Balatsky and Salkola Reply:

Abstract

Balatsky and Salkola Reply: In their Comment, Aristov and Yashenkin [1] claim that we [2] have calculated the impurity-induced wave function incorrectly and hence missed a power-law term that decays as 1͞r for all directions. Contrary to their claim, we have explicitly noted the presence of this term in the impurity wave function as is evident from the discussion following Eq. (1) in [2]. Moreover, we have explored the spatial dependence of the impurity state in [3], where all of the terms were kept (see also Ref. [4]). That there is a 1͞r term for all directions is a simple observation which follows, for instance, from the calculation with the linearized quasiparticle spectrum near the nodes. Nonetheless, we decided to neglect it in our subsequent discussion of localization for the following reasons.First, if we take into account the 1͞r decay of the wave function for all directions, the impurity state is not normalizable. The fact that the single-particle states are already extended makes the whole issue of localization more subtle. However, we did not address this question, because the hybridization of impurity states with the quasiparticle continuum regularizes the problem naturally by introducing a cutoff at large distances.Second, for D 0 ø E F , the numerical solution of the impurity state clearly shows the strong cross-shaped anisotropy. Indeed, with good approximation, the impurity wave function is concentrated in the cross-shaped tails as shown in Fig. 2 Third, at nonzero impurity concentrations, the lifetime effects cut off the power-law behavior at finite lengths. As an illustration, consider a d-wave superconductor close to half-filling so that the Fermi surface is square. Let n be the normal to the two sheets of the Fermi surface which are most nearly normal to r, and define r Ќ jn ? rj and r k j͑e 3 3 n͒ ? rj; by definition, r Ќ $ r k . Thenwhere K 1 is the modified Bessel function of the first order, j Ќ a͑W ͞D 0 ͒ is the coherence length, and l Ќ j Ќ ͑D 0 ͞g͒ and l k a͑D 0 ͞g͒ are the cutoff length scales.Here, g is the inverse lifetime, W is the half-bandwidth, and a is the lattice spacing. Note that, for D 0 ø W , the anisotropy of the impurity state is fully developed because j Ќ ¿ a. Furthermore, the decay rate along the nodal directions is much slower than in the other directions: l Ќ ¿ l k . For r Ќ , l Ќ and r k , l k , we obtain the usual power-law behavior, and, for r Ќ . l Ќ or r k . l k , the impurity state is exponentially small to all directions due to the factors e 2r Ќ ͞l Ќ and e 2r k ͞l k . Thus, we may conclude that the power-law decay for all but the nodal directions is less important. As a consequence, the original problem of long-range hoppings with the exponential cutoff is mapped in the limit of a dilute concentration of unitary impurities onto an effective tight-binding model with the hopping range l Ќ ¿ j Ќ . In this model, weak localization leads to an extremely long localization length compared to the bare one in the absence of the impurity-induced quasiparticle states. ...

Chirald-wave superconductivity in doped graphene

“…In fact, it has even been shown that impurities can help to produce additional subdominant orders. For example, magnetic impurities in a d x 2 −y 2 -wave superconductor has been shown to locally induce an id xy component [110]. This is a consequence of the coupling of the impurity spin to the orbital moment of the condensate.…”

“…The key thing is not the equal weight but simply the presence of the additional order parameter and the relative π/2 phase shift between the two order parameters. A subdominant complex d xy order has been proposed to appear in the high-temperature d x 2 −y 2 -wave cuprate superconductors in the presence of, for example, magnetic fields [50,109,51], impurities [110], and surfaces [47,48,111].…”

Vortex state of ad-wave superconductor at low temperatures

“…(A41)), making the Zeeman orbital effects very small for general density of states. It is therefore worth pointing out that analyses of the d x 2 −y 2 -d xy mixing problem based solely on a symmetry-based analysis of the orbital moment [20] may lead to unrealistic results. Besides, it is important to note the very weak temperature dependence of the coefficient Q OZ of F OZ , as shown in Fig.…”

“…(20), there are two terms, F OZ and F 4 , competing in driving the induced d xy component. The first term F OZ , the so-called orbital Zeeman term, [12] can be rewritten as…”

“…This scenario is similar to one proposed by Movshovich et al [19] in the related case of magnetic impurities in a d-wave superconductor in zero field. Finally, Balatsky [20] has recently investigated the effect of the orbital Zeeman term near the upper critical field, and argued that the d-wave state is always unstable to a d x 2 −y 2 + id xy mixture. An unusual collective clapping mode in association with the relative phase of d xy to d x 2 −y 2 was predicted in this superconductor.…”