pythtb.TBModel.quantum_geometric_tensor#

TBModel.quantum_geometric_tensor(k_pts, occ_idxs=None, plane=None, *, cartesian=False, non_abelian=False, param_periods=None, diff_scheme='central', diff_order=2, use_tensorflow=False, **params)[source]#

Quantum geometric tensor at a list of k-points via Kubo formula.

The quantum geometric tensor is computed from the derivatives of the Bloch Hamiltonian \(\partial_\mu H_k\), where \(\mu\) is the direction in k-space, and is given by (when non_abelian=True):

\[Q_{\mu \nu;\ mn}(k) = \sum_{l \notin \text{occ}} \frac{ \langle u_{mk} | \partial_{\mu} H_k | u_{lk} \rangle \langle u_{lk} | \partial_{\nu} H_k | u_{nk} \rangle }{ (E_{mk} - E_{lk})(E_{nk} - E_{lk}) }\]

The Abelian quantum geometric tensor (when non_abelian=False) is obtained by taking the trace over occupied bands:

\[Q_{\mu \nu}(k) = \sum_{m \in \text{occ}} Q_{\mu \nu;\ mm}(k)\]

By specifying the plane parameter, we choose a particular \((\mu, \nu)\) pair of the quantum geometric tensor to return.

Added in version 2.0.0.

Parameters:

k_pts(Nk, dim_k) array-like: Array of k-points with shape (Nk, dim_k), where Nk is the number of points and dim_k is the dimensionality of the k-space.
occ_idxs1D array, optional: Indices of the occupied bands. Defaults to the first half of the states.
planetuple of int, optional: Tuple of two integers specifying the plane in k-space for which to compute the curvature. If None (default), computes all components of the Berry curvature tensor. This will affect the shape of the returned array.
cartesianbool, optional: If True, computes the velocity operator in Cartesian coordinates. Default is False (reduced coordinates).
non_abelianbool, optional: If True, returns the full tensor (non-abelian case). If False, returns the band-trace of the tensor (abelian case). Default is False.
param_periodsdict[str, float], optional: Optional map {param_name: period} for swept parameters. When supplied, assumes the parameter is cyclic and trims any duplicated endpoints, or endpoints equal to the start plus the given period, before building finite-difference stencils. This can improve numerical accuracy by using centered differences throughout the parameter range. Otherwise, the function will use a backward difference at the endpoint and a forward difference at the start.
diff_schemestr, optional: Finite difference scheme to use for parameter derivatives. Options are “central” (default) or “forward”. This parameter is only relevant when passing varying parameters.
diff_orderint, optional: Order of accuracy for finite difference lambda derivatives. Must be an even integer for “central” scheme (default is 4), and a positive integer for “forward” scheme. This parameter is only relevant when passing varying parameters.
use_tensorflow: bool, optional: If True, will use TensorFlow to speed up linear algebra routines. Requires TensorFlow to be installed. Default is False.
**params: Keyword arguments mapping parameter names to value(s). Each value can be a scalar or a 1D array of values. If any values are array-like, the QGT is evaluated at all combinations of parameter values, and the final array is stacked with the k-axis leading, followed by each parameter axis in the order of given parameter names.

Returns:

Qarray: Quantum geometric tensor at the specified k-points. If plane is None, shape is (dim_k, dim_k, Nk, n_orb, n_orb). If plane is a tuple, shape is (Nk, n_orb, n_orb) and the returned tensor is restricted to the specified directions. If non_abelian=False, returns the band-trace of the quantum geometric tensor and the last two axes are not present. If parameter sweeps are performed via params, parameter axes are added after the k-point axis.