Bernevig et al. Reply:We provide a Reply to the preceding Comment [1] by Greiter and Schuricht (GS). Let us first stress that there is no doubt about the mathematical correctness of our derivation. We now show that our interpretation of the physics of spinons is correct. Against the existence of spinon attraction, GS recall that spinons constitute an ''ideal gas of half fermions'' [2]. In reality, spinons feel a statistical interaction associated with a rule for progressively filling single-particle states. The fully dressed S matrix for spinons [3] in the Haldane-Shastry model (HSM) [4] takes the trivial form S i times the identity I. (This is well known and we derived it on p. 9 of the second paper of Ref. [5], which is equivalent to computing the S matrix associated with exchange statistics of semions-which can also be done by asymptotic Bethe ansatz -there is an extra i in our formula: it comes from a misprint in the final version of the Letter.) Triviality of the S matrix means that spinons are alleged asymptotic states of the HSM. Their long-distance behavior is not affected by their dynamical short-range attraction. In our Letter, we explicitly show the short-distance effects of the interaction on two-spinon wave functions [5]. We now consider the points raised by GS separately.GS claim our interpretation of the p mn e i as twospinon relative wave function is ambiguous, because the p mn 's are defined by expanding the (overcomplete) set of states in the (basis of) energy eigenfunctions mn . This statement is false. In real space, spinons are nonlocal excitations, with typical size of the order of the lattice step [5]. The 's should be thought of as the lattice version of two-particle coherent states. The overcompleteness of the 's is, therefore, the lattice version of the usual overcompleteness of coherent states. Two spinons at the same site correspond to a localized spin-1 excitation, unambiguously described by z 1 ; . . . ; z N=2ÿ1 [5]. The physical meaning of p mn 1 as the probability enhancement when the two spinons are at the same site (and form the spin-1 excitation) is hence absolutely unambiguous, at odds to GS's statement that the jp mn j 2 's ''cannot be interpreted as probability distribution.'' We also find a similar short-range attraction between a spinon and a holon (in the supersymmetric t ÿ J model with 1=r 2 interaction) although the states of a localized spinon and a localized holon do not form an overcomplete set [5]. We wish to remark that using overcomplete sets of states to treat nonlocal excitations as quantum-mechanical particles is in fact widely used for Laughlin quasiholes, a fact which Greiter correctly and explicitly points out in a prior publication [6]. The spinon situation is no different.GS assert that ''the second argument of [Bernevig, Giuliano, and Laughlin] is that the last term in their expression for the energy of the two-spinon states represents 'a negative interaction contribution that becomes negligibly small in the thermodynamic limit'.'' We clearly state that th...

By deriving and studying the coordinate representation for the two-spinon wavefunction, we show that spinon excitations in the Haldane-Shastry model interact. The interaction is given by a shortrange attraction and causes a resonant enhancement in the two-spinon wavefunction at short separations between the spinons. We express the spin susceptibility for a finite lattice in terms of the resonant enhancement, given by the two-spinon wavefunction at zero separation. In the thermodynamic limit, the spinon attraction turns into the square-root divergence in the dynamical spin susceptibility.

Experimental evidence was recently obtained for topological superconductivity in spin-orbit coupled nano wires in a magnetic field, proximate to an s-wave superconductor. When only part of the wire contacts the superconductor, a localized Majorana mode exists at the junction between superconducting and normal parts of the nanowire. We consider here the case of a T-junction between the superconductor and two normal nanowires and also the case of a single wire with two (or more) partially filled bands in the normal part. We find that coupling this 2-channel Luttinger liquid to the single Majorana mode at the junction produces frustration, leading to a critical point separating phases with perfect Andreev scattering in one channel and perfect normal scattering in the other.

We show that a finite Josephson Junction (JJ) chain, ending with two bulk superconductors, and with a weak link at its center, may be regarded as a condensed matter realization of a two-boundary Sine-Gordon model. Computing the partition function yields a remarkable analytic expression for the DC Josephson current as a function of the phase difference across the chain. We show that, in a suitable range of the chain parameters, there is a crossover of the DC Josephson current from a sinusoidal to a sawtooth behavior, which signals a transition from a regime where the boundary term is an irrelevant operator to a regime where it becomes relevant.

We show that, for pertinent values of the fabrication and control parameters, an attractive finite coupling fixed point emerges in the phase diagram of a Y-junction of superconducting Josephson chains. The new fixed point arises only when the dimensionless flux $f$ piercing the central loop of the network equals $\pi$ and, thus, does not break time-reversal invariance; for $f \neq \pi$, only the strongly coupled fixed point survives as a stable attractive fixed point. Phase slips (instantons) have a crucial role in establishing this transition: we show indeed that, at $f = \pi$, a new set of instantons -the W-instantons- comes into play to destabilize the strongly coupled fixed point. Finally, we provide a detailed account of the Josephson current-phase relationship along the arms of the network, near each one of the allowed fixed points. Our results evidence remarkable similarities between the phase diagram accessible to a Y-junction of superconducting Josephson chains and the one found in the analysis of quantum Brownian motion on frustrated planar lattices.Comment: 32 pages, 8 .eps figures. Accepted versio