The Fermi-Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. With a simplistic form of the Hamiltonian, the model can potentially provide explanation to interesting phenomena in correlated electrons. The interest of our current work is the model in one dimension. The one-dimensional Fermi-Hubbard model has been extensively studied and there exist analytical results for determining its ground-state energy in the thermodynamic limit. We want to find the ground-state from the perspective of quantum computing. In particular, we employ the protocol of quantum annealing and perform high performance simulations of the same using the Juelich Quantum Computer Simulator (JUQCS) for systems with upto 40 qubits. We learn that the time required to find the ground-state would scale sublinearly with system size for the half-filling cases considered.
The Fermi-Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. With a simplistic form of the Hamiltonian, the model can potentially provide explanation to interesting phenomena in correlated electrons. The interest of our current work is the model in one dimension. The one-dimensional Fermi-Hubbard model has been extensively studied and there exist analytical results for determining its ground-state energy in the thermodynamic limit. We want to find the ground-state from the perspective of quantum computing. In particular, we employ the protocol of quantum annealing and perform high-performance simulations of the same using the Jülich Quantum Computer Simulator (JUQCS) for systems with up to 40 qubits. We learn that the time required to find the ground-state would scale sublinearly with system size for the half-filling cases considered.
The Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. This model provides insight into a range of phenomena in correlated electron systems. We wish to examine the paradigm of quantum algorithms for solving such many-body problems. The focus of our current work is on the one-dimensional model that is integrable, meaning that there exist analytical results for determining its ground state. In particular, we demonstrate how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian. We perform simulations for systems with up to 40 qubits to study the scaling of required annealing time for obtaining the ground state. We find that for the half-filled cases considered, there is a polynomial quantum speedup over algorithms based on the Bethe-ansatz equations.
The Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. This model provides insight into a range of phenomena in correlated electron systems. We wish to examine the paradigm of quantum algorithms for solving such many-body problems. The focus of our current work is on the one-dimensional model that is integrable, meaning that there exist analytical results for determining its ground state. In particular, we demonstrate how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian. We perform simulations for systems with up to 40 qubits to study the scaling of required annealing time for obtaining the ground state. We find that for the half-filled cases considered, there is a polynomial quantum speedup over algorithms based on the Bethe-ansatz equations.
The use of quantum computers to calculate the ground state (lowest) energies of a spin lattice of electrons described by the Fermi-Hubbard model of great importance in condensed matter physics has been studied. The ability of quantum bits (qubits) to be in a superposition state allows quantum computers to perform certain calculations that are not possible with even the most powerful classical (digital) computers. This work has established a method for calculating the ground state energies of small lattices which should be scalable to larger lattices that cannot be calculated by classical computers. Half-filled lattices of sizes 1x4, 2x2, 2x4, and 3x4 were studied. The calculated energies for the 1x4 and 2x2 lattices without Coulomb repulsion between the electrons and for the 1x4 lattice with Coulomb repulsion agrees with the true energies to within 0.60%, while for the 2x2 lattice with Coulomb repulsion the agreement is within 1.50% For the 2x4 lattice, the true energy without Coulomb repulsion was found to agree within 0.18%.
Since the advent of high-
Tc cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent
α for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report
α≳1, which would imply
Kρ≲0.17, a value which cannot be reconciled with the bare Hubbard model where
KρH≥0.5, i.e.,
αH≲1/8. We develop new accurate numerical methods to obtain
Kρ and investigate how to obtain such small values for
Kρ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate
Uc. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.