1 Introduction
An electron has both charge and spin properties. The active manipulation of spin degrees of freedom in solidstate systems is known as spintronics
Žutić et al. (2004); Gomonay and Loktev (2014). Most of researches have been focused on ferromagnets (FMs) for GHzfrequency magnetization dynamics in the past decades and the domain wall velocity can reach . In early days, antiferromagnets (AFMs), however, are thought to be less effective for spintronic manipulations since they are robust against the magnetic perturbation and produce no stray field Baltz et al. (2018). In fact, it was latter realized that the robustness of AFMs with respect to the magnetic perturbation makes them better candidates for spintronic applications due to the high stability of domain wall structures. In addition, one striking feature in AFMs is the femtosecond magnetization dynamics due to the antiferromagnetic exchange coupling, which has been employed to generate THzfrequency magnetization dynamics Puliafito et al. (2019); Ivanov (2014); SanchesTejerina et al. (2019), and boosts the domain wall velocity in AFMs to Gomonay et al. (2016a, b); Shiino et al. (2016). Moreover, roomtemperature antiferromagnetic order has been found over a broad range of materials, such as metal, semiconductor, and insulator, which can be used in spintronics devices such as racetrack memories, memristors, and sensors SanchesTejerina et al. (2019); Baltz et al. (2018); Wadley et al. (2016); Fiebig et al. (2008).From the modeling perspective, magnetization is the basic quantity of interest. In ferromagnetic materials, its dynamics is modeled by the LandauLifshitzGilbert (LLG) equation Landau and Lifshitz (1935); Gilbert (1955) with the property that where is the saturation magnetization. In general, an AFM contains magnetic sublattices and the magnetization with . Most common AFMs, such as FeMn and NiO, have two sublattices and , associated with two magnetization fields and satisfying . For each sublattice, its magnetization satisfies a LLG equation with an antiferromagnetic exchange term which couples these two equations. In the physics community, by introducing another order parameter , reduced models for AFMs are developed for describing magnetization dynamics Ivanov (2014); Baltz et al. (2018).
In this work, we focus on numerical methods for AFMs and ferrimagnets described by the coupled system of LLG equations. By ferrimagnets such as FeO, we mean . A large volume of methods have been proposed for FMs; see Kruzik and Prohl (2006); GarcíaCervera (2007); Cimrák (2008) for reviews and references therein. For the temporal discretization, there are explicit schemesFrançois and Pascal (2006); Romeo et al. (2008), implicit schemesH and N (2004); Bartels and Andreas (2006); Fuwa et al. (2012), and semiimplicit schemesWang et al. (2001); Li et al. (2019); E and Wang (2000); Chen et al. (2019); Cimrák (2005); Xie et al. (2019). However, there is no work on numerical methods for AFMs in the literature.
At a first glance, numerical methods for FMs can be directly applied to AFMs and ferrimagnets with minor modifications. However, pros and cons of different methods for FMs may not be directly transferred. For example, explicit methods require subpicosecond stepsize in micromagnetics simulation. For AFMs or ferrimagnets, the stepsize becomes subfemtosecond for explicit methods while the time scale of interest is of nanoseconds. The underlying reason is that the antiferromagnetic exchange term poses an characteristic time scale of femtoseconds on magnetization dynamics. Even unconditionally stable implicit and semiimplicit schemes have to resolve the magnetization dynamics at femtosecond scales in order to capture the correct physics. Implicit schemes solve a nonlinear system of equations at each step. From FMs to AFMs, the dimension of the nonlinear system is doubled with possibly more solutions (locally stable magnetic structures). In semiimplicit schemes, the nonlinear structure of the coupled system does not bring any difficulty in an explicit way and only the computational complexity is doubled. Therefore, semiimplicit methods provide the best compromise between stability and efficiency. Gaussseidel projection methods (GSPMs) Wang et al. (2001); Li et al. (2019) are of our first choice since only linear systems of equations with constant coefficients needs to be solved at each step. As shown in the paper, the coupled system of LLG equations can be solved by GSPMs with the computational complexity doubled at each step. Due to the antiferromagnetic exchange, the stepsize in GSPMs is femtosecond as expected.
The rest of this paper is organized as follows. We introduce the model for AFMs and ferrimagnets in Section 2. The corresponding GaussSeidel projection methods are described in Section 3. Their accuracy with respect to temporal and spatial stepsizes is validated in Section 4. Femtosecond magnetization dynamics, Néel wall structures, and phase transition in presence of a magnetic field are simulated for AFMs in Section 5. Conclusions are drawn in Section 6.
2 Model for antiferromagnets: A coupled system of LandauLifshitzGilbert equations
Consider a bipartite collinear magnetic material occupied by . Its magnetization structure can be a ferromagnetic phase (fig. 1(a)), a ferrimagnetic phase (fig. 1(b)), or an antiferromagnetic phase (fig. 1(c)). The dyanmics of the twosublattices system is modeled by a coupled system of phenomenological LLG equations Baltz et al. (2018)
(1) 
is the dimensionless damping parameter, is the gyromagnetic ratio, and is the magnetic permeability of vacuum. For each sublattice , we have magnetization with . For ferrimagnets, is different for and . with being the magnetic energy density of an antiferromagnetic or ferrimagnetic system including magnetic anisotropy, ferromagnetic exchange, antiferromagnetic exchange, and Zeeman energy (external magnetic field) Puliafito et al. (2019); SanchesTejerina et al. (2019)
(2) 
(1) can be viewed as two sets of LLG equations for and with the antiferromagnetic exchange coupling in (2).
Details of the four terms in (2) are described as follows:

Anisotropy energy: Magnetization usually favors an easyaxis direction of the form
where is a smooth function. Suppose the easyaxis direction is the xaxis for a uniaxial material, the total anisotropy energy of two sublattices is
where is the material parameter.

Ferromagnetic exchange: Magnetization of each sublattice experiences a ferromagnetic exchange energy of the form
where presents the exchange constant of the material.

Antiferromagnetic exchange: Magnetization of sublattice A and sublattice B favors alignment along an antiparallel direction, thus the exchange energy is of the form
(3) where is the antiferromagnet exchange parameter and is the atomic lattice constant. For a positive , the system favors a ferromagnetic state. For a negative , however, an antiferromagnetic state is preferred.

Zeeman energy: In the presence of an external magnetic filed , the interaction energy is of the form
Thus, the free energy of an antiferromagnetic or ferrimagnetic material is explicitly written as
(4) 
The system of LLG equations (1) can be rewritten equivalently as
(5) 
and the effective fields are
with and .
To ease the description, we now nondimensionlize (5). Defining , , with the diameter of , and , we have
(6) 
where , , and . Upon rescaling time , (5) can be rewritten as
(7) 
where
(8)  
(9) 
Homogeneous Neumann boundary conditions are used
(10) 
where and
is the unit outward normal vector along
. It’s worth mentioning that the above model is also used for ferrimagnetic materials with one of magnetization, .It is easy to check from (7) that the following statement is true.
Lemma 1.
For , we have
The antiferromagnetic exchange (3) plays an important role in AFMs. Consider the case when and the system (7) reduces to
Combining the above two equations with , we get an equation of Bernoulli type
(11) 
Lemma 2.
, if the antiferromagnetic exchange parameter , the system favors the antiferromagnetic state, and if , the system favors the ferromagnetic state.
Proof.
Therefore, when , , we have , . When , , we have , .
The definition of yields
As a consequence, and for and , respectively. ∎
3 GaussSeidel projection methods for antiferromagnetics
In this section, we introduce three GaussSeidel projection methods for (7). The finite difference method is used for spatial discretization with unknowns in 1D and in 3D, where , , and and represent the number of segments for each direction.
3.1 Original GaussSeidel Projection Method
This is a direct generalization of the original GSPM Wang et al. (2001) to the antiferromagnetic or ferrimagnetic case. For (7), the GSPM works as follows. Define the vector field for the splitting procedure:
where and . The GSPM solves (7) in three steps:

Implicit GaussSeidel:

Heat flow without constraints:

Projection onto :
Note that within the definition . is for AFMs and is for ferrimagnetics.
3.2 Scheme A
Both Scheme A and Scheme B are based on improved GSPMs for ferromagnetics Li et al. (2019). In Scheme A, we do not treat the gyromagnetic term and the damping term separately. Instead, the implicit GaussSeidel method is applied to the gyromagnetic term and the damping term simultaneously, and a projection step follows up.

Implicit GaussSeidel step:

Projection step:
3.3 Scheme B
Scheme B reduces the computational cost further by the introduction of two sets of approximations. At each step, one set of solution is updated in the implicit GaussSeidel step and the other is updated in the projection step.

Implicit GaussSeidel step:

Projection step:
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