Add doc for spin data format (#3762)

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## Summary by CodeRabbit

- **Documentation**
- Updated documentation to clarify data format differences and detailed
file structures for TensorFlow and PyTorch/DP backends.
<!-- end of auto-generated comment: release notes by coderabbit.ai -->

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

doc/model/train-energy-spin.md

@@ -109,6 +109,57 @@ The options {ref}`start_pref_e <loss[ener_spin]/start_pref_e>`, {ref}`limit_pref
 
 If one does not want to train with virial, then he/she may set the virial prefactors {ref}`start_pref_v <loss[ener_spin]/start_pref_v>` and {ref}`limit_pref_v <loss[ener_spin]/limit_pref_v>` to 0.
 
-## Data preparation
+## Data format
 
-(Need a documentation for data format for TensorFlow and PyTorch/DP.)
+:::{note}
+Note that the spin data format is different between TensorFlow and PyTorch/DP.
+:::
+
+### Spin data format in TensorFlow
+
+In the TensorFlow backend, the spin system data format may contain the following files:
+
+```
+type.raw
+set.*/box.npy
+set.*/coord.npy
+set.*/energy.npy
+set.*/force.npy
+```
+
+This system contains `Nframes` frames with the same atom number `Natoms` and magnetic atom number `Nspins`, the total number of element and virtual types contained in all frames is `Ntypes`. The `box` and `energy` files are the same as those in [standard formats](../data/system.md). The `type` file contains the types of both real atoms and virtual atoms. In `coord` and `force` files, virtual atomic coordinates are integrated with real atomic coordinates, and magnetic forces are combined with atomic forces. Specifically, magnetic forces are obtained from [DeltaSpin](https://github.com/caizefeng/DeltaSpin) and virtual atomic coordinates are given by:
+
+$$\bm{R}_{i^p} = \bm{R}_i + \frac{\eta_{\zeta_i}}{\mu_{\vert \bm{S}_i \vert}} \cdot \bm{S}_i$$
+
+where $\bm{R}_{i^p}$, $\bm{R}_i$, and $\bm{S}_i$ denote the virtual atomic coordinate, atomic coordinate and spin, respectively. $\eta_{\zeta_i}$ and $\mu_{\vert \bm{S}_i \vert}$ correspond to the `virtual_len` and `spin_norm` defined in [spin settings](#spin-settings-in-tensorflow).
+
+We list the details about spin system data format in TensorFlow backend:
+
+| ID     | Property                   | Raw file   | Unit | Shape                             | Description                                                                                                                                               |
+| ------ | -------------------------- | ---------- | ---- | --------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| type   | Atom type indexes          | type.raw   | \    | Natoms + Nspins                   | Integers that start with 0. The first `Natoms` entries represent real atom types, followed by `Nspins` entries representing virtual atom types.           |
+| coord  | Coordinates                | coord.raw  | Å    | Nframes \* (Natoms + Nspins) \* 3 | The first `3 \* Natoms` columns represent the coordinates of real atoms, followed by `3 \* Nspins` columns representing the coordinates of virtual atoms. |
+| box    | Boxes                      | box.raw    | Å    | Nframes \* 3 \* 3                 | in the order `XX XY XZ YX YY YZ ZX ZY ZZ`                                                                                                                 |
+| energy | Frame energies             | energy.raw | eV   | Nframes                           |
+| force  | Atomic and magnetic forces | force.raw  | eV/Å | Nframes \* (Natoms + Nspins) \* 3 | The first `3 \* Natoms` columns represent atomic forces, followed by `3 \* Nspins` columns representing magnetic forces.                                  |
+
+### Spin data format in PyTorch/DP
+
+In the PyTorch backend, spin and magnetic forces are listed in seperate files, and the data format may contain the following files:
+
+```
+type.raw
+set.*/box.npy
+set.*/coord.npy
+set.*/spin.npy
+set.*/energy.npy
+set.*/force.npy
+set.*/force_mag.npy
+```
+
+This system contains `Nframes` frames with the same atom number `Natoms`, the total number of element contained in all frames is `Ntypes`. Most files are the same as those in [standard formats](../data/system.md), here we only list the distinct ones:
+
+| ID             | Property         | Raw file      | Unit    | Shape                  | Description                                                         |
+| -------------- | ---------------- | ------------- | ------- | ---------------------- | ------------------------------------------------------------------- |
+| spin           | Magnetic moments | spin.raw      | $\mu_B$ | Nframes \* Natoms \* 3 | Spin for magnetic atoms and zero for non-magnetic atoms.            |
+| magnetic force | Magnetic forces  | force_mag.raw | eV/Å    | Nframes \* Natoms \* 3 | Magnetic forces for magnetic atoms and zero for non-magnetic atoms. |