Electronic states near a square Fermi surface are mapped onto quantum chains. Using boson-fermion duality on the chains, the bosonic part of the interaction is isolated and diagonalized. These interactions destroy Fermi-liquid behavior. Nonboson interactions are also generated by this mapping, and give rise to an alternate perturbation theory about the boson problem. A case with strong repulsions between parallel faces is studied and solved. This solution discards irrelevant operators in a new interaction Hamiltonian. There is spin-charge separation and the square Fermi surface remains square under doping. At half-filling, there is a charge gap and insulating behavior together with gapless spin excitations. This mapping appears to be a general tool for understanding the properties of interacting electrons on a square Fermi surface.
We calculate zero-temperature correlation functions for a model of two-dimensional interacting electrons with short-range interactions and a square Fermi surface. The model was arrived at by mapping electronic states near a square Fermi surface with Hubbard-like interactions onto one-dimensional quantum chains, retaining terms that can be written in terms of bosonic density operators. Interactions between orthogonal chains, corresponding to orthogonal faces of the square Fermi surface, are neglected. The correlation functions become sums of Luttinger-type correlation functions due to the bosonic model. However, the correlation function exponents differ in form from those of the Luttinger model. As a consequence, the simple scaling relations found to exist between the Luttinger model exponents, do not carry over to the leading exponents of our model. We find that for repulsive effective interactions, charge-density wave/spin-density wave instabilities are dominant. We do not consider d-wave instabilities here.
We study, via bosonization, the Landau fixed point for the problem of interacting spinless fermions near the Fermi surface in dimensions higher than one. We rederive the bosonic representation of the Fermi operator and use it to find the general form of the fermion propagator for the Landau fixed point. Using a generalized Bogoliubov transformation we diagonalize exactly the bosonized Hamiltonian for the fixed point and calculate the fermion propagator (and the quasiparticle residue) for isotropic interactions (independently of their strength). We reexamine two well-known problems in this context: the screening of long-range potentials and the Landau damping of gauge fields. We also discuss the origin of the Luttinger fixed point in one dimension in contrast with the Landau fixed point in higher dimensions.
We take an in depth look into the celebrated Fermi-Hubbard model for strongly interacting fermions, beginning with a two-site exact diagonalization demonstration of the Mott ground state. We continue with measurements of compressibility and susceptibility using Determinantal Quantum Monte Carlo (DQMC) methods on a larger square lattice 6x6 over the parameter space of interaction strength U/t and doping, controlled by the chemical potential, mu/t. We construct the Fermi surface for a 16x16 cluster size using the retarded Green's function and the single particle spectral function (derived from the Green's function using an analytic continuation procedure) and find conclusive evidence of Fermi surface restructuring (FSR) as a result of strong interactions freezing out states from contributing to the Luttinger count, n, at intermediate temperatures, T = t/2. FSR is found for strongly interacting, doped systems for dopings p < 0.2. This phase is dubbed the Luttinger's rule breaking (LB) phase, and we provide a discussion of the consequences of the deviation from Luttinger's theorem. We strongly suggest a topological nature underlying the restructuring of the Fermi surface by eliminating competing ordering in spin or charge channels.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (
qc/
kF)dF0 [1+
F0${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here
qc is the cutoff of the interaction in wave-vector space,
kF is the Fermi wave vector, \ensuremath{\mu} is the chemical potential,
F0 is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff
qc is small compared with
kF, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.