Superconductivity and mean field distribution theory on a Hubbard model with local symmetries

Kainth, Manjinder ORCID: 0000-0002-5906-3888 (2020). Superconductivity and mean field distribution theory on a Hubbard model with local symmetries. University of Birmingham. Ph.D.

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Abstract

At the core of thesis is the following particular Hubbard model
\[
H = -t_1\sum_{\langle ij \rangle \sigma} (t^{\dagger}_{i\sigma} + b^{\dagger}_{i\sigma}) (t_{j\sigma} + b_{j\sigma}) - t_0 \sum_{i\sigma} (t^{\dagger}_{i\sigma}b_{i\sigma} + b^{\dagger}_{i\sigma}t_{i\sigma})
+ U \sum_{i} (t^{\dagger}_{i\uparrow}t_{i\uparrow} t^{\dagger}_{i\downarrow}t_{i\downarrow} + b^{\dagger}_{i\uparrow}b_{i\uparrow} b^{\dagger}_{i\downarrow}b_{i\downarrow}),
\]
for which there are two large investigations.
In both of these a particular symmetry plays a central role: explicitly the set of local transformations \(t^{\dagger}_{i\sigma} \leftrightarrow b^{\dagger}_{i\sigma}\), one for each site \(i\) in the system.
As is with any local symmetry of a Hamiltonian, the basis Hilbert space must be divided into symmetry sectors \(\mathbb{Z}_2\) i.e. symmetric or anti-symmetric under the transformation.
We exclusively examine systems for which all basis states are either symmetric or anti-symmetric and only consider mixed systems as an afterthought.
The first investigation concerns an effectively exact examination of superconductivity in the limit \(U = \infty\).
Pairing is shown with binding energy and correlation length calculated exactly.
We find that phase diagram of many unconventional superconductors is qualitatively recreated, and in particularly we find a ferromagnetic phase coexisting with a superconducting phase.
The limit of \(U = \infty\) is lifted perturbatively which gives rise to an anti-ferromagnetic phase at the Mott point.
The second investigation is a new approach for dealing with strongly correlated problems: we create a non-orthogonal basis which is self consistently solved for.
With this we can calculate particle dispersions, hole dispersions, and the occupation factor.
All three are compared to exact diagonalisation and show great agreement.