Hubbard model | EPFL Graph Search

The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring atoms, while the other pushes it away from its neighbors. Its Hamiltonian thus has two terms: a kinetic term allowing for tunneling ("hopping") of particles between lattice sites and a potential term reflecting on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, in which case the model is referred to as the "Bose–Hubbard model". The Hubbard model is a useful approximation for particles in a periodic potential at sufficiently low temperatures, where all the particles may be assumed to be in the lowest Bloch band, and long-range interactions between the particles can be ignored. If interactions between partic

Novel materials and algorithms for quantum technologies

Francesco Libbi

The enormous advancements in the ability to detect and manipulate single quantum states have lead to the emerging field of quantum technologies. Among these, quantum computation is the most far-reaching and challenging, aiming to solve problems that the classic computers could never address because of the exponential scaling, while quantum sensing exploits the ability to address single quantum states to realize ultra-sensitive and precise detectors. Defect centers in semiconductors play a primary role in these fields. The possibility to store information in the spin of their ground state, manipulate it through microwaves, and read it optically allows to use them as qubits. In addition, the very sharp dependence of their properties on temperature, strain and magnetic fields makes them very promising quantum sensors. In this Thesis we aim at contributing to the progress of quantum technologies both at the hardware and software level. From a hardware point of view, we study a key property of defect centers in semiconductors, the phonon-assisted luminescence, which can be measured to perform the readout of the information stored in a quantum bit, or to detect temperature variations. We predict the luminescence and study the exciton-phonon couplings within a rigorous many-body perturbation theory framework,an analysis that has never been performed for defect centers.In particular, we study the optical emission of the negatively-charged boron vacancy in 2D hexagonal boron nitride, which currently stands out among defect centers in 2D materials thanks to its promise for applications in quantum information and quantum sensing. We show that phonons are responsible for the observed luminescence, which otherwise would be dark due to symmetry. We also show that the symmetry breaking induced by the static Jahn-Teller effect is not able to describe the presence of the experimentally observed peak at 1.5 eV.The knowledge of the coupling between electrons and phonons is fundamental for the accurate prediction of all the features of the photoluminescence spectrum. However, the large number of atoms in a defect supercell hinders the possibility use density functional perturbation theory to study this coupling. In this work we present a finite-differences technique to calculate the electron-phonon matrix elements, which exploits the symmetries of the defect in such a way to use the very same set of displacement needed for the calculation of phonons. The computation of electron-phonon coupling thus becomes a simple post-processing of the finite-differences phonons calculation. On the quantum software side, we propose an improved quantum algorithm to calculate the Green's function through real-time propagation, and use it to compute the retarded Green's function for the 2-, 3- and 4-site Hubbard models. This novel protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the 2-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the

\ket{N\pm 1}

states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.

EPFL2022

Correlations and renormalization of the electron-phonon coupling in the honeycomb Hubbard ladder and superconductivity in polyacene

Ronny Thomale

We have performed extensive density matrix renormalization group (DMRG) studies of the Hubbard model on a honeycomb ladder. The band structure (with Hubbard U = 0) exhibits an unusual quadratic band touching at half-filling, which is associated with a quantum Lifshitz transition from a band insulator to a metal. For one electron per site, nonzero U drives the system into an insulating state in which there is no pair-binding between added electrons; this implies that superconductivity driven directly by the repulsive electron-electron interactions is unlikely in the regime of small doping, x < 1. However, the divergent density of states as x -> 0, the large values of the phonon frequencies, and an unusual correlation induced enhancement of the electron-phonon coupling imply that lightly doped polyacenes, which approximately realize this structure, are good candidates for high-temperature electron-phonon driven superconductivity.

Amer Physical Soc2013

Band structure effects on the superconductivity in Hubbard models

Ronny Thomale

We study the influence of the band structure on the symmetry and superconducting transition temperature in the (solvable) weak-coupling limit of the repulsive Hubbard model. Among other results we find that (1) as a function of increasing nematicity, starting from the square-lattice (zero nematicity) limit where a nodal d-wave state is strongly preferred, there is a smooth evolution to the quasi-1D limit, where a striking near-degeneracy is found between a p-wave- and a d-wave-type paired states with accidental nodes on the quasi-one-dimensional Fermi surfaces-a situation that may be relevant to the Bechgaard salts. (2) In a bilayer system, we find a phase transition as a function of increasing bilayer coupling from a d-wave to an s(+/-)-wave state reminiscent of the iron-based superconductors. (3) When an antinodal gap is produced by charge-density-wave order, not only is the pairing scale reduced, but the symmetry of the pairs switches from d(x2-y2) to d(xy); in the context of the cuprates, this suggests that were the pseudogap entirely due to a competing CDW order, this would likely cause a corresponding symmetry change of the superconducting order (which is not seen in experiment).

Amer Physical Soc2013

Novel materials and algorithms for quantum technologies

Francesco Libbi

\ket{N\pm 1}

states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.

EPFL2022