Spatial and Spin Symmetries

Electronic structure depends on both real-space coordinates and electronic spin. These degrees of freedom are often described using the same Cartesian labels \(x,y,z\), but they belong to different spaces and need not rotate together. The relation between their rotations is determined by the physics: without spin–orbit coupling they are independent, while spin–orbit coupling retains only their joint action.

This chapter establishes that structure before applying it to real-space fields or LCAO operators. Proper rotations carry the main argument. Inversion and time reversal are added at the end because they have different roles.

Two Rotation Spaces

Let \(R\in\mathrm{SO}(3)_{\mathrm{space}}\) rotate spatial coordinates. It acts on positions, atomic centers, orbital angular dependence, and ordinary spatial tensors. Spatial Cartesian indices will be written as \(i,j\in\{x,y,z\}\).

Let \(S\in\mathrm{SO}(3)_{\mathrm{spin}}\) be a global spin rotation. It rotates the spin degree of freedom without moving the atoms or the spatial arguments of a function. Spin-vector indices will be written as \(\alpha,\beta\in\{x,y,z\}\).

The two groups are isomorphic, but their actions are different. A spatial vector \(v^i\) and a spin vector \(m^\alpha\) both have three components, yet under a general pair \((R,S)\) they transform as

\[ {v'}^i=R^i{}_jv^j, \qquad {m'}^\alpha=S^\alpha{}_\beta m^\beta. \]

The distinction lies in which group acts, not in the number or names of the components.

Spinors and Pauli Components

A spin-\(\tfrac12\) state transforms under \(\mathrm{SU}(2)\). For every proper spin rotation \(S\) there are two lifts \(\pm U_{\frac12}(S)\in\mathrm{SU}(2)\), and the two signs represent the same rotation of spin observables.

Consider an operator on the two-dimensional spin space. Its Pauli decomposition is

\[ X=X^0\sigma_0+\sum_{\alpha=x,y,z}X^\alpha\sigma_\alpha. \]

Under \(X'=U_{\frac12}(S)XU_{\frac12}(S)^\dagger\), the scalar component \(X^0\) is unchanged and the three-vector \(X^\alpha\) transforms by \(S\):

\[ {X'}^{\alpha'}=\sum_\alpha S^{\alpha'}{}_{\alpha}X^\alpha. \]

Thus the operator space decomposes under spin rotations as a scalar plus a vector, \(V_0^{\mathrm{spin}}\oplus V_1^{\mathrm{spin}}\). Spinor states still require the double-valued \(\mathrm{SU}(2)\) action; Pauli components carry the induced integer-degree \(\mathrm{SO}(3)_{\mathrm{spin}}\) action because the sign of the spinor lift cancels in conjugation.

Independent Rotations Without Spin–Orbit Coupling

At the level of spatial tensors and Pauli components, a Hamiltonian with no spin–orbit coupling, fixed external magnetic field, or other interaction that ties spin to the spatial frame has rotation structure

\[ G_0 = \mathrm{SO}(3)_{\mathrm{space}} \times \mathrm{SO}(3)_{\mathrm{spin}}. \]

A group element is a pair \((R,S)\). A pure spatial rotation \((R,I)\) moves the geometry and orbital shape while leaving the spin components unchanged. A pure global spin rotation \((I,S)\) rotates every spin together while leaving the spatial geometry fixed. These actions commute.

An object with spatial angular degree \(\ell\) and integer spin-tensor degree \(s\) belongs to

\[ V_\ell^{\mathrm{space}}\otimes V_s^{\mathrm{spin}}, \]

which is conventionally labelled \((\ell,s)\) as an irreducible representation of the product group. In particular, a spatial degree-\(\ell\) object carrying one Pauli-vector index has type \((\ell,1)\).

This tensor product is not the angular-momentum coupling of two representations of one rotation group. Its action is \(\mathcal D^\ell(R)\otimes\mathcal D^s(S)\) with independently chosen \(R\) and \(S\), so there is no Clebsch–Gordan decomposition between the two factors.

Joint Rotations With Spin–Orbit Coupling

Spin–orbit coupling links spin orientation to spatial orientation. The familiar central-field form \(H_{\mathrm{SOC}}\propto\boldsymbol L\mathbin{\cdot}\boldsymbol s\) makes this visible: under independent rotations it becomes \((R\boldsymbol L)\mathbin{\cdot}(S\boldsymbol s)\), which is unchanged for arbitrary states only when \(S=R\).

At the level of Pauli components and other integer-degree tensors, the surviving proper-rotation action is therefore the diagonal subgroup

\[ G_{\mathrm{joint}} = \left\{(R,R):R\in\mathrm{SO}(3)\right\} \simeq \mathrm{SO}(3). \]

Spinor states carry the corresponding \(\mathrm{SU}(2)\) double-cover action, as before.

Space and spin have not become the same vector space. Rather, a physical rotation now acts on both through the same group element. Once the product-group representation \((\ell,s)\) is restricted to this joint subgroup, ordinary angular-momentum coupling applies:

\[ \left. \left(V_\ell^{\mathrm{space}}\otimes V_s^{\mathrm{spin}}\right) \right|_{\mathrm{joint}} \cong \bigoplus_{J=|\ell-s|}^{\ell+s}V_J. \]

For a Pauli vector, \(s=1\), so \(\ell\otimes1\) produces \(J=\ell-1,\ell,\ell+1\), with only \(J=1\) when \(\ell=0\). The Clebsch–Gordan coefficients implement this change from separate spatial and spin components to components of total angular degree \(J\).

The joint subgroup is also a valid subgroup when spin–orbit coupling is absent. Using only that subgroup gives a correct but weaker equivariance: it omits the additional transformations in which space and spin rotate independently.

Transformation Laws and Symmetries of Particular Objects

An equivariant transformation law describes how a family of physical objects is carried into another member of the same family. It does not imply that each object is unchanged. For example, without spin–orbit coupling, a global spin rotation maps a magnetic solution to a degenerate rotated solution; a generic magnetized solution is not itself invariant under that rotation.

Likewise, rotating a geometry and all of its electronic quantities gives a transformed sample even when the original geometry has no nontrivial rotational symmetry. The symmetry group of a particular crystal or molecule is the subgroup that leaves that particular object unchanged, whereas the rotation group above governs covariance across rotated objects.

Inversion and Time Reversal

For an improper spatial transformation \(Q\in\mathrm{O}(3)\), a polar spatial vector transforms by \(Q\) and an axial spatial vector by \(\det(Q)Q\). Spin is axial. Writing \(Q=P^\epsilon R\) with \(P=-I\) and \(R\in\mathrm{SO}(3)\), its joint spin action is therefore \(S=\det(Q)Q=R\). Spatial inversion changes the parity of orbital factors but does not reverse a spin vector.

At the Pauli-component level, the independent no-SOC rotation group can consequently be extended to \(\mathrm{O}(3)_{\mathrm{space}}\times\mathrm{SO}(3)_{\mathrm{spin}}\), while the joint action embeds \(Q\) as \((Q,\det(Q)Q)\).

Time reversal is different from both spatial inversion and spin rotation. It is antiunitary, reverses spin, and acts by complex conjugation as well as a spin-space matrix. Reality and time-reversal constraints are therefore separate from rotational equivariance.

Consequence for Electronic-Structure Data

The same Pauli-component organization can describe data with or without spin–orbit coupling. What changes is the group action assigned to the spatial and spin indices:

without spin–orbit coupling, keep the product-group type \((\ell,0)\) or \((\ell,1)\);
with spin–orbit coupling, restrict to joint rotations and, when useful, couple the spatial and spin factors into total-\(J\) irreps.

The component labels \(0,x,y,z\) identify the spin scalar and spin vector. They do not by themselves determine whether the physical symmetry is independent or joint.