# Theoretical Background: Chiral Phase Transitions via Phonon Irreducible Representations

This document provides the theoretical foundation for identifying chiral phase transitions through phonon modes of specific irreducible representations.

## 1. Crystallographic Chirality

### 1.1 Definition of Chirality in Crystals

A crystal structure is **chiral** if it cannot be superimposed on its mirror image by any combination of rotations and translations. Chirality in crystals arises when the atomic arrangement breaks all improper symmetry operations (mirror planes, inversion centers, glide planes, and rotoinversion axes).

### 1.2 Sohncke Space Groups

The **Sohncke space groups** are the 65 space groups that contain only **proper** symmetry operations (rotations and screw axes). These are the only space groups that can support chiral crystal structures.

**Mathematical criterion**: A space group is Sohncke if and only if all its rotation matrices have determinant +1:
$$\det(R) = +1 \quad \forall R \in G$$

Any space group containing at least one operation with $\det(R) = -1$ (improper operation) cannot host a chiral structure.

### 1.3 Classification of Sohncke Groups

The 65 Sohncke groups are classified into two classes:

| Class | Description | Count | Example |
|-------|-------------|-------|---------|
| **Class II** | Enantiomorphous pairs - always chiral | 11 pairs (22 groups) | P4₁ (76) ↔ P4₃ (78) |
| **Class III** | Chiral-supporting - chirality depends on atomic positions | 43 groups | P1 (1), P2 (3) |

**Class II (Enantiomorphous pairs)**: These space groups form 11 pairs where each member is the mirror image of the other. Any structure crystallizing in these groups is inherently chiral regardless of atomic positions.

**Class III (Chiral-supporting)**: These space groups can host chiral structures, but chirality depends on the specific atomic arrangement. A structure may be achiral if atoms lie on special positions.

### 1.4 The 11 Enantiomorphous Pairs

| Pair | Space Groups | Crystal System |
|------|--------------|----------------|
| 1 | P4₁ (76) ↔ P4₃ (78) | Tetragonal |
| 2 | P4₁22 (91) ↔ P4₃22 (95) | Tetragonal |
| 3 | P4₁2₁2 (92) ↔ P4₃2₁2 (96) | Tetragonal |
| 4 | P3₁ (144) ↔ P3₂ (145) | Trigonal |
| 5 | P3₁2 (151) ↔ P3₂2 (153) | Trigonal |
| 6 | P3₁21 (152) ↔ P3₂21 (154) | Trigonal |
| 7 | P6₁ (169) ↔ P6₅ (170) | Hexagonal |
| 8 | P6₂ (171) ↔ P6₄ (172) | Hexagonal |
| 9 | P6₁22 (178) ↔ P6₅22 (179) | Hexagonal |
| 10 | P6₂22 (180) ↔ P6₄22 (181) | Hexagonal |
| 11 | P4₃32 (212) ↔ P4₁32 (213) | Cubic |

---

## 2. Group Theory of Displacive Phase Transitions

### 2.1 Landau Theory Framework

In a displacive phase transition, the high-symmetry parent phase transforms to a lower-symmetry daughter phase through atomic displacements. According to **Landau theory**, the transition is characterized by:

1. **Order parameter**: A quantity $\eta$ that is zero in the parent phase and non-zero in the daughter phase
2. **Free energy expansion**: $F(\eta) = F_0 + \frac{1}{2}a(T-T_c)\eta^2 + \frac{1}{4}b\eta^4 + ...$
3. **Symmetry breaking**: The order parameter transforms according to an irreducible representation (irrep) of the parent space group

### 2.2 The Little Group and Star of **k**

For a phonon at wavevector **k**, the **little group** (or little co-group) $G_{\mathbf{k}}$ consists of all space group operations $\{R|\mathbf{t}\}$ that leave **k** invariant modulo a reciprocal lattice vector:

$$R\mathbf{k} = \mathbf{k} + \mathbf{G}$$

where $\mathbf{G}$ is a reciprocal lattice vector.

The **star of k** is the set of all distinct wavevectors generated by applying the full space group to **k**:

$$\{R_i\mathbf{k}\} \text{ for } R_i \in G$$

### 2.3 Small Representations

The phonon eigenvectors at wavevector **k** form a **small representation** (or little group representation) of $G_{\mathbf{k}}$. The small representation matrices $D^{(\mathbf{k},\nu)}(g)$ satisfy:

$$D^{(\mathbf{k},\nu)}(g_1) D^{(\mathbf{k},\nu)}(g_2) = D^{(\mathbf{k},\nu)}(g_1 g_2) \quad \forall g_1, g_2 \in G_{\mathbf{k}}$$

where $\nu$ labels the irreducible representation.

### 2.4 Character Table

For each irrep, the **character table** gives the trace of each representation matrix:

$$\chi^{(\mathbf{k},\nu)}(g) = \text{Tr}\left[D^{(\mathbf{k},\nu)}(g)\right]$$

The character table is the fingerprint used to identify which irrep a phonon mode belongs to.

---

## 3. Isotropy Subgroups

### 3.1 Definition

The **isotropy subgroup** $I_{\eta}$ of an order parameter $\eta$ is the subgroup of the parent space group $G$ that leaves the order parameter invariant:

$$I_{\eta} = \{g \in G : D(g)\eta = \eta\}$$

where $D(g)$ is the matrix representation acting on the order parameter.

### 3.2 Kernel vs. Image Subgroup

- **Kernel**: The subgroup that leaves ALL directions in the irrep space invariant (acts as identity). This is the highest-symmetry isotropy subgroup.
- **Image subgroup**: The subgroup that leaves a SPECIFIC order parameter direction invariant.

### 3.3 Order Parameter Directions (OPD)

The **order parameter direction** is a specific vector in the irrep space that determines which isotropy subgroup is obtained. For an $n$-dimensional irrep, the OPD is written as:

$$\vec{\eta} = (\eta_1, \eta_2, ..., \eta_n)$$

Common notation uses symbols like $(a,0,0)$, $(a,a,0)$, $(a,b,0)$ to denote the pattern of components:
- $(a,0,0)$: One free parameter, direction along axis 1
- $(a,a,0)$: One free parameter, equal components along axes 1 and 2
- $(a,b,0)$: Two free parameters

### 3.4 Maximal Isotropy Subgroups

The **maximal isotropy subgroups** are the highest-order subgroups that can be obtained for a given irrep. They correspond to the "directions" of maximum symmetry breaking. These can be systematically enumerated using the **Chain Rule** and **Subgroup Criterion**.

### 3.5 Complex Order Parameter Directions for Screw Axes

Standard real-valued OPDs like $(a, 0)$ or $(a, a)$ are sufficient for many transitions. However, identifying transitions to Sohncke groups that only contain screw axes (like $P4_1$ or $P4_3$) from parents with $4_2$ screw axes at the zone boundary (e.g., $P4_2/m$ or $P4_2/n$ at $Z=(0,0,1/2)$) requires **complex OPDs**.

For a 2D irrep, the following complex OPDs are tested:
- $(a, ia) / \sqrt{2}$: Represents a helical displacement pattern with $a + i a$ phase shift.
- $(a, -ia) / \sqrt{2}$: Represents the opposite helical handedness.

These complex directions correspond to the two members of an enantiomorphous pair (Class II). For example, at the $Z$ point of SG 84 ($P4_2/m$):
- OPD $(a, 0)$ preserves $C_2$ rotation $\rightarrow$ Daughter $P2$ (#3).
- OPD $(a, ia)$ preserves $4_1$ screw axis $\rightarrow$ Daughter $P4_1$ (#76).
- OPD $(a, -ia)$ preserves $4_3$ screw axis $\rightarrow$ Daughter $P4_3$ (#78).

Without these complex directions, the transition to the 4-fold screw axis daughter would be missed by simplified search algorithms.

---

## 4. Domain Analysis

### 4.1 Domain Multiplicity

When a crystal undergoes a symmetry-breaking transition, multiple equivalent orientations (domains) of the daughter phase can form. The number of domains is:

$$n_{\text{domains}} = \frac{|G_{\text{parent}}|}{|G_{\text{daughter}}|}$$

where $|G|$ denotes the order (number of symmetry operations) of the space group.

### 4.2 Types of Domains

1. **Orientation domains**: Related by lost rotation operations
2. **Translation domains (antiphase domains)**: Related by lost translation operations
3. **Enantiomeric domains**: Related by lost improper operations (mirror, inversion)

### 4.3 Enantiomeric Domain Pairs

For chiral transitions, the most important domains are **enantiomeric pairs** - left-handed and right-handed versions of the structure. Each lost improper operation creates an enantiomeric domain pair:

- Loss of inversion center: Creates 2 enantiomeric domains
- Loss of mirror plane: Creates 2 enantiomeric domains per mirror

The enantiomeric domains are:
- Related by the lost improper operation
- Cannot interconvert without breaking and reforming bonds
- Have identical physical properties except for optical rotation

---

## 5. Computational Strategy

### 5.1 Algorithm Overview

```
For each parent space group (non-Sohncke):
    For each special q-point:
        For each irrep at that q-point:
            1. Get small representation matrices
            2. Enumerate maximal isotropy subgroups
            3. For each isotropy subgroup + OPD:
                a. Identify daughter space group
                b. Check if daughter is Sohncke
                c. If Sohncke: record chiral transition
```

### 5.2 Key Computational Tools

| Task | Library | Function |
|------|---------|----------|
| Space group symmetry | spglib | `get_symmetry_from_database()` |
| Irrep tables | irrep + irreptables | `SpaceGroupIrreps`, `IrrepTable` |
| Small representations | spgrep | `get_spacegroup_irreps()` |
| Isotropy subgroups | spgrep-modulation | `IsotropyEnumerator` |
| Daughter space group | spglib | `get_spacegroup_type_from_symmetry()` |

### 5.3 Sohncke Detection Algorithm

```python
def is_sohncke(spg_number):
    """Check if space group is Sohncke using spglib database."""
    for hall_number in range(1, 531):
        sg_type = spglib.get_spacegroup_type(hall_number)
        if sg_type.number == spg_number:
            sym = spglib.get_symmetry_from_database(hall_number)
            # Check if all operations are proper (det > 0)
            for rotation in sym['rotations']:
                if np.linalg.det(rotation) < 0:
                    return False  # Contains improper operation
            return True  # All operations are proper
```

---

## 6. Example: Rutile Structure (P4₂/mnm, SG 136)

### 6.1 Parent Space Group Properties

- Space group: P4₂/mnm (No. 136)
- Crystal system: Tetragonal
- Point group: 4/mmm
- Contains: Inversion, mirrors, glides
- Not Sohncke → Can undergo chiral transition

### 6.2 Special q-points

| Label | Coordinates | Little Group |
|-------|-------------|--------------|
| Γ (GM) | (0, 0, 0) | Full group (4/mmm) |
| Z | (0, 0, ½) | 4/mmm (zone boundary) |
| X | (½, 0, 0) | mmm |
| M | (½, ½, 0) | mmm |
| A | (½, ½, ½) | mmm |

### 6.3 Example Chiral Transition

**Parent**: P4₂/mnm (136) → **Daughter**: P4₁ (76)

- **q-point**: Z = (0, 0, ½)
- **Irrep**: Z₄ (2-dimensional)
- **Order parameter direction**: (a, 0) or (0, a)
- **Domain multiplicity**: |G₁₃₆| / |G₇₆| = 16 / 4 = 4
- **Lost operations**: Inversion, 2 mirrors, 2 glides
- **Enantiomeric partner**: P4₃ (78)

The two enantiomeric daughters P4₁ and P4₃ correspond to OPD directions (a, 0) and (0, a) respectively.

---

## 7. Mathematical Details

### 7.1 Representation Theory Summary

For a little group $G_{\mathbf{k}}$ of order $g$, the irreps satisfy:

$$\sum_{\nu} d_\nu^2 = g$$

where $d_\nu$ is the dimension of irrep $\nu$.

### 7.2 Orthogonality Relations

For two irreps $\mu$ and $\nu$:

$$\frac{1}{g}\sum_{g \in G_k} D^{(\mathbf{k},\mu)}(g)^*_{ij} D^{(\mathbf{k},\nu)}(g)_{i'j'} = \frac{1}{d_\mu}\delta_{\mu\nu}\delta_{ii'}\delta_{jj'}$$

### 7.3 Projection Formula

To find the irrep decomposition of a phonon mode, use the **projection operator**:

$$P^{(\mathbf{k},\nu)}_{ij} = \frac{d_\nu}{g}\sum_{g \in G_k} D^{(\mathbf{k},\nu)}(g)^*_{ij} D(g)$$

### 7.4 Isotropy Subgroup Enumeration

The isotropy subgroups are found by solving for $\vec{\eta}$ such that:

$$(D(g) - I)\vec{\eta} = 0 \quad \forall g \in I_{\eta}$$

This is equivalent to finding the null space of $(D(g) - I)$ for each subgroup.

---

## 8. References

1. **Sohncke Groups**: International Tables for Crystallography, Volume A
2. **Landau Theory**: Landau, L.D. & Lifshitz, E.M., Statistical Physics
3. **Isotropy Subgroups**: Stokes, H.T. & Hatch, D.M., Isotropy Subgroups of the 230 Crystallographic Space Groups
4. **BCS Irrep Tables**: Bilbao Crystallographic Server, https://www.cryst.ehu.es/
5. **spgrep Documentation**: https://phonopy.github.io/spgrep/
6. **spgrep-modulation**: https://phonopy.github.io/spgrep-modulation/

---

## 9. Glossary

| Term | Definition |
|------|------------|
| Chirality | Property of an object that cannot be superimposed on its mirror image |
| Enantiomorph | One of a pair of mirror-image chiral structures |
| Improper operation | Symmetry operation with det < 1 (inversion, mirror, glide, rotoinversion) |
| Irreducible representation (irrep) | A representation that cannot be decomposed into smaller representations |
| Isotropy subgroup | Subgroup leaving a specific order parameter invariant |
| Little group | Subgroup of space group leaving a wavevector invariant |
| Order parameter | Quantity characterizing the degree of symmetry breaking |
| Small representation | Representation of the little group at a wavevector |
| Sohncke group | Space group containing only proper operations |
| Star of k | Set of all wavevectors equivalent to k by space group symmetry |
