What is "staggered magnetization"?

Question

I find that the term "staggered magnetization" appears frequently, but often the authors do not give a precise and universal definition (it seems that everyone knows what this term mean?). In 1D spin-chain I think it is

$$ M_\text{stag} = m_1 - m_2 + m_3 - m_4 + \cdots $$

where $m_i$ is the magnetization for the $i$th spin. In 2D it seems that it is a checkerboard-like pattern: on a (say $4\times 4$ square lattice labelled by $(i,j)$ (row and column), it is

$$ \begin{align*} M_\text{stag} &= + \ m_{1,1} - m_{1,2} + m_{1,3} - m_{1,4} \\ &\quad - m_{2,1} + m_{2,2} - m_{2,3} + m_{2,4} \\ &\quad + m_{3,1} - m_{3,2} + m_{3,3} - m_{3,4} \\ &\quad - m_{4,1} + m_{4,2} - m_{4,3} + m_{4,4} \end{align*} $$

Is my understanding correct? How to generalize it to other 2D lattices (triangular, honeycomb etc) and higher dimensional systems? i.e. how is "staggered magnetization" defined exactly (in equations, not in words)?

Answer 1

Although mathematics (and therefore equations) is indeed the language of physics, equations only are hardly understandable without definitions in words. For this reason, let me start with words, leaving equations at the end.

The presence of a staggered magnetization in a lattice means the crystal may be partitioned into two or more interpenetrating sublattices, each with its own ordered magnetic moment. It is a phenomenon present in antiferromagnetism and ferrimagnetism. In the former case, the sublattices magnetic moments sum to zero, while in the latter, a residual total magnetic moment remains.

The definition in words of staggered magnetization is important to understand why in the simplest cases (1D lattice or square lattice), it can be mathematically expressed as a sum of the magnetic moment at each site with an alternating sign and what could be a more general definition. It should be clear that the main requirement is to use the site magnetic moments on the sublattices to build a quantity that is not zero when antiferromagnetic order is present.

Insisting on the sublattices is important. For example, in a rectangular lattice in 2D, we could have an antiferromagnetic order made by opposite first neighbor magnetic moments but aligned moments in the direction of the longer side of the rectangles. In such a case, the alternation of the sign should be only along the direction of the shorter sides.

Having understood this point, the general recipe should be clear. A general definition of staggered magnetization, valid for any lattice in any dimension, is a sum of the average magnetization per site of all the sublattices, each suitably rotated to align all in the same direction.

If one likes a formula, the staggered magnetization could be defined as proportional to the modulus of the vector $$ {\bf M}_s=\sum_{\alpha,{\bf R}_{\alpha}} {\cal R}_{\alpha}{\bf m}_{{\bf R}_{\alpha}} $$ where the index $\alpha$ selects one sublattice, ${\bf m}_{{\bf R}_{\alpha}}$ represents the magnetic moment at each lattice position ${\bf R}_{\alpha}$ of the $\alpha$-th sublattice, and ${\cal R}_{\alpha}$ represents a different rotation matrix for each sublattice.

In the simplest case of two alternating magnetic moments, ${\cal R}_{\alpha}$ reduces to $\pm 1$. For more complex patterns in 2D or 3D, one has to use a representation of the relevant rotations.