Theory overview

Interacting problems

In physics, one often encounters problems where a system of multiple particles interacts with each other. In this package, we consider a general electronic system with density-density interparticle interaction:

(1)\[\hat{H} = \hat{H_0} + \hat{V} = \sum_{ij} h_{ij} c^\dagger_{i} c_{j} + \frac{1}{2} \sum_{ij} v_{ij} c_i^\dagger c_j^\dagger c_j c_i\]

where \(c_i^\dagger\) and \(c_i\) are the creation and annihilation operators respectively for fermion in state \(i\). The first term \(\hat{H_0}\) is the non-interacting Hamiltonian which by itself is straightforward to solve on a single-particle basis by direct diagonalizations made easy through packages such as kwant. The second term \(\hat{V}\) is the density-density interaction term between two particles, for example Coulomb interaction. To solve the interacting problem exactly, one needs to diagonalize the full Hamiltonian \(\hat{H}\) in the many-particle basis which grows exponentially with the number of particles. Such a task is often infeasible for large systems and one needs to resort to approximations.

Mean-field approximation

The first-order perturbative approximation to the interacting Hamiltonian is the Hartree-Fock approximation also known as the mean-field approximation. The mean field approximates the quartic term \(\hat{V}\) in (1) as a sum of bilinear terms weighted by the expectation values of the remaining operators:

(2)\[\hat{V} \approx \hat{V}_{\text{MF}} \equiv \frac12 \sum_{ij} v_{ij} \left[ \braket{c_i^\dagger c_i} c_j^\dagger c_j - \braket{c_i^\dagger c_j} c_j^\dagger c_i \right],\]

where we neglect the constant offset terms and the superconducting pairing (for now). The expectation value terms \(\langle c_i^\dagger c_j \rangle\) are due to the ground state density matrix and act as an effective field on the system. The ground state density matrix reads:

(3)\[\rho_{ij} \equiv \braket{c_i^\dagger c_j } = \text{Tr}\left(e^{-\beta \left(\hat{H_0} + \hat{V}_{\text{MF}} - \mu \hat{N} \right)} c_i^\dagger c_j\right),\]

where \(\beta = 1/ (k_B T)\) is the inverse temperature, \(\mu\) is the chemical potential, and \(\hat{N} = \sum_i c_i^\dagger c_i\) is the number operator. Currently, we neglect thermal effects so \(\beta \to \infty\).

Finite tight-binding grid

To simplify the mean-field Hamiltonian, we assume a finite, normalised, orthogonal tight-binding grid defined by the single-particle basis states:

\[ \ket{n} = c^\dagger_n\ket{\text{vac}} \]

where \(\ket{\text{vac}}\) is the vacuum state. We project our mean-field interaction in (2) onto the tight-binding grid:

(4)\[V_{\text{MF}, nm} = \braket{n | \hat{V}_{\text{MF}} | m} = \sum_{i} \rho_{ii} v_{in} \delta_{nm} - \rho_{mn} v_{mn},\]

where \(\delta_{nm}\) is the Kronecker delta function.

Infinite tight-binding grid

In the limit of a translationally invariant system, the index \(n\) that labels the basis states partitions into two independent variables: the unit cell internal degrees of freedom (spin, orbital, sublattice, etc.) and the position of the unit cell \(R_n\):

\[ n \to n, R_n. \]

Because of the translational invariance, the physical properties of the system are independent of the absolute unit cell position \(R_n\) but rather depend on the relative position between the two unit cells \(R_{nm} = R_n - R_m\):

\[ \rho_{mn} \to \rho_{mn}(R_{mn}). \]

That allows us to re-write the mean-field interaction in (4) as:

(5)\[V_{\text{MF}, nm} (R) = \sum_{i} \rho_{ii} (0) v_{in} (0) \delta_{nm} \delta(R) - \rho_{mn}(R) v_{mn}(R),\]

where now indices \(i, n, m\) label the internal degrees of freedom of the unit cell and \(R\) is the relative position between the two unit cells in terms of the lattice vectors.