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arXiv:2008.07674 · Nuclear Physics B

Tunable Kondo screening length at a Y-junction of three inhomogeneous spin chains

Domenico Giuliano, Andrea Kazemi Nava, Pasquale Sodano

Published Sep 23, 2020• 14 citations
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Abstract

We derive the topological Kondo Hamiltonian describing a Y junction of three XX-spin chains connected to outer quantum Ising chains with different tilting angles for the Ising axis. We show that the tilting angles in the spin models play the role of the phases of the superconducting order parameters at the interfaces between bulk superconductors and one-dimensional conducting normal electronic wires. As a result, different tilting angles induce nonzero equilibrium spin (super)currents through the junction. Employing the renormalization group approach to the topological Kondo model, we derive the scaling formulas for the equilibrium spin currents. We argue that, by monitoring the crossover in the currents induced by the Kondo effect, it is possible to estimate the Kondo screening length. In particular, we prove how it is possible to tune the Kondo length by acting on the applied phases only; this enables us to map out the scaling properties by just tuning the tilting angles and the Kondo length accordingly.

DOI
10.1016/j.nuclphysb.2020.115192

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We derive the topological Kondo Hamiltonian describing a Y junction of three XX-spin chains connected to outer quantum Ising chains with different tilting angles for the Ising axis. We show that the tilting angles in the spin models play the role of the phases of the superconducting order parameters at the interfaces between bulk superconductors and one-dimensional conducting normal electronic wires. As a result, different tilting angles induce nonzero equilibrium spin (super)currents through the junction. Employing the renormalization group approach to the topological Kondo model, we derive the scaling formulas for the equilibrium spin currents. We argue that, by monitoring the crossover in the currents induced by the Kondo effect, it is possible to estimate the Kondo screening length. In particular, we prove how it is possible to tune the Kondo length by acting on the applied phases only; this enables us to map out the scaling properties by just tuning the tilting angles and the Kondo length accordingly.
We consider the Kondo effect in Y-junctions of anisotropic XY models in an applied magnetic field along the critical lines characterized by a gapless excitation spectrum. We find that, while the boundary interaction Hamiltonian describing the junction can be recasted in the form of a four-channel, spin-1/2 antiferromagnetic Kondo Hamiltonian, the number of channels effectively participating in the Kondo effect depends on the chain parameters, as well as on the boundary couplings at the junction. The system evolves from an effective four-channel topological Kondo effect for a junction of XX-chains with symmetric boundary couplings into a two-channel one at a junction of three quantum critical Ising chains. The effective number of Kondo channels depends on the properties of the boundary and of the bulk. The XX-line is a “critical” line, where a four-channel topological Kondo effect can be recovered by fine-tuning the boundary parameter, while along the line in parameter space connecting the XX-line and the critical Ising point the junction is effectively equivalent to a two-channel topological Kondo Hamiltonian. Using a renormalization group approach, we determine the flow of the boundary couplings, which allows us to define and estimate the critical couplings and Kondo temperatures of the different Kondo (pair) channels. Finally, we study the local transverse magnetization in the center of the Y-junction, eventually arguing that it provides an effective tool to monitor the onset of the two-channel Kondo effect.
We analyze the crossover from Kondo to weak-link regime by means of a model of tunable bond impurities in the middle of a spin-1/2 XXZ Heisenberg chain. We study the Kondo screening cloud and estimate the Kondo length by combining perturbative renormalization group approach with the exact numerical calculation of the integrated real-space spin-spin correlation functions. We show that, when the spin impurity is symmetrically coupled to the two parts of the chain with realistic values of the Kondo coupling strengths and spin-parity symmetry is preserved, the Kondo length takes values within the reach of nowadays experimental technology in ultracold-atom setups. In the case of nonsymmetric Kondo couplings and/or spin parity broken by a nonzero magnetic field applied to the impurity, we discuss how Kondo screening redistributes among the chain as a function of the asymmetry in the couplings and map out the shrinking of the Kondo length when the magnetic field induces a crossover from Kondo impurity to weak-link physics.
Topological Kondo insulators are strongly correlated materials where itinerant electrons hybridize with localized spins, giving rise to a topologically nontrivial band structure. Here, we use nonperturbative bosonization and renormalization-group techniques to study theoretically a one-dimensional topological Kondo insulator, described as a Kondo-Heisenberg model, where the Heisenberg spin-1/2 chain is coupled to a Hubbard chain through a Kondo exchange interaction in the p-wave channel (i.e., a strongly correlated version of the prototypical Tamm-Schockley model). We derive and solve renormalization-group equations at two-loop order in the Kondo parameter, and find that, at half filling, the charge degrees of freedom in the Hubbard chain acquire a Mott gap, even in the case of a noninteracting conduction band (Hubbard parameter U=0). Furthermore, at low enough temperatures, the system maps onto a spin-1/2 ladder with local ferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-1 chain with frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypical interacting topological spin model, and features two magnetic spin-1/2 end states for chains with open boundary conditions. Our analysis allows us to derive an insightful connection between topological Kondo insulators in one spatial dimension and the well-known physics of the Haldane chain, showing that the ground state of the former is qualitatively different from the predictions of the naive mean-field theory.
We study the boundary phase diagram and the low-temperature heat and magnetization transport at a YYY junction of XX spin chains. Depending on the magnetization axis anisotropy between the magnetic exchange interactions at the junction, the system exhibits two different strong-coupling regimes at low energies/temperatures, hosting a remarkable realization of, respectively, the overscreened (topological) four- and to the two-channel Kondo fixed points, both robust against anisotropies in the chain parameters and/or in the boundary couplings. Using renormalization group arguments combined with boundary conformal field theory methods, we show the instability of the former under any XY-type anisotropy at the junction. Our system allows for using the low-temperature spin and the heat conductances to evidence the fractionalization of the elementary excitations at the four-channel Kondo fixed point by means of the magnetic Wiedemann-Franz law. We caution that the instability under XY anisotropy may hinder the detection of the phenomenology related to the four-channel Kondo effect, therefore requiring careful control in experimental realizations.
In this paper, I present a conjecture for the Bethe ansatz equations for the model describing a star junction of M quantum critical Ising chains. For such a model exhibits the so-called topological Kondo effect (Beri and Cooper 2012 Phys. Rev. Lett. 109 156803) related to the existence of Majorana zero energy modes at the junction. These modes are of a topological nature; they non-locally encode an SO(M) 'spin' which is screened by the collective excitations of the chains. For certain values of M , the model is equivalent to the Kondo models with a known exact solution. These cases are used to check the validity of the conjecture. It is demonstrated that the model behaves differently for M even and odd; in the former case the model has a Fermi liquid and the latter case corresponds to a non-Fermi liquid infrared fixed point.
We show that- in the quantum regime- a Josephson junction rhombi chain (i.e. a Josephson junction chain made by rhombi formed by joining 4 Josephson junctions) may be effectively mapped onto a quantum Hamiltonian describing Ising spins in a transverse magnetic field with open boundary conditions. Then, we elucidate how a Y-shaped network fabricated with 3 Josephson Junction Rhombi chains may be used as a quantum device realizing the two channel Kondo model recently proposed by Tsvelik. We point out that the emergence of a 2 channel Kondo effect in this superconducting network may be probed through the measurement of a pertinent Josephson current.