Magnon Orbital Nernst Effect in Honeycomb Antiferromagnets without Spin–Orbit Coupling

Recently, topological responses of magnons have emerged as a central theme in magnetism and spintronics. However, resulting Hall responses are typically weak and infrequent, since, according to present understanding, they arise from effective spin–orbit couplings, which are weaker compared to the exchange energy. Here, by investigating transport properties of magnon orbital moments, we predict that the magnon orbital Nernst effect is an intrinsic characteristic of the honeycomb antiferromagnet and therefore, it manifests even in the absence of spin–orbit coupling. For the electric detection, we propose an experimental scheme based on the magnetoelectric effect. Our results break the conventional wisdom that the Hall transport of magnons requires spin–orbit coupling by predicting the magnon orbital Nernst effect in a system without it, which leads us to envision that our work initiates the intensive search for various magnon Hall effects in generic magnetic systems with no reliance on spin–orbit coupling.


A model Hamiltonian of an antiferromagnet
In momentum space representation, our model Hamiltonian is where κ ± = (2K ± gµ B B/S)/J and f k = j e ik•a j .As discussed in Ref. 1 , the Hamiltonian is composed of two block diagonal matrices.Explicitly, we have and H II,k = (H I,k ) * .To diagonalize the Hamiltonian, we find the paraunitary matrix Following the notation used in Ref. 2 , we write the eigenvalue equations where is the Pauli matrix acting on the particle-hole space.Here where and κ = 2K/J.Also, the paraunitary matrix U k is given by and Note that the applied magnetic field only splits energy eigenvalues of the two magnonic states and does not change U k .

Calculation details of Berry curvature and magnon orbital Berry curvature
We start from the Berry connection of the Bogoliubov-de Gennes Hamiltonian 3,4 By inserting the identity operator I = m (σ 3 ) mm |u m,k ⟩⟨u m,k |σ 3 into Eq.(S7) and using we have From Ref. 2 , we read the expression of the magnon orbital Berry curvature where ∂k i is the velocity operator and j L z,y = 1 4 (v y σ 3 Lz + Lz σ 3 v y ) is the orbital current operator.By using the identity operator, we rewrite the interband matrix element in Eq. (S10) as Then, By utilizing calculation procedure in Ref. [5][6][7] , we obtain the matrix element of the magnon orbital moment

Induced polarization
Here, we develop a phenomenological model for the electric polarization induced by the magnon orbital Nernst effect, which is intended to provide the order-of-magnitude estimation, not quantitative predictions.To obtain the magnon orbital moment profile, we adopt the drift-diffusion formalism on a two-dimensional sample.For a temperature gradient ∂ x T , the spin-polarized magnon orbital moment profile is obtained by where τ is the magnon orbital relaxation time, ρ L s = ρ L α − ρ L β is the difference of magnon orbital moment density between two magnonic modes, and is spin-polarized magnon orbital current density with a corresponding orbital Nernst conduc- , and a diffusion coefficient D. By assuming the uniform temperature gradient and solving the bulk equation (S14) with the boundary conditions J L y (y = 0) = 0 and J L y (y = W ) = 0, where W is the width of the sample, we obtain the steady-state solution where λ = √ τ D is the magnon orbital diffusion length.Fig. S1(a) shows the exponential decay of the magnon orbital moment accumulation.
Using this, we compute the electric potential due to the magnon orbital accumulation in the following method.For the magnonic spin current, we can invoke to compute the induced polarization since the characteristic time scale of the magnon is generally much longer than that of the electron hopping process.The spin current carried by a single magnon is given by I s = −S(v/a)ẑ, which leads to the electric polarization where S = ±ℏ is the magnon spin and v = v e 12 is the magnon velocity.Then, a magnon with S = ℏ moving in a velocity v l ≡ v(r l ) at r l induces an electric dipole and the corresponding electric potential For a simple estimation, we consider that the sample consists of square unit cells with lattice constant a and assume that the magnons circulate along the lattice bonds of the each cell with a fixed speed [see Fig. S1(b)], for example, magnon revolves on k-th unit cell with a fixed speed v k .Fig. S1(c) shows schematic illustration of the resulting electric polarizations by Eq. (S19).We first calculate the electric potential of single square cell in the presence of the spin-polarized magnon orbital moment.If we consider a magnon as a point-like particle, the magnon's orbital moment is va/2 ẑ, where a/2 is the radius of orbital motion and v is the velocity.To caputure the position dependence of the magnon orbital distribution [Eq.(S16)] in the lattice model, we take cell averaged value of ρ L s (r) and write it as ρL s (r ′ k ) where r ′ k is position center of k-th cell.By replacing the magnon orbital moment va/2 to ρL s (r ′ k )a 2 , we obtain the electric potential due to the magnon orbital moment on k-th cell, where C k is the closed path along lattice bonds of the unit cell centered at r ′ k .Here we take the line average of the lattice bonds, because we assume the magnons circulate along the lattice bonds of the cell with fixed speed.The function f is resulted from the integration along the unit cell, i.e., f (x, y, z) = p,q=±1 Eq. ( S22) is in a form of the summation of the function f e (x, y, z) = shifted onto the each lattice site of the cell with the alternating sign.By summing up the local contributions from each cell [as given in Eq. (S21)] across the entire sample, we obtain the electric potential profile

Longitudinal spin current
Here we compute the longitudinal spin current with a sample length L in the case where the magnon spin current is blocked at the sample boundaries.In the presence of the temperature gradient the longitudinal spin current is given by where ρ s is the magnon accumulation and D s is the magnon diffusion constant.To solve the diffusion equation, we take an ansatz ρ s = Ae −x/lm + Be x/lm , where l m is the magnon are the left-and right-eigenvectors of the pseudo-Hermitian Hamiltonian σ 3 H k , respectively.The orthonormal relation of the eigenvectors reads ⟨u L n,k |u R m,k ⟩ = ⟨u R n,k |σ 3 |u R m,k ⟩ = (σ 3 ) nm and the pseudo-eigenvalue satisfies εn,k = (σ 3 ϵ k ) nn .The energy eigenvalues of the magnon bands are Figure S1: a Magnon orbital accumulation profile.b Schematic illustration of magnon current circulation (green and orange) and the polarization charge (red and blue) due to the magnon orbital accumulation.c Schematic illustration of the electric polarization.d Electric potential profile in the vicinity of y = 0 edge.Here, N denotes the number of cells along the y-direction, with a total length of 1 µm.e Schematic representation of the electric potential profile.

Figure S2 :
Fig.S2shows the potential profiles in the continuum limit for various values of magnon orbital relaxation time τ and magnon orbital diffusion length λ.

Figure S3 :
Figure S3: Longitudinal spin current profiles with different values of l m /L.