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/ These are the examples of ABACUS program.
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# Berry Phase Examples

This directory contains examples for performing Berry phase calculations in ABACUS. The Berry phase method is used to calculate polarization, orbital magnetization, and other topological properties of materials.

## What is Berry Phase?

The Berry phase is a geometric phase acquired by a quantum system as it evolves adiabatically around a closed path in parameter space. In condensed matter physics, it is used to calculate:

- **Polarization**: Electric polarization of crystals
- **Orbital magnetization**: Magnetization due to orbital motion of electrons
- **Topological invariants**: Such as Chern numbers and Z₂ invariants
- **Anomalous Hall effect**: Hall conductivity in topological materials
- **Piezoelectric response**: Relationship between stress and electric polarization

## Berry Phase Calculations in ABACUS

ABACUS implements the Berry phase method for calculating electric polarization. The calculation requires two steps:

1. **Self-consistent field (SCF) calculation**:
   - Set `calculation` to `scf`
   - Set `out_chg` to `1` to output charge density information

2. **Non-self-consistent field (NSCF) calculation**:
   - Set `calculation` to `nscf`
   - Set `init_chg` to `file` to read the charge density from the SCF calculation
   - Set `berry_phase` to `1` to enable Berry phase calculation
   - Set `gdir` to `1`, `2`, or `3` to specify the direction of polarization (x, y, or z)
   - For plane wave basis, set `pw_diag_thr` to a suitable value for better convergence

## Examples Included

### 1. 01_pw_PbTiO3
- **System**: Lead titanate (PbTiO3)
- **Basis**: Plane wave (PW)
- **Purpose**: Demonstrates Berry phase calculation for polarization using plane wave basis
- **Input Files**:
  - `INPUT-scf`: SCF calculation to generate charge density
  - `INPUT-nscf-a`: NSCF calculation for polarization along direction 1 (x)
  - `INPUT-nscf-b`: NSCF calculation for polarization along direction 2 (y)
  - `INPUT-nscf-c`: NSCF calculation for polarization along direction 3 (z)
  - Corresponding KPT files for each calculation

### 2. 02_lcao_PbTiO3
- **System**: Lead titanate (PbTiO3)
- **Basis**: LCAO (Linear Combination of Atomic Orbitals)
- **Purpose**: Demonstrates Berry phase calculation for polarization using LCAO basis
- **Input Files**:
  - `INPUT-scf`: SCF calculation to generate charge density
  - `INPUT-nscf-a`: NSCF calculation for polarization along direction 1 (x)
  - `INPUT-nscf-b`: NSCF calculation for polarization along direction 2 (y)
  - `INPUT-nscf-c`: NSCF calculation for polarization along direction 3 (z)
  - Corresponding KPT files for each calculation

## Key Parameters in INPUT

### For SCF Calculation:
- `calculation`: Set to `scf`
- `out_chg`: Set to `1` to output charge density
- `basis_type`: Set to `pw` or `lcao` depending on the basis set

### For NSCF Calculation:
- `calculation`: Set to `nscf`
- `init_chg`: Set to `file` to read charge density from SCF calculation
- `berry_phase`: Set to `1` to enable Berry phase calculation
- `gdir`: Set to `1`, `2`, or `3` to specify the polarization direction
- `pw_diag_thr`: Set to a small value (e.g., 1e-10) for plane wave basis

## How to Run

1. Navigate to the example directory:
   ```bash
   cd /abacus/examples/26_berryphase/01_pw_PbTiO3
   ```

2. Run the calculation using the provided script:
   ```bash
   bash run.sh
   ```

3. The script will automatically run the SCF calculation followed by the NSCF calculations for all three directions.

## Output Files

After running the calculation, you will find the following key output files in the `OUT.ABACUS` directory:

- `berryphase.out`: Contains the Berry phase and polarization results
- `running_scf.log`: Log file for the SCF calculation
- `running_nscf.log`: Log file for the NSCF calculation

## Interpreting Results

The polarization is calculated as:

 P = rac{e}{(2i)^3} um_{n,k} angle u_{n,k} | i 
abla_k | u_{n,k} 
angle 

where  u_{n,k}  are the periodic parts of the Bloch wavefunctions. The result is typically reported in units of  e/	ext{Å}^2  or  u	ext{C}/	ext{cm}^2 .

## Notes

- Berry phase calculations require a well-converged charge density, so ensure that the SCF calculation is properly converged
- For accurate results, use a sufficiently dense k-point mesh in both SCF and NSCF calculations
- The choice of basis set (PW or LCAO) affects the accuracy and computational cost of the calculation
- Polarization is a bulk property, so ensure that the unit cell is properly defined
- The examples use PbTiO₃, which is a ferroelectric material with spontaneous polarization
- For systems with symmetry, you can use symmetry operations to reduce the number of k-points needed