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Odd-parity altermagnetism through sublattice currents: From Haldane-Hubbard

model to general bipartite lattices

Yu-Ping Lin1 and Marc Vila2

1 Department of Physics, University of California, Berkeley, California 94720, USA

D epartment of Physics, University of California, Berkeley, California 94720, USA
2
C atalan Institute of Nanoscience and Nanotechnology - ICN2 (CSIC and BIST),
Campus UAB, Bellaterra, E-08193, Barcelona, Spain
(Dated: August 7, 2026)

arXiv:2503.09602v3 [cond-mat.str-el]

6

Aug 2026

We propose the sublattice currents in a compensated collinear magnetic system as a feasible route to odd-parity altermagnetism (ALM), where nonrelativistic collinear spin splitting occurs in the bands as an odd function of momentum. In contrast to previously classified ALMs, the sublattice currents break the time-reversal symmetry in the nonmagnetic crystal structure and allow for such odd-parity spin splitting. A representative example is the Haldane-Hubbard model at half filling. Although the compensated collinear magnetic ground state was previously recognized as antiferromagnetism, we show that it is actually an odd-parity ALM. Interestingly, its topological version serves as an example of an ALM Chern insulator. We further generalize the Haldane- Hubbard model to common two- and three-dimensional bipartite lattices. With spin splitting allowed by sublattice currents, the compensated collinear magnetic ground states at half filling are generally odd-parity ALM. Introduction— Recent studies of unconventional mag- netism discovered altermagnetism (ALM) as the third type of collinear magnetic orders [1–9]. This com- pensated magnetic order resembles antiferromagnetism (AFM) for its zero net magnetization. Meanwhile, the lack of inversion-time-reversal PT symmetry breaks the Kramers degeneracy, leading to nonrelativistic spin splitting in the bands as ferromagnetism (FM). Under crystal symmetry breaking, the spin splitting manifests momentum-dependent alternating sign structures. With this unconventional combination of AFM and FM fea- tures, ALM is recognized as a promising platform for exciting phenomena, including innovative spintronics ap- plications [10], anomalous Hall effects [11], optical control of magnetization [12, 13], and intricate interplay with su- perconductivity [14–20]. In a previous classification with nonrelativistic spin groups [6], it was assumed that collinear magnetic orders always have the spin group symmetry [C2 T ||T ], where the left (right) side of the bracket indicates a symme- try operation in the spin (orbital or lattice) space, with C2 being a 180◦ rotation perpendicular to the spin axis. This symmetry leaves the spin invariant since both C2 and T flip the spin, as illustrated in Fig. 1(a). On the other hand, because the right-side T acts not only in real space but also in momentum space, it flips momentum (k → −k). Therefore, the presence of such symmetry im- plies that the band structure follows ε(k, s) → ε(−k, s), that is, the spin splitting, if any, is even in momentum. However, there are situations where the [C2 T ||T ] sym- metry is not preserved and consequently such even-parity splitting need not occur. While the breaking of the [C2 T | part occurs in noncoplanar systems [21], here we realize that the orbital or lattice symmetry |T ] can already be broken in systems with collinear magnetic order. For in- stance, Fig. 1(b) shows that, when orbital currents or magnetization are present, they break T in real space (i.e., spin independent), and hence [C2 T ||T ] is also bro- ? [C2 T ||T] [C2 T ||T] [C2 T ||T] [C2 T ||T] [C2 T ||T] [C2 T ||T] FIG. 1. Schematic representation of the [C2 T ||T ] symmetry in a spinful system without (a) and with (b) time-reversal symmetry breaking in real space, as in the case of, e.g., orbital currents (orange arrow). When [C2 T ||T ] is preserved, the spin splitting is even in momentum. Red and blue arrows denote up and down spins, respectively. The green font shows what symmetry operation is applied. ken. In that case, it is currently unknown how the spin splitting changes with momentum. While nonrelativistic odd-parity spin splittings were explored in the phenomenology of spin Pomeranchuk in- stability [22–25], lattice models with spin-bond orders [22, 26], or noncollinear magnetic orders [21, 27, 28], the current classification of collinear magnetic systems does not contemplate such a scenario. Increasing the types of materials possessing ALMs with a different type of split- ting would be important not only to broaden our classifi- cation and understanding of magnetic materials but also to enrich the potential spintornic applications brought about by spin splittings, such as magnetoresistance [29– 31] or spin torques [32, 33]. Altogether, a natural ques- tion arises: Can we find odd-parity ALM, that is, odd- parity spin splitting in a compensated collinear magnetic system, by inducing time-reversal symmetry breaking in the nonmagnetic crystal structures? In this Letter, we answer this question by demonstrat- ing repulsion-driven odd-parity ALM through sublattice currents. Unlike relativistic spin-orbit coupling [34], sub-

2 lattice currents are nonrelativistic effects that break the time-reversal symmetry. A representative example of this scenario is the classic Haldane-Hubbard model at half filling [35–41]. Previous studies discovered a compen- sated collinear magnetic ground state and classified it as an AFM. Although spin splitting in the bands was no- ticed [36, 38], its momentum-dependent symmetry struc- ture was not studied. Here we revisit this spin split- ting from the perspective of sublattice imbalance and spin group symmetry, thereby reclassifying the magnetic ground state as an odd-parity ALM. Interestingly, its topological version serves as an example of ALM Chern insulator. We further generalize the model to common two- and three-dimensional (2D and 3D, respectively) bi- partite lattices, where odd-parity ALMs are consistently confirmed. Our work opens the avenue to the system- atic search for odd-parity ALMs in correlated quantum systems, reveals sublattice currents as a mechanism for ALMs, and broadens our current classification of ALMs. Haldane-Hubbard model— We consider single-orbital models on 2D and 3D bipartite lattices, which are the minimal models with multisublattice structures. Our starting point is the classic Haldane-Hubbard model [42] H = −t1 ⟨ij⟩1 σ c†i σ cj σ − t2 ⟨ij⟩2 σ c†i σ ei ϕi j cj σ + 1 2 U0 iσσ′ c†i σ c†i σ′ ci σ′ ci σ (1) on the 2D honeycomb lattice [Fig. 2(a)] at half filling. Here c( †) i σ annihilates (creates) a fermion at site i with spin σ =↑↓; t1 = 1 and t2 = 0.1 represent the hoppings between first- and second-neighbor sites ⟨ij⟩1 ,2, respec- tively; and U0 = 4 is the on-site repulsion. The second- neighbor hoppings carry complex phases ϕi j = ±π/2, leading to uniform charge currents within each sublat- tice. Because of the opposite flows on the two sublattices, we refer to them as sublattice currents. Note that the el- ementary triangles in each sublattice host loop currents with three-1-in-1-out (3+−) corners. These loop currents may be related to auxiliary staggered fluxes through the triangles, with zero net flux on the whole lattice [42]. Using Hartree-Fock theory [43, 44], we obtain the inter- acting ground state on a finite-size (182 × 2) lattice with a periodic boundary condition. With the on-site repul- sion sufficiently large, the ground state manifests oppo- site FMs on the two sublattices [Fig. 2(b)]. This magnetic ground state remains robust under weak extended repul- sion [45, 46]. In the previous literature [35–41], the mag- netic order was recognized as an AFM. However, sublat- tice currents break the PT symmetry and allow for spin splitting in the bands [36, 38]. This important property suggests that the ground state is not an AFM in modern classification [6]. As the central point of our work, we explain how the sublattice currents actually turn the ground state into an odd-parity ALM. We begin with the effect of sublattice currents in the noninteracting Haldane model. In the absence of sublattice currents with t2 = 0, the bands have equal weights in the two sublattices. This balance can be quantified by a zero sublattice weight wn k = ⟨un k| τ 3 σ0 |un k⟩ , (2) where |un k⟩ is the nth-band Bloch state at momentum k and τ µ = (12 , τ ) with τ = (τ 1 , τ 2 , τ 3 ) (same for σµ ) are the Pauli matrices in the sublattice (spin) sector. Since ωn k measures a trivial identity σ0 in the spin space, it is basis independent in the bands with double spin degener- acy. At the Brillouin-zone (BZ) corners ±K, Dirac points with band crossings appear. The additional double de- generacy at these points can be understood from degen- erate sublattice-polarized states [47]. When the sublat- tice currents are turned on with t2 = 0.1, the bands be- come sublattice-imbalanced wn k ̸= 0 [Fig. 2(c)(i)] [48]. This imbalance is maximal at ±K, where the sublattice- polarized states are split. We further inspect the symme- try structure of this sublattice imbalance. The sublattice currents exhibit the momentum-space form factor jk = 2t2 3 m=1 sin(k · a2 m) (τ 3 σ0 ), (3) where a2 (m=1,2,3) are the C3 - related Bravais-lattice unit vectors at second neighbors. The resulting sublattice im- balance is C3 symmetric, as confirmed by a BZ map of to- tal sublattice weights in the occupied bands [Fig. 2(c)(i)] wk = 2 n=1 wn k. (4) Note that it takes an odd-parity form wk = w− k since inversion switches the sublattices. We now turn on the on-site repulsion U0 = 4 and study the mean-field band structure [43, 44] of the mag- netic ground state. Under sublattice imbalance, the op- posite sublattice FMs naturally drive spin splitting in the bands. Indeed, our computation finds band splitting with nonzero spin polarization [Fig. 2(c)(ii)] sn k = ⟨un k| τ 0 ˆσ|un k⟩ , (5) where ˆσ = ˆs · σ measures the spin polarization along the collinear magnetic direction ˆs. The spin splitting should inherit the odd-parity C3 - even structure from the sub- lattice imbalance. This symmetry structure is confirmed by a BZ map of the spin-splitting energy in the occupied bands [Fig. 2(c)(ii)] Esk = 2 n=1 En ks n k, (6) where En k is the mean-field dispersion energy. The dis- covery of this nonrelativistic spin splitting recognizes the ground state as an odd-parity ALM. Notably, its odd- parity nature goes beyond the previous classification [6],

KM

(a) (c) (i) 2 0 2 E k (b) (d)(i) 2 0 2 E k 0 + −

MK

-M -K 0 + −

-M -K 4 2 0

MK (ii) 0 +

MK (ii)

Ek K M -M -K

4 2 0 E k 3 MK

M K (iii) C =0 K (iii) 0 + − 0 + −

M K (iii) C = −2 0 + −

-M -K

FIG. 2. Haldane-Hubbard model and repulsion-driven odd-parity ALM. (a) Haldane model on the honeycomb lattice (gray), where opposite currents flow on the second-neighbor bonds (lighter green and pink) in the two sublattices (green and pink). (b) Repulsion-driven ground state at half filling, where opposite FMs develop on the two sublattices. Here we show the on-site spin orders with colors representing z components. (c) Haldane-Hubbard model with t2 = 0.1. (i) Left: sublattice imbalance wn k occurs in the noninteracting bands. Right: BZ map of total sublattice imbalance wk in the occupied bands. (ii) Spin splitting in the ALM at U0 = 4. Left: the bands split with nonzero spin polarization sn k. Right: BZ map of the spin-splitting energy in the occupied bands Esk . (iii) BZ map of Berry curvatures in the ALM. (d) Haldane-Hubbard model with t2 = 0.3 and U0 = 4.8. The colors in (b)–(d) represent the respective data and follow the respective color bars. which is enabled by the nonmagnetic time-reversal sym- metry breaking from sublattice currents. The discov- ery of odd-parity ALM opens promising opportunities in spintronics: because the symmetry of the spin split- ting is similar to that of spin-orbit-coupled split bands (i.e., odd in momentum), we expect similar phenomenol- ogy but with substantially larger magnitudes [49] as the splitting arises from exchange interaction rather than rel- ativistic effects. Although the noninteracting Haldane model is topo- logical with the Chern number C = −2 [42], the ALM is topologically trivial with C = 0 under a sufficiently large repulsion [Fig. 2(iii)] [35–38]. Interestingly, previ- ous mean-field studies also found a topological regime with C = −2 near the phase transition to normal in- sulator [35–38]. The ground state in this regime serves as an example of ALM Chern insulator. We briefly ex- plain these topological properties. In the noninteracting model, the negative Berry curvatures accumulate around ±K. In the strong ALM regime, where U0 is large enough compared to t2 , a band inversion occurs between the spin- down (up) bands around sublattice-polarized K (−K) [36, 38]. This process leads to positive Berry curvatures around ±K in the occupied bands [Fig. 2(c)(iii)]. The remaining negative Berry curvatures migrate to the BZ edge centers Ms and neutralize the newcoming positive ones. Therefore, the ground state becomes topologically trivial with C = 0. On the other hand, a weak ALM may not induce band inversion, thus keeping the ground state topological with C = −2. We demonstrate this situation under stronger sublattice currents with t2 = 0.3, where the noninteracting band gap at ±K becomes larger than the one at Ms [Fig. 2(d)(i)] [48]. Interestingly, the nega- tive Berry curvatures now accumulate around Ms, some of which migrate back to ±K when ALM reduces the spin-down (up) band gap at K (−K). As ALM remains weak compared to the band gap, such as at U0 = 4.8 [Fig. 2(d)(ii)], the band inversion does not occur. The ground state remains a Chern insulator with negative Berry curvatures in the occupied bands [Fig. 2(d)(iii)]. General sublattice-current model—Through the study of Haldane-Hubbard model, we have recognized the sub- lattice currents as a feasible route to odd-parity ALM. Building upon this knowledge, we now establish a system- atic construction of lattice models that host repulsion- driven odd-parity ALM. Our construction generalizes the Hamiltonian (1) of the Haldane-Hubbard model with t1 = 1, t2 = 0.1, ϕi j = ±π/2, and U0 = 4 to the 2D and 3D bipartite lattices at half filling. This model intro- duces sublattice currents to the second-neighbor bonds, with uniform currents flowing oppositely in the two sub- lattices. Under sublattice currents, the noninteracting bands acquire sublattice imbalance. The strongest imbal- ance occurs at high-symmetry points, lines, or surfaces on the BZ boundary, where degenerate sublattice-polarized states [47] are split. Under on-site repulsion, the half- filling ground states naturally develop opposite sublattice FMs. With the aid of sublattice imbalance, this com- pensated collinear magnetic order induces nonrelativistic spin splitting in the bands and makes the ground states ALM. The sublattice currents exhibit the momentum-space form factor jk = 2t2 N2 a m=1 ηm sin(k · a2 m) (τ 3 σ0 ). (7) Here a2 ms and ηm = ±1 are N2 a Bravais-lattice unit vectors and current directions along different second- neighbor axes, respectively. Note that some lattices host

4 (a) (i) (ii) X M -X -M 4 0 4 E k X M Y (iii) (iv) X M -X -M 2 0 2 4 6 E k X M Y (b) (i) (ii) MK -M -K 2 0 2 E k M K (iii) (iv) MK -M -K 0 2 4 E k M K 0 + − 0 + −

FIG. 3. 2D bipartite-lattice models and repulsion-driven odd-parity ALMs. For (a) checkerboard and (b) honeycomb lattices, we show (i) model structures, (ii) sublattice imbalances in the noninteracting bands, (iii) opposite FMs in the two sublattices, and (iv) ALM-induced spin splittings in the bands. Note that (b) the honeycomb-lattice model exhibits a different sublattice- current configuration (1 + +, 1 + −, 1 − −) from the Haldane model. The formats of the figures follow Fig. 2. multiple inequivalent configurations for sublattice cur- rents. As we discuss later, different configurations sup- port different types of spin splittings on the same lattice. Importantly, the ALMs in our systematic construction are all odd-parity, which expands the class of ALM be- yond the previous classification [6]. This expansion is attributed to the nonmagnetic time-reversal symmetry breaking from the sublattice currents. 2D lattices—We first investigate the bipartite lattices in 2D. As a starting point, we consider the simplest checkerboard lattice. There is only one inequivalent con- figuration for sublattice currents under lattice symmetry [Fig. 3(a)(i)]. Note that the squares in each sublattice carry zero auxiliary fluxes. According to the form factor (7) of sublattice currents with N2 a = 2, the sublattice im- balance manifests an odd-parity Rx - odd-Ry -even struc- ture with maxima along the BZ boundary [Fig. 3(a)(ii)]. Here Rx ,y represent the reflections in the x and y direc- tions, respectively. We confirm the repulsion-driven odd- parity ALM on a finite-size (162 ×2) lattice [Fig. 3(a)(iii)]. The band structure acquires spin splitting under ALM, following the same odd-parity structure as the sublattice imbalance [Fig. 3(a)(iv)]. Notably, the ground state re- spects an effective time-reversal symmetry T Tx ,y , where Tx ,y are the one-site translations along the x and y di- rections, respectively. This symmetry enforces trivial topology in the ground state with zero Berry curvatures. We also note that such T Tx ,y symmetry is important to realize odd-parity magnetism in noncollinear systems [21, 27]. However, it is not necessary in our collinear magnetic order where the lack of [C2 T ||T ] is what allows for the odd-parity spin splitting. Although we have studied the honeycomb lattice in the Haldane-Hubbard model, there actually exists a second inequivalent configuration for the sublattice currents. By reversing the sublattice currents along one of the three second-neighbor directions, this configuration manifests the (1 + +, 1 + −, 1 − −) in-out corners in the elemen- tary triangles of each sublattice [Fig. 3(b)(i)]. The form factor (7) with N2 a = 3 and η1 = −η2 = η3 = 1 breaks the C3 symmetry, leading to a twofold symmetric RK - odd-R⊥ K- even sublattice imbalance [Fig. 3(b)(ii)]. The repulsion-driven ALM [Fig. 3(b)(iii)] then induces spin splitting in the bands with the same odd-parity struc- ture [Fig. 3(iv)]. Note that the noninteracting model also hosts a topological ground state with C = −2, while the ALM trivializes the nontrivial topology with C = 0. 3D lattices—The Haldane-Hubbard model can also be generalized to 3D bipartite lattices. Here we consider the common examples of body-centered-cubic (BCC), 3D checkerboard, and diamond lattices (Fig. 4). Our finite- size (83 ×2) computation at half filling confirms the onset of repulsion-driven opposite FMs in the two sublattices. The spin splittings in the bands again recognize these compensated collinear magnetic ground states as odd- parity ALMs. The BCC lattice stands on a simple-cubic Bravais lat- tice, which supports only one inequivalent configuration for sublattice currents [Fig. 4(a)(i)]. Similar to the 2D checkerboard lattice, the sublattice imbalance is odd par- ity [Fig. 4(a)(ii)] with maxima on the whole BZ boundary [Fig. 4(a)(iii)]. The repulsion-driven ALM [Fig. 4(a)(iv)] then induces spin splitting in the bands with the same odd-parity structure [Fig. 4(a)(v)]. The 3D checkerboard and diamond lattices [Figs. 4(b) and 4(c)] both stand on the face-centered-cubic (FCC) Bravais lattices. Because of their frustrated nature, the FCC Bravais lattices support more abundant structures for sublattice currents. There exist three inequivalent configurations with (1+++, 3+−−), (1+++, 1++−, 1+ −−, 1 − −−), and (2 + +−, 2 + −−) [Figs. 4(b)(i) and 4(c)(i)] in-out corners in the elementary tetrahedrons of each sublattice. Different configurations exhibit the N2 a = 6 form factors (7) with different symmetry struc- tures. For example, the (1 + ++, 3 + −−) configuration obeys a C3 symmetry, which is absent in the other two configurations. The sublattice imbalances on these two lattices are again odd parity [Figs. 4(b)(ii) and 4(c)(ii)],

5 (a) (i) (iii) X M R (ii) XMR -X -M -R 8 0 8 E k (iv) (v) XMR -X -M -R 4 0 4 8 E k (b)(i) (ii) L WX -L-W - X 4 0 4 E k (iv) (v) L WX -L-W - X 4 0 4 8 E k (c) (i) (iii) L W X (ii) L WX -L-W - X 4 0 4 E k (iv) (v) L WX -L-W - X 2 0 2 4 6 E k 0 + − 0 + − FIG. 4. 3D bipartite-lattice models and repulsion-driven odd-parity ALMs. For (a) BCC, (b) 3D checkerboard, and (c) diamond lattices, we show (i) model structures, (ii) sublattice imbalances in the noninteracting bands, (iii) BZs, (iv) opposite FMs in the two sublattices, and (v) ALM-induced spin splittings in the bands. For (b) 3D checkerboard and (c) diamond lattices, we show the results for the sublattice-current configuration (2 + +−, 2 + −−). For clear illustration of (i) the model structures, we show the sites and currents only in one sublattice. The formats of the figures follow Fig. 2. which are maximal along L-W and X-W lines on the BZ boundary [Figs. 4(b)(iii) and 4(c)(iii)], respectively. The repulsion-driven ALMs [Figs. 4(b)(iv) and 4(c)(iv)] inherit these symmetry structures and induce odd-parity spin splittings in the bands [Figs. 4(b)(v) and 4(c)(v)]. Similar to the 2D checkerboard lattice, the ALMs on the BCC and 3D checkerboard lattices respect the ef- fective time-reversal symmetry T T1 m. Here T1 m is the one-site translation along any first-neighbor directions. The discovery of these odd-parity ALMs in 3D can again be compared with recent studies of coplanar odd-parity magnetism [21, 27, 28]. Discussion— We have identified the sublattice currents in compensated collinear magnets as a feasible route to achieve odd-parity ALM by nonmagnetic time-reversal symmetry breaking. Starting from the Haldane-Hubbard model, we have established a systematic construction of 2D and 3D bipartite-lattice models, where on-site repulsion drives odd-parity ALMs at half filling. The search for practical materials with sublattice currents is presently an active line of research. While the kagome metals have been the most promising candidates for loop currents [50–53], recent experiments have observed pos- sible evidences for the Haldane-model-like sublattice cur- rents in Mn3 S i2 T e6 [54]. Meanwhile, ultracold atoms rep- resent another attractive platform for the realization of odd-parity ALM [55]: they can realize Hubbard antifer- romagnets [56] as well as the Haldane model under Flo- quet drives [57]. While this work uses sublattice currents as the mechanism for odd-parity ALM, the same physics is expected in systems possessing orbital magnetization [50, 51, 53] or complex charge density waves [58–60], as these effects also break time-reversal symmetry indepen- dently of the spin degree of freedom. Note added—We have become aware of follow-up work that investigated the spin-group analysis [61, 62] and other possible origins of odd-parity ALM, such as Flo- quet drives [63–66] and orbital orders [67]. Acknowledgments—M.V thanks O.A. Ashour for help- ful discussions. Y.-P.L. acknowledges the fellowship sup- port from the Gordon and Betty Moore Foundation through the Emergent Phenomena in Quantum Systems (EPiQS) program. M.V. is supported by the Beatriu de Pin´os Grants Programme of the Department of Re- search and Universities, Government of Catalonia (2024 BP 00155). The ICN2 is funded by the CERCA Pro- gramme/Generalitat de Catalunya. The ICN2 is sup- ported by the Severo Ochoa Centres of Excellence pro- gramme, Grant No. CEX2021-001214-S, funded by MCIN/AEI/10.13039.501100011033. [1] S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- dependent spin splitting by collinear antiferromagnetic ordering, J. Phys. Soc. Jpn. 88, 123702 (2019). [2] L. ˇSmejkal, R. Gonz´alez-Hern´andez, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous hall effect in collinear antiferromagnets, Sci. Adv. 6, eaaz8809 (2020). [3] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Giant momentum-dependent spin splitting in centrosymmetric low-z antiferromagnets, Phys. Rev. B 102, 014422 (2020). [4] H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current, Nat. Commun. 12, 2846 (2021). [5] I. I. Mazin, K. Koepernik, M. D. Johannes, R. Gonz´alez- Hern´andez, and L. ˇSmejkal, Prediction of unconventional magnetism in doped fesb2 , Proc. Natl. Acad. Sci. U.S.A. 118, e2108924118 (2021). [6] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven-

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8 noninteracting states are occupied under Fermi statistics. This state can be represented by the density matrix Pa b = ⟨c†b ca ⟩ (9) with the projector condition P 2 = P . The energy of the state can be derived by the Wick’s theorem and rewritten in a mean-field form E[P ] = 1 2 Tr[P (H0 + HH F[ P ])]. (10) Here the Hartree-Fock Hamiltonian HH F[ P ]a b = H0 ,ab + cd (Ua cdb − Ua cbd) Pd c (11) captures an effectively noninteracting theory at the mean-field level HH F[ P ] = ab HH F[ P ]a bc †a c b . (12) The mean-field ground state |ΨG S⟩ is obtained by min- imizing the energy E[P ] under the projector condition. This process yields the self-consistent equation for the ground-state density matrix PG S [PG S, HH F[ PG S] ] = 0, (13) under the condition that the ground-state density matrix PG S is formed by the N lowest-energy eigenstates of the Hartree-Fock Hamiltonian HH F[ PG S] . In practice, the numerical computation is achieved by an iterative algo- rithm. The algorithm begins with a random or designed density matrix. In each iteration, we obtain the Hartree- Fock Hamiltonian and update the density matrix from the lowest-energy eigenstates. The algorithm ends when the desired convergence of energy and density-matrix el- ements is achieved. Magnetic order—Once the ground-state density matrix is obtained, we can compute the on-site spin orders to understand the magnetic order of the ground state [43]. In our lattice model, the basis index a = iσ contains the indices of site i and spin σ. The computation of on-site spin orders is straightforward si = Tr Pi σ 2 , (14) where Pi is the 2 × 2 on-site density matrix at site i and σ = (σ1 , σ2 , σ3 ) is the Pauli-matrix vector. Band-structure analysis—We can study the band structures of both the noninteracting Hamiltonian and the Hartree-Fock Hamiltonian [44]. In our study, the former tells us the effects of sublattice currents, while the later shows us the occurrence of odd-parity collinear spin splitting. Consider the noninteracting Hamiltonian ˜H = ˜ i ˜i ′ ˜τ˜τ ′ σσ′ ˜H˜i ˜i ′ ˜τ ˜τ ′ σσ′ c†˜ i˜τ σ c˜i ′ ˜τ ′ σ′ , (15) which represents either the noninteracting Hamiltonian or the Hartree-Fock Hamiltonian. Here˜ i is the Bravais- lattice-site index under the periodicity of the Hamilto- nian, ˜τ is the sublattice index within the unit cell˜ i, and σ is the spin index. We map the Hamiltonian to momen- tum space by the Fourier transform c˜i ˜τ σ = 1 ˜N 1/2 B L k ck ˜τ σ ei k·r˜i ˜τ , (16) where ˜NB L is the number of Bravais lattice sites, k is the momentum, and r˜i ˜τ is the position of the lattice site˜ i˜τ . The mapping yields the momentum-space representation of the Hamiltonian ˜H = k˜τ ˜τ ′ σσ′ ˜Hk ˜τ ˜τ ′ σσ′ c†k ˜τ σ ck ˜τ ′ σ′ (17) with the Bloch Hamiltonian ˜Hk ˜τ ˜τ ′ σσ′ = ˜i′ ˜H˜0 ˜i ′ ˜τ ˜τ ′ σσ′ e− ik·(r˜0 ˜τ −r˜i ′ ˜ τ ′ ) . (18) Diagonalizing the Bloch Hamiltonian ˜Hk at each momen- tum k, we get the nth-band Bloch states |un k⟩ and their dispersion energies En k.