Abstract:
Within bosonization theory we introduce in this paper a new definition of "quasi-particles" for interacting fermions at arbitrary space dimenions. In dimensions higher than one we show that the constructed quasi-particles are consistent with quasi-particle descriptions in Landau Fermi liquid theory whereas in one-dimension the quasi-particles" are non-perturbative objects (spinons and holons) obeying fractional statistics. The more general situation of Fermi liquids with singular Landau interaction is discussed.

Abstract:
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 processes 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 $ ( \frac{q_{c} }{ k_{F}} )^d F_{0} [ 1 + F_{0} ]^{-1} \frac{\mu}{\nu^{\alpha}} | \frac{ \partial \nu^{\alpha} }{ \partial \mu} |$ are small compared with unity for all patches $\alpha$. Here $q_{c}$ is the cutoff of the interaction in wave-vector space, $k_{F}$ is the Fermi wave-vector, $\mu$ is the chemical potential, $F_{0}$ is the usual dimensionless Landau interaction-parameter, and $\nu^{\alpha} $ is the {\it{local}} density of states associated with patch $\alpha$. 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 for the density-density correlation function is exact in $d=1$, exists also in $d> 1$, provided (1) the interaction cutoff $q_{c}$ is small compared with $k_{F}$, and (2) the energy dispersion is locally linearized at the Fermi the Fermi surface. Finally, we suggest a new 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.

Abstract:
We present an exact mapping of models of interacting fermions onto boson models. The bosons correspond to collective excitations in the initial fermionic models. This bosonization is applicable in any dimension and for any interaction between fermions. We show schematically how the mapping can be used for Monte Carlo calculations and argue that it should be free from the sign problem. Introducing superfields we derive a field theory that may serve as a new way of analytical study.

Abstract:
The single-particle Green's function of an interacting Fermi system with dominant forward scattering is calculated by decoupling the interaction by means of a Hubbard-Stratonowich transformation involving a bosonic auxiliary field $\phi^{\alpha}$. We obtain a higher dimensional generalization of the well-known one-dimensional bosonization result for the Green's function by first calculating the Green's function for a fixed configuration of the $\phi^{\alpha}$-field and then averaging the resulting expression with respect to the probability distribution ${\cal{P}} \{ \phi^{\alpha} \} \propto \exp [ - S_{eff} \{ \phi^{\alpha} \} ]$, where $S_{eff} \{ \phi^{\alpha} \}$ is the effective action of the $\phi^{\alpha}$-field. We emphasize the approximations inherent in the higher-dimensional bosonization approach and clarify its relation with diagrammatic perturbation theory.

Abstract:
It is shown that the criticism presented in the Comment by Galanakis et al \cite{1} on the paper by Efetov et al \cite{2} is irrelevant to the bosonization approach.

Abstract:
A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time $t\to t/\lambda^2$ and performing the limit $\lambda\to0$ (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations.

Abstract:
Using methods of functional integration, and performing simple Gaussian integrals, I show that an interacting system of electrons can be bosonized in arbitrary dimensions, in terms of the electrostatic potential which mediates the interaction between them. Working with the bosonic field, the sytem is shown to exhibit localized structures reminiscent of striping in the cuprates.

Abstract:
We discuss interacting fermion models in two dimensions, and, in particular, such that can be solved exactly by bosonization. One solvable model of this kind was proposed by Mattis as an effective description of fermions on a square lattice. We review recent work on a specific relation between a variant of Mattis' model and such a lattice fermion system, as well as the exact solution of this model. The background for this work includes well-established results for one-dimensional systems and the high-Tc problem. We also mention exactly solvable extensions of Mattis' model.

Abstract:
We demonstrate that the skeleton of the Fermi surface S_{F;s} pertaining to a uniform metallic ground state (corresponding to fermions with spin index s) is determined by the Hartree-Fock contribution to the dynamic self-energy. The Fermi surface S_{F;s} consists of all points which in addition to satisfying the quasi-particle equation in terms of the Hartree-Fock self-energy, fulfill the equation S_{s}(k) = 0, where S_{s}(k) is defined in the main text; the set of k points which satisfy the Hartree-Fock quasi-particle equation but fail to satisfy S_{s}(k) = 0, constitute the pseudo-gap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum-distribution function n_{s}(k) for k in the vicinity of S_{F;s} and show that whereas for the uniform metallic ground states of the conventional Hubbard Hamiltonian n_{s}(k) is greater/less than 0.5 for k approaching S_{F;s} from inside/outside the Fermi sea, for interactions of non-zero range these inequalities can be violated (without thereby contravening the condition of the non-negativity of the possible jump in n_{s}(k) on k crossing S_{F;s} from directly inside to directly outside the Fermi sea). We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the `frustration' of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds. [Short abstract]