Official Resources
Overview
TRACK (TRAnsport properties for Correlated materials using Kubo formalism) is a Python 3 code designed to calculate temperature-dependent transport coefficients in solids. It utilizes the linear-response Kubo formalism to compute electrical conductivity, thermal conductivity, Seebeck coefficient, and the Lorenz number. A key feature of TRACK is its careful handling of current operators in interacting systems, making it suitable for Hamiltonians derived from Hartree-Fock or hybrid functionals where non-local potentials affect the velocity operator.
Scientific domain: Correlated Electrons, Thermoelectrics, Transport Theory
Target user community: Theorists working on transport in complex oxides and strongly correlated metals
Theoretical Methods
- Kubo Formalism: Calculation of the current-current correlation function $\Pi(\omega)$ in the DC limit.
- Interactions: Correct implementation of the velocity operator $\mathbf{v} = \frac{i}{\hbar} [H, \mathbf{r}]$ for non-local potentials (e.g., Fock exchange).
- Integration: Tetrahedron method or dense mesh integration over the Brillouin Zone.
- Scattering: Constant relaxation time approximation ($\tau$) or energy-dependent scattering rates.
Capabilities
- Coefficients:
- Electrical Conductivity Tensor ($\sigma_{\alpha\beta}$).
- Electronic Thermal Conductivity ($\kappa_{e}$).
- Seebeck Coefficient ($S$).
- Lorenz Number ($L = \kappa_e / (T \sigma)$).
- Analysis:
- Temperature dependence of transport ($T$-scans).
- Band-by-band decomposition of currents.
- Optical conductivity (AC limit).
Key Strengths
- Correlations: Specifically addresses the "Peierls substitution failure" in non-local Hamiltonians, ensuring gauge-invariant transport results for correlated models.
- Pythonic: Easy to inspect and modify, leveraging NumPy for tensor operations.
- Thermoelectrics: Direct calculation of power factors and efficiency metrics.
Inputs & Outputs
- Inputs:
- Eigenvalues and Eigenvectors (from DFT or TB).
- Velocity matrix elements (critical for interacting parts).
- k-mesh definitions.
- Outputs:
- Text files containing $\sigma(T)$, $S(T)$, $\kappa(T)$.
Interfaces & Ecosystem
- Hamiltonians: Can interface with output from tight-binding codes or DFT codes (if matrix elements are provided).
- Ecosystem: Relies on standard Python scientific stack (NumPy, SciPy).
Performance Characteristics
- Speed: Python overhead is minimal for dense matrix operations; bottleneck is the number of k-points and bands.
- Parallelism: Easy to parallelize over temperature or k-points (multiprocessing).
Comparison with Other Codes
- vs. BoltzTrap: BoltzTrap uses semi-classical Boltzmann theory (group velocities); TRACK uses the fully quantum mechanical Kubo formula, which captures interband transitions (optical conductivity) and can treat scattering more rigorously.
- vs. LinReTraCe: Similar scope (Kubo); TRACK has a specific emphasis on the velocity operator distinctions in interacting systems.
Application Areas
- Bad Metals: Violation of the Wiedemann-Franz law in correlated systems.
- Thermoelectrics: High-throughput screening of $S$ and $\sigma$.
- Optical Response: Drude weight and interband optical transitions.
Community and Support
- Development: Drexel University / University of Pennsylvania (R. J. M. Venderbos).
- Source: GitHub.
Verification & Sources