This repository contains code for calculating the Levy-Lieb density functional,
- qiskit
- qiskit_nature
- qiskit_algorithms
- sklearn
- numpy
- matplotlib
The Fermi-Hubbard Model is a lattice model in which electrons experience an interaction potential, on-site potential and can tunnel between lattice sites. Each site has two spin orbitals, corresponding to up and down. Therefore, each site can possess a maximum of two electrons. In this project, we consider the two site Fermi-Hubbard lattice whose hamiltonian is as follows:
Where:
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$\hat{c}_{i,\sigma}^\dagger$ and$\hat{c} _{i, \sigma}$ are the creation and annihilation operators for electrons with spin$\sigma$ at site$i$ . -
$\hat{n}_{i,\sigma}$ is the number operator for electrons with spin$\sigma$ at site$i$ . -
$t$ is the tunnelling amplitude between adjacent sites. -
$U$ is the interaction term between two electrons occupying the same site. -
$v_{i}$ is the on-site potential energy at site$i$
We will refer to the tunnelling term in the Hamiltonian as
Now, we may introduce the Levy-Lieb density functional:
We will, therefore, calculate
where the parametrised state,
where
This process also yields the Levy-Lieb embedding,
The ground state energy can be calculated as:
Therefore, we can use a classical optimiser to minimise
In this repository, we will also investigate the Levy-lieb kernel defined in Ref. 1 as:
We will pre-compute the Levy-Lieb kernel matrix using states
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C. D. Pemmaraju and A. Deshmukh, “Levy-lieb embedding of density-functional theory and its quantum kernel: Illustration for the hubbard dimer using near-term quantum algorithms,” Physical Review A, vol. 106, no. 4, 2022. doi:10.1103/physreva.106.042807.
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D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, “The Hubbard Dimer: A density functional case study of a many-body problem,” Journal of Physics: Condensed Matter, vol. 27, no. 39, p. 393001, Sep. 2015. doi:10.1088/0953-8984/27/39/393001