mf_notes.md

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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:

:::{math} 🏷️ hamiltonian \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 {eq}hamiltonian as a sum of bilinear terms weighted by the expectation values of the remaining operators: :::{math} 🏷️ mf_approx \hat{V} \approx \hat{V}{\text{MF}} \equiv \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: :::{math} 🏷️ density \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 {eq}mf_approx onto the tight-binding grid:

:::{math} 🏷️ mf_finite 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 {eq}mf_finite as:

:::{math} 🏷️ mf_infinite 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.