Quantum Entangled Fractional Topology and Curvatures

We propose a two-spin quantum-mechanical model with applied magnetic fields acting on the Poincar\'e-Bloch sphere, to reveal a new class of topological energy bands with Chern number one half for each spin-1/2. The mechanism behind this fractional topology is a two-spin product state at the north pole and a maximally entangled state close to the south pole. The fractional Chern number of each spin can be measured through the magnetizations at the poles. We study a precise protocol where the spin dynamics in time reflects the Landau-Zener physics associated with energy band crossing effects. We show a correspondence between the two-spin system and topological bilayer models on a honeycomb lattice. These models describe semimetals with a nodal ring surrounding the region of entanglement.

models of interacting spins with applied magnetic fields acting on the Poincaré-Bloch sphere to reveal a new class of topological states with rational-valued Chern numbers for each spin.
We present a geometrical definition of this fractional topology related to the formation of a product state at the north pole and a maximally entangled state at the south pole. We study a driving protocol in time and the spin magnetizations at the poles to reveal the stability of the fractional topological numbers towards various forms of interactions in the adiabatic limit. We elucidate a correspondence between a two-spin system with one-half topological number for each spin and a topological bilayer model on a honeycomb lattice, which describes semimetals with a nodal ring at one Dirac point encircling a region of entanglement and a topological bandgap at the other Dirac point revealing a π Berry phase. Such materials belong to a novel topological class with a Z 2 layer symmetry. We discuss the bulk-edge correspondence applied to quantum transport and light-matter coupling.
In recent years, rising interest in topology travels from mathematics to physics related to advancing quantum science and technology. This allows for the direct observation of the Chern 1 arXiv:2002.11823v5 [cond-mat.mes-hall] 2 Apr 2021 number, a measure that distinguishes topological insulators and superconductors 1,2 . The properties of these systems can be revealed from the reciprocal or momentum space showing how the topology is already encoded in a spin-1/2 particle or two-state system when equivalently applying a magnetic field that acts radially on the sphere 3 with polar angle θ and azimuthal angle φ. Upon adiabatically sweeping from the north to south pole, along a curved path with fixed angle φ, the Chern number C of this two-state system represented by a vector of Pauli matrices σ = (σ x , σ y , σ z ) is equal to one. Incredibly this topological quantity can be measured directly from the spin magnetizations at the poles: [4][5][6] C ≡ 1 2π 2π 0 dφ π 0 dθF φθ (1) The angles θ = 0 and θ = π refer to the north and south poles of the sphere, respectively. We have introduced the Berry curvature and the Berry connection A, defined from the gradient of the ground state |ψ according to 7 A α = i ψ|∂ α |ψ .
The associated Berry phase represents an important foundation of quantum physics 8 . In the quantum Hall effect, such a geometrical description in terms of curvatures plays a key role in the link with electronic transport properties such as the quantum Hall conductivity 9,10 . Here, the integer Chern number C of a given spin-1/2 is related to a topological charge -the degeneracy point of the Hamiltonian -contained within the sphere spanned by the magnetic field vector. The spin-1/2 orientation then measures directly this topological charge [11][12][13] . A recent experiment 12 has studied two spin-1/2s, σ 1 , σ 2 , under the influence of the radial fields H 1 and H 2 forming the surface of the sphere. The two spins interact through a transverse coupling (σ x 1 σ x 2 + σ y 1 σ y 2 ). Their resulting topological phase diagram consists of integer C = 0, 1 and 2 phases, corresponding to topological charges located outside both spheres, inside one sphere, and inside both spheres respectively. To show the possibility of entangled states with a stable fractional Chern number for each spin, we add a crucial ingredient corresponding to adjustable constant magnetic fields on the sphere. In the following, we introduce a model with two spins σ 1 = (σ x 1 , σ y 1 , σ z 1 ) and σ 2 = (σ x 2 , σ y 2 , σ z 2 ) interacting through an Ising coupling, to reveal half-topological numbers for each spin on the sphere. The topology is defined on each sub-system, here a spin-1/2, directly from the poles. We show applications of the spheres with C = 1/2 per spin for the characterization of topological semimetallic phases in bilayer honeycomb systems showing one topological Dirac point associated with a π Berry phase and another Dirac point revealing a nodal entangled ring. In the Supplementary Information, we show that this is just one example of a large class of models, including XY couplings and higher numbers of spins that all reveal the same effect.

Model with Two Spheres
The Hamiltonian for two spheres reads 3 The magnetic field H i acts on the same sphere parameterized by (θ, φ) and may be distorted along theẑ direction with the addition of the uniform field M i according to 4 : for i = 1, 2. We show below through energetics arguments that the fields M i are indeed important to stabilize a fractional Chern number. We also consider a generic θ-dependent couplingrf (θ) withrf (θ) > 0. The ± denote two distinct classes of models. It is important to highlight here that in the case where a spin-1/2 is coupled to an environment, the topological number associated to the spin may vary continuously from C = 1 to C = 0 dependently on the coupling strength between the two systems. In the present case, we show that the fractional Chern numbers are stable towards smooth deformations of the geometry and towards the form of the interactions. In experiments, the magnetizations may be measured for each spin independently, such that the Chern number also has a well-defined component corresponding to each subsystem. Therefore, we find it important to first generalize Eqs. (1) and (3) for subsystem or spin j in the interacting model. The corresponding Chern number C j will provide a robust topological number related to the quantum Hall conductivity and will also represent a measure of entanglement. The spin system we consider here provides a nice platform for understanding how topology can be partitioned between subsystems.

5
In the Supplementary Information, we show that the Chern number for the jth spin can be written as This form is gauge invariant as shown in the Supplementary Information through the Stokes' theorem and the introduction of smooth fields. It is interesting to observe that a similar correspondence is useful to describe the 'quantized' topological response of one pseudospin-1/2 when coupling with circularly polarized light then referring to quantized circular dichroism of light [15][16][17] . Here, we also show that Eq. (7) defining the topology at the poles only, is related to the charge polarization and the quantum Hall conductivity for the sub-system j itself; see Supplementary Information.
Then, we have the general result From the Pauli operator σ j z = | ↑ jj ↑ | − | ↓ jj ↓ |, and from the normalization equation of the state |ψ , we also find the equality ψ|σ j z |ψ = 1 + 2A j φ , leading to Eq. (9) is an interesting generalization of Eq.
(1) because this shows that one can yet define and measure for these interacting models in curved space the topology from the magnetizations of a given spin j at the poles. Now, we consider the specific system of interest whose ground state evolves from a product 6 state at θ = 0 to an entangled state at θ = π: The non-zero coefficients are |c ++ (0)| 2 = 1, |c +− (π)| 2 = |c −+ (π)| 2 = 1 2 , for which The presence of entanglement at one pole leads to a fractional Chern number of 1/2 for each spin. This value is in agreement with σ j z (θ = 0) = 1 and with σ j z (θ = π) = 0, reflecting the formation of a maximally entangled Bell pair at the south pole 14 . The norm of each spin effectively shrinks at the south pole, leading to a ln 2 entanglement entropy 18 . In the case where the two spins would form a product state that follows the magnetic field, then from c ++ (0) = 1 and c −− (π) = 1, we verify C j = 1. In the case where the two spins would be entangled at both poles then C j = 0.
To show that our model in Eq. (4) does indeed fulfill the necessary prerequisites to observe C 1 = C 2 = 1 2 , we study the topological phase diagram which is entirely determined by the energetics at the poles. For clarity, we analyse the H + Hamiltonian hereafter (the H − Hamiltonian reveals a similar fractional phase). At the poles, the ground state is readily determined, and the resulting topological phase for each spin is shown in Fig. 1a for a constant interaction f (θ) = 1.
Allowing for a non-constant interaction does not change this phase diagram significantly, though it does open up the intriguing possibility of a direct transition from In the presence of Z 2 symmetry between the two spins corresponding to σ 1 z ↔ σ 2 z when (4), the ground state at the north pole with θ = 0, is | ↑ 1 | ↑ 2 provided that rf (0) < H + M . At the south pole with θ = π, the ground state is | ↓ 1 | ↓ 2 forrf (π) < H − M , but it is degenerate between the anti-aligned configurations forrf (π) > H − M . In that case, the presence of the transverse fields in the Hamiltonian along the path over the sphere will then produce the analogue of resonating valence bonds 19 . Indeed in Sec. 3, we will see that the singlet state is decoupled from the rest, while the triplet state 1 showing the resonance between the states | ↑ 1 | ↓ 2 and | ↓ 1 | ↑ 2 , is the one adiabatically connected to the θ = 0 ground state. As a result, we obtain half-integer Chern numbers (11). For the simple constant interaction f (θ) = 1, this occurs within the range indicated by the gold line in Fig. 1a. This line can be considered as a critical point between two distinct topological phases of a given spin. In the limit M → 0, it becomes the quantum critical point between the total-Chern-number 2 and total-Chern-number 0 phases. We find that the H − Hamiltonian also contains a line of fractional Chern numbers with C 1 = −C 2 = 1 2 . In the Supplementary Information, we show that the fractional phase with C j = 1/2 can be stabilised and in fact spreads in the presence of an XY coupling.
We also verify that the fractional Chern number may be generalized for N > 2 spins; starting from a product state at the north pole, spins may evolve via the transverse field to an entangled state at the south pole with C j = 1/2 for an even number of spins or C j = N +1 2N for a frustrated system with an odd number of spins. The spin model of Fig. 3 Brillouin zone torus can be mapped onto the parameter space discussed above. It follows that a stack of two Haldane layers may be represented by a two-spin model.

Lattice model
We consider a plane realization of Eq. (4) consisting of two AA and BB-stacked graphene lattices 32 and show how to find a fractional magnetization representing C j . Here, θ = 0 and θ = π map onto the K and K points of the first Brillouin zone respectively (see Supplementary Information). The spin degrees of freedom now describe the momentum-space sublattice magnetization for each layer j. A correspondence between the spheres model and the lattice model, that will be developed below Eq. (16), can be formulated through the identification σ j z ↔ n j kB − n j kA . Here, n j kα represents the density of particles associated to sublattice α = A or B for a wavevector k, in a given layer j. Here, we will discuss the situation with equal fluxes. The mapping suggests that we need an unusual interaction -one that is local in k-space -to produce a momentum-dependent Ising interaction. Such interactions have been studied in relation to Weyl semimetals 35,36 . In fact, we can achieve the same result with an interlayer coupling r between neighbouring sites.
All of this motivates the following lattice model in momentum-space: where ψ † ki ≡ (c † kAi , c † kBi ) and is represented in terms of the Pauli matrices σ, the 2 × 2 identity matrix I, and the k-dependent The eigenvalues and eigenvectors of this matrix are readily found at the K and K points where the gap closes, respectively, for the values of r: For the case of asymmetric Semenoff masses M 1 = M 2 , the gap closes and reopens at r − c . When Importantly, the states c † A1 c † B1 |0 and c † A2 c † B2 |0 do not modify the pseudo-spin magnetization in each plane as they favor an equal particle density on the two sublattices, but they will participate in the entanglement entropy maximum in the nodal ring region. Therefore, from the point of view of the pseudo-spin magnetization at the Dirac points or equivalently at the poles on the sphere then only four states intervene. Explicitly, the correspondence between states in the lattice model and states in the sphere model is given by These are the states that enter in the evaluation of the topological properties. The pseudo-spin magnetic structure around the K point is therefore related to the reduced wave-function 1 in |ψ g which corresponds then to the same entangled state as for the two spheres around the south pole. The topological properties of this semimetal can then be described through Eqs.
(7) and (9). Thus, through the magnetization related to the particle densities on the two sublattices of each layer at the K and K points, we introduce the lattice version of C j (Eq. (9)) where j = 1, 2 refers to the layer basis. The magnetization for a single layer is shown over the unit cell of the reciprocal lattice in Fig. 2b.
Alternatively, we may represent the ground state at half-filling in terms of the occupancy in 13 each layer (comprising two sub-lattices with a given ket |ij , i + j = 1, such that |10 refers to sublattice A occupancy and |01 to sublattice B occupancy respectively): |ψ = i+j+k+l=2 c ijkl |ij 1 |kl 2 , from which we get the reduced density matrix ρ 1 by tracing out the second layer. From this the entanglement entropy is computed numerically (see the Supplementary Information) and shown for the case of symmetric masses in Fig. 2c. For r < r − c , the entanglement entropy is identically zero. Above r − c , we verify that the system shows a maximum entanglement entropy of ln 4 located in the band crossing region, in agreement with the form of |ψ g . One Dirac point is characterized by a nodal ring enclosing the entangled region. Since the two Dirac points map to the two poles on the spheres, this emphasizes the correspondence between the two-spins and the lattice model.
We highlight here that even though we have a band crossing effect in the nodal ring region, the spheres' formalism allows us to conclude that the topological number defined through Eq. (7) is yet applicable in this situation showing then that C j = 1/2 is measurable through the quantum Hall conductivity, with j referring to one layer. From Stokes' theorem on the sphere, it is important to emphasize here that the C j = 1/2 topological number can also be interpreted as a π Berry phase encircling just one Dirac point associated to the topology (Eq. (33) of Supplementary Information).
Regarding the bulk-edge correspondence, the edge spectrum in the reciprocal space produces one  Fig. 1b), then this mode progressively redistributes in one plane only. The nodal ring gives rise to delocalized bulk gapless modes in real space, and yet the robust topology can also be measured from the particles' densities associated to each layer resolved in momentum space at the two Dirac points from Eq. (18). We also find that the layer magnetization numberC j varies smoothly across the transition, in contrast to the sharp change in C j that occurred in the spin model. See Supplementary Information, for further details related to these facts and proofs.

Protocol in Time
Here, we show the occurrence of stable half-topological numbers in a real-time protocol, in the adiabatic limit. We also illustrate energy bands interferometry effects and deviations from these rational values when increasing the speed of the protocol. One experimental protocol for measuring C j in a spin system is to perform a linear sweep, θ = vt, t ∈ [0, π/v] for some velocity v, of the magnetic field along the meridian φ = 0, measuring σ j z at the endpoints of the path 12 , i.e. at the north and south poles. Any finite velocity will lead to non-adiabatic transitions via the Landau-Zener-Majorana mechanism 43-45 , which describes a time-dependent two-state model of the form The amplitudes for the | ↑ and | ↓ components of the wavefunction were derived by Zener 43 for the asymptotic case t → ∞. Here we are actually interested in the values at t = 0, which are derived in the Supplementary Information. There we also show that the quasiadiabatic regime of our two-spin system is described by an effective two-state Hamiltonian where the basis for the Pauli matrices is now given by two of the triplet states (1, 0) T = |1, 0 and (0, 1) T = |1, −1 . We see that the entangled state |1, 0 is indeed the unique ground state at θ = π forr sufficiently large. More precisely, the window in which the ground state evolves from the product state at the north pole to the entangled state at the south pole, and therefore has C j = 1/2, is given by Returning to the dynamics of Eq. (20), we expand near θ = π, such that t → t − π/v. With this new time variable, the important dynamics takes place near t = 0 such that we approximate f (θ) = f (π) close to the south pole, but we find that relaxing this condition does not affect the result noticeably, as shown in Fig. 3f. We then rotate the Pauli matrices about the y-axis. In the rotated basis, the effective Hamiltonian takes the Landau-Zener form, with and adiabaticity parameter γ = ∆ 2 /λ. The amplitude for measuring the |1, −1 state is then , while the amplitude for measuring the entangled state is 1 former results in C j = 1, upon sweeping to the south pole (now at t = 0) while the latter gives The product A(0)B(0) * is evaluated in the Supplementary Information, which yields 16 in terms of the gamma function Γ(z). We check the adiabatic limit of this formula, v → 0 (γ → ∞) and find In addition, we have verified that the additional fractional phases found for N > 2 spins are also stable from the time evolution of these models in the quantum circuit simulator Cirq 46 , illustrating that these phases can indeed be seen through the action of unitary gates in a generic quantum computer; See Supplementary Information.

Discussion
Our analysis shows that one can realize quantum states with fractional topology from the interplay between Berry curvatures and resonating valence bond states 19,22,47 . Quantum entanglement between two spins can produce a Chern number of one-half for each spin. We have provided a geometrical and physical interpretation of this result through the derivation of Eq. (7). We have shown the stability of the fractional Chern number regarding various forms of interactions in the adiabatic limit. We have formulated a correspondence with topological lattice models respecting Z 2 (layer) symmetry, which form nodal ring semimetals in momentum space around a Dirac point.
The one-half topological number arises from a π Berry phase around the Dirac point that shows the topological band gap and also reveals one protected low-energy edge mode in the reciprocal space. This prediction can be measured from momentum-resolved tunneling i.e. when injecting a charge e resolved in energy and wave-vector 48 . In real space, we verify that this mode equally redistributes between the two planes with 1/2 probabilities as if a charge e would equilibrate as two averaged charges e/2 in the two layers. It is important to highlight here that for in the presence of a band-crossing effect around the nodal semimetallic ring, we have shown that the one-half topology of each spin or each plane in the bilayer model can be defined from the spin magnetizations at the poles, the bulk charge polarization and the quantum Hall conductivity which can also be reinterpreted as an effective charge e/2. Since the ground state wavefunction is a direct product state on the sphere, defining the operatorĈ j = 1/2(σ z j (0)−σ z j (π)), we obtain the standard which is a result of the formation of an entangled Bell pair at the south pole. Related to circular dichroism of light, we have verified that at the topological Dirac point the response is similar to the Haldane model 15,16 and that in the semimetallic region there is no light response, such that when averaging on both light polarizations the response at the two Dirac points is also in agree-  Competing Interests The authors declare that they have no competing financial interests.
Additional Information Supplementary Information is available for this paper.
Author details and Contributions The two authors (joel.hutchinson@polytechnique.edu and karyn.le-hur@polytechnique.edu) have contributed to the elaboration of ideas and the establishment of results. They have also participated in the writing of the manuscript.
Correspondence The corresponding author is karyn.le-hur@polytechnique.edu.
Methods The methodology begins from general quantum arguments to show the possibility of a fractional Chern number for an interacting spin-1/2 particle, leading to Eqs. (7), (9) and (11). Then, we analyze the ground state energetics of a particular model and show how to observe a Chern number 1/2. Furthermore, we formulate a mathematical correspondence between the spin-1/2 and topological bilayer lattice models. We find a relation between the Chern number measurement and the quantum Hall conductivity, the polarization and the light response in a given plane. We perform numerical evaluations in the bilayer model of the Berry curvature, magnetization, entanglement entropy as well as the band structure in a finite and infinite system.
For the time-dependent protocol, we check, through numerical evaluation of the Schrödinger equation, that our results are very similar for various forms of spin interaction in curved space. We also study the effect of increasing the speed of the protocol related to Landau-Zener-Majorana interferometry effects.
In the Supplementary Information, we present in the first section two proofs for the gauge invariance of Eq. In this Supplementary Information, we provide two alternative derivations of Eqs. (6) and (7) in the manuscript and through smooth fields on the sphere we derive the relations between the topological geometrical responses and measurable observables in transport properties. We also discuss the relation between the one-half topological number on each sphere (associated to a plane) and a π Berry phase at one pole (Dirac point). We show the gauge invariance associated to the topological response (Sec. 1). We then discuss generalized spin models with XY coupling (Sec. 2) and larger numbers of spins (Sec. 3) which allow us to identify other forms of entangled spin states. Then, we provide definitions on the lattice geometry and on the calculation of entanglement entropy, as well as results for the edge modes and local density of states for a finite system (Sec. 4). We also discuss the measure of the topological number with a driving protocol in time and study transition amplitudes in time (Sec. 5). We address a time-evolution protocol for a situation with N > 2 spins to show the stability of these fractional topological numbers for larger systems.
1 Gauge invariance and non-quantization of C j First in Sec. 1.1, we provide a geometrical proof of Eqs. (6) and (7) in the article from vector theorems. This illustrates the intriguing fact that the topological response can be encoded in the poles of the Bloch sphere, which holds for both the regular Chern number C and the partial Chern number C j . In Sec. 1.2, we show the relation between the smooth fields in Stokes' theorem and quantum transport properties related to the charge polarization and quantum Hall conductivity. Then, in Sec. 1.3 we provide a proof based on the specific class of wavefunctions we study. In Sec. 1.4, we verify our arguments for various forms of wave-functions.

Topological Response
In the article, the Berry connection A j is well-defined on the abstract parameter space {θ, φ}. The goal in this section is to reinterpret A j as a vector on the surface of a sphere S 2 , and then to make use of Stokes's theorem in three dimensions to evaluate the Chern number. We start from Eq. (1) of the article: Here, A is computed in an arbitrary gauge, n is the normal vector to the sphere, and we have used Note that we have dropped the j superscript since this proof applies equally to the Chern number C and the partial Chern number C j . We decompose the hemisphere into north and south hemispheres demarcated by a fixed θ = θ c (which need not be at the equator) such that In mapping the space {θ, φ} to S 2 , we have to take special care of the poles because while A(0, φ) and A(π, φ) have well-defined φ-components that contribute to the Chern number, any smooth vector field on S 2 must have vanishing φ-components at the poles. The case where these components do vanish and the mapping is faithful, is precisely the case when the Chern number is trivial. The use of Stokes's theorem requires that we have a smooth smooth vector field over the relevant manifold. Here, we would like to show the form of this smooth field as well as the form of C in terms of the Berry connections at the poles. We hypothesize that we can build a piecewise smooth field A on the north and south hemispheres, such that on each hemisphere, for all values of the azimuthal and polar angles. Now, we show the form of this field, as follows. Looking at Eq. (1), we can subtract infinitesimally small areas encircling the poles to define a surface S 2 which is no longer a closed manifold. Since these areas are infinitesimally small and the Berry curvature is finite, this will not affect the Chern number so we can write The surface S 2 can be decomposed into a north (north') hemisphere defined by 0 < θ < θ c and south (south') hemisphere defined by θ c < θ < π on which the field A is smooth, such that On north', we have from Stokes' theorem: where θ − c means we approach θ c from the north. This form assumes that the field is uniquely defined on the boundary path at the north pole with A φ (0) = A φ (0, φ). The right-hand side then corresponds to the two boundary paths encircling north'. Similarly, we have for south': Again, θ + c means we approach θ c from the south, and the field is uniquely defined on the boundary path at the pole with A φ (π) = A φ (φ, π). These expressions suggest a natural definition for the smooth fields on the full north and south hemispheres: whereφ refers to the unit vector associated with the azimuthal angle. We satisfy ∇ × A = ∇ × A over each hemisphere, and also ensures that Note that these identities further imply that if we fix θ and perform a closed path in φ on a given hemisphere, then which can be viewed as the integral of a flux through the disk at fixed θ. The last term then places the pole information inside a cylinder with an infinitesimally small radius inside the hemisphere. Returning to Eqs. (7), and (8), we show the form of C in terms of Berry connections as follows. Suppose we move the boundary very close to the north pole such that θ c → 0 (the same relation would be obtained with θ c → π), then Therefore, by summing these two lines, we obtain This corresponds to Eq. (7) in the article which shows that the topology can be encoded through the poles.

Smooth Fields and Topological Observables
Here, we show a relation between the formulae (7)-(9), (14) and quantum transport properties associated to the quantum Hall conductivity and the charge polarization, directly from the surface of the sphere. We apply an electric field E = Ee x in real space along the axis associated to the (wave-vector) coordinate k in the reciprocal space. Then, the velocity of the particle v satisfies k = mv and from Newton's equation with the force F = eE we have ma = k = ∂k /∂t = eE, where m refers classically to the mass of a particle and a to the acceleration. Quantum mechanically, we have used the de Broglie principle. Then, we introduce the map (k , k ⊥ ) = f (θ, φ) such that the two Dirac points now correspond to the poles. The direction of the electric field here corresponds to a line crossing the two Dirac points, from north to south poles, e.g. at azimuthal angle φ = 0 and characterized through θ(t). In flat space, we can fix effectively the lattice spacing such that the (relative) distance between the poles is equal to π as for the unit sphere. For this situation, we observe for a charge e thatθ = (e/ )E and = h/(2π) being the reduced Planck constant. At time t = 0, the particle is at the north pole and at small time dt, the particle has moved to the position dθ, such that we have θ(t) 0 dθ = θ(t) = t 0 dtθ = eEt/ . The velocity of a particle in real space can be written asṙ = (ẋ ,ẋ ⊥ ) and below we study one component of the velocity vector, and more precisely the transverse component to the electric field which is related to topological properties. To show the relation between the transport from the reciprocal space and the Berry fields, one can start from quantum mechanics laws. We have a simple relation between real and reciprocal or wave-vector k-space from the Parseval-Plancherel with here x = x ⊥ , k = k ⊥ and T referring to the final time in the protocol. The left-hand side has the dimension of an (averaged) current density in a one-dimensional pumping protocol and on the right-hand side we introduce the wave-function of interest, e.g., the ground-state wavefunction associated to a given sub-system j resolved in the reciprocal space. On the sphere, we study the perpendicular electron current and the time T is related to the angle θ through the equality θ = eET / = v * T , referring to a specific point along the path at φ = 0. From the analogy between the left and right-hand side, we can define a current density from the reciprocal space. Then, we define for a fixed angle φ such that We identify the important relation Therefore, we observe a relation between the transverse current density and the smooth fields: Here, A φ,θ<θc (θ, φ) refers to the azimuthal angle component related to the smooth field defined in (9) with an angle θ in the north hemisphere such that θ < θ c . This shows a relation between the perpendicular current and the smooth fields. These definitions agree with general definitions, which are also applicable for many-body systems, between current density and charge polarization in a pump-geometry. The current density in a one-dimensional topological pump geometry reads with the anomalous velocity v(q) = −Ω qt being related to the Berry curvature and BZ refers to the Brillouin zone. We have defined F µν = F φθ and here we have the relation and For the sphere geometry, the polarization takes the form and here we use periodic boundary conditions in the direction of the azimuthal angle. This relation is in principle independent of the value of the electric field, but we assume the adiabatic limit i.e. the polarization is defined from the ground state. We also have the relation between current and polarization and in the present protocol the current density is uniformly distributed in the time period ∆P = jT = J ⊥ T , such that any choice of T will measure the same polarization.
To define a gauge-invariant current density, we can now take the point of view of the south pole and take into account the current associated to a charge which has possibly traveled from south pole to the measurement angle θ during the protocole. Due to the application of the electric field, this charge now corresponds to a charge −e. We verify below that this protocole is also equivalent to measure directly the flow of a charge e from north to south pole, such that the protocole is in agreement with physical laws. The choice of south pole as an upper limit for the polar angle here comes from the fact that θ ∈ [0; π] from the definition of the area and topological properties on the sphere. The current linked to the charge −e is with A φ,θ>θc now referring to the azimuthal component of the smooth field in (9) with an angle θ in the south hemisphere such that θ > θ c . Related to the definition of the smooth fields, we can now choose the measurement angle to be θ c , such that we observe a relation between the definitions of north and south hemispheres. Using Eqs. (7) and (8), the current density measured at the position (θ c , φ) now reads with the identifications A φ,θ>θc (θ, φ) = A φ (θ + c , φ) and A φ,θ<θc (θ, φ) = A φ (θ − c , φ). Then, we obtain the quantized polarization on a sphere associated to a physical plane This analysis is yet valid for multispheres or planes when we refer to the polarization in a given plane as a response to the electric field and we have not used a specific form of wave-functions in agreement with the generality of Stokes' theorem. For the specific case where θ c → π, we obtain the relation with T * = π/v * . This formula shows that the topological number can be equivalently measured when driving from north to south poles which is already visible from Eq. (20) when setting θ → π.
To make a bridge with the quantum Hall conductivity, we start from Eq. (21) in the reciprocal space where we can define the current density for a fixed value of q = φ with v(q) = v * ∂A φ (θ, φ)/∂θ. The quantum Hall conductivity can be then calculated from the reciprocal space integrating this current density on all the possible values of φ and θ associated to the reciprocal space and respecting the measure in flat space Then, we identify the transverse current density and therefore the quantum Hall conductivity σ xy = (e 2 /h)C defined from Eq. (14). This formula implies for multispheres' or multiplanes' systems, that applying an electric field on a given subsystem j, one can now measure C related to the Berry fields at the Dirac points. The related arguments developed in the article then show that the one-half topological numbers can be observed from a charge polarization protocol and from the quantum Hall conductivity. From Eq. (9) in the article, the topological number can also be measured when driving from north to south pole measuring the magnetizations. To the best of our knowledge, the relation between the smooth fields on the sphere, the Berry connections at the Dirac points and quantum transport properties was not previously mentioned in the literature. For the specific case C = 1/2, the Stokes' theorem allows us to show that we may re-interpret all the topology as a π Berry phase encircling the Dirac point associated to the edge structure and to the hemisphere encircling the topological charge. More precisely, Eqs. (7) and (8) are then equivalent to If we now move the boundary very close to the north pole encircling the Dirac point, then from our definitions A φ (θ − c , φ) = 0 and we obtain The left-hand side of this equation agrees with the fact that in the Haldane model a Dirac point is characterized by a π Berry phase, and in the present case, this particular point has a well defined energy bandgap such that a local π Berry phase interpretation remains meaningful. The right-hand side of this equation emphasizes that one can transport the topological response from one pole to another, as discussed in the previous section such that only the quantity A φ (0) − A φ (π) defined from the poles is gauge-invariant. The situation is different for the blue phase of the phase diagram in Fig. 1b) (in the main article) as in that case C 1 = 1 and C 2 = 0 associated with the two planes, implying that we do not have a quantum Hall conductivity σ xy = 1 2 e 2 h per plane. It is important to mention here for a comparison that when a sphere develops a unit quantized Chern number, from Stokes' theorem, the Berry phases at the two Dirac points can be transported at one pole (or one Dirac point in the plane) when setting θ c → 0 or π, and in that case the sum of the two Berry phases englobing the two hemispheres reads 2π = π + π.

Vector Potential and Wavefunction
Here, we discuss gauge arguments relating the north and south poles to the two-particle wavefunction.
There exists a set of gauge choices for which the ground state is single-valued (i.e. independent of φ) at θ = 0; we denote any of these choices by |ψ N . Likewise, there are gauges for which the ground state is single-valued at the θ = π, denoted |ψ S . We then define A j N φ (θ, φ) and A j Sφ (θ, φ) such that Note the behaviour of these functions at the poles. By definition, |ψ N is independent of φ at θ = 0 and |ψ S is independent of φ at θ = π. Thus, so that Furthermore, in any particular gauge, the Berry connection (not the wavefunction) is a single-valued function over the entire parameter space, so that This is easily checked from the form of wavefunction used in the article.
Proof. Here, we refer to the class of wavefunctions in the article: Since the wavefunction must take this form at all points on the sphere, we need only consider the set of gauge transformations that preserve this form (as pointed out in the article, the jth Berry connection is well-defined as long as we stay in this sector), which means Note that the decomposition of χ into χ 1 and χ 2 is not unique. Each different decomposition should be regarded as a different gauge choice. It suffices to just consider the φ component of the Berry connection: In the last line we used the normalization of the wavefunction kl |c kl (θ)| 2 = 1.
There exists a north gauge such that A j N φ (0, φ) = 0 and a south gauge such that A j Sφ (π, φ) = 0.
Proof. For any given north gauge, |ψ N with Berry connections A j N φ , we know that |ψ N (θ = 0) is independent of φ by definition. Choose Then using Eq. (45), the new connection given by this gauge transform is The first line is zero by Eq. (37). Since χ = χ 1 + χ 2 = 0, we have not left the north gauge sector. The same construction can be used for the south gauge, so indeed we can always find north and south gauges such that Note that these particular gauges are the ones used to define A in the previous section. Lemma 1.1 gives a simple expression for the partial Chern number: For the traditional Chern number C, integer quantization follows from the fact that χ(φ) = χ(φ + 2π) + 2πn for integer n. The same condition applies here, only now it's χ 1 (φ) + χ 2 (φ) = χ 1 (φ + 2π) + χ 2 (φ + 2π) + 2πn, so that fractional values of C j are allowed. There is another way to write C j . Starting from Eq. (3) where the sphere has been split along a line of constant θ = θ c , and using the particular gauges of Eq. (49), we have where we applied Stokes's theorem since the Berry connections in the north and south gauges are smooth over their respective hemisphere. Since the lemma guarantees that the difference between the two Berry connections of different gauges is independent of θ, we may set θ = π (we could also choose θ = 0, the proof goes the same either way): where we used Eq. (39) in the third line. This is possibly the simplest expression for the Chern number, but it requires computing everything in a specifically defined gauge. We prefer to write an expression that is explicitly gauge-independent relating to the geometrical argument of the preceding section. We can do this by adding zero to the above expression in the form Now we have an expression that involves the difference between a Berry connection at two different angles. But from the lemma, we know that such a difference is independent of gauge. Therefore we again obtain in any gauge.

Examples
It's worthwhile checking some of the above equations in some simple examples.

Product state
Consider ther = 0 case and M i = 0 where the ground state is the product state. The north and south gauges are: In the article, we present the proof with A j N φ = − sin 2 (θ/2) implying that A j N φ = 0 at the north pole and A j N φ = −1 at the south pole. At the equator, we also find A j N φ = −1/2. In either case, the Berry curvature is F φθ dθdφ = sin θ 2 dθdφ, which gives partial Chern numbers of This equation naturally relates to the magnetizations at the poles since If we move to a different gauge |ψ = e iχ0(θ) e iχ1(φ) cos(θ/2) e i(χ1(φ)+φ) sin(θ/2) ⊗ e iχ2(φ) cos(θ/2) e i(χ2(φ)+φ) sin(θ/2) , in agreement with Eq. (45). Finally, at the poles, we have So that both Eq. (55) and Eq. (57) give C j = 1.

Fractional topology state
Now consider the case where the ground state evolves from the product state | ↑ 1 | ↑ 2 to the entangled state 1 . In this case, we don't know the form of the wavefunction over the whole sphere, but we can still check the poles. The gauge used in the article corresponds to a north gauge since the ground state is single-valued at the north pole while at the south pole it is given by so that Both Eqs. (55) and (57) then give C 1 = C 2 = 1/2. This leads to the physical interpretation of the fractional Chern number in Fig. 1 of the article if we use the arguments of the preceding sub-section. The system is topologically in a superposition of two geometries, one enclosing the topological charge and another geometry which is topologically trivial.
On the other hand, we can find a south gauge by multiplying |ψ N by e −iφ , i.e. χ = −φ. Now we have the additional gauge freedom of choosing the decomposition into χ 1 (φ) and χ 2 (φ). For example, suppose we choose χ 1 = −φ, χ 2 = 0. Then so that which of course gives C 1 = C 2 = 1/2. Note that A 1 Sφ (θ = π) + A 2 Sφ (θ = π) = 0 in accordance with Eq. (38), but this is not the particular south gauge that satisfies Eq. (49). To obtain that gauge we follow the construction in Eq. (46): where c is an arbitrary integration constant. With this transformation, the wavefunction becomes for which which satisfies both Eq. (49) and C 1 = C 2 = 1/2. This choice of gauge leads to the physical picture of Fig. 2 (top) in the article where the massive Dirac point carries the π Berry phase and the semimetal ring region participates in the entanglement entropy, and leads to a "zero" topological response. These examples have all used north or south gauges, but it's important to emphasize that Eq. (57) holds for any gauge. As a final example, consider starting with (66) and (67), and applying the transform is clearly not single-valued at any pole, but its Berry connections still give C 1 = C 2 = 1/2.

Generalized interactions
We now consider a general spin model with both in-plane and out-of-plane coupling: We anticipate that fractional invariants occur only along the line of symmetry M 1 = M 2 , and we set , focusing on sweeps along the φ = 0 meridian. This Hamiltonian admits a nice singlet-triplet representation: where  Table 1: Energies at the poles for each of the singlet and triplet states.
We choose the definition H, M > 0. Of the 16 possible transitions between these four states, six are ruled out by the singlet-triplet decoupling, and five are ruled out by energetics. The remaining five are Let's focus on the C j = 1/2 transition. At the north pole, this requires while at the south pole, this requires Generically then, we find the fractional Chern number in the region in agreement with the r xy = 0 case studied in the main text. We now study other special cases of this model.
• XY model: r z = 0. In this case, the condition for C j = 1/2 becomes We see that the XY model admits a fractional Chern number, but only for ferromagnetic in-plane coupling. This system was studied experimentally in Ref. [12] in the article for the case M = 0, which did not allow for the observation of a fractional Chern number. Without the inversion symmetrybreaking term, the fractional Chern phase collapses to a point at r xy = −H and therefore was not observed in that experiment.
• Heisenberg model: r z = r xy ≡ r. Condition (90) yields three conditions The last one is impossible, so we see that the Heisenberg model does not admit a fractional Chern number. However, anisotropy rescues the C j = 1/2 phase. For example, if we take r xy = r = −r z , then condition (90) becomes which is readily satisfied.
The topological phase diagram in the plane of the couplings is shown in Supplementary Fig. 1. For M < H, the fractional Chern number phase forms a wedge between the C j = 0 and C j = 1 phases, while for M > H, the C j = 1 phase is taken over by C j = 0.
In the main article, we found that we could extend the fractional Chern region by breaking inversion symmetry, either through the inclusion of a constant offset M = 0, or an asymmetric coupling r z → r z f (θ). If we do the same in the generic anisotropic model (with r xy → r xy g(θ)), we can also get an extended region of C j = 1/2. It is redundant to include both inversion-symmetry breaking mechanisms, so we set M = 0 in the following. In that case, the condition for a |1, 1 → |1, 0 transition relaxes to r xy g(π) < r z f (π) − H.
This unusual phase diagram is shown in Supplementary Fig. 2.

N spins
Here we discuss the fractional phase for more than two spins. Without specifying any details of the model, we know that the fractional Chern number arises when the ground state changes from a ferromagnet at the north pole to a degenerate antiferromagnet at the south pole. This can be achieved through inversionsymmetry-breaking masses or θ-dependent interactions as we have seen. The key observation in the case of two spins was that the presence of an infinitesimal transverse field breaks the ground-state degeneracy near the south pole and favours the entangled state. Specifically, in the subspace of south-pole ground states for the Ising-coupled model with M 1 = M 2 : D ≡ {| ↑↓ , | ↓↑ }, with energy E D = −r z = −r, we apply second-order degenerate perturbation theory where P ≡ α∈D |α α| is the projection operator on the south-pole ground-state subspace, In the Hilbert space formed by {| ↑↓ , | ↓↑ }, this yields the effective perturbation whose unique ground state 1 √ 2 (| ↑↓ + | ↓↑ ). In H eff , we take into account the two possible states | ↑↑ and | ↓↓ in (1 − P ) summing their two contributions.
We can generalize this reasoning to chains with more spins. For four spins, all antiferromagnetically coupled, the south-pole ground states are the two Néel states. The degeneracy is preserved at second order in the perturbation, but fourth order spin-flip terms will choose the ground state 1 √ 2 (| ↑↓↑↓ + | ↓↑↓↑ ). This is a maximally entangled state, and an analogue of the Greenberger-Horne-Zeilinger state which again has C j = 1/2 for all j. A chain of 2N spins requires 2N orders of perturbation theory to lift the degeneracy, so the gap near θ = π will be reduced as (H sin θ) 2N .
Let us now consider a different four-spin model corresponding to two Chern one-half systems with a weak transverse coupling. We would like to know if the fractional invariant is robust to this coupling. In the absence of the perturbative coupling, the ground state at the south pole is If we couple the two systems at just one site via a term H = r σ x 2 σ x 3 with r = r x as shown in Supplementary  Fig. 3 (a), then the second order perturbation is In Supplementary Fig. 3  . This is due to virtual excitations of the of the fully polarized state aligned with the magnetic field. In this case the fractional Chern number is destroyed and C j = {1, 0, 0, 1}. This is because fractional C j state is only protected by exchange symmetry of the spins. However, we can also construct a perturbative coupling that respects this symmetry, for example, by adding r σ x 1 σ x 4 as shown in Fig. 3 Upon diagonalization, this gives the unique ground state 1 √ 2 (| ↑↓↓↑ + | ↓↑↑↓ ), with C j = 1/2. Other fractional Chern numbers can appear if we introduce frustration in the system without breaking spin-exchange symmetry. For three antiferromagnetically coupled spins as shown in Fig. 3 (c) of the Supplementary Material, the ground state at θ = π is The transverse field −H sin θ 3 i=1 σ x i , yields the second-order perturbation Upon diagonalizing, the ground state is 1 . The corresponding partial Chern numbers are C j = 2 3 for all j. This reasoning can be applied to higher numbers of spin as well. For odd N > 1 spins, the N -fold degenerate space of frustrated antiferromagnets at the south pole is Each of these states can be written as |α where α indexes the site location of the ferromagnetic pair. At second order in perturbation theory, for N > 3, the transverse field has a diagonal contribution from flipping a single spin twice. The only off-diagonal contributions come from flipping one spin in the ferromagnetic pair along with its neighbour outside the pair. This is equivalent to shifting the pair by two sites: Thus the effective Hamiltonian describes a single particle hopping on a lattice. This is easily solved by Fourier transform, from which we get the ground state 1 √ N N α=1 |α . The corresponding spin expectation value for each site j is which gives the sequence of rational partial Chern numbers with integer total Chern number C tot = N +1 2 . Note that in the large N limit, the even and odd sectors converge to give C j = 1/2.

Monolayer and bilayer Haldane models on the honeycomb lattice
We employ the following definitions in our lattice model. We set the lattice spacing to a = 1. The honeycomb graphene Bravais lattice consists of A and B sites with primitive vectors nearest-neighbour vectors and next-nearest-neighbour vectors as shown in Supplementary Fig. 4.
The reciprocal lattice has a primitive cell defined by The diamond formed by v 1 , v 2 result in Fig. 2b) and 2c) in the article. Some important points in the Brillouin zone are as shown in Supplementary Fig. 5. For a single layer i, we start with the tight-binding Hamiltonian for graphene with nearest-neighbour hopping t 1 and Semenoff mass M i , which is given by To construct the Haldane model, we add next-nearest-neighbour hopping t 2 with flux φ oriented as in Supplementary Fig. 6, via the term Fourier transforming H 1 + H 2 gives the single-layer Hamiltonian where At the Dirac points K, K , we have Our bilayer model consists of two copies of the Haldane model for which we fix φ = π/2 for simplicity and couple them with an interlayer hopping r: This corresponds to Eq. (14) in the article. From this we get the energy spectrum at the K point, The eigenvectors corresponding to these bands are where We may obtain the energies at the K point through the transformation |d z | with −|d z | in Eqs. (129)-(132). In the bilayer model, we may represent the ground state at half-filling in terms of the occupancy of each layer: In this representation, a ket |ij n is defined for layer n with the two sublattices A and B such that |10 n refers to a state with sublattice A occupied in layer n and |01 n to a state with sublattice B occupied in layer n. The subset of the Hilbert space with each layer half-filled corresponds to the constraint i + j = 1.
We get the reduced density matrix ρ 1 by tracing out one layer: where the 2 × 2 block ρ red 1 describes the space of states where each layer is half-filled: From these coefficients, the entanglement entropy: is computed numerically.

Edge states
Here, we study the edge states in this bilayer model. We consider a finite ribbon geometry with 30 sites in the y-direction and 50-sites in the x-direction. Boundary conditions are open in y and periodic in x. We evaluate the band structure and wavefunctions of the edge modes. We confirm that the topological phase diagram (Fig. 1b in the main text) has a region with two chiral edge modes, a region with one chiral edge mode and a fully gapped phase as shown in Supplementary Fig. 7. Along the line of symmetry M 1 = M 2 , we find that that the bulk gap closes due to the nodal ring, but a single chiral edge mode persists in the reciprocal space as shown in Supplementary Fig. 8. By evaluating the probability density of the wavefunction, we find that these modes are indeed uniformly split between the two layers. Deviating very slightly from the critical line M 1 = M 2 , we find that the edge mode remains delocalized in the two planes and then progressively redistributes in one plane only when increasing the mass asymmetry related to the blue region of the phase diagram of Fig. 1b) in the article. In Supplementary Fig. 9, we also show the density of states coming from the nodal ring region in real space giving rise to additional delocalized modes for M 1 = M 2 .

Single spin-1/2 model in time
In this paper, we are interested in transition amplitudes of a two-state system at finite times, since the linear sweep protocol on a sphere takes place over a finite time. To that end, it is worth deriving the full time-dependent amplitudes for different states of the spin-1/2 Hamiltonian using Ref.
[46] as a guide. The instantaneous eigenenergies and eigenstates of this system are: It is important to note that these eigenstates change character as t goes from −∞ to +∞. Since E → ±λt as t → ±∞, we have In other words, if the evolution is adiabatic (i.e. we track the ground state as t increases), then the spin will necessarily flip. The Landau-Zener result says that if the evolution is not completely adiabatic (in a sense we will soon make precise), then there is a significant probability of ending up in the excited state where the spin has not flipped. Also note that exactly at t = 0, the eigenstates are equal combinations of up and down: We wish to solve the time-dependent Schrödinger equation. We represent the quantum state as so that we have two coupled differential equationṡ Differentiating the second equation gives Substituting Eqs. (149) and (150) into this gives We can put this differential equation in the form of the Weber equation 1 , by using the dimensionless quantity so that which has the linearly independent solutions Here we have defined ν ≡ i ∆ 2 2λ , and D ν (z) are the parabolic cylinder functions. From Eq. (150), we also get the solution for A: The initial condition ensures that the spin begins in the ground state at t = −∞, which means that B(t = −∞) = 0 according to Eq. (145). The second initial condition (|A(t = −∞)| 2 = 1) will not be used just yet.
One has to be careful with the asymptotics of the parabolic cylinder functions, since they have different behaviours depending the direction in which their argument goes to infinity. For t → −∞, Eq. (153) shows that arg(z) = 3π/4 and arg(−iz) = π/4. One can check that the first term in Eq. (155) diverges along the former axis, while the second term decays along the later. Thus c 1 = 0. We then have Using the identity this simplifies to Instead of solving for c 2 using the other initial condition, it is easier to use the probability normalization at time t = 0: where the parabolic cylinder functions take the analytic form D n (0) = 2 n/2 √ π Γ( 1−n 2 ) .
It is also useful to define the parameter in terms of which the coefficient becomes Using the asymptotic expansions for the parabolic cylinder function, one can show that the probability of a non-adiabatic transition (i.e. spin up at t = ∞) is e −πγ . This is the Landau-Zener result, and shows that γ is the appropriate adiabaticity parameter. For γ 1, the system remains in the ground state, while for γ 1 the system transitions to the excited state. For our purposes however, we are interested in the case t = 0 for which we make use of Eqs. (172). Supplementary Fig. 10 shows the time-dependence of the transition for small γ. It is important to note that changes to the distribution of probability only begin very close to t = 0.

Application to the two interacting spins
Even though our model, is a four-state system, we now show that the dynamics of this model are well captured by Eqs. (170), (171). We consider the symmetric case M 1 = M 2 = M < H. For simplicity, here it suffices to treat just the H + Hamiltonian. First, we write H + in the singlet-triplet basis with s = σ 1 + σ 2 . In this basis, we have H + = −rf (θ)|0, 0 0, 0| + H + trip .
Here |0, 0 refers to the singlet state and The singlet component is completely decoupled from the equations of motion, provided we initialize the spins in the ground state at the north pole, which is the triplet state |1, 1 forr < (H + M )/f (0); the ground state is |1, 0 otherwise. The total Chern number is encoded in the spin magnetization value according to To simplify the dynamics further we assume that any transitions to excited states occur near θ = π. This assumption is justified because for a broad class of interactions, the gap at θ = 0 is much larger than at θ = π. Near θ = π the |1, 1 state always has the highest energy, so we may project it out and write an effective two-state model to match the Landau-Zener model:          Figure 11: Numerically determined Chern number C j (solid) and reversed Chern number C j r (dashed) of a single spin vsrf (π)/H for different interactions with v = 0.05H; Θ refers to the Heaviside step function. The dashed black line shows the analytic approximation of C j r (Eq. (184)) which is universal for a given speed v.  Figure 12: Partial Chern numbers as a function of the couplingr measured in a five-spins quantum circuit simulation with nearest-neighbour Ising interactions and periodic boundary conditions. To time-evolve the spins (qubits), we use a Trotter decomposition with 800 time steps and sweep velocity v = 0.03H. The bias field for all qubits is fixed to M = 0.6H.