Giant magnetic field from moiré induced Berry phase in homobilayer semiconductors

Abstract When quasiparticles move in condensed matters, the texture of their internal quantum structure as a function of position and momentum can give rise to Berry phases that have profound effects on the material’s properties. Seminal examples include the anomalous Hall and spin Hall effects from the momentum-space Berry phases in homogeneous crystals. Here, we explore a conjugate form of the electron Berry phase arising from the moiré pattern: the texture of atomic configurations in real space. In homobilayer transition metal dichalcogenides, we show that the real-space Berry phase from moiré patterns manifests as a periodic magnetic field with magnitudes of up to hundreds of Tesla. This quantity distinguishes moiré patterns from different origins, which can have an identical potential landscape, but opposite quantized magnetic flux per supercell. For low-energy carriers, the homobilayer moirés realize topological flux lattices for the quantum-spin Hall effect. An interlayer bias can continuously tune the spatial profile of the moiré magnetic field, whereas the flux per supercell is a topological quantity that can only have a quantized jump observable at a moderate bias. We also reveal the important role of the non-Abelian Berry phase in shaping the energy landscape in small moiré patterns. Our work points to new possibilities to access ultra-high magnetic fields that can be tailored to the nanoscale by electrical and mechanical controls.

The nature of the moiré pattern as a spatial texture of atomic configurations suggests that the Berry-phase effect can be an indispensable part of moiré superlattice physics. In a smoothly varying crystalline environment like a long-period moiré, the Berry phase has several important manifestations in the electron's equations of motion [20] where E , R and k are the energy, position and momentum of an electron wavepacket. k ≡ i ∂u ∂k |×| ∂u ∂k and R ≡ i ∂u ∂R |×| ∂u ∂R are Berry curvatures from the momentum-space and real-space textures, respectively, in the electron's spin and pseudospin wavefunction u. Homogeneous crystals can only have the momentum-space curvature k and the resultant anomalous velocity −k × k is responsible for the various Hall effects [20]. Spatial inhomogeneity can give rise to the real-space curvature R that plays the same role as a magnetic field. Such an emergent C   magnetic field also generates a Hall current, which has attracted remarkable interest in explorations of magnetization skyrmions and domains [21][22][23].
Here, we show that the real-space Berry phase from moiré patterns realizes a nanoscale patterned magnetic field for the massive Dirac fermions in TMD homobilayers. The field is normal to the plane, with the full symmetry of the moiré pattern. The magnetic flux per moiré supercell has a quantized value, while its sign distinguishes moirés introduced by a uniaxial strain from those by a twisting or biaxial strain, although they have the same potential landscape. For low-energy carriers, the complex hopping in the moiré potential as determined by the Berry phase realizes the topological flux lattices hosting the quantum-spin Hall effect. Strain control of the moiré period (b) can dramatically tune the moiré magnetic field with the b −2 scaling, while preserving the flux per supercell. The field magnitude can reach hundreds of Tesla at a moiré size of b ∼ 10 nm. The profile of the moiré magnetic field can also be electrically tuned through an interlayer bias and a topological transition occurs at a moderate bias where the magnetic flux per supercell has a quantized jump. We also reveal that the geometric scalar potential due to the non-Abelian nature of the Berry phase can qualitatively change the potential landscape in a small moiré.

RESULTS
TMD monolayers have their spin-valley locked conduction and valence band edges at the ±K corners of the Brillouin zone, which are described by the massive Dirac model [24]. We focus here on the physics at these band edges in a homobilayer moiré, formed by a small twisting or strain applied to one layer in R-stacking [25] (cf. Fig. 1 and Table 1). The interlayer hopping of the valley electron is sensitive to the atomic registry between the layers [18,26]. The smooth variation of the interlayer registry in a long-period moiré thus leads to the dependence of the hopping on the position R [27,28]. In the basis of |t c , |b c , |t v , |b v , denoting conduction (c) and valence (v) band-edge states from the top (t) and bottom (b) layers, respectively, the Hamiltonian for the two coupled massive Dirac cones reads: describes the spatially varying interlayer-coupling effects in valence bands.σ is the layer pseudospin. h is the hopping matrix element between the valence band edges from the two layers. The interlayer coupling also leads to energy shifts of the band edges accounted for by δ (R) and (R) [19,26].Ĥ c describes similar interlayer-coupling effects in the conduction bands. TheĤ c andĤ v blocks are coupled through the ∂ ± ≡ ∂ ∂ X ± i ∂ ∂Y term of the monolayer Dirac Hamiltonian. Equation (1) is for the −K valley, which has spin-up states only at the band edge, and the K valley is its time reversal.

Non-Abelian Berry connection in real space
We derive an effective Hamiltonian for holes in the bilayer moiré, whereas that for the electron is similar. It is convenient to go to the basis spanned by the eigenstates of interlayer couplinĝ . Through a unitary transformation byÛ (R), the bilayer Hamiltonian in Equation (1) becomes: In the off-diagonal block, the matrixÂ (R) is the non-Abelian real-space Berry connection on the RESEARCH ARTICLE Table 1. Comparison of flux superlattices in three types of homobilayer moiré. One monolayer has the primitive lattice vectors 2 e x − a 2 e y and the other layer is rotated by a twisting angle θ or subject to a biaxial or uniaxial tensile strain, respectively. The uniaxial strain is applied along the armchair or zigzag direction, with the area of the strained unit cell conserved. The magnetic flux per moiré supercell is given for the spin-up carrier (−K-valley). The local registry is parameterized by the in-plane displacement r between a near-neighbor pair of atoms from the opposite layers (cf. Fig. 1 inset). Ed is the interlayer electric bias and its critical value 2 0 /e = 0.044 V.
basis of the layer-pseudospin eigenstates: where x and e y are the unit vectors along the x and y directions.
At the three high-symmetry locales in the moiré ( Fig. 1a and b), rotational symmetry dictates that h vanishes at B and C, while vanishes at A and takes opposite values at B and C (Fig. 2b). These determine a real-space texture in the two branches of layer-pseudospin eigenstates. For example, in the |+ state, σ is in-plane at A and points out-of-plane in opposite directions at B and C, as illustrated in Fig. 1b. Such a spatial texture of the layer pseudospin gives rise to a non-trivial Berry connection.
With the large gap E g , the coupling to the conduction states can be perturbatively eliminated and the hole Hamiltonian keeping its leading effects becomes: where the dropped terms are is the Dirac mass.

Periodic magnetic fields in different types of moiré
We focus first on the diagonal Berry connection A ±± . The main effect of off-diagonal connection A ±∓ is a correction to the scalar potential landscape that is negligible in a sufficiently large moiré, which we will revisit later. If A ±∓ is simply dropped in Equation (4), the two branches of layer-pseudospin eigenstates are decoupled, each described by an effective Hamiltonian: The moiré effects manifest as the scalar superlattice potential −E ± (R) and, more intriguingly, the vector potential A ±± that generates a real-space magnetic field: x ∂Y e z . We establish below the quantitative forms of the superlattice potential E ± (R) and the periodic magnetic field B ± . In a long-period moiré, the interlayer-coupling parameters at location R are determined by the local atomic registry, which can be parameterized by the in-plane displacement r between a near-neighbor pair of atoms from opposite layers (Fig. 1, inset). Under the two-centerapproximation and keeping only the leading Fourier components [26], we find the dependences of , δ and h on the registry r: where 0 , δ 0 , h 0 and h 1 are real constants.
K are, respectively, the three corners of the monolayer  Table 1).
Brillouin zone related by 2π/3-rotations. On the other hand, the split valence band edges E ± = δ ± |h| 2 + 2 can be obtained from ab initio band structures of lattice-matched bilayers of various registry r. In Fig. 2a, the ab initio calculated E ± as functions of r are shown as the symbols, which are remarkably well fitted by the solid curves from Equation (5) The registry r is a function of position R in the moiré. The mapping r (R), together with Equation (5), gives the position dependences of the interlayer-coupling parameters inĤ v (R), i.e. (R) = (r (R)), etc. The resultant spatial texture of layer-pseudospin eigenstates, the scalar potential −E ± (R) and the moiré magnetic field B ± can then be determined. Table 1 summarizes the mapping function r (R) for three different types of moiré pattern formed, respectively, by applying a twisting, a biaxial strain and an area-conserving uniaxial strain to one layer. All three moiré patterns realize hexagonal superlattices, with the identical scalar potential landscape −E ± (R) (except for a 90 o rotation of the superlattice in the twisting case, cf. Table 1). Figure 2d and e shows the spatial profile of layerpseudospin vector σ + and a moiré magnetic field B + in a moiré supercell in the lower-energy hole branch. Remarkably, while the twisting moiré and biaxial moiré have identical pseudospin textures and magnetic-field distribution, the uniaxial moiré is of a distinct texture that leads to an opposite magneticfield distribution. The magnetic flux per supercell SC e z · B + d R is a quantized value: 2π for the twisting moiré and biaxial moiré and −2π for the uniaxial moiré, corresponding to the fact that the pseudospin texture is of a skyrmion [19] and anti-skyrmion configuration, respectively. Thus, the three different origins give rise to two topologically distinct types of moiré patterns. The moiré magnetic field B + is peaked between the B and C points (Fig. 2d  and e). In a moiré of period b = 10 nm, this magnetic field reaches a peak value of 200 Tesla, comparable in size to the giant pseudo magnetic field from an inhomogeneous strain in a graphene nano-bubble [29][30][31].

Topological flux lattice and quantum-spin Hall effect
The periodic scalar potential and magnetic field generated by the moiré together define a flux superlattice. Some remarkable features include: (i) the flux per supercell is quantized and independent of the moiré period; (ii) the two layer-pseudospin branches |+ and |− have opposite magnetic fields B + = −B − and distinct trapping locations in their potentials −E + (R) and −E − (R) (Fig. 2a); (iii) spin-up and -down carriers experience opposite magnetic fields as required by the presence of timereversal symmetry.
Holes can be trapped in the low-energy pseudospin branch |+ at the three high-symmetry locales A, B and C in a moiré supercell (cf. Fig. 2c). Low-energy holes hopping between these trapping sites can then be described by a three-orbital tight-binding model (Fig. 3a): Here,Â l ,B l andĈ l are annihilation operators for the hole trapped at A, B and C sites in lth supercell, and . . . runs over nearest-neighbor pairs of sites. The sum of hopping phases φ 1 and φ 2 around any closed loop equals the magnetic flux enclosed. The equilateral triangle A-B-C-A loop in Fig. 3a encloses a flux of π/3 (−π/3) for the spin-up (-down) hole, for the example of a twisting moiré or a biaxial moiré.
The out-of-plane mirror symmetry dictates that ε B = ε C and t 1 = t 2 . Because of the higher barrier between B and C, t 3 < t 1,2 . The onsite energy ε A is larger than ε B ,C in the scalar potential shown in Fig. 2a. In a sufficiently large moiré, where the hopping becomes exponentially small, t 1,2,3 ε A − ε B ,C , we can adiabatically eliminate the A sites. The resultant two-orbital tight-binding model consisting of the B and C sites becomes (Fig. 3b): runs over next-nearest-neighbor pairs. φ l ,m B , φ l ,m C = ± 2π 3 and the positive phase hopping directions are indicated by the arrows shown in Fig. 3b. This realizes the Kane-Mele model [32] and the Haldane model in each spin subspace [33], which explains the quantized spin Hall conductance found in mini-band calculations in the TMD homobilayer moiré [19,25]. Figure 3c shows the dispersions and the momentum-space Berry curvatures k of the three mini-bands, calculated with the parameters ε A − ε B ,C = 5 meV, t 1,2 = 1 meV and t 3 = 0.5 meV. The topological numbers of the three bands are C = 1 2π e z · mBZ k d k = −1, +1 and 0. This three-band tight-binding model reproduces well The overall scalar potential −E ± + G in the two pseudospin branches at different moiré periods. (d) The schematic-level scheme for the non-Abelian Berry-phase effect in a homobilayer moiré. |+ is coupled to the conduction state of the same pseudospin |c + and the second-order perturbative correction E (2) s accounts for the moiré magnetic field B + effects, including the Zeeman energy shown in (b). The coupling of |+ to the conduction state of the opposite pseudospin |c − leads to another correction E (2) o and its sum with the Zeeman energy term is the overall geometric scalar potential G shown in (a). the band dispersions, the k distributions and topological numbers from the direct mini-band calculations in the twisted MoTe 2 bilayer [19].

Geometric scalar potential
The off-diagonal Berry connection can become increasingly important with the decrease in the moiré period b. In the Hamiltonian in Equation (4), A ±∓ plays two roles. First, it contributes G (R) ≡ 2 2m * A +− · A −+ to the diagonal element of Hamiltonian. As a result, the scalar superlattice potential is corrected from −E ± (R) to −E ± (R) + G (R). Second, it introduces a residue coupling between the layer-pseudospin branches |+ and |− , with the coupling form i ∂ ∂R + A ++ · A +− + However, in the lower branch |+ , low-energy states are detuned from the branch |− by a gap independently of b (cf. Fig. 2a), which quenches the off-diagonal effect except for a very small moiré (b ≤ 5 nm). Therefore, the lowenergy holes are well described by the effective Hamiltonian: G (r) here is an additional energy due to the precession of the layer pseudospin in the adiabatic transport [34], which repels electrons (holes) from the position where the pseudospin changes quickly. This effective potential of the geometric origin is known as the geometric scalar potential. In a homobilayer moiré, G (R) ∝ b −2 , increasing rapidly with the decrease in the moiré period b and peaking between the B and C points of the supercell (Fig. 4a). Figure 4c plots the overall scalar potential at different moiré periods b. The correction by the geometric contribution becomes significant at b ≤ 20 nm. The local out-of-plane mirror symmetry at A dictates that G (R) vanishes at this location. The geometric scalar potential therefore pushes ε B ,C towards ε A (see Fig. 4c), so the A orbital can no longer be perturbatively eliminated in a small moiré. From the three-orbital tight-binding calculation using Equation (6), we find that a topological phase transition occurs when t 3 is increased to (ε A − ε B ,C ) /3. In Fig. 3d, we show the results with parameters ε A = ε B ,C = 0, t 1 = t 2 = 5 meV and t 3 = 2 meV, where the topological numbers of the three bands become C = −1, −1 and +2, respectively. The tight-binding calculations shown in Fig. 3c and d reproduce the distinct moiré band dispersions and topologies in MoTe 2 bilayers at twist angles 1.2 • and 2 • , respectively. The change in the energy landscape caused by the geometric scalar potential accounts for the topological phase transition as a function of the twist angle.

Quantized switch of magnetic flux by interlayer electric bias
An interlayer bias can add a tunable term 1 2 e Edσ z toĤ v (R), where d is the interlayer distance and E is the perpendicular electric field. The effect is to replace (R) by E (R) ≡ (R) − 1 2 e Ed , which changes the profiles of both the potential landscape E ± (R) and the magnetic field B ± (R). The electric bias can thus be exploited to continuously tune the moiré magnetic field and the flux superlattice.
The lower panel of Fig. 5a shows the magneticfield distribution in the pseudospin branch |+ under an interlayer bias Ed = 0.026 V in the twisting moiré. The magnetic-field distribution is significantly changed compared to the zero-bias distribution shown in Fig. 5a upper panel, having been pushed from the B-C center towards the C point, where the difference between E + (R) and E − (R) is minimal. The magnetic flux per supercell remains 2π . With a further increase in bias to the critical value Ed = 2 0 /e = 0.044 V, E crosses 0 at C point, where h (R) also vanishes (cf. Fig. 2b). Therefore, the gap between E + (R) and E − (R) RESEARCH ARTICLE closes, whereupon a topological transition of the layer-pseudospin texture occurs (Fig. 5b). Figure 5c shows the magnetic-field distribution after this transition (Ed = 0.06 V), where the magnetic flux per supercell becomes 0.

DISCUSSION
Massive Dirac fermion features an orbital magnetic moment M = 2 2m * e z arising from the momentumspace Berry-phase effect [35], which leads to Zeeman shifts in the band edges of the monolayer TMDs in an external magnetic field [36][37][38]. It is interesting to ask whether the emergent moiré magnetic field B + , as a manifestation of the real-space Berry phase, can couple to the orbital magnetic moment as well. In the perturbative expansion from Equation (2) to Equation (7), we can separate two contributions in the second-order perturbative correction:Ĥ + = −E + (R) + E (2) s + E (2) o , is from the coupling between |+ = U (R) |t v and the conduction state of the same pseudospin orientation |c + =Û (R) |t c (see Fig. 4d), which can be rewritten as: E In addition to the kinetic-energy term, it does contain the coupling of the orbital magnetic moment to the moiré magnetic field B + , which contributes to the scalar potential landscape (Fig. 4b). Because of the non-Abelian nature of the Berry-phase effect here, there is also a contribution E (2) o from the coupling between |+ and the conduction state of opposite pseudospin |c − =Û (R) |b c . The Zeeman energy partially cancels with E (2) o , with the net effect being the geometric scalar potential G (R).
In contrast to the twisting moiré, the moiré introduced by a biaxial or uniaxial strain allows engineering a moiré magnetic field and flux superlattice through mechanical controls. In a rotationally aligned homobilayer, the application of a relative strain η between the two layers, e.g. through the substrate, can create a moiré pattern with period b = a/η. A modest tensile strain can thus tune b over several orders of magnitude, where the magnetic-field scales as b −2 . This homobilayer moiré points to unprecedented opportunities to explore physics in an ultra-high magnetic field that can be tailored on the nanoscale by both electrical and mechanical means.
It is also interesting to compare with twisted bilayer graphene, where the interplay of a sublattice pseudospin and a layer pseudospin leads to a more complex gauge structure for the massless Dirac fermion [28,39]. Only at the AA stacking locales [39] can the gauge structure in a graphene moiré be simplified to a form equivalent to that of a magnetic field ∝ b −1 (in contrast to the b −2 scaling underlying the b-independent flux quantization here). This gives rise to pseudo Landau levels of the massless Dirac fermion at the AA locales, which can be connected to the flat mini-bands in the bilayer graphene at the magic twist angle [28,39]. For massive particles here, the strongly inhomogeneous magneticfield profile in a moiré supercell of exactly quantized flux points to manifestations of ultra-high magnetic fields that are remarkably different from the wellstudied Landau-level physics.
The flux lattices and quantum-spin Hall effects need to be explored at low doping in a relatively small moiré, since the topological gap (determined by hopping integrals between superlattice sites) decreases exponentially with b. The other limit, i.e. relatively high doping in a large moiré, is equally interesting to explore, where the nanoscale patterned magnetic field in the associated potential landscape (Fig. 5) points to a new realm of magneto transport. Electrostatic doping and interlayer bias control of the moiré magnetic flux can both be