Chern Kondo Insulator in an Optical Lattice
We propose to realize and observe Chern Kondo insulators in an optical superlattice with laser-assisted and orbital hybridization and synthetic gauge field, which can be engineered based on the recent cold atom experiments. Considering a double-well square optical lattice, the localized orbitals are decoupled from itinerant bands and are driven into a Mott insulator due to strong Hubbard interaction. Raman laser beams are then applied to induce tunnelings between and orbitals, and generate a staggered flux simultaneously. Due to the strong Hubbard interaction of orbital states, we predict the existence of a critical Raman laser-assisted coupling, beyond which the Kondo screening is achieved and then a fully gapped Chern Kondo phase emerges, with the topology characterized by integer Chern numbers. Being a strongly correlated topological state, the Chern Kondo phase is different from the single-particle quantum anomalous Hall state, and can be identified by measuring the band topology and double occupancy of orbitals. The experimental realization and detection of the predicted Chern Kondo insulator are also proposed.
Introduction.– The search for new quantum states with nontrivial topology has become an exciting pursuit in condensed matter physics. The recent important examples are the theoretical prediction and experimental discovery of the 2D and 3D topological insulators (TIs) hasan10 (); qi11 (). Recently, a few heavy-fermion materials (e.g. ) were predicted to be candidates of time-reversal-invariant topological Kondo insulator (TKI), which originates from the hybridization between itinerant conduction bands and strong correlated electrons Dzero10 (); Alexandrov13 (); lu13 (). It was shown that screening the local moments by conduction electrons leads to an insulating gap in the Kondo regime. The proposed scenario of TKIs is consistent with the transport measurement wolgast13 (); Kim14 (); Luo15 (), angle-resolved photoemission spectroscopy jiang13 (); Neupane13 (); Xu14 () and scanning tunneling spectroscopy Ruan14 (); Roessler14 (). Note that the essential difference between a TKI and conventional TIs is not topological classification, but the strong correlation physics for electrons which are absent in conventional TIs. To measure such strong correlation physics can directly distinguish a TKI from conventional TIs, while this might be a challenging task for condensed matter systems.
In this letter, we propose to realize and observe a strongly correlated quantum anomalous Hall (QAH) phase, called Chern Kondo (CK) insulator, in an optical lattice, motivated by the recent rapidly developing new technologies for cold atoms. These technologies include the Raman coupling scheme Ruseckas2005 (); Osterloh2005 (); Liu2009 () used to create spin-orbit interactions Lin (); Pan (); MIT (); Wang (), laser-assisted tunneling or shaken lattices to generate synthetic gauge fields Bloch2011 (); Bloch2013 (); Ketterle2013a (); Struck2013 (), and optical control of Feshbach resonance Walraven1996 (); Lett2000 (); Denschlag2004 (); Takahashi2008 (); Bauer2009 (); Ye2011 (); ChengChin2015 (). Compared with solid state systems, the cold atoms can offer extremely clean platforms with full controllability to study many-body physics and topological phases Chuanwei2008 (); Sato2009 (); Liu10 (); Goldman2010 (); Zhu2011 (); Liu2014 (); Goldman2014 (); Zhai2015 (). Here, we consider a double-well square lattice with Raman-coupling-assisted - orbital hybridization to observe CK insulating phases. Due to the strong Hubbard interaction of orbital states, the Kondo screening is achieved when the applied Raman coupling exceeds a critical value, and then a nonzero renormalized - orbital hybridization drives the system into a fully gapped CK phase. The parent state of orbital is a Mott insulator which has no double occupancy due to the Hubbard gap, and this property keeps in the CK regime, signifying an essential difference from single-particle QAH insulators. With this unique feature the CK insulators can be identified by measuring band topology and double occupancy of orbital in experiment.
Model.–For the realization we introduce a novel bipartite square optical lattice created with the setup in Fig. 1(a), which has a staggered energy difference between the neighboring A and B sites. This double-well checkerboard lattice can be realized based on NIST experiments sebby06 (); SI (), and more details will be presented later. The minimal model for realizing the CK insulators includes several basic ingredients. First, the sublattices A and B are anisotropic along the local coordinates . The relevant orbitals for our consideration are the orbital at A sites and orbital at B sites [Fig. 1(b)], with the intraorbital AA/BB hopping couplings along the diagonal directions. In the moderate to deep lattice regime we can verify that is much less than , leading to a relatively flat band for s-orbital. Accordingly, in such regime the transverse tunneling between -orbitals is much weaker than axial tunneling lsacsson05 (); liu06 (). Thus for the practical consideration, we neglect the -bond hoppings of the and orbitals by setting , and for the -bond hopping we have . Secondly, similar as the recent experimental studies Bloch2011 (); Bloch2013 (); Ketterle2013a (), we consider an onsite energy difference to suppresses the bare tunneling couplings between and orbitals. On the other hand, the neighboring - hybridization can be induced by a standard two-photon Raman coupling in the following form , where is the coupling amplitude, and represent the differences of frequencies and wave vectors between two Raman photons, respectively [Fig. 1(c)] Bloch2011 (); Bloch2013 (); Ketterle2013a (). The Raman potential drives the neighboring hopping between and orbitals when compensates their energy difference . Moreover, this laser-assisted hopping along direction is associated with a phase if the hopping is toward (away from) B sites with A being lattice constant, rendering a staggered flux pattern depicted in Fig. 1 (b) with the flux . Finally, we consider a (pseudo)spin- two-copy version of the - band model, so the Hubbard interactions can be introduced for the spinful fermions (e.g. Li or ) and denoted by () for the () orbitals. With these ingredients we can write down the effective Hamiltonian by , where
Here is the two-photon detuning for the Raman coupling [Fig. 1(c)], and ( and ) are annihilation (creation) operators of and orbitals, respectively, and The Raman coupling induced hybridization satisfies for and for , with being the induced hybridization strength.
In the single particle regime the present setup can realize QAH effect Liu10 (). We rewrite in the space with . The Bloch Hamiltonian takes the form
where , , and the Pauli matrices act on orbital space. The single-particle spectra, given by , has node points if with , and is gapped when and . The gapped single-particle system is in the QAH phase if Liu10 (). On the other hand, as studied below, a positive detuning will be applied to reach the CK phase.
Strongly interacting regime and Kondo transition.–To study the Kondo phase, we consider the strongly interacting regime for -orbital states with , and a weak interaction for orbital states, satisfying . To treat the interaction appropriately, the weak terms are decoupled by mean-field approximation . In the strong interacting limit , the state of double-occupation on orbital is prohibited. The system can be studied by the slave boson mean field theory with vanishing doublon Kotliar86 (). We introduce the slave-boson operator () at -th site to describe the creation(annihilation) holon state . Moreover, we consider instead of in eq. (1) with . Here the slave operators in uniform mean field approximation are replaced by time- and site- independent numbers, and . The hopping is also renormalized by a factor of , which preserves Luttinger volume and captures the correct band renormalization. To eliminate the redundant many-particle configurations of orbital, the constraints and with being the Lagrange multipliers are introduced. The functional for free energy per site takes the form SI ()
where , the quasiparticle spectra of the lower and upper hybridized bands are given by , respectively, with , and . The minimization of the free energy with respect to the mean field parameters are determined by with , which yields several coupled saddle-point equations
where is the Fermi-Dirac distribution function with the chemical potential. The half-filling condition for the entire system is fixed by the equation: , which sets the chemical potential in the self-consistent calculation.
The Kondo transition can be studied in the regime with , which implies that the onsite energy of the -orbital is lower than the Fermi energy of the half-filled -band. The direct hopping from orbital to orbital is however forbidden due to the Hubbard gap in the states, while the process of “cotunneling” which is a second-order virtual process can occur between and orbitals. This process is known as the Kondo coupling. In the weak Raman coupling regime, the -orbital states stay in a Mott insulator decoupled from the itinerant band, since the spectral weight of -band vanishes at the Fermi energy. We thus expect that the emergence of the mean-field effective hybridization, namely, the Kondo coupling between and orbitals requires that the laser-assisted tunneling exceeds a critical value, beyond which the Kondo screening is achieved and a fully gapped CK phase emerges.
We show the numerical results in Fig. 2 by solving the mean-field equations (Chern Kondo Insulator in an Optical Lattice) self consistently. We can see that the effective hybridization between and orbitals is strongly renormalized by a factor due to the Hubbard interaction in the orbital states. A critical laser-assisted tunneling is observed and when , the effective - hybridization renormalizes to zero [see the red curves in Fig. 2(a)]. This is in sharp contrast to a single-particle system. Accordingly, increasing to exceed , the coherent hybridization is gradually developed, leading to the formation of heavy quasi-particle bands as shown in Fig. 2 (b) and (c). The local moments are screened by spin flipping through the cotunneling process. A direct bulk gap opens between the heavy quasi-particle bands for . In contrary, if the gauge flux , the in eq. (3) vanishes and the bulk has gapless nodal points.
To analyze the topology of the bands, the Chern number for the -th band is explicitly calculated by
where are velocity operators along directions. Fig. 2(a) shows nonzero Chern number for . The nontrivial topological property originates from the band inversion between the and bands. The phase diagrams versus gauge field and are shown in Fig. 3 (a,b), with the subfigure showing the gauge field dependence of and Chern number at a line cut of . The and the direct gap show symmetric symmetric behavior with respect to , while Chern number changes sign across this point.
Experimental realization and detection.–Now we turn to the model realization and detection of the CK insulating phase. The square superlattice potential can be generated through the setup shown in Fig. 1 (a) with an incident laser field SI (). Neglecting all the irrelevant phase factors, the in-plane polarized () components generate the standing wave as . The out-of-plane polarized () component generates the light field as . Here we set that the component light is partially reflected with a ratio by the mirror . The phase is the difference of phases acquired by and components of the laser beam traveling from the lattice center to mirrors and , respectively, and back to the lattice. From and one can obtain the total lattice potential with the amplitudes . Hence the amplitudes can be readily controlled by tuning and . For example, taking that and , we have and if setting . This regime gives , and , which well meet the former two ingredients of realization.
The configuration for a strong interaction at A sites and weak at B sites can be reached with optical control of Feshbach resonance, which manipulates the atomic interactions by optical field strength Walraven1996 (); Lett2000 (); Denschlag2004 (); Takahashi2008 (); Bauer2009 (); Ye2011 (); ChengChin2015 (). The staggered modulation of interactions at A and B sites can be tuned with a periodic optical potential which minimizes (maximizes) at A (B) sites. Interestingly, in our case the -terms in the optical lattice potential match the required profile and can play such role. We note that such a spatial modulation of interactions has been achieved in the recent experiment with a long life time ChengChin2015 ().
The full controllability of cold atoms can enable us to distinguish the CK insulating phase from single-particle QAH states. The present CK insulator can be identified by three characteristic features, namely, a critical - coupling strength for CK phase transition, the nontrivial topology in the bulk band, and the Mott behavior of -orbital. The former two features can be confirmed by band topology measurement through Hall effect Esslinger2014 () which exists for but is absent for , while the last one is a key feature which makes the CK phase be exceptional. Since the parent state of -orbital is a Mott insulator, the double occupancy of -orbital is suppressed by the Hubbard gap. This property keeps both before and after the CK phase transition. Therefore, the third feature of CK phase can be detected by measuring the double occupancy of the and orbitals, which can be carried out in standard cold experiments Jordens2008 (); Takahashi2012 ().
The double occupancy of -orbital can be derived in the atomic limit. The partition function reads , where and is the energy of atoms occupying the -orbital at an A site. We have that and the double occupancy of -orbital . Note that should be determined self-consistently with the total number of atoms in and states being fixed. On the other hand, the double occupancy of -orbital can be obtained by . From the numerical results shown in Fig 4 (a), one can find that is greatly suppressed both before and after the CK phase transition, while is unchanged versus in the decoupled phase and increases in the CK phase. The later property reflects that the emergent hybridization in the CK phase leads to a net pumping of atoms from -orbital to orbital. For a comparison, in Fig 4 (b) we show the results for a single-particle QAH phase. With the same single-particle parameters, the double occupancy is much larger and exhibits qualitatively different behavior upon increasing .
In conclusion we have proposed to realize and observe CK insulators in an optical superlattice with laser-assisted and orbital hybridization and synthetic gauge field, which can be engineered based on the recently fast developing techniques in cold atoms. The predicted CK insulator can be identified by three characteristic features, namely, the existence of a critical - coupling strength for CK phase transition, the nontrivial topology in the bulk band, and the Mott behavior of the -orbital. These features distinguish the strongly correlated CK insulating phase from the single-particle QAH states.
Acknowledgements.We thank Fa Wang for valuable discussions. X.J.L. also thanks Kai Sun, Hui Zhai, and Andriy Nevidomskyy for insightful comments. This work is supported by National Basic Research Program of China (Grants No. 2015CB921102) and National Natural Science Foundation of China (No. 11534001 and No. 11574008). XJL is also supported in part by the Thousand-Young-Talent Program of China.
Supplementary Material – Chern Kondo Insulator in an Optical Lattice
.1 Double-well optical superlattice
We provide here more details for the generation of the square superlattice potential through the setup shown in Fig. 1 (a) of the main text, with an incident laser field . The initial phases are irrelevant and have been neglected. The light beam is reflected by the mirrors, with the forward and backward traveling fields intersecting and forming lattice in the middle region. In particular, the in-plane polarized () components are reflected by mirrors , , and , and can form the standing wave in the form
Here represents the phase acquired by the in-plane light beam through the path from lattice center (intersecting point) to the mirror , then to , and finally back to the lattice center, and represents the phase acquired through the path from lattice center to the mirror [Fig. 1 (a)]. On the other hand, after traveling via and , the out-of-plane polarized () component is reflected by a PBS (polarization beam splitter) and then partially reflected by with a reflection ratio . By controlling the optical path, we can set that the -component beam acquires an additional phase relative to by the in-plane polarized component traveling through the path from lattice center to PBS and then to . This gives the standing wave for the -component light
It is easy to see that the phase factors can be absorbed by shifting the position and . Then the light fields read
The above formulae imply that the generated standing wave fields are stable against phase fluctuations, which at most give rise to global shifts of the lattice and do not affect the model realization. From the above results we can get light field strength that (neglecting the constant terms)
Note that the lattice potential is proportional to the light field strength, with the proportional factor being the dipole matrix element. Therefore, the total lattice potential takes the following form
where the amplitudes . Hence the amplitudes can be readily controlled by tuning and . This is the lattice potential considered in the main text. Taking that and , we have and if setting . This parameter regime gives the lattice potential file shown in Fig. 1(b), with a large anisotropy along direction. The relevant parameters are obtained as , and . It is noteworthy that the present realization is of topological stability, namely, the phase fluctuations in the incident laser beam cannot affect the configuration of the present double square optical lattice.
.2 Slave boson theory
The local correlation for the mixed and orbital lattice takes usual on-site Hubbard interaction
When the interaction for orbitals is weak compared with band width of the dispersive orbitals, the Hartree approximation is adequate to capture local correlation effects. The strong interaction is treated more seriously by introducing auxiliary slave particles to describe the many-body configurations due to the localized nature of orbitals. In the slave-boson theory Kotliar (), the spin- fermion operator is written as
where the boson operators describe the holon , singly occupied states , doublon , and is a fermion operator. The explicit form of renormalization operators and takes the form
recovering the Gutzwiller approximation Gutzwiller65 () as a saddle point solution of Hubbard model.
To eliminate unphysical states, the Lagrange multipliers and are then introduced to ensure the local constraint of the completeness of the enlarged Hilbert space
and two equivalent ways to count the local fermion occupancy for each spin projection between original fermion and slave boson representations
The partition function within path integral formulism over coherent states of Fermi and Bose fields has the following form
where and the corresponding Lagrangian is given by
In the uniform mean-field slave boson approximation with infinite on-site Coulomb interactions , the doublon vanishes to avoid the unphysical divergence and the boson operators are replaced by time- and site-independent numbers , and . The resulting renormalization factor has a concise expansion
The functional for free energy per site is easily obtained from the partition function after integration of Fermi fields and takes the following form
Here, are the lower and upper hybridized bands for the quasi particle with spin projection , respectively: , , and . The extremization of the free energy with respect to the mean field parameters are determined by with , which yields several coupled saddle-point equations
where is the Fermi-Dirac distribution function.
Note that in the main text, the substituted notion and , keeping the formalism complete in essence, is adopted without expanding the main text to introduce the auxiliary number and operator .
.3 Effective model
The standard periodic Anderson model is shown to connect the Kondo lattice via the well known Schrieffer-Wolff transformation Schrieffer66 (). Below we shall derive the effective Kondo Hamiltonian describing the essential Kondo screening at and limit. The original Hamiltonian takes the form
Atomic projectors operating on -th site orbital are then introduced as follow