Abstract
Ultracold atoms in optical lattices serve as a tool to model different physical phenomena appearing originally in condensed matter. To study magnetic phenomena one needs to engineer synthetic fields as atoms are neutral. Appropriately shaped optical potentials force atoms to mimic charged particles moving in a given field. We present the realization of artificial gauge fields for the observation of anomalous Hall effect. Two species of attractively interacting ultracold fermions are considered to be trapped in a shaken two dimensional triangular lattice. A combination of interaction induced tunneling and shaking can result in an emergent Dice lattice. In such a lattice the staggered synthetic magnetic flux appears and it can be controlled with external parameters. The obtained synthetic fields are nonAbelian. Depending on the tuning of the staggered flux we can obtain either anomalous Hall effect or its quantized version. Our results are reminiscent of Anomalous Hall conductivity in spinorbit coupled ferromagnets.
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Introduction
Due to its unusual features such as quantized conductance and dissipationless edge states, the Quantum Hall effect (QHE)^{1} has various possible applications in quantum information sciences. In practical implementation, the standard QHE needs strong external magnetic fields and high mobility samples to occur. Therefore, it is particularly desirable to realize Hall effects without external magnetic fields.
In 1881, Hall^{2} observed that in ferromagnetic materials there are unusually large Hall currents at low fields when compared to nonmagnetic conductors^{3}. Since then, theoretical explanation of this effect was a subject of a debate and it has taken a century until the physics of the phenomena were explained. This effect, known now as the anomalous Hall effect (AHE), originates from spontaneous magnetization in spinorbit coupled ferromagnets^{4,5,6}. The magnetization breaks the time reversal symmetry while the spinorbit coupling induces nontrivial topology of the bands^{7,8}. Conductivty is not quantized for a metal, giving AHE and quantized for insulators when Fermi energy lies in the bandgap, giving quantum anomalous Hall effect (QAHE). AHE and its quantized version can occur even in zero magnetic fields and they have been observed in various systems^{5,9,10,11,12}.
Haldane^{13} in 1988 gave a theoretical proposal of an AHE without spin orbit coupling. He presented a quantized Hall effect without Landau levels in a system with circulating currents on a honeycomb lattice where the time reversal symmetry is broken only locally. Since then concentrated effort have been put forward to simulate AHE without the presence of a magnetic field.The key point of such proposals is to engineer nontrivial topology of energy bands where the Hall conductance is related to the integral of the Berry curvature of the filled part of the band. To tune the band structure in order to change its topology and induce the anomalous Hall effect, we need to create nonAbelian synthetic gauge fields^{14,15,16,17,18}. In the case of Ferromagnets it is done by spinorbit coupling, in Haldane model — by circulating currents. In all of those proposals regarding AHE without magnetic field, one important ingredient is the presence of strong nextnearest neighbor tunneling with certain complex amplitude. Such a tunneling is in general hard to realize in normal lattices due to the exponential suppression of tunneling amplitudes with the distance. This presents another pertinent question: is it possible to generate AHE in a lattice with only the nearest neighbor tunneling? In the present paper we present such a lattice model leading to AHE in the quantum regime.
We show that in twodimensional lattice, the combined effect of interaction induced tunneling and shaking can induce AHE and QAHE (used in 1D, this ingredients can also lead to topological phenomena^{19}). We focus on the systems of ultracold gases that provide versatile platform to simulate and engineer novel forms of matter^{14}. Our proposal consists of attractive twospecies fermions (as in^{20}) trapped in a periodically shaken triangular lattice. Triangular lattice introduces geometrical frustration while the shaking can resonantly enhance the interactioninduced sporbital nearest neighbor tunneling. In effect, an emergent Dice lattice is formed accompanied by a strong staggered flux which, due to inclusion of porbitals, leads to a formation of synthetic nonAbelian fields. The system shows spontaneous magnetization accompanied by appearance of anomalous Hall conductivity forming an ultracold gas analogue of spinorbit coupled ferromagnetic insulators. Furthermore, we show that, in presence of a strong staggered field, one can reach the regime of quantized Hall conductivity. This is a proposal of an experimentally realizable system with AHE without spinorbit coupling.
Results
The model
Consider an unequal mixture of twospecies attractive ultracold fermions (denoted by ↑, ↓) trapped in a triangular lattice with fillings n^{↑} = 1/3 and n^{↓} > 1/3. A strong attractive contact interaction between atoms leads to pairing — formation of composites between the ↑ and ↓ fermions, as studied experimentally for different lattice geometries^{21,22,23}. We define a composite creation operator with the corresponding number operator . are the creation and annihilation operators of the σ fermions in the respective sbands. The composites are hardcore bosons which anticommute at the same site, and commute for different sites, for i ≠ j^{24}.
We consider three lowest bands of the triangular twodimensional (2D) lattice (we assume some tight trap in the third direction as in typical 2D cold atoms experiments^{25,26}). For sufficiently deep optical lattices the structure of the bands may be understood using a harmonic approximation for separate sites. The lowest band is the sband with two close in energy porbitals forming the excited bands. Typically fermions (for low filling) reside in the sband. However, once the composite occupies a given site an additional fermion coming to this site must land in the excited band due to the Pauli exclusion principle.
The harmonic approximation typically underestimates the tunneling coefficients (for a discussion see a recent review^{27}). This is of no importance for the following since we assume that by using the well developed lattice shaking techniques, one can tune the standard intraband tunneling to negligible values^{25,26,28}. Such a shaking simultaneously makes the intraband interaction induced tunneling^{29,30,31,32,33,34} (called also bondcharge tunneling) vanishingly small. The only remaining tunneling mechanism is then the spinterband interaction induced tunneling^{20} which can be resonantly enhanced adjusting the shaking frequency (note that the standard singlebody tunneling between sp orbitals vanishes in Wannier function representation). Therefore, the system at lowenergies consists of the composites and the excess ↓ fermions with filling n^{↓} − n^{↑}. Note that the ↓  or ↑ fermions of the composites cannot undergo sptunneling without breaking the strong pairing  which costs energy. Similarly, as discussed in detail in^{22,24}, the tunneling of the composites to a neighboring vacant site as a whole is extremely small, (see Methods section) so it is neglected. Thus, the lowenergy local Hilbert subspace is spanned by and states, where denote the excess ↓fermions operators in the s and porbitals. The latter are written in the chiral representation . Within this subspace, one can show that the composite number operator equals the ↑ fermions number operator, and the densities n^{c} = n^{↑} = 1/3. Other important relations are: i) — a composite and an excess sfermion cannot occupy the same site due to the Pauliexclusion principle; ii) and iii) for i ≠ j.
The effective Hamiltonian for the composites and the excess ↓ fermions consists of three parts (see Methods for more details: H_{sp} describing interactioninduced sptunneling, H_{onsite} describing energies and local contact interactions and H_{shaking} describing the driving force. First of them reads,
where vectors connecting nearestneighbors in the triangle lattice are δ = ±δ_{0}, ± δ_{±}with . Due to the angles created by the different δ vectors, in the chiral representation an additional phase factor f_{δ} appears. In the harmonic approximation of the triangular lattice potential, this phase factor is given by f_{δσ} = exp[−iσtan^{−1}(δ_{y}/δ_{x})]. The tunneling J_{sp} is given in terms of the s and pband Wannier functions and as
with the contact interaction strength g_{2D} adjusted for a quasi2D geometry (with a tight harmonic confinement along z)^{35}. The second part gives the onsite Hamiltonian including higher band energy contribution and contact interactions. It reads^{20},
where U_{2} denotes the energy of the composites and U_{sp} is the additional interaction energy to occupy the porbital of a composite filled site. E_{1} is the singleparticle excitation energy of the pband. Shaking with elliptical periodic driving force leads to^{25},
with the shaking force . We consider the case where J_{sp} U_{2},U_{sp} ≤ Ω. That allows us to use rotatingwave approximation and Floquet theorem and to average terms fast oscillating in time (see Methods).
The sptunneling will be resonantly enhanced when the energy to occupy the pbands is an integer multiple of the shaking frequency. This translates into the condition that E_{1} + U_{sp} = NΩ for integer N. The resonance order, N, can be controlled by varying either the lattice depth, interaction strength or the driving frequency. The timeaveraged Hamiltonian then becomes,
where
with being the Bessel function of the first kind with integer order N. The amplitudes are and . The phase factor α_{δ} = 0 for δ = ±δ_{0} and for δ = ±δ_{±}. The effective tunneling strength may be characterized by . Moreover, lattice shaking also induces phases to the sptunnelings (6) as illustrated in Fig.1(a) (see also Methods).
The ground state composite structure
The composite number operator commutes with the Hamiltonian (5), . Therefore, we can characterize a site by the presence or the absence of a composite, i.e. which makes the Hamiltonian, (5), quadratic in operators for a particular realization of composite configuration. For a given composite configuration, we then diagonalize the quadratic Hamiltonian and fill up the energy levels depending on the excess ↓ fermions filling n^{↓} − n^{↑} ≤ 1/3. We find the ground state composite structure by comparing the energies of different composite configurations using simulated annealing on 6 × 6 up to 20 × 12 lattices with periodic boundary conditions. For details about the parameters of simulated annealing, we refer to Ref. 20. The resulting ground state selforganized structure of the composites resembles a Dice lattice and is shown in Fig. 1(a). Its basis consists of three sites denoted A, B and C. The A site consists of two orbitals p_{+} and p_{−} whereas the sites B and C have only sorbitals. The basis vectors for the Dice lattice are given by and . For any deviation from the 1/3 filling of the composites, the excess composites or vacancies will show up as impurities on top of the Dice lattice as long as the density of such impurities is small (n_{imp} 1/3).
To understand the origin of the Dice structure, consider first a composite at some chosen site A. The energy is minimized when all the neighboring sites (forming hexagon with the site A in the center) are without composites. This facilitates the sp tunneling from a p orbital at site A to the neighboring sites. Any composite on these neighboring sites increases the energy by J_{sp}F_{avg,δ}. Thus, for a composite filling of n^{c} = 1/3, the delocalization area is maximized by filling the lattice with hexagons with a composite at their center.
Assuming the ground state configuration is fixed, the effective Hamiltonian for the excess fermions is quadratic and thus easily diagonalized to yield the band structure. The behavior of the excess fermions is then that of an ideal Fermi gas with such band structure, which is easily computed.
Creation of the staggered field
For the Dice lattice considered here, one can construct two kind of plaquettes: i) The three plaquettes as shown in Fig. 1(b) where the particle traverses the closed path involving p_{x} ↔ s↔ p_{x} ↔ s orbitals. Such a path does not mix the p_{x} and p_{y} orbitals. Due to the phases of the tunneling amplitudes, a particle going through each of those plaquettes (denoted by φ_{1},φ_{2} and φ_{3}, see Fig.1(b) and calculated along the direction of the arrow) acquire fluxes due to AharonovBohm effect. We find that the induced flux is staggered in nature as the phases obey the constraint mod(φ_{1} + φ_{2} + φ_{3},2π) = 0. These fluxes are calculated by taking into account a single porbital. Figure 1(c) shows the flux strengths for N = 1. In particular, for Φ = 0, φ_{1} = φ_{2} = φ_{3} = 2π/3 which is equivalent to a uniform magnetic flux of the same magnitude. With growing Φ, fluxes change with all fluxes vanishing at Φ = π/2.
ii) The other kind of plaquette involves the A sites containing p_{x}, p_{y} orbitals as given by the parallelogram shown in Fig. 1(d). A particle going around such plaquette picks up a nonAbelian flux. Consider the transport from site “1” to “2”. The process can go either via the upper or the lower path with two consecutive sp tunnelings with strengths and . The effective amplitude becomes . The kinetic energy term around the plaquette for A sites may be written as
where the array Ψ_{l} = (p_{+}, p_{−})_{l} denotes porbitals at site l. The corresponding link variables connecting the neighboring A sites along the clockwise direction are given by 2 × 2 matrix L_{mn} with
L_{23} () is given by changing π/3 → −π/3 and δ_{+} → δ_{−} in the expression for L_{12} (). Moreover, we find that L_{12} = L_{34} and L_{23} = L_{41} (with similar relations for ) and they depend on the staggered flux through the phase of the tunneling amplitudes. The link variables are not unitary, which makes it not straightforward to describe them as synthetic nonAbelian fields. Nonetheless one can polar decompose them, , where . Such decompositions are possible as the L_{nm} matrices are positive semidefinite. Then one can define a corresponding Wilson loop parameter^{36}
The Wilson loop parameter has (i) an intrinsic contribution (appearing at Φ = π/2), with no staggered flux through an individual plaquete in Fig. 1(b) due to the appearance of the spband tunneling and (ii) an extrinsic contribution due to the external staggered flux induced by shaking. As a result, the link matrices are of nonAbelian nature (W ≠ 2 ) for any shaking phase Φ.
Spontaneous magnetization
First, we study the behavior of the system in the absence of staggered flux realized for N = 1,Φ = π/2 Fig. (1). The effective nonAbelian field is intrinsic in nature and the corresponding dispersion relation for the lowest energy band is shown in Fig. 2a. The main characteristic of the dispersion relation is the appearance of two nonequivalent Dirac cones and disappearance of flat bands. This is in contrast to the dispersion relation in a normal Dice lattice where the dispersion relation contains an intersecting Dirac cone and a flat band. Moreover, above a certain Fermi energy (of the excess fermions), the first two bands are degenerate. When we introduce the staggered flux, the dispersion changes and the gap opens at the band touching points (Fig. 2b). Once the Fermi energy is higher than the gap, the two bands become degenerate again. This is in a stark contrast to other situations with nearest neighbor tunneling where staggered flux leads only to the movement of the Dirac cones^{37,38} and to opening a gap one either needs longdistance tunneling^{13,17}, uniform magnetic field or synthetic nonAbelian fields along with magnetic field^{39}.
Anomalous Hall effect
Consider the local magnetization () in position space as well as magnetization in momentum space defined as
A nonzero local magnetization characterizes the breaking of timereversal symmetry as the particles acquire local angular momentum due to the particle number difference between the p_{+} and p_{−} orbitals. First, we find that the presence of nonzero staggered flux immediately results in nonzero and in opening of the gap. Thus, appearance of nonzero can be used as an indirect evidence for the presence of a gap in our system. The local magnetization is shown in Fig.2c (dashed line) for a small staggered flux. It vanishes only when the first two bands are totally filled. The presence of spontaneous magnetization (spontaneous timereversal symmetry breaking) is reminiscent of spinorbit coupled ferromagnets^{4}. Moreover, to look into the topological nature of the system, we define the intrinsic Hall conductivity,
The Berry curvature, Ω_{n}(k), for the nth band is given by Ω_{n}(k) = ∇_{k} × 〈u_{nq}  ∇_{k}  u_{nq}〉 where u_{nq}〉 denotes an eigenvector for the nth band. The total Hall conductivity σ_{xy} then depends on the Fermi energy ε_{F} of the system as shown in [Fig.2c (solid line)]. We find that the local Berry curvature is concentrated near the Dirac points which results in a nonzero contribution to when ε_{F} is in the band. As ε_{F} enters the band gap, we find that flattens at a value >1/2. This can be ascribed to the presence of two Dirac cones near the band gap. As we increase ε_{F}, the contribution from the next band begins to play a role and eventually the conductivity changes sign. The second peak appears when the Fermi energy reaches the maximum of the first band. Such structures in conductivity have been predicted to arise due to the presence of magnetic monopoles in the momentum space^{40}.
Quantum Anomalous Hall effect
Finally, consider the strong flux limit, e.g the case of Φ = 0 where the flux through each plaquette is 2π/3. Strong flux results in lifting the degeneracy between the first two bands (Fig. 3, top plot). The middle two bands still touch each other in the form of Dirac cones. With the degeneracy lifted, one can define Chern numbers given by ν = (2,−4,2) resulting in the appearance of quantum Anomalous Hall effect. We have also calculated the Hall conductivity and when the Fermi energy of the excess fermions resides in the band gap, conductivity becomes integer valued (Fig. 3, bottom plot). The magnitude of the band gap is ≈ J_{sp}. For a triangular lattice (lattice constant a = 500 nm)with lattice depth of 6E_{R} and transverse frequency of 10E_{R}, the sp tunneling strength in the harmonic approximation is given by J_{sp} ~ 0.008E_{R} assuming the scattering length of −400 Bohr radius. This corresponds to a band gap of about ~10 nanoKelvin which determines the temperature regime where the Hall phase can be observed. For the Dice lattice with dilute impurities, the Hall conductivity presented in this paper remain unchanged due to the topological nature of the Berry curvature for the dispersion bands^{4}. The band topology discussed here can be measured in principle by using recently proposed methods of Ramsey interferometry and Bloch oscillations^{41,42}, or from momentum distribution from TimeofFlight images^{43}. Moreover, the generation of local orbital angular momentum due to broken timereversal symmetry in the chiral porbitals can also be detected by timeofflight measurements^{44}.
Conclusions
To summarize, we considered an unequal mixture of attractively interacting fermions in a shaken triangular lattice. Pairing produces immobile composites that gives rise to dice lattice for the excess fermions. Adjustments of shaking frequency and amplitude allow to make intraband tunnelings negligible while resonantly enhancing interactioninduced sptunnelings for the excess fermions. Moreover, shaking leads to the controlled staggered magnetic field and induces (on the porbitals) nonAbelian character of the system. Their joint effect leads to spontaneous chiral magnetization (due to time reversal symmetry breaking) along with appearance of Anomalous Hall effect. Many fascinating question related to the findings here can be investigated further including the role of impurities, longrange interaction etc. Moreover, by using dipolar atoms, one can further study manybody effects like superconductivity^{45,46}, densitywaves in presence of the artificial nonAbelian gauge fields presented here.
Methods
The model Hamiltonian
We consider an unequal mixture of twospecies ultracold fermions (denoted by ↑, ↓ ) assuming strong attractive interactions between two species. It is energetically favorable for fermions to pair, the low energy system is then effectively composed of paired composites and the excess ↓ fermions. We denote the creation and annihilation operators for ↑ fermions as and . For the more abundant ↓ fermions we include both s and p orbitals denoting the corresponding operators as . In the main text, for simplicity, we have neglected ↑ fermion tunneling and all the intraband tunnelings for ↓ fermions from the beginning. Here, let us derive the Hamiltonian without these assumptions and show that, indeed, these effects may be neglected.
The full timedependent Hamiltonian H(t) consists of three parts H(t) = H_{tun} + H_{onsite} + H_{shaking}. The first, H_{tun} describes the tunnelings, H_{onsite} describes the onsite interactions and H_{shaking} describes the shaking. Together they read:
Here, , denote number operators of ↑ (↓ ) fermions respectively while are number operators for the ↓ pfermions with ±chirality. The same amplitude, J_{0} corresponds to the standard tunneling between s orbitals, the corresponding tunneling in the pband is described by . Moreover, we include density induced (bondcharge) intraband tunneling for porbitals with strength . J_{sp} is the amplitude of the hopping between s and p bands which is also induced by the interaction with ↑ fermions. The various tunneling processes in Hamiltonian (12) are shown in Fig. 4. The tunneling amplitudes are given by
where is the Wannier function of the sband and with σ = ± are the Wannier functions corresponding to p_{+} and p_{−}bands in the harmonic approximation for the triangular lattice potential. The single particle Hamiltonian for the triangular lattice is denoted by H_{latt}.
Note that the Hamiltonian (12) does not contain tunnelings of the composites themselves. Such a pair tunneling term can arise due to interaction^{28} but is 34 orders of magnitude smaller than other tunneling terms. The composites can also tunnel via higherorder processes (discussed in^{22}). The leading term of this collective tunneling is of the second order^{24} with the corresponding amplitude being proportional to , i.e. very small assuming strong attraction. The effect is further reduced by assumed shaking  modification of effective J_{0}  so such tunnelings can be safely neglected.
Lowenergy and resonant subspaces
Now we define the lowenergy subspace and the resonant subspace which are coupled by the driving (shaking). First we assume the strong interaction limit i.e. . Yet larger energy scale is set by single particle energy of the p band E_{1} and the shaking frequency. Thus we assume U_{2},U_{sp} E_{1} ~ Ω. U_{2}  the strength of attraction between ↑ and ↓ fermions sets the lowenergy scale, thus we restrict the analysis to the subspace of Hilbert space where all ↑ minority fermions are paired with their ↓ partners. Thus the lowenergy local subspace is spanned by states. As we will show below, due to the sp tunneling and periodic driving this subspace is resonantly connected to the subspace where a paired site can be occupied by porbital fermions, with energy E_{1} + U_{01}. Therefore, from now on our Hilbert space will consists of states.
We now apply the unitary transformation, transferring the timedependence in the total Hamiltonian H(t) into the tunneling amplitudes. The new Hamiltonian is given by
where . We expand the exponential functions in (14) as: . Then as U_{2} Ω, after rotatingwave approximation (RWA) and projecting on our local Hilbert space, the first term of Hamiltonian (14) may be resonant only if U_{2} contribution vanishes. Since this term corresponds to ↑ fermion tunneling (which appear paired only in our subspace) this process is possible only if a paired state and a ↓ fermion in sorbital are neighbors (Fig. 1(a)). Otherwise the pair (composite) is pinned. Similarly, the second term may be resonant () when a ↓ fermion in sorbital tunnels to a neighboring empty site. The third term gives a resonant contribution via the tunneling process depicted in Fig.4(b). After RWA, all the timeindependent tunneling amplitudes of the above intraband tunnelings are changed by a factor . We see that to minimize the ss and pp tunnelings we have to tune the shaking amplitude such that and . This assures that for the shaking phase Φ = 0, there is no intraband tunneling along the δ_{+} direction as .
In the last term of Hamiltonian (14) the fast oscillation with E_{1} + U_{01} frequency must be compensated by appropriate Fourier component yielding the sp resonant condition E_{1} + U_{01} = NΩ. Inspecting the tunneling term we see that, the tunneling in pband is resonantly enhanced only when the composite density in neighboring sites i and i + δ follows the relation . Due to the type of sp coupling in Hamiltonian, (14), pfermions may appear only in composite occupied sites. This may occur only from a site occupied by a lonely ↓ fermion (if there were a composite at that site, an additional energy difference, U_{2}, the pair energy would appear bringing the system out of the chosen resonance). After carrying RWA and in the limit of vanishing intraband tunneling, the effective Hamiltonian reads,
where defines Bessel function of orderN. We see that, one can control the different tunneling amplitudes by tuning the shaking amplitude, frequency and interaction strength.
When the shaking phase Φ ≠ 0, along δ_{0} and δ_{−} directions the intraband tunneling still vanishes, but remains nonzero along δ_{+} direction. Amplitude of the latter can be tuned to values smaller than the sptunneling amplitude by changing the interaction strength. Moreover, once the Dice structure of the composites is created, the only possible tunneling along δ_{+} direction is the interband sp tunneling (compare Fig.1(a) in the main text). So, adding small intraband tunneling due to a finite shaking phase will not destabilize the Dice structure.
Effects of tunneling on the emergent lattice
In this section, we discuss the effect of the tunneling on the Dice lattice structure. As discussed before, a composite can tunnel to a vacant site only via higher order processes^{22,24} which are negligible for large U_{2}. So the only way a composite can tunnel is if the minority fermion tunnels to a site already occupied by a majority fermion in sorbital site as shown in the first figure in Fig.4. Such a process can be described by an effective tunneling for the composite coupled to the tunneling of the excess fermions in the opposite direction. To investigate the effect of such a tunneling we use a onedimensional minimal model,
where at each site i we have only s and porbitals and 〈ij〉 denotes the nearest neighbors. We have introduced operators as the composite annihilation and creation operators. The first term denotes the composite density dependent sp tunneling of the excess fermions and the last term just denotes the composite tunneling and excess fermion tunneling. When J_{0} = 0, the ground state is given by the composite structure, when composite filling is n^{c} = 1/2. Such a density wave structure is equivalent to the Dice lattice structure we study in a triangular lattice. Due to the hardcore bosonic nature of the composites, we use a factorized variational composite wavefunction, Φ_{c} 〉= Π_{i}Φ_{ci}〉, where and Φ_{c}〉_{2i+1} = cosθ0〉_{c} + sinθ1〉_{c} and 1〉_{c},0〉_{c} denote a composite occupied or empty site. In the composite wavefunction ansatz, θ is the variational parameter. The density wave state at J_{0} = 0 is obtained for θ = 0. Using such an ansatz, we can integrate over the composite subspace and get an effective Hamiltonian,
Then we write the energy for excess fermion filling n = 1/4 (this is 1/2 of the previous value due to the doubling of number of degrees of freedom) for θ ≪ 1 and J_{sp} ≫ J_{0} as , which is independent of composite tunneling. From that we conclude that the energy is minimized for θ = 0. For larger tunneling strength J_{0}, we have compared the energy of the homogenous state with θ = π/4 and the density wave state with θ = 0 finding that the density wave state has lower energy as long as J_{sp} > 3J_{0}/4. Though the present calculation is onedimensional, the essential physics also applies to the more complicated situation of triangular lattice, where we expect the Dice lattice density wave structure to be stable even in the presence of small composite tunneling.
Additional Information
How to cite this article: Dutta, O. et al. Spontaneous magnetization and anomalous Hall effect in an emergent Dice lattice. Sci. Rep. 5, 11060; doi: 10.1038/srep11060 (2015).
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Acknowledgements
We thank M. Lewenstein and K. Sacha for enlightening discussions. This work was realized under National Science Center (Poland) project No. DEC2012/04/A/ST2/00088. Support of EU QUIC FET project is also acknowledged. A.P. is supported by the International PhD Project “Physics of future quantumbased information technologies”, grant MPD/20093/4 from Foundation for Polish Science and by the University of Gdansk grant BW 5385400B169131E.
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O.D., A.P., J.Z. conceived the idea, performed derivations and calculations, discussed the results and wrote the manuscript.
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Dutta, O., Przysiężna, A. & Zakrzewski, J. Spontaneous magnetization and anomalous Hall effect in an emergent Dice lattice. Sci Rep 5, 11060 (2015). https://doi.org/10.1038/srep11060
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DOI: https://doi.org/10.1038/srep11060
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