We develop a systematic approach to bosonization and vertex algebras on quantum wires of the form of star graphs. The related bosonic fields propagate freely in the bulk of the graph, but interact at its vertex. Our framework covers all possible interactions preserving unitarity. Special attention is devoted to the scale invariant interactions, which determine the critical properties of the system. Using the associated scattering matrices, we give a complete classification of the critical points on a star graph with any number of edges. Critical points where the system is not invariant under wire permutations are discovered. By means of an appropriate vertex algebra we perform the bosonization of fermions and solve the massless Thirring model. In this context we derive an explicit expression for the conductance and investigate its behavior at the critical points. A simple relation between the conductance and the Casimir energy density is pointed out.
We study scattering in noninteracting multi-terminal quantum wire junctions and show that junctions with dihedral symmetry can exhibit exact zero-reflection points for
N≥4 terminals. By analyzing the scattering matrix, we identify these reflectionless points in the
(E,t′) parameter space, where
E is the incident particle energy and
t′ is the junction hopping amplitude. These points exhibit an even-odd dependence on
N and converge asymptotically to a common limiting value in the large-
N limit. We show that the reflectionless points are characterized by an integer winding number associated with the phase of the reflection amplitude, providing a topological description for their stability against weak on-site disorder. We also consider junctions with broken time-reversal symmetry and find that a magnetic flux can induce additional reflectionless points, including for the
N=3 case. For a four-terminal junction threaded by a
π-flux, we identify a unique parameter regime in which the reflection amplitude vanishes over the entire energy band. Finally, we discuss experimental signatures through the behavior of Friedel oscillations and examine the stability of these reflectionless points in the presence of weak interactions.
We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.
We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.
We investigate Luttinger junctions of quantum wires away from criticality. The one-body scattering matrix, corresponding to the off-critical boundary conditions at the junction, admits, in general, antibound and/or bound states. Their contribution to the theory is fixed by causality. The presence/absence of bound states determines the existence of two different regimes with inequivalent physical properties. A scattering matrix without bound states defines an isolated equilibrium system. Bound states instead drive the system away from equilibrium, giving raise to nontrivial incoming or outgoing energy flows in the junction. We derive in both regimes and in explicit form the electromagnetic conductance tensor, pointing out the different impacts of bound and antibound states.