quantum wells [12] and we take parameters such that h ¼ 2tðcoskx þ cosky Þ, h ¼ 2tso ðsinkx ; sinky ; 0Þ. In the absence of interactions, these parameters give zero Chern number. However, by computing the gaps in Eq. (13), we find for the filling factor (1, 2) with t=U 0:01 and an additional on-site spin splitting of =U 0:01 that point to the existence of a quantum spin Hall effect, whereas the sum of the Chern numbers is zero for this model. An Abelian spin-orbit coupling.—We now consider the spin-orbit coupling of Ref. [3] by taking a Hamiltonian that at low energy is specified by hk ¼ k2 =2m and hk ¼ ðx ; kx ; 0Þ, where m is the effective mass [22], is the spin-orbit coupling, and x is an applied detuning. We plot the quasiparticle dispersions in Figs. 1(a) and 1(b) and sk in Fig. 1(c). The in-plane components of sk are qualitatively the same as hk , but interactions generate an out-of-plane component. Although the noninteracting Berry curvature vanishes, h^ @x h^ @y h^ ¼ 0, the interacting Berry electric ~ z ¼ vy E x , respectively, are field and curvature E xk and B k k k
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pffiffiffiffiffiffiffiffiffiffiffi FIG. 1 (color online). For the filling fraction (1, 0): Dispersions with energy in units of U with m=U ¼ x =U ¼ 0:01 for (a) band sp ¼ffiffiffiffiffiffiffiffiffiffi quasiparticles and (b) band s ¼ quasiholes. (c) Ground state momentum distribution of spin density (in units of @) sk for ffi m=U ¼ x =U ¼ 0:04. The vector field represents the in-plane components, and the colored density plot represents the z x and interacting Berry curvature B ~ z of the spin-up quasiparticle band, plotted component. (d) The ground state Berry electric field Ep ffiffiffiffiffiffiffiffiffiffiffi as a function of dimensionless wave vectors k0 ¼ k= 2mU, in the region jk0x;y j < 0:2.
nonzero, which we plot in Fig. 1(d). The Chern number is still zero because h has zero in-plane winding, and this is ~ z is antisymmetric verified in our computation because B k under inversion and hence vanishes upon integration. There are still nontrivial physical consequences due to the ~ z ðr z^ Þ, quasiparticle anomalous velocity r_ ksn ¼ B ksn which gives an edge current in the presence of a trapping potential . Furthermore, it was shown in Ref. [17] for the Fermi liquid that the quasiparticle weight is renormalized by a factor Z0 E r . Since the only key assumption was a Hermitian self-energy matrix, which is true for our case, we expect this to be valid. Experimental signatures.—There are many experimental signatures of the interaction-induced Berry curvature and topological transport that we studied in this Letter. The ground state particle and spin density can be measured using rf spectroscopy [23] and time of flight measurements [24]. The quasiparticle dispersions and spectral weights enter into the density-density response function, which can be measured with Bragg spectroscopy [25]. A nonzero
anomalous Hall conductivity causes transverse oscillations in collective modes which can be seen by absorption imaging of the density [26]. Furthermore, bulk and edge quasiparticle transport can be probed directly by exciting the particles and holes with lasers. Conclusion and outlook.—We have shown that, in a spinful bosonic Mott insulator, interactions significantly modify the quasiparticle’s spin-orbit momentum-space texture. In our strong-coupling perturbative approximation, we have computed the interacting Green function for a general spin-orbit coupled hopping Hamiltonian and show that the ground state can in principle support an integer Hall conductivity given by the sum of the hole Chern numbers. For the optical lattice version of the experiment in Ref. [3], we computed quasiparticle dispersions, Berry curvature, and momentum distributions of spin density. We hope this will motivate experiments to measure these quantities and to create the spin-orbit couplings with in-plane vortex textures which we have predicted will exhibit topological transport.
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It seems to be a generic property of two-band models, which have opposite signs of the Berry charge, that their Chern numbers tend to cancel. For future work, we will investigate whether it is possible to find a model with a nonzero sum of hole Chern numbers, for example, in a three-band model. According to the arguments presented in Ref. [11], the Hall conductivity for bosons should be quantized in even integers. Further research is needed to investigate the mutual consistency of these arguments with our approximation and the zero temperature Hall conductivity that follows from it. By choosing the (1, 0) filling fraction, we have in a sense broken time-reversal symmetry ‘‘by hand.’’ In the presence of interspin interactions Uþ , time-reversal symmetry will be broken spontaneously because the ground state will have magnetic order [27]. This can readily be included in our theory, where Eq. (9) still holds but g^ 1 acquires offdiagonal components [28]. A uniform ferromagnetic state gives qualitatively similar results to the case studied in this Letter. Furthermore, even in the presence of magnetization textures, our momentum-dependent results will be approximately valid for momenta larger than the inverse of the typical length scale of the texture. In general, textures will result in interplay between momentum-space and realspace Berry curvature, which motivates further study. Last, we leave the analysis of the Mott insulatorsuperfluid transition to future work. Note that, in the latter regime, the Green function in the Bogoliubov approximation has a trivial frequency dependence and does not contain the physics discussed in this Letter, which stems from the nontrivial frequency dependence in the Green function due to the on-site propagator in Eq. (8). We also intend to investigate the bulk and edge state transport properties of these systems in more detail. We thank Henk Stoof, Allan MacDonald, Yaroslav Tserkovnyak, and Ari Turner for valuable discussions. This work was supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM), by the Netherlands Organization for Scientific Research (NWO), and by the European Research Council (ERC) under the Seventh Framework Program (FP7).
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