Abstract
Spinorbit interaction has established itself as a key player in the emergent phenomena in modern condensed matter, including topological insulator, spin liquid and spindependent transports. However, its function is rather limited to adding topological nature to band kinetics, leaving behind the growing interest in the direct interplay with electron correlation. Here, we prove by our spinor line graph theory that a very strong spinorbit interaction realized in 5d pyrochlore electronic systems generates multiply degenerate perfect flat bands. Unlike any of the previous flat bands, the electrons in this band localize in real space by destructively interfering with each other in a spin selective manner governed by the SU(2) gauge field. These electrons avoid the Coulomb interaction by selforganizing their localized wave functions, which may lead to a flatband state with a stiff spin chirality. It also causes perfectly trimerized charge ordering, which may explain the recently discovered exotic lowtemperature insulating phase of CsW_{2}O_{6}.
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Introduction
Electronic flat bands in momentum space are an ideal platform for achieving the highest correlation in a zerobandwidthlimit^{1,2,3}. A long history tells us that such flat bands naturally arise in a class of geometrically frustrated lattices like kagome, pyrochlore, and checkerboard lattices, which are wellunderstood based on the line graph theory and its analogs^{4}. Recently, the importance of having flat bands in real correlated materials is highlighted in twisted bilayer graphene^{5,6,7}, where the relationship between superconductivity and magnetism has been extensively discussed. On top of finely tuning a magic ’twisting’ angle, a flat band arises by structurally introducing a pseudo magnetic field onto a graphene layer^{8}.
There is another trend to add some topological nature to these flat bands^{9,10,11}, expecting emergent fractional quantum Hall states without a magnetic field, as they have a nonzero Chern number and mimic the Landau levels. A small spin–orbit coupling (SOC) helps to realize such nearly flat bands^{12}, which are experimentally observed in kagome lattice materials like CoSn^{13} and twisted multilayer silicene^{14}. Unfortunately, all these examples show that the perfect flatness of bands is sacrificed if the system gains topological properties^{15}.
Indeed, SOC rather enhances an itinerancy of electrons. Its major role had been to introduce some topological nature to the kinetic motion of particles. In SOC electronic systems^{16,17,18}, Berry phase is introduced to energy bands, which had been serving as a source of spindependent transports like anomalous Hall effect^{19} and spin Hall effect^{20,21}. A surface state of topological insulator^{22,23} is a Dirac state, which is another distinguishing feature of energy bands induced by a weak SOC. When strong electronic interactions are present, the topological band insulator is transformed into a topological Mott insulator with a gapless surface spinon excitations^{24}. In Kitaev materials^{25}, a very strong SOC creates a more exotic spin liquid phase^{26} hosting Majorana fermions, and antiferromagnets with topological magnons^{27,28}. Despite all these hallmark studies, there had been no example that the SOC gives an impact on the electronic correlation effect.
Here, we prove analytically that a SOCinduced spindependent hopping, which previously made the bands dispersive, perfectly flattens the energy bands of pyrochlore and kagome lattices when it becomes comparable to other transfer integrals. Most importantly, the SOC generates an SU(2) gauge field^{29} and strictly selects the relative angles of electron spins. When electron wave functions have these spin angles, they destructively interefere^{30} and localize in real space. We obtain an analytical form of such spintwisted flat band wave function, allowing us to access the important but most unreachable physical regime, the strongest correlation. In analogy to the flat band ferromagnetism, the SOC flat band may select its form by polarizing its spins in a sitedependent manner avoiding the loss of onsite Coulomb energy, resulting in a stiff spin chirality. When the nearest neighbor Coulomb energy is introduced at quarterfilling, the wave function further optimizes its form to a trimerized shape by fully occupying half of the flat band wave functions, and becoming a spinsinglet state. This mechanism may explain the exotic trimerized charge ordering found in 5d pyrochlore CsW_{2}O_{6}^{31}, where onequarter of the pyrochlore sites become perfectly vacant. The present model may provide a platform for testing the interplay of strong correlation and spin topology.
Results
Model system
We introduce a minimal microscopic model for 5d pyrochlore oxides^{32} (see Fig. 1a) with CsW_{2}O_{6} as a specific example. A metallic W^{5.5+} ion on a pyrochlore lattice is surrounded by a slightly distorted oxygen octahedron, and its electronic state is understood by considering the lowest Kramers doublet of this ion (E_{2} in Fig. 1b). The E_{2} doublet comes out as the mixture of t_{2g} triplet in a trigonal crystal field by introducing the strong SOC typical of the 5d electrons^{33}. Its effective momentum deviates from the values of the regular octahedra, \({J}_{{{{{\rm{eff}}}}}}=3/2,{J}_{{{{{\rm{eff}}}}}}^{z}=\pm 1/2\), by more than 10%. However, as in the case of Iridates, the J_{eff}picture works well^{34}. In the present quarterfilled case, the doublet carries 0.5 electrons on an average, where the energy levels are well separated as E_{1} − E_{2} ~ 200 meV (see Supplementary A and B). For such doublet described by a pseudospin, α = ↑, ↓, a conventional Hubbard type of Hamiltonian^{35,36} is written as a sum of hopping terms with spatially uniform transfer integral t and Coulomb interaction V between the nearest neighbor sites, 〈i, j〉, as well as the onsite U
where c_{jα} annihilates an electron with pseudospin α at sitej, and n_{jα} and n_{j} = n_{j↑} + n_{j↓} are their number operators. Eq. (1) has the same shape as an effective Hamiltonian for Iridates targeting E_{3} doublet with J_{eff} = 1/2 and \({J}_{{{{{{{{\rm{eff}}}}}}}}}^{z}=\pm 1/2\)^{36}. This is because both E_{3} and E_{2} consist of a_{1g} and \({e}_{g}^{\pi }\) orbitals, and their difference appears only in the value of λ/t (see Supplementary A Eq. (S8) and C). We note that due to small trigonal distortion, t and λ become slightly bonddependent. For simplicity, we first approximate them as uniform and finally examine the effect of distortion. A bare atomic SOC which may amount to ζ ~200–300 meV manifests as a spindependent hopping integral λ. A vector ν_{ij} is a coefficient of Pauli matrices, σ = (σ_{x}, σ_{y}, σ_{z}), which is bonddependent and is determined by the crystal symmetry. For a uniform pyrochlore lattice, we find \({{{{{{{{\boldsymbol{\nu }}}}}}}}}_{ij}=\sqrt{2}\frac{{{{{{{{{\boldsymbol{b}}}}}}}}}_{ij}\times {{{{{{{{\boldsymbol{d}}}}}}}}}_{ij}}{ {{{{{{{{\boldsymbol{b}}}}}}}}}_{ij}\times {{{{{{{{\boldsymbol{b}}}}}}}}}_{ij} }\) with vectors b_{ij} and d_{ij} pointing from the center of the tetrahedron to the bond center and along the bond, respectively (see Fig. 1a). A meanfield phase diagram of a model similar to Eq. (1) is studied at halffilling for Iroxides^{37} showing that a strong SOC generates a topological band insulator, a topological semimetal, and a topologically nontrivial Mott insulator in increasing U. There, an overall evolution of energy band structures in varying λ/t and U/t is studied in the context of finding a good Weyl point near the Fermi level^{35,36}. In the present work, we notice that the SOC can drive another exotic phenomena, a perfect flat band and a trimerized charge ordering.
Let us first set \({{{{{{{{\mathcal{H}}}}}}}}}_{I}=0\) and write down the energy bands by varying λ/t in Fig. 1c. One finds a perfect flat band at the bottom when λ/t = −2. There is another case, λ/t = 0, with a flat band at the top, which is understood from the line graph theory. Introducing the SOC is known to destroy the perfectness of this top flat band^{12} as one can see from the band structure for λ/t = −0.5. In the same context, it is shown that a perfect flat band cannot have a nonzero Chern number^{15}. Notice that among the 32 bands, half contribute to the top flat band at λ/t = 0 which gradually gains a bandwidth by λ < 0, while at the same time the other dispersive half starts to shrink and finally becomes perfectly flat at λ/t = −2.
The flat bands at both λ/t = 0 and −2 touch the other dispersive bands at Γpoint. This band touching is neither an accidental degeneracy nor a typical symmetryprotected band degeneracy^{38}. It is necessitated by the perfect flatness of bands, combined with some symmetry of the lattice^{39,40}. When the perfect flatness of bands is lost at λ < −2t, the band touching disappears and a gap opens (see Supplementary D), and at halffilling, the system becomes a topological insulator.
We also show in Fig. 1d the band structure of a hyperkagome lattice at the same λ/t = −2, obtained by depleting 1/4 of the pyrochlore sites, where we also find an 8fold degenerate flat band at the same location.
Phase diagram
The SOCinduced flat band clarifies the origin of the trimerized charge ordering observed in CsW_{2}O_{6}^{31}. Figure 1e shows a meanfield phase diagram at quarterfilling, corresponding to two electrons per tetrahedron. We approximate the Coulomb interaction terms \({{{{{{{{\mathcal{H}}}}}}}}}_{I}\) using a Hartreetype of meanfield and denote the solutions with nchargerich sites per tetrahedron as nin(4 − n)out. A trivial paramagnetic metallic state with uniform charge and spin distribution are dominant when the Coulomb interaction is small. There is an emergent 3in1out state extending at around λ/t = −2, which has 2/3 electrons per hyperkagome site, keeping 1/4 of the site almost perfectly empty (see the inset of Fig. 1e). The 2in2out phase with about 0.35:0.15 charge disproportionation is stabilized only at λ/t ≲ −2.
The reason why 3in1out is stable is understood by comparing the band energies \(\langle {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{\rm{kin}}}}}}\rangle\) in Fig. 1f when pyrochlore and hyperkagome lattices host 8N_{c} electrons, where N_{c} is the number of unit cells. A bandenergygain is always larger for a pyrochlore lattice with a larger coordination number and thus having the larger bandwidth. Indeed, the λ > 0 region of the phase diagram is dominated by a trivial metallic phase even for large U and V. However, at λ/t = −2, the pyrochlore and hyperkagome band energies become degenerate because all the electrons fill the bottom flat bands for both cases. The meanfield interaction energy is roughly evaluated by hand as \({E}_{I}^{metal}=U+12V\) and \({E}_{I}^{3i1o}=4U/3+32V/3\) per unit cell for metal and 3in1out, respectively, which is consistent with our numerical evaluation based on a meanfield approximation(see Supplementary E). Then, the introduction of V/t ≳ 1 stabilizes the 3in1out state against the metallic phase.
Spinor line graph theory
The perfect flat band at λ/t = −2 cannot be explained within any of the previous frameworks. Here, we develop a spinor line graph theory to prove the existence of SOCinduced flat bands, which can be applied to general linegraphrelated lattices. To this end, we first overview the flat band theory for line graphs. Figure 2a, b shows the relationships between the original lattice and its dual lattice described by red circles. The pyrochlore lattice is a line graph of its dual lattice, a diamond lattice, and by connecting pyrochlore and diamond sites and deleting pyrochlore bonds, one reaches a bipartite graph with blue bonds. The same relationship holds between the kagome–honeycomb lattices.
Let us introduce an incidence matrix of a graph theory, T_{OD}, to describe the relationship between the original lattice and its dual lattice. It is an N × N_{D} matrix and has one row for each pyrochlore site and one column for each diamond site, where N = 16N_{c} and N_{D} = 8N_{c} denote the number of pyrochlore and diamond lattice sites, respectively. The entry in rowi and columnm is 1 if pyrochloresitei and diamondsiteC_{m} are connected by a blue bond. If we take a product of the incidence matrix with its transpose matrix T_{DO} = ^{t*}T_{OD} as (T_{OD}T_{DO}), its ijentry becomes 1 when there is a connection between ith and jth pyrochlore sites mediated via the diamond site through two blue bonds. The diagonal element of (T_{OD}T_{DO}) has entry2 since each pyrochlore site can be transferred to its two neighboring diamond sites and come back. Using this product form, a matrix representation of a tightbinding Hamiltonian of the pyrochlore lattice is written as
where \(\hat{I}\) is a unit matrix. According to this equation, if there is an Ndimensional vector φ_{l} that fulfills T_{DO}φ_{l} = 0, it also satisfies \({\hat{H}}_{{{{{{{{\rm{pyrochlore}}}}}}}}}{{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{l}=2t{{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{l}\). A set of such vector forms a kernel (nullspace) {φ_{l}} of T_{DO}. Since T_{DO} is nonsquare, the number of independent φ_{l}, namely the dimension of the kernel is at least N − N_{D} = 8N_{c}(> 0). It means that there exist at least (N − N_{D})/N_{c} = 8 flat bands in the pyrochlore lattice with an energy 2t, which is the one found in Fig. 1c at λ/t = 0, where considering the spin degeneracy, the number of flat bands is doubled.
The extension of the line graph theory to λ ≠ 0 is not straightforward, since the hopping term is rewritten as
and includes a nonAbelian SU(2) gauge field U_{ij}^{29}, where \({{{{{{{{\boldsymbol{c}}}}}}}}}_{j}^{{{{\dagger}}} }=({c}_{j\uparrow }^{{{{\dagger}}} },{c}_{j\downarrow }^{{{{\dagger}}} })\) and \(\hat{{{{{{{{\boldsymbol{\nu }}}}}}}}}={{{{{{{\boldsymbol{\nu }}}}}}}}/ {{{{{{{\boldsymbol{\nu }}}}}}}}\) is a unit vector. The gauge field along j → i enforces an SU(2) spin rotation about the ν_{ij}axis by an angle \(\theta =2\arctan\; (\sqrt{2}\lambda /t)\). We want to construct another incidence matrix \({\tilde{T}}_{OD}\), whose jnentry represents a spinrotating hopping of an electron from the jth pyrochlore site to the C_{n} th diamond site. It should be such that the ijentry of \(({\tilde{T}}_{OD}{\tilde{T}}_{DO})\) will reproduce the complex hopping of Eq. (3). In hopping twice along the blue bonds, electron spin is rotated twice, ending up with the same state as rotated by θ about the νaxis. As we show in Fig. 2c, considering the symmetry of the tetrahedron, the rotation axis in hopping 1 → C_{1} is uniquely chosen along the bond pointing from the vertex to the center of the tetrahedron, which we denote as \({{{{{{{{\boldsymbol{r}}}}}}}}}_{{C}_{1}1}\). The rotation angle is also uniquely chosen as π. Resultantly, an incidence matrix \({\tilde{T}}_{OD}\) including the effect of SU(2) gauge field for λ ≠ 0 is given as
As shown in the caption of Fig. 2, we take \( {{{{{{{{\boldsymbol{r}}}}}}}}}_{j{C}_{n}} =\sqrt{3}\) for convenience, while this value only influences the coefficient of the second term of Eq. (5). Since the spin degrees of freedom is explicitly included, the matrix has twice as large dimension as T_{OD}, and fulfills \({\tilde{T}}_{DO}{ = }^{t* }\tilde{{T}_{OD}}\).
In the similar manner as Eq. (2), the incidence matrix is related to a hopping matrix \({\hat{H}}_{{{{{{{{\rm{pyrochlore}}}}}}}}}\), i.e., a realspace matrix representation of Eq. (3), as
when and only when λ/t = −2. To understand why λ/t needs to take this value, we show in Fig. 2d an example; consider a spin at site1 pointing inside the 1 − C_{1} − 2 triangular plane with angle −φ. For the present geometry of the pyrochlore lattice, we have an angle \(\theta ^{\prime} =\arccos (\sqrt{2/3})\) spanned by 1 → 2 and 1 → C_{1}. When the spin is transferred by (\({\tilde{T}}_{OD}{\tilde{T}}_{DO}\)) it rotates by π twice, takes the angle \((\varphi +2\theta ^{\prime} )\) at siteC_{1} and points to \((\varphi 4\theta ^{\prime} )\) at site2. When \(\theta ^{\prime} =\theta /4\), this operation replaces the θrotation about the νaxis. This geometrical condition gives λ/t = −2, and is a unique solution to fulfill Eq. (5). A kernel of \({\tilde{T}}_{DO}\) is a manifold of eigenstate of \({\hat{H}}_{{{{{{{{\rm{pyrochlore}}}}}}}}}(\lambda /t=2)\) with a constant energy −6t, and has a dimension 2(N − N_{D}). Therefore, we find 2(N − N_{D})/N_{c} = 16 flat bands at the energy bottom −6t.
A guide to design such SOC flat band is simple. The above mentioned geometrical condition for angle θ can be generalized to
which is schematically shown in Fig. 2e. Using Eq. (6), one may search for a lattice geometry that gives a reasonable vaule of λ/t. Another expression for this condition uses a Wilson loop operator \({{{{{{{{\mathcal{A}}}}}}}}}_{j{C}_{n}i}\) around the closed loop i → C_{n} → j. Eq. (6) is equivalent to having \({{{{{{{{\mathcal{A}}}}}}}}}_{2{C}_{1}1}={U}_{12}{U}_{2{C}_{1}}{U}_{{C}_{1}1}={{{{{{{{\rm{e}}}}}}}}}^{i\frac{\theta }{2}{\hat{{{{{{{{\boldsymbol{\nu }}}}}}}}}}_{12}\cdot{{{\boldsymbol{\sigma }}}} }{{{{{{{{\rm{e}}}}}}}}}^{i\frac{\pi }{2}{\hat{{{{{{{{\boldsymbol{r}}}}}}}}}}_{2{C}_{1}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}{{{{{{{{\rm{e}}}}}}}}}^{i\frac{\pi }{2}{\hat{{{{{{{{\boldsymbol{r}}}}}}}}}}_{{C}_{1}1}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}=I\). The condition means that for two dimensional lattices, the SOC vector ν shall point in the outofplane direction, and also when θ = π, the ijbond takes t = 0 and λ ≠ 0. Since such parameter values may not easily be realized, the edgeshared lattices like square, checkerboard, and honeycomb lattices may not be considered as realistic examples.
The spinor line graph theory is applied to kagome and hyperkagome lattices. For a kagome lattice, as shown in Fig. 2c, a usual λ = 0flat band at +2t starts to gain bandwidth with λ ≠ 0, while a dispersive bottom band shrinks and becomes a SOC flat band at −4t when \(\lambda =\pm \sqrt{3}\). (See Supplementary F for details).
Destructive interference
Although treating a quantum manybody model beyond a meanfield level is too challenging in general, our case with a zerobandwidth at λ = −2t may become simpler since it practically corresponds to a strong coupling limit which can be partially treated analytically. Among the onebody flat bands orbitals, φ_{lα} (l = 1, ⋯ 16N_{c}, α = ↑, ↓), half are filled when we consider CsW_{2}O_{6}. The set of onebody flat bands is chosen as their linear combinations, such that they minimize the interaction energy loss in total when they are combined to form a manybody flat band wave function.
The mth onebody flatband eigenstate of \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{\rm{kin}}}}}}\) including SOC is written as \(\left{\psi }_{m}\right\rangle ={\sum }_{j,\alpha }{\varphi }_{j\alpha }^{m}{c}_{j\alpha }^{{{{\dagger}}} }\left0\right\rangle\), where the complex coefficients \({\varphi }_{j\alpha }^{m}\) are the elements of 32N_{c} dimensional vector \({\tilde{{{{{{{{\boldsymbol{\varphi }}}}}}}}}}_{m}\) that fulfills \({\tilde{T}}_{DO}{\tilde{{{{{{{{\boldsymbol{\varphi }}}}}}}}}}_{m}=0\). This condition is factorized to the condition for each tetrahedron; it prohibits a net propagation of electrons from four pyrochlore sites labeled by j ∈ n to an nth diamond site as
which should be fulfilled for all tetrahedra n = 1, ⋯, N_{D}. In visualizing this equation, we first set a fictitious SU(2) spinor χ_{n} (twodimensional vector) at the nth tetrahedron center pointing somewhere as in Fig. 3a. Suppose that the spins on four pyrochlore sites point in the directions rotated by π from this spinor about the bluebonds. Among these four spins, if some have finite weight \({\varphi }_{j\alpha }^{m}\) in the wave function, they need to be canceled out by Eq. (7).
When considering the two adjacent tetrahedra, a spin shared by them should fulfill the two conditions. This spin shares the same πrotation axis in hopping to the diamond sites on both sides. Therefore, if it has a finite population in the wave function, the two fictitious spinors on both sides are enforced to point in the same direction. One example of \(\left{\psi }_{m}\right\rangle\) is given as such that they form a closed loop consisting of an even number of bonds, shown in Fig. 3b. By assigning +1 and −1 weights alternatively along the loop while fixing their spin direction in a way mentioned above, a single electron is perfectly localized on the loop. This is because if it wants to hop outside the loop, its weights are canceled out by Eq. (7), which is the physical meaning of a destructive interference^{30} or a kinetic frustration. The product of onebody flat band wave functions becomes an eigenstate of \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{\rm{kin}}}}}}\), which is also an eigenstate of \({{{{{{{{\mathcal{H}}}}}}}}}_{I}\), namely of the whole Hamiltonian.
We now consider a trimerized chargeordered state based on a flat band wave function. There are (16N_{c} + 2) linearly independent onebody states that belong to the pyrochlore flat band including the pseudoup/downspins and band touching ones. Among them, one can choose 4N_{c} × 2independent ones, forming a loop consisting of ten sites that belong to the hyperkagome lattice which we call loop10 as shown in Fig. 3c (see Supplementary G). A 3in1out many body flat band wave function is thus given in a factorized form, \(\left{{{\Psi }}}_{3{{{{{{{\rm{in1out}}}}}}}}}\right\rangle \propto {\prod }_{n,\sigma }{\hat{\psi }}_{n,\sigma }^{{{{{{{{\rm{10}}}}}}}}}\left0\right\rangle\), using a single electron operator of loop10, where \(\left{\psi }_{n\sigma }^{10}\right\rangle ={\hat{\psi }}_{n\sigma }^{10}\left0\right\rangle\). The index σ = ↑, ↓ of \({\hat{\psi }}_{n,\sigma }^{{{{{{{{\rm{10}}}}}}}}}\) corresponds to χ_{n} = (1, 0) and (0, 1). In the present quarterfilled case, since we need to put two electrons per tetrahedron, namely 8N_{c} electrons on 4N_{c} × 2independent loop10 states, they accomodate both pseudoup and down spins and are fully occupied. Therefore, \(\left{{{\Psi }}}_{3{{{{{{{\rm{in1out}}}}}}}}}\right\rangle\) is a nonmagnetic singlet state. These loop10’s have finite overlap and distribute uniformly over the whole hyperkagome lattice with all sites having the same electron occupancy of 2/3.
Apart from the case of CsW_{2}O_{6}, there is purely theoretical interest in lower fillings. For no more than halffilling of flat bands, one can prepare a manybody wave function consisting of a product of loops, e.g., loop6 state written in Fig. 3b that fulfill Eq. (7). Here, by selecting the spin orientation for each, the whole wave function is constructed as such that it gives the lowest \(\langle {{{{{{{{\mathcal{H}}}}}}}}}_{I}\rangle\). When all these constituent onebody functions have finite overlap with some others and cannot be disconnected into two groups, one can fully avoid the double occupancy of electrons on all sites by polarizing χ_{n} for all n in the same direction, which gives 〈Un_{i↑}n_{i↓}〉 = 0. When V = 0, this wave function becomes the exact and unique ground state of the Hamiltonian. This context is analogous to the mechanism of flat band ferromagnetism of a Hubbard model^{1,2}; electrons choose which of the localized onebody flatband wave functions to occupy by fully polarizing their spins at finiteU since Pauli’s principle helps the electrons to avoid double occupancy in space.
When χ_{n} for all tetrahedra point in the same direction, the manybody flat band state exactly keeps the relative angles of the spins on four sublattices, which indicates the stiff chiral ordering. As shown in Fig. 3d there are eight species of triangles in a unit cell, whose spin orientations are shown for the case where the fictitious spinor points in the +zdirection. These pseudospins are exposed to an internal magnetic field generated by an SU(2) gauge field, and its flux equals half of the solid angle Ω_{ijk} subtended by the spin directions around the triangle. We evaluated Ω_{ijk} = n_{i} ⋅ (n_{j} × n_{k}) for four independent triangles in Fig. 3d as a function of angle Θ of the fictitious spinor about the +zaxis. We define a unit vector n_{j} parallel to the pseudo spins with the righthand rule about +zaxis. At Θ = 0, π we find maximum amplitude, Ω_{321} = ± 16/27. In this case, this scalar chirality contributes to a xycomponent of an anomalous thermal Hall conductivity for insulators or it might affect σ_{xy} for metals^{41,42}.
Discussion
Concerning the experimental findings, an important question is whether the actual material parameters really fit to our scenario. It is known that the 5d electrons are more extended in space with a reduced value of onsite Coulomb repulsion U ~ 1 − 2 eV^{5} and an enhanced bandwidth, which may favor a metallic state^{32,34}. However, a large atomic SOC, ζ, comparable to transfer integral(t) usually dominates the t_{2g} orbitals and splits them into higher J_{eff} = 1/2 doublet and lower 3/2 quartet. A Mott insulating Sr_{2}IrO_{4} is reported to have t ~ 0.3 eV and ζ ~ 0.5 eV^{43}, and parameters of a honeycomb Kitaev material Na_{2}IrO_{3} are evaluated as t ~ 0.27 eV and ζ ~ 0.39 eV from the first principles calculation^{44}. In CsW_{2}O_{6}, the value of SOC should be ζ ~ 200 − 300 meV, which is considered to be about half of that of 5d Iridates. A trigonal distortion of the crystal further splits the J_{eff} = 3/2 quartet into two, and the lowest E_{2} doublet with \({J}_{{{{{\rm{eff}}}}}}^{z} \sim \pm 1/2\) and J_{eff} ~3/2 is focused(see Fig. 1b).
In CsW_{2}O_{6} the distortion angle, α = 55.71°, is slightly larger than the regular octahedron 54.74°. Based on this information, we examined in detail the energylevel splitting of W5d in a trigonal crystal field in Supplementary A and B, and by associating the results with the energy band structure of the first principles calculations without SOC, we estimated a set of material parameters as t ~ 0.06 eV, 10Dq ~ 2 eV, and Δ_{1} ~ 0.23 eV. By introducing ζ ~ 0.1 − 0.15(10Dq), the energy levels of the three doublets are obtained and we find E_{1} − E_{2} ~ 0.1(10Dq) = 0.2 eV, which is reasonably large to justify our approximation dealing with only E_{2} doublets.
At ζ = 0 and in a trigonal crystal field, the E_{2} doublet has a character of a_{1g}, while with increasing ζ the contribution from \({e}_{g}^{\pi }\) levels becomes the same order as a_{1g}. The spindependent hopping integral λ originates from the direct and oxygenmediated indirect hoppings between \({e}_{g}^{\pi }\) and a_{1g}, and has different signs from t coming from the a_{1g}–a_{1g} and \({e}_{g}^{\pi }\)–\({e}_{g}^{\pi }\) hoppings. We made a microscopic evaluation of λ/t of CsW_{2}O_{6} based on the SlaterKoster parameter and found that the ratio ranges between λ/t ~ −3 to −1 depending on the ratio of direct hopping against indirect hopping(see Supplementary C). Our SOCinduced flat band can thus be reasonably realized in the material. We also notice that in our theory, one does not need strictly λ/t = −2 to have a trimerization, as the phase diagram shows that there is some sort of pinning effect to the flat bands when the electronic interactions are finite.
In the J_{eff}picture the t_{2g} orbital momentum L^{eff} = 1 resembles the porbital representation with its sign taken as minus, where we find J_{eff} = − L^{eff} + S as good quantum numbers^{45}. Then, the magnetic moment M = 2S − L^{eff} becomes zero for the undistorted octahedron, while for the present case the admixture of levels coming from small trigonal distortion gives finite moment 〈M_{z}〉 still about half of that of the full moment of the electron, while it is difficult to compare this directly with the available experimental results.
In the lowtemperature phase II, we expect the trimerized flat band state, which has a Mott gap. This explains the sharp increase of the resistivity at the transition temperature^{31}. The manybody flat band state on a hyperkagome lattice we obtained is nonmagnetic, which may explain a finite spin gap.
Before the recent discovery of trimerized charge ordering that keeps the Anderson condition^{31}, CsW_{2}O_{6} was considered to undergo a Peierlstype of metalinsulator transition^{46}. This was partially because the DFT calculation showed a large enhancement of the density of states near the Fermi level^{47}, which was ascribed to the electronically driven structuralmetalinsulator transition to a zigzaglike onedimensional structure. Other firstprinciples calculations supported this picture arguing that the SOC enhances the nesting instability^{48}. Also, a certain amount of lattice distortion takes place at the transition, and a hyperkagome lattice based on the chargerich sites shows breathing into large and small triangles with the difference in their bond length by 2%^{31}, which seemingly supports the Peierls transition.
To clarify that the SOC is the driving force of the trimerized charge ordering, we finally show that it is difficult to attain such perfect charge disproportionation solely by the lattice distortion and without λ. Considering the type of structural distortion taking place in the material, we modify the originally uniform t to three classes: \(t^{\prime}\) shown in broken lines that connect the chargerich and poor sites, and t ± δ which form small/large triangles of a hyperkagome lattice. Figure 4a shows the density plot of charges on the plane of λ and \(t^{\prime}\) for δ = 0. Only near λ ~−2t, one can attain a nearly perfect (2/3: 0)ratio of charge disproportionation at \(t^{\prime} \;\lesssim\; t\). Notice that in general, \(t^{\prime}\) can never be smaller than even half of t with such lattice distortion, although we examined the whole range of \(t^{\prime} /t=0\) to 1. Figure 4b, c is the variation of rich/poor 〈n_{i}〉 as functions of \(t^{\prime}\) and δ, and a bandgap at the Fermi level. There are two notable features. The charge density can be very close to the flat band ones even though λ is off −2t, once we decrease \(t^{\prime}\) slightly from 1. In contrast, the breathing effect, δ, typical of the “Peierls transition”, does not change the charge density, even when the bandgap increases as we see for the case of λ = 0; the gap opening at δ = 0.1 with the disappearance of the Fermi surface on the left panel is shown in Fig. 4d.
In revisiting the aforementioned previous works, the enhanced density of states does not mean the Peierls instability but may rather fit the scenario of possible SOC induced flat band, which may not be perfect, but would be enough to drive the system to a trimerized charge ordering. According to our theory, this charge order is different from the conventional ones driven mostly by the Coulomb interaction V. The interplay of SOC and transfer integral is its main source. U and V only indirectly support it, since the flatband wave function has an advantage over trivial electronic states in that, they could selforganize their shape freely within the manifold of flatband eigenstates and optimize their charge configuration to avoid the Coulomb interactions.
The present picture might be examined by an anomalous thermal Hall measurement in the insulating phase or an anomalous Hall electronic transport in the metallic state by the holedoping to the material. In the previously known cases of the intrinsic anomalous Hall effect, often the SOC acting on the conducting electrons^{42} or the localized moments working as spatially coplanar internal field onto the conducting electrons^{41} was considered as a source of the emergent gauge field. In our case, the SOC is playing a more crucial role, as it works to kill their momentum k and strictly selects the orientation of pseudospin moments. These electrons may virtually propagate in space since it is on a flat band. It is thus beyond the scope of the present transport theories on how such features may appear in the transport phenomena.
Data availability
The data that support the findings of this study are available on request from the authors.
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Acknowledgements
We appreciate Youichi Yamakawa for useful information on the first principles band structure. We thank Takahisa Arima, Yoshihiko Okamoto, and Masataka Kawano for discussions. The work is supported by JSPS KAKENHI Grants Nos. JP17K05533, JP18H01173, JP21H05191, and JP21K03440 from the Ministry of Education, Science, Sports, and Culture of Japan.
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C.H. designed the project and wrote the paper. H.N. constructed the model and performed the meanfield calculation. Both constructed the theory together and equally contributed to this work.
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Nakai, H., Hotta, C. Perfect flat band with chirality and charge ordering out of strong spinorbit interaction. Nat Commun 13, 579 (2022). https://doi.org/10.1038/s4146702228132y
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DOI: https://doi.org/10.1038/s4146702228132y
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