We derive a solvable resonant-level-type model, to describe an impurity spin coupled to zero-energy bound states localized at the edge of a one-dimensional d-wave superconductor.This results in a two-channel Kondo effect with a quite unusual low-temperature thermodynamics.For instance, the local impurity susceptibility yields a finite maximum at zero temperature (but no ln-divergence) due to the splitting of the impurity in two Majorana fermions.Moreover, we make comparisons with the Kondo effect occurring in a two-dimensional d-wave superconductor.
We derive a solvable resonant-level-type model, to describe an impurity spin coupled to zero-energy bound states localized at the edge of a one-dimensional d-wave superconductor.This results in a two-channel Kondo effect with a quite unusual low-temperature thermodynamics.For instance, the local impurity susceptibility yields a finite maximum at zero temperature (but no ln-divergence) due to the splitting of the impurity in two Majorana fermions.Moreover, we make comparisons with the Kondo effect occurring in a two-dimensional d-wave superconductor.
We consider a band of fermions in two space dimensions with a flux phase (relativistic) dispersion relation coupled to a local magnetic impurity via an s-d interaction. This model describes spinons of a flux phase and it is also a qualitative model of the quasiparticles in a ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$ superconductor. We find a zero-temperature phase transition at a finite coupling constant between a weak coupling unscreened impurity state and a strong coupling regime with a Kondo effect. We use large-N methods to study the phase transition in this Kondo system away from marginality. The Kondo energy scales linearly with the distance to the transition. The zero-field magnetic susceptibility at zero temperature diverges linearly. Similar behavior is found in the T matrix which shows a resonance at the Kondo scale. However, in addition to this simple scaling, we always find the presence of logarithmic corrections to scaling. Such behavior is typical of systems at an upper critical dimension. We derive an effective fermion model in one space dimension for this problem. Unlike the usual Kondo problem, this system has an intrinsic multichannel nature which follows from the spinor structure of (2+1)-dimensional relativistic fermions. \textcopyright{} 1996 The American Physical Society.
We study a quantum impurity coupled to the edge states of a two-dimensional helical topological superconductor, that is, to a pair of counterpropagating Majorana fermion edge channels with opposite spin polarizations. For an impurity described by the Anderson impurity model, we show that the problem maps onto a variant of the interacting resonant two-level model which, in turn, maps onto the ferromagnetic Kondo model. Both magnetic and nonmagnetic impurities are considered. For magnetic impurities, an analysis relying on bosonization and the numerical renormalization group shows that the system flows to a fixed point characterized by a residual ln2 entropy and anisotropic static and dynamical impurity magnetic susceptibilities. For nonmagnetic impurities, the system flows instead to a fixed point with no residual entropy and we find diamagnetic impurity response at low temperatures. We comment on the Schrieffer-Wolff transformation for problems with nonstandard conduction band continua and on the differences which arise when we describe the impurities by either Anderson or Kondo impurity models.
We present a thermodynamic description of a single magnetic impurity at the edge of a superconducting wire. The impurity exhibits four phases $\unicode{x2014}$ Kondo, Yu-Shiba-Rusinov (YSR) I and II, and local moment $\unicode{x2014}$ a phase diagram richer than in the gapless case, contrary to the expectation that the effects of impurities in gapped hosts are less consequential. We derive the impurity contribution to free energy Fimp(T) and entropy in each phase: in Kondo phase, the entropy flows monotonically from ln2 (UV) to 0 (IR) with critical exponents same as that of the conventional Kondo model; in YSR phases, thermal activation of a midgap bound state produces entropy overshoots above ln2, saturating to ln2 at high T and approaching either 0 or ln2 at low T depending on whether impurity is screened or not; in the local-moment phase the impurity remains effectively decoupled, with entropy near ln2, with some intermediate-temperature features that progressively fade as δ→0. These behaviors, including the entropy overshoots in the YSR and local-moment phases, stem from a splitting of the Hilbert space into distinct excitation towers: one in the Kondo phase, two in YSR~I, and three in YSR~II and the local-moment phase. Resolving these tower structures and thereby going beyond conventional TBA yields closed-form analytic expressions for the impurity contribution to the free energy and entropy across the entire phase diagram.
We demonstrate that a boundary defect in the single spin-21 quantum XX chain exhibits two-channel Kondo physics. Due to the presence of the defect, the edge spin fractionalizes into two Majorana fermions, out of which one decouples, and one is overscreened by the free fermion in bulk, leading to non-trivial boundary behavior characteristic of the two-channel Kondo model. When the ratio of boundary to bulk coupling exceeds a critical value of 2, a massive boundary-bound mode is exponentially localized near the impurity site for strong impurity coupling. This leads to unusual behavior in physical quantities, such as the g-function not being monotonic. We compute the g−function of the impurity from both thermodynamic and entanglement entropy calculations and show that it takes a non-integer value of 2 just as in the two-channel Kondo problem.
Abstract We investigate the boundary phenomena that arise in a finite-size XX spin chain interacting through an XX interaction with a spin − 1 2 impurity located at its edge. Upon Jordan–Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. In more detail, using both Bethe Ansatz and exact diagonalization techniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling J imp J . When the ratio falls below the critical value 2 , the impurity is screened via the Kondo effect. However, when the ratio between the coupling exceeds the critical value 2 an exponentially localized bound mode is formed at the impurity site which screens the spin of the impurity in the ground state. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility of the impurity. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling J imp exceeds the critical value 2 J , the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.
We discuss the dynamics of magnetic moments in d-wave superconductors. In particular, we focus on moments induced by doping nonmagnetic impurities into cuprates. The interaction of such moments with the Bogoliubov quasiparticles of the superconductors can be described by variants of the pseudogap Kondo model, characterized by a power-law density of states at the Fermi level. The numerical renormalization-group technique is employed to investigate this Kondo problem for realistic band structures and particle-hole asymmetries, both at zero and finite temperatures. In particular, we study the boundary quantum phase transition between the local-moment and asymmetric strong-coupling phases, and argue that this transition has been observed in recent nuclear magnetic resonance experiments. We determine the spectral properties of both phases and the location of the critical point as a function of Kondo coupling and doping, and discuss quantum-critical crossovers near this phase transition. In addition, changes in the local density of states around the impurity are calculated as functions of temperature, being relevant to scanning tunneling microscopy experiments.