Real-Space Fields

A real-space field combines two kinds of structure: its argument has a spatial shape, and its value may itself transform. Treating these separately gives one construction that covers scalar fields, ordinary spatial-vector fields, and spin-vector-valued fields such as magnetization density.

Fields With Transforming Values

Let a field take values in a finite-dimensional space \(V\):

\[ f:\mathbb R^3\longrightarrow V. \]

Choose a basis of \(V\) and write its components as \(f^a(\boldsymbol r)\). Under a spatial rotation \(R\), a spin rotation \(S\), and a translation \(\boldsymbol t\), the general active action is

\[ {f'}^{a'}(\boldsymbol r) = \sum_a \rho_V(R,S)^{a'}{}_a f^a\!\left(R^{-1}(\boldsymbol r-\boldsymbol t)\right). \]

The inverse rotation inside the argument transforms the spatial shape. The matrix \(\rho_V(R,S)\) transforms the field value. Which of \(R\) and \(S\) appears in that matrix depends on what kind of value the field carries.

Choose a center \(\boldsymbol R_i\) and define the local field \(f_i^a(\boldsymbol x)=f^a(\boldsymbol R_i+\boldsymbol x)\). When the center moves with the system to \(\boldsymbol R'_{\pi(i)}=R\boldsymbol R_i+\boldsymbol t\), translation cancels from the relative coordinate:

\[ {f'}_{\pi(i)}^{a'}(\boldsymbol x) = \sum_a \rho_V(R,S)^{a'}{}_a f_i^a(R^{-1}\boldsymbol x). \]

At fixed radius, expand the angular dependence in real spherical harmonics,

\[ f_i^a(r,\widehat{\boldsymbol r}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{i\ell m}^a(r) Y_{\ell m}(\widehat{\boldsymbol r}). \]

The centered coefficients then transform as

\[ {c'}_{\pi(i)\ell m'}^{a'}(r) = \sum_{m,a} \mathcal D^\ell_{m'm}(R) \rho_V(R,S)^{a'}{}_a c_{i\ell m}^a(r). \]

This equation is the common starting point. The spherical index \(m\) always describes the spatial shape around the center; the value index \(a\) retains its own physical origin.

The remaining radial dependence can be expanded as \(c_{i\ell m}^a(r)=\sum_n c_{in\ell m}^aR_{in\ell}(r)\). The radial label \(n\) adds multiplicity without changing either the spatial or value transformation type.

Scalar Fields

For a scalar field, \(V=V_0\) and \(\rho_V=1\). Charge density, a scalar potential, and any individual scalar coefficient field follow this case. The coefficients simply obey

\[ {c'}_{\pi(i)\ell m'}(r) = \sum_m \mathcal D^\ell_{m'm}(R) c_{i\ell m}(r). \]

Thus each fixed center, radius, and \(\ell\) gives one spatial irrep \(V_\ell\). Angular degrees \(\ell>0\) describe anisotropy in the spatial shape; they do not turn the field value itself into a vector or tensor.

Degree-One Values

Both a spatial Cartesian vector and a spin vector carry a degree-one rotation representation. In the real spherical-harmonic convention used by ELFES, the order \(m=(-1,0,1)\) corresponds to Cartesian components \((y,z,x)\). The fixed conversion from \((x,y,z)\) is

\[ T^{1\leftarrow\mathrm{cart}} = \begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&0 \end{pmatrix}, \qquad \mathcal D^1(A) = T^{1\leftarrow\mathrm{cart}} A \left(T^{1\leftarrow\mathrm{cart}}\right)^{-1} \]

for any \(A\in\mathrm{SO}(3)\). This is a change of component basis. It can be applied to a spatial vector with \(A=R\) or to a spin vector with \(A=S\); using the same numerical conversion does not identify the two vector spaces.

Spatial-Vector Fields

An ordinary spatial-vector field \(v^i(\boldsymbol r)\) takes values in \(V_1^{\mathrm{space}}\). Under a proper spatial rotation,

\[ {v'}^i(\boldsymbol r) = \sum_j R^i{}_j v^j\!\left(R^{-1}(\boldsymbol r-\boldsymbol t)\right). \]

After angular expansion and Cartesian-to-spherical conversion of the value index, the coefficients carry \(V_\ell^{\mathrm{space}}\otimes V_1^{\mathrm{space}}\). The same rotation \(R\) acts on both factors, so the product decomposes as

\[ V_\ell\otimes V_1 \cong \bigoplus_{L=|\ell-1|}^{\ell+1}V_L. \]

If \(c_{i\ell m_1m_2}(r)\) denotes the coefficients after converting the Cartesian value index to \(m_2=-1,0,1\), their coupled components are

\[ c_{i\ell LM}(r) = \sum_{m_1,m_2} C^{LM}_{\ell m_1,1m_2} c_{i\ell m_1m_2}(r). \]

Each fixed parent degree \(\ell\) and output degree \(L\) then transforms as one \(V_L\). The retained \(\ell\) distinguishes different coupling paths that produce the same output type.

Spin-Vector-Valued Fields

Magnetization density is naturally a field \(m^\alpha(\boldsymbol r)\) whose argument lies in real space and whose value lies in spin space. Without spin–orbit coupling its transformation law is

\[ {m'}^{\alpha'}(\boldsymbol r) = \sum_\alpha S^{\alpha'}{}_\alpha m^\alpha\!\left(R^{-1}(\boldsymbol r-\boldsymbol t)\right). \]

Its angular coefficients therefore transform by \(\mathcal D^\ell(R)\otimes\mathcal D^1(S)\) and have product-group type

\[ (\ell,1) = V_\ell^{\mathrm{space}} \otimes V_1^{\mathrm{spin}}. \]

Because \(R\) and \(S\) are independent, the spatial-shape index and the spin-vector index must remain uncoupled. They are not the two inputs of an ordinary Clebsch–Gordan decomposition.

With spin–orbit coupling, or whenever the action is deliberately restricted to joint rotations, set \(S=R\). The same coefficient array then carries \(V_\ell\otimes V_1\) under a single rotation group and may be coupled to \(L=\ell-1,\ell,\ell+1\) by the preceding CG map. The algebra is identical to the spatial-vector case only after this restriction; its two indices still have different physical origins.

For a spinful local density, the charge component \(n(\boldsymbol r)\) is a spin scalar and the magnetization components \(m^\alpha(\boldsymbol r)\) form a spin vector. At each spatial degree \(\ell\), the Pauli decomposition therefore produces \((\ell,0)\oplus(\ell,1)\) under independent rotations, or the corresponding total-\(L\) blocks under joint rotations.

Parity

Improper spatial transformations add parity without changing the proper-rotation distinction above. Write \(Q=P^\epsilon R\) with \(P=-I\). The angular shape of degree \(\ell\) contributes \((-1)^\ell\). A scalar value has parity \(+1\), a polar spatial-vector value has parity \(-1\), and an axial value has parity \(+1\).

Consequently, scalar fields produce types \((\ell,(-1)^\ell)\). Coupled spatial-vector fields produce parity \((-1)^\ell p_v\), where \(p_v=-1\) for polar values and \(p_v=+1\) for axial values. A spin vector is axial under spatial inversion, so its product-group type \((\ell,1)\) carries spatial parity \((-1)^\ell\); the same parity is retained after joint coupling.

From Fields to Irrep Features

The construction has one continuous-to-discrete step and, when allowed by the group action, one coupling step:

\[ f^a(\boldsymbol r) \longrightarrow c_{i\ell m}^a(r) \longrightarrow c_{in\ell m}^a \longrightarrow \text{coupled irrep components}. \]

Centering removes the global translation, spherical harmonics resolve the spatial shape, and the radial basis adds multiplicity. Whether the final value index can be coupled with the spatial degree is decided by the symmetry group: spatial values already share the spatial rotation, while spin values share it only under the joint action.