In a previous paper, we showed that the problem of tunneling in quantum wires was integrable in the isotropic case ${g}_{\ensuremath{\sigma}}=2$. In the present work, we continue the exploration of the general phase diagram by looking for other integrable cases. Specifically, we discuss in detail the manifold ${g}_{\ensuremath{\rho}}{+g}_{\ensuremath{\sigma}}=2$, where the associated ``double sine-Gordon'' model is integrable. Transport properties are exactly computed. Surprisingly, the IR fixed points, while having complete reflection of charge and spin currents, do not correspond to two separate leads. Their main characteristic is that they are approached along irrelevant operators of dimension ${1+(1/g}_{\ensuremath{\rho}})$ and ${1+(1/g}_{\ensuremath{\sigma}})$, corresponding to transfer of one electron charge but no spin, or one spin
1/2 but no charge.
We develop a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model. They are both boundary integrable quantum field theories with a quantum-group spin at the boundary which takes values, respectively, in standard or cyclic representations of the quantum groupSU(2) q. This unification is powerful, and allows us to find new results for both models. For the anisotropic Kondo problem, we find exact expressions (in the presence of a magnetic field) for all the coefficients in the Anderson-Yuval perturbative expansion. Our expressions hold initially in the very anisotropic regime, but we show how to continue them beyond the Toulouse point all the way to the isotropic point using an analog of dimensional regularization. The analytic structure is transparent, involving only simple poles which we determine exactly, together with their residues. For the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid, we find the nonequilibrium conductance for all values of the Luttinger coupling. This is an intricate computation because the voltage operator and the boundary scattering do not commute with each other.
We study a system consisting of a junction of
N wires, where the junction is characterized by a scalar
S-matrix, and an impurity spin is coupled to the electrons near the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group (RG) equations for the Kondo couplings of the spin to the electrons on different wires. We analyze the RG flows and fixed points for different values of the initial Kondo couplings and of the junction
S-matrix, such as the decoupled
S-matrix and the Griffiths (connected)
S-matrix. We find that the Kondo couplings flow either towards a ferromagnetic (FM) fixed point or towards large and antiferromagnetic (AFM) values in one of two ways. For the Griffiths
S-matrix, one of the strong coupling flows is towards a FM fixed point with decoupled wires; this is seen by a perturbative analysis. Thus if we start with a system of connected wires with an AFM coupling to the spin impurity, the flow at large distances is towards a system of disconnected wires at the FM fixed point. For the decoupled
S-matrix, the flow is either to a FM fixed point or to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. Strong interactions between the electrons with ${K}_{\ensuremath{\rho}}<N∕(N+2)$ can stabilize a multichannel fixed point in which the coupling between different channels goes to zero. We also study the temperature dependence of the conductance at the decoupled and Griffiths
S-matrices.
The problem of a local impurity in a Luttinger liquid, just like the anisotropic Kondo problem (of which it is technically a cousin), describes many different physical systems. As shown by Kane and Fisher, the presence of interactions profoundly modifies the physics familiar from Fermi liquid theory, and leads to non-intuitive features, best described in the Renormalization Group language (RG), such as flows towards healed or split fixed points. While this problem has been studied for many years using more traditional condensed matter approaches, it remains somewhat mysterious from the point of view of entanglement, both for technical and conceptual reasons. We propose and explore in this paper a new way to think of this important aspect. We use the realization of the Kane Fisher universality class provided by an XXZ spin chain with a modified bond strength between two sites, and explore the difference of (Von Neumann) entanglement entropies of a region of length \ell ℓ with the rest of the system - to which it is connected with a modified bond - in the cases when \ell ℓ is even and odd. Surprisingly, we find out that this difference \delta S\equiv S^e-S^o δ S ≡ S e − S o remains of O(1) O ( 1 ) in the thermodynamic limit, and gives rise now, depending on the sign of the interactions, to “resonance” curves, interpolating between -\ln 2 − ln 2 and 0 0 , and depending on the product \ell T_B ℓ T B , where 1/T_B 1 / T B is a characteristic length scale akin to the Kondo length in Kondo problems. \delta S δ S can be interpreted as a measure of the hybridization of the left-over spin in odd length subsystems with the “bath” constituted by the rest of the chain. The problem is studied both numerically using DMRG and analytically near the healed and split fixed points. Interestingly - and in contrast with what happens in other impurity problems - \delta S δ S can, at least to lowest order, be tackled by conformal perturbation theory.
We explicitly evaluate the infinite series of integrals that appear in the "Anderson-Yuval" reformulation of the anisotropic Kondo model in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo model of arbitrary spin with the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid and tunneling in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
The phase diagram of the multichannel Kondo Hamiltonian with an
XXZ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-
21 impurity and
k>2 conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling
Jz∗>0 and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-
21 local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary
k and
s obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here,
Jz is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and
s is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for
s>1∕2 and arbitrary
k. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of
2s, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a
k-channel, spin-
21 Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-
21 local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in
1∕Jz, and explicit numerical renormalization-group calculations.