In this thesis we derive a set of bosonization rules for the problem of a fermionic interacting system living on the half line coupled to a quantum degree of freedom at a boundary. This is an appropriate model to describe the physics of an electronic system in quantum wires with impurities. First we derive the bosonization rules for free fermions on a half-line with physically sensible boundary conditions for Luttinger fermions. We use path-integral methods to calculate the bosonized fermionic currents on the half-line and derive their commutation relations for a system with a boundary. We compute the fermion determinant of the fermionic fluctuations for a system with a boundary using Forman's approach. We find that the degrees of freedom induced at the boundary do not modify the commutation relations of the bulk. We give an explicit derivation of the bosonization rules for the fermion operators for a system with boundaries. We derive a set of bosonization rules for the Fermi operators which include the explicit effect of the boundaries and of boundary degrees of freedom. As a byproduct, we calculate the one-particle Green's function and determine the effects of the boundaries on its analytic structure.
We revisit the dynamic spin susceptibility $\ensuremath{\chi}(q,\ensuremath{\omega})$ of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\ensuremath{\chi}(q,\ensuremath{\omega})$ at $q\ensuremath{\ll}{k}_{F}$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ is nonzero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ due to backscattering is larger than that due to finite mass by a factor of
kF/q. We derive the renormalization group equations for the coupling constants of the $g\ensuremath{-}\text{ology}$ model at finite $\ensuremath{\omega}$ and
q and find the corresponding expression for $\ensuremath{\chi}(q,\ensuremath{\omega})$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.
We revisit the issue of the temperature dependence of the specific heat
C(T) for interacting fermions in one dimension. The charge component
Cc(T) scales linearly with
T, but the spin component
Cs(T) displays a more complex behavior with
T as it depends on the backscattering amplitude,
g1, which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that
Cs(T) is strictly linear in
T to order
g12 as it contains the renormalized backscattering amplitude not on the scale of
T, but at the cutoff scale set by the momentum dependence of the interaction around
2kF. The running amplitude
g1(T) appears only at third order and gives rise to an extra
T∕log3T term in
Cs(T). This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in
g1 within the sine-Gordon model.
We study a spin-dependent Tomonaga-Luttinger model in one dimension, which describes electron transport through a single barrier. Using the Fermi-Bose equivalence in one dimension, we map the model onto a massless Thirring model with a boundary interaction. A field theoretical perturbation theory for the model has been developed, and the chiral symmetry is found to play an important role. The classical bulk action possesses a global U A (1)4 chiral symmetry because the fermion fields are massless. This global chiral symmetry is broken by the boundary interaction, and the bosonic degrees of freedom, corresponding to a chiral phase transformation, become dynamical. They acquire an additional kinetic action from the fermion path-integral measure and govern the critical behaviors of the physical operators. On the critical line where the boundary interaction becomes marginal, they decouple from the fermi fields. Consequently, the action reduces to the free-field action, which contains only a fermion bilinear boundary mass term as an interaction term. By using a renormalization group analysis, we obtain a new critical line, which differs from the previously known critical lines in the literature. The result of this work implies that the phase diagram of the one-dimensional electron system may have a richer structure than previously thought.