Magnetic disorder in superconductors: Enhancement by mesoscopic fluctuations

# Magnetic disorder in superconductors: Enhancement by mesoscopic fluctuations

I. S. Burmistrov Skolkovo Institute of Science and Technology, 143026 Moscow, Russia L. D. Landau Institute for Theoretical Physics, Kosygina street 2, 117940 Moscow, Russia Laboratory for Condensed Matter Physics, National Research University Higher School of Economics, 101000 Moscow, Russia    M. A. Skvortsov Skolkovo Institute of Science and Technology, 143026 Moscow, Russia L. D. Landau Institute for Theoretical Physics, Kosygina street 2, 117940 Moscow, Russia
July 14, 2019
###### Abstract

We study the density of states (DOS) and the transition temperature in a dirty superconducting film with rare classical magnetic impurities of an arbitrary strength described by the Poissonian statistics. We take into account that the potential disorder is a source for mesoscopic fluctuations of the local DOS, and, consequently, for the effective strength of magnetic impurities. We find that these mesoscopic fluctuations result in a non-zero DOS for all energies in the region of the phase diagram where without this effect the DOS is zero within the standard mean-field theory. This mechanism can be more efficient in filling the mean-field superconducting gap than rare fluctuations of the potential disorder (instantons). Depending on the magnetic impurity strength, the suppression of by spin-flip scattering can be faster or slower than in the standard mean-field theory.

## I Introduction

The properties of superconductors in the presence of impurities have remained at the focus of intense theoretical and experimental research during the past half-century. It is generally accepted that the potential scattering in -wave superconductors affects neither the transition temperature, , nor the density of states (DOS), . This statement usually referred to as Anderson’s theorem AG0a; AG0b; Anderson1959 is valid for sufficiently good metals. As the potential disorder increases, the emergent inhomogeneity due to the interplay of quantum interference (Anderson localization) and interaction leads to modification of Maekawa1982; Maekawa1984; Muttalib1983; Fin1987; Feigelman2007; Feigelman2010; FS2012; BGM2012; BGM2015 and Maki1970; Abrahams1970; Castro1990; BGM2016, with the effect being controlled by the parameter (where is the Fermi momentum and is the mean free path).

Magnetic impurities violating the time-reversal symmetry affect superconductivity much stronger, already at . Classical magnetic impurities lead both to suppression of and to reduction of the superconducting gap in with the increase of their concentration AG2. Beyond the Born limit, magnetic impurities produce degenerate subgap bound states (see Fig. 1a). Their hybridization results in the formation of an energy band giving rise to a nontrivial DOS structure Yu; Soda; Shiba; Rusinov. The account for the Kondo effect MHZ1971; Matsuura1977; Bickers1987, the indirect exchange interaction between magnetic impurities Ruvalds1981, or the spin-flip scattering assisted by the electron-phonon interaction Jarrell1990 can lead to the reentrant behavior of vs.  (see Ref. Balatsky for a review).

A hard gap in obtained for superconductors with magnetic impurities in the mean-field approximation is smeared by inhomogeneity. This can be due to rare fluctuations of a potential disorder LamS-1; LamS-2; MS; MaS, SilvaIoffe, or superconducting order parameter LO. A combined theory of these mechanisms has been developed in Refs. SkF; FS2016.

In this Letter we describe a novel mechanism for smearing of the superconducting gap. We reconsider the problem of rare classical magnetic impurities with the Poissonian statistics in a dirty superconductor. The key point that distinguishes our work from the previous ones is that we take into account mesoscopic fluctuations of the local DOS in a potential disorder. Physically, this implies that the energies of subgap bound states become dependent on the spatial positions of magnetic impurities (see Fig. 1b). Averaging over these bound states results in a non-zero homogenous DOS at all energies in the region of the phase diagram where in the absence of this effect is zero within the mean-field approximation. Motivated by the recent experiment on magnetic Gd impurities in superconducting MoGe films Rogachev2012, in this Letter we develop the theory of the enhancement of magnetic disorder by mesoscopic fluctuations in the case of a dirty superconducting film.

The outline of the paper is as follows. In Sec. II we present description of dirty superconductors with rare magnetic impurities in terms of the nonlinear sigma model and its renormalization. Our results for the renormalized spin-flip rate, superconducting transition temperature, and the density of states are given in Sec. III. We end the paper with discussions (Sec. IV) and conclusions (Sec. V) Some details of calculations are presented in Appendices.

## Ii Nonlinear sigma model for paramagnetic impurities

We consider a two-dimensional (2D) dirty -wave superconductor in the presence of both potential (spin-preserving) and magnetic disorder. Scattering off the former is responsible for the dominant contribution to the momentum relaxation rate . Much weaker spin-flip scattering rate is related with the exchange interaction between magnetic impurities and electrons described by the Hamiltonian

 Hmag=J∑jψ†(rj)Sjσψ(rj). (1)

We shall treat rare magnetic disorder under standard assumptions Yu; Soda; Shiba; Rusinov; MaS: (i) impurity positions have the Poisson distribution; (ii) impurity spins are classical statistically independent vectors with the flat distribution over their orientations, .

The low energy description of two-dimensional disordered superconductors with rare paramagnetic impurities can be conveniently formulated in terms of the replicated verion of a nonlinear sigma-model Fin; BK. Its action can be written as

 S=SD+SΔ+Smag. (2)

Here is the standard diffusive action

 SD=πν8∫d2rtr[D(∇Q)2−4(ετ3+Δτ1)Q], (3)

where and denote the density of states at the Fermi energy (per one spin projection) and the diffusive coefficient in the normal state, respectively. The matrix operates in the spin, Nambu, replica, and Matsubara energy spaces. It is subject to the following constraints Houzet_Skvortsov:

 Q2=1,Q=¯¯¯¯Q≡τ1σ2QTτ1σ2. (4)

Here the transposition acts in both the Matsubara energy space and the replica space. The Pauli matrices () act in the Nambu (spin) spaces. The matrix is the diagonal matrix with the elements .

The superconducting correlations are described by the order-parameter matrix which is diagonal in the Nambu space with matrix elements . In the absence of a supercurrent, is chosen to be real. The action reads

 SΔ=νλT∫d2rN∑a=1|Δa(r)|2. (5)

Here stands for the number of replica and denotes the attraction amplitude in the Cooper-channel.

We consider the case of rare classical magnetic impurities with the concentration [the precise condition on see below], when the magnetic part of the action, , becomes separable in the individual magnetic impurities MS:

 Smag≈∑js(j)mag=−12∑jtrln(1+i√αQ(rj)τ3σnj). (6)

Here stands for the three-dimensional unit vector and the dimensionless parameter is expressed in terms of the impurity spin and exchange constant . We note that approximation (6) of the full action is equivalent to the self-consistent -matrix approximation for magnetic scattering which treats all orders in scattering off a single magnetic impurity but neglects diagrams with intersecting impurity lines.

Performing the Poisson averaging over positions of the magnetic impurities with the help of the following relation Friedberg1975

 ⟨exp∑jf(rj)⟩=exp{ns∫d2r[ef(r)−1]}, (7)

we find that the contribution to the nonlinear sigma model action due to magnetic impurities becomes

 Smag→−ns∫d2r(⟨e12trln(1+i√αQ(r)τ3σn)⟩n−1). (8)

Here stands for the averaging over direction of the unit vector . Expanding in powers of , we find

 Smag= −ns∫d2r[∞∑m=1(−1)m−12m^CTm +12!∞∑m,n=1(−1)m+n4mn^CTmn +13!∞∑m,n,p=1(−1)m+n+p−18mnp^CTmnp+…]. (9)

Here we introduced the operators

 ^CTm= Ci1…imtr(QAi1…QAim), ^CTmn= Ci1…im+ntr(QAi1…QAim) ×tr(QAim+1…QAim+n), (10)

and so on. The operator acts as the symmetric tensor: . For convenience we defined the self-dual matrix . Since operators are symmetric with respect to its indices, the expansion can be written in the following form:

 Smag= ns∫d2r[14^CT2−18^CT11+18^CT4−112^CT31 −132^CT22+132^CT211−1384^CT1111+…]. (11)

The nonlinear sigma model action (2) with the magnetic part given by Eq. (11) provides full description of quantum effects for a dirty superconductor in the diffusive regime. These effects (weak localization and Aronov-Altshuler-type corrections) are responsible for the renormalization of system’s parameters, e.g. the diffusion coefficient and the attraction amplitude. In the 2D case, the magnitude of quantum corrections at the energy scale is governed by the parameter

 t(ε)=1πgln1|ε|τ, (12)

where is the bare dimensionless conductance of the film. In a superconductor, renormalization stops at . Assuming that the transition temperature is not too low, , one can neglect the renormalization of the conductance and interaction parameters between the energy scales and (see Refs. Fin; BK for a review). In contrast, renormalization of the magnetic-impurity part of the nonlinear sigma model is essential.

Treating this renormalization in the one-loop approximation, we find that after the renormalization this part of the action can be written as (see Appendix A)

 Smag=ns∫d2r[γ2^CT2+γ11^CT11+γ4^CT4+γ31^CT31 +γ22^CT22+γ211^CT211+γ1111^CT1111+…]. (13)

Here the coefficients , where , are given as follows:

 γk1k2…kq(t)=γk1k2…kq(0)en(n−1)t, (14)

where initial values of the coefficients follow from Eq. (11).

In what follows we are interested in the singlet sector of the theory. Therefore, one can operate with matrix which is the unit matrix in the spin space, . Then we can average over directions of the impurity magnetization in operators . Then the renormalized magnetic-impurity part of the nonlinear sigma-model action (13) can be written in the following convenient short-hand notation (see Appendix B):

 Smag=−ns∫d2r⟨exp{14trln[1+a(Qτ3)2]}−1⟩a, (15)

where the averaging is defined with respect to the following log-normal distribution function:

 (16)

Comparing Eq. (15) with Eq. (8) we may interpret the effect of renormalization as follows: Now instead of a single value of there is a log-normal distribution of the effective strength of impurity a, schematically shown in Fig. 1.

## Iii Results

In the mean-field approximation, a dirty superconductor in the diffusive regime is described by two coupled equations: the self-consistency equation for the superconducting order parameter and the Usadel equation for the quasiclassical Green’s function Usadel; Beltzig; Kopnin. These equations can be derived as the saddle-point equations of the nonlinear sigma model described in the previous section. The mean-field solution for the matrix can be parametrized as

 Q=τ1sinθ+τ3cosθ. (17)

Performing variation of the action on the configuration (17) with respect to , we find the following mean-field self-consistency equation:

 Δ=πλT∑εsinθε. (18)

For the study of space-averaged configurations at the mean-field level, it is sufficient to retain the term of the first order in trace only in the renormalized action (15):

 (19)

Since the eigenvalues of are , we find explicitly

 (20)

Performing variation of the action [with given by Eq. (20)] on the configuration (17) with respect to , we find the following modified Usadel equation:

 εsinθε−Δcosθε+nsπν⟨asin2θε1+a% 2+2acos2θε⟩a=0, (21)

where is the energy-dependent spectral angle, is the normal DOS at the Fermi energy per one spin projection, and denotes the fermionic Matsubara frequencies. We remind that the averaging in Eq. (21) is defined with respect to the log-normal distribution function (16). Since , Eq. (21) at coincides with the standard Usadel equation in the case of magnetic impurities AG2; Shiba; Rusinov; MaS. The linearity of Eq. (21) in is justified for small concentration of magnetic impurities: , where is the dirty superconducting coherence length.

The quantity a in Eq. (21) plays a role of the renormalized impurity strength. Since the bare impurity strength is proportional to the local DOS which is subjected to mesoscopic fluctuations, reflects the log-normal distribution of the local DOS in 2D weakly disordered systems AKL1986; Lerner1988. Contrary to naive expectations, one should average over a the Usadel equation rather than physical observables, e.g. the DOS. This is a consequence of the Poisson distribution of impurity positions .

### iii.1 Effective spin-flip rate

In the vicinity of the thermal transition, and we can linearize Eq. (21) with respect to . This procedure yields

 θε≈Δ/(ε+1/τs), (22)

where the effective spin-flip rate is given by

 1τs=2nsπν⟨a(1+a)2⟩a. (23)

At , one recovers the standard expression for the bare spin-flip rate due to magnetic impurities, Rusinov. In the limiting cases and , the spin-flip rate (23) becomes enhanced in comparison with the bare one: and , respectively. For an arbitrary value of , the asymptotic expansion at has the form (see Appendix C.1):

 τs0τs≈1+2−16α+6α2(1+α)2t(Tc)+O(t2). (24)

At small , the spin-flip rate is suppressed (enhanced) for (otherwise), where . The overall behavior of the ratio as a function of and is illustrated in Fig. 2.

### iii.2 Transition temperature

Since the spin-flip rate (23) is the only characteristic of magnetic disorder that enters the linearized solution for , we obtain the standard equation for the superconducting transition temperature (see Appendix C.2):

 lnTc0Tc=ψ(12+12πTcτs)−ψ(12), (25)

where denotes the transition temperature in the absence of magnetic impurities, and stands for the di-gamma function. Equation (25) was derived by Abrikosov and Gor’kov (AG) in the Born limit () AG2, and later was shown to describe the suppression of for arbitrary values of Rusinov. For a scale-independent spin-flip time, , Eq. (25) defines a universal function shown by the black dashed line in Fig. 3. Superconductivity is eventually destroyed at the critical spin-flip rate AG2. This standard approach corresponds to the limit , when mesoscopic fluctuations can be neglected.

An essential modification introduced by the log-normal distribution of the impurity strength (16) is that now the spin-flip rate depends on the parameter , i.e. on the conductance and the transition temperature itself. This leads to an unusual behavior illustrated in Fig. 3, where we present the numerical solutions of Eq. (25) for fixed values of and and for various values of . At finite , dependence of on renders the curves sensitive to a particular value of . In the range , the spin-flip rate decreases monotonously down to zero with increasing . Therefore the reduction of with the increase of is slower than for . This agrees qualitatively with the slowdown of suppression with increasing the film resistance measured in Ref. Rogachev2012. In the opposite case, for and , the dependence of on is qualitatively different since the ratio can be larger than unity and is a non-monotonous function of . Since the spin-flip rate is enhanced, the reduction of with the increase of is faster than in the case . The non-monotonicity of results in the existence of two solutions of Eq. (25) for . Formally, it admits the solution with nonzero for any value of the parameter . However, we remind that our approach is valid provided the inequality holds.

The dependence of the spin-flip rate on transforms into the dependence of on the film conductance. To illustrate this effect, we fix the value of the parameter and plot the ratio on the film resistance for some values of in the inset to Fig. 3. Since for the spin-flip rate decreases monotonously with the increase of , is enhanced with respect to . The non-monotonous dependence of on obtained for and leads to the reentrant behavior of on .

It is worthwhile to mention that not only the suppression of by magnetic impurities but also the reduction of is modified at finite due to the log-normal distribution of the effective impurity strength elsewhere.

### iii.3 Density of states

Consider now the superconducting phase with a finite . The DOS can be obtained from the solution of Eq. (21) after analytic continuation to real energies : . It is convenient to parametrize the spectral angle as . Without renormalization (), the angle should be determined from equation , where AG2; Shiba; Rusinov; MaS

 FE(ψ)=sinhψ−EΔcoshψ−[αns/(πνΔ)]sinh2ψ1+α2−2αcosh2ψ. (26)

This leads to a complicated structure of the DOS at energies , which depends on the values of and (see Ref. FS2016 for a review). In the case , the impurity band touches the Fermi energy, leading to a finite DOS at . Below we shall consider the opposite regime, , in which has a finite gap for . The gap opens since possesses only real solutions at energies .

Typical modification of the DOS at finite is illustrated in Fig. 4, where we plot obtained by numerical solution of Eq. (21) at . For , mesoscopic fluctuations of magnetic disorder affect the DOS in two ways. (i) On the perturbative level, they shift the position of the gap: , with , but the gap remains hard. (ii) A finite DOS below the renormalized gap is then generated nonperturbatively in , due to the tail of the distribution . In Fig. 4, the smearing of can be clearly seen for , whereas for larger the smearing and gap shift cannot be separated. A profound feature of the DOS is its finite value right at the Fermi energy.

In the limit of weak renormalization, , the DOS can be obtained analytically. The general expression is quite cumbersome (see Appendix C.3), so we present here only the results in the regime of weak magnetic impurities (). The gap smearing at is described by

 ρ(E)/2ν=√2/3η−2/3Re√ϵ+iϵ∗, (27)

where and

 ϵ∗=η2/3√2π16√t(η2/3α)3/4exp(−116tln2η2/34α). (28)

The subgap DOS (27) decays as power law. The residual DOS at the Fermi energy it is determined by the probability to find and in the limit reads

 ρ(0)2ν=√πη8α3/4√texp(−116tln21α). (29)

This result is non-perturbative in both and . The dependence of on for some values of is shown in the inset to Fig. 4. Its non-monotonicity is related to that of as a function of . At a fixed value of , behaves non-monotonically with the impurity strength at a given value of .

## Iv Discussions

Our main Eq. (21) could be derived for a toy model of Poissonian magnetic impurities with the strength independently distributed according to . We emphasize however that in a disordered film the log-normal distribution is generated intrinsically due to mesoscopic fluctuations of the local DOS.

The log-normal distribution predicts an exponentially small probability for realization of very small and very large values of the effective impurity strength a. As well-known from the theory of mesoscopic fluctuations of the local DOS and wave function multifractality, this implies that typically the impurity strength is realized Mirlin2000. Using results of Ref. Foster2009, we obtain the following estimate for the termination points: (see Appendix D). In order our result (29) for were applicable to a typical sample, vicinity of should be inside the interval . It is fulfilled provided .

We emphasize the difference with the instanton analysis MaS; FS2016, where the effect of mesoscopic fluctuations on magnetic disorder was not taken into account: (i) in our approach, the DOS is modified already at the mean-field level and (ii) our results (27)–(29) involve a spreading resistance  which is parametrically larger than the sheet resistance emerging in the instanton analysis. As a result, our mechanism predicts a larger DOS at the Fermi level, whereas near it prevails provided (see Appendix C.3).

Although our results were derived for a weak disorder, , they can be extended to the case of a moderate disorder, (provided ) elsewhere. In this situation the mean-field equation for remains the same as Eq. (21), but the distribution function must be found taking Fermi-liquid renormalizations into account.

The enhancement of magnetic disorder due to mesoscopic fluctuations is not restricted to classical magnetic impurities. It is known Kettemann2006; Micklitz2006; Kettemann2007; Micklitz2007 that the Kondo effect in the disordered electron systems is also modified by mesoscopic fluctuations of the local DOS. Therefore, the theory for the interplay of the Kondo effect and superconductivity developed in Refs. MHZ1971; Matsuura1977; Bickers1987 needs to be modified for disordered films elsewhere.

The dependence of on the film conductance can be caused by a variety of reasons, among which are the dependence of the DOS at the Fermi energy on disorder, renormalization of the Cooper channel attraction in ballistic and diffusive regimes, Berezinskii-Kosterlitz-Thouless transition etc. Fin; BK The sensitivity of the spin-flip rate on the conductance is a new mechanism providing a nontrivial dependence of on .

## V Conclusions

To summarize, we reconsidered the problem of rare classical magnetic impurities with the Poissonian statistics in a dirty superconducting film. We took into account renormalization of the multiple spin-flip scattering due to mesoscopic fluctuations of the local DOS in a potential disorder. This effect results in the log-normal distribution of the effective magnetic impurity strength rendering the energies of quasiparticle bound states position dependent (see Fig. 1). In the superconducting state, this results in the smearing of the hard gap (obtained in the absence of spin-flip renormalization) and emergence of a non-zero DOS for all energies already at the mean-field level. Depending on the bare magnetic impurity strength, the superconducting transition temperature is suppressed by the spin-flip scattering slower or faster than in the absence of renormalization. Finally, we mention that our results can be extended to the model with an arbitrary distribution of magnetic impurities, the vicinity of a superconductor-insulator transition, the case with Coulomb repulsion in addition to attraction, the presence of Zeeman splitting, etc. elsewhere.

###### Acknowledgements.
We thank M. V. Feigel’man, Ya. V. Fominov, and A. D. Mirlin for useful discussions. The research was supported by Skoltech NGP Program (Skoltech–MIT joint project).

## Appendix A Renormalization of the action Smag

In this Appendix we present details of the one-loop renormalization of the magnetic-impurity part of the action.

To renormalize we write , where obeys two linear constraints: and , and the convergency condition . The matrix is assumed to be self-dual: . Then the quadratic part of the action reads:

 S(2)D[W]=−πνD8∫drtr(∇W)2. (30)

The quadratic part of the action determines the following contraction rules:

 ∂t⟨trAWtrBW⟩=tr[AB−AΛBΛ+A¯¯¯¯B−AΛ¯¯¯¯BΛ], (31a) ∂t⟨trAWBW⟩=trAtrB−trAΛtrBΛ −tr[A¯¯¯¯B−AΛ¯¯¯¯BΛ], (31b)

where with denoting the infrared length scale.

Next we write the matrix as where the slow field obeys the condition . Using contraction rules (31), we find

 ∂ttrAQBQ=∂t⟨trUAU−1ΛWUBU−1ΛW⟩ =[trAtrB−trAQtrBQ]−tr[A¯¯¯¯B−AQ¯¯¯¯BQ], (32a) ∂ttrAQtrBQ=∂t⟨trUAU−1ΛWtrUBU−1ΛW⟩ =−tr[AQBQ−AB−A¯¯¯¯B+AQ¯¯¯¯BQ]. (32b)

The action for the slow modes after integration over fast modes can be found as

 Smag→−ln⟨e−Smag⟩W=⟨Smag⟩W−⟨⟨S2% mag⟩⟩W/2+… (33)

Here denotes the averaging over fast modes . In what follows, in the expansion in the right-hand side of Eq. (33) we neglect all terms except the lowest order one in the impurity concentration, . The smallness of the omitted terms is controlled by the condition , where is the superconducting coherence length in the dirty limit. As we shall see below, it is the first term in the right-hand side of Eq. (33) that is responsible for the logarithmic renormalization of .

### a.1 Operators of the second order in Q

In the Born approximation (first order in ) we need to consider the operators and with two matrices involved. Their contribution to is controlled by the coefficients and with the initial conditions following from (11): and .

Using the contraction rules (32), we find

 ∂t(^CT2^CT11)=M2(^CT2^CT11),M2=(1−1−20). (34)

The operators and transform into each other under the renormalization. We note that under renormalization the operators with the same or fewer number of matrices are generated only. The eigenvalues of are equal to and . The eigenvalue corresponds to the operator :

 ∂t(^CT2−12^CT11)=2(^CT2−12^CT11). (35)

The operator is known to be a pure scaling operator beyond the lowest order perturbation theory Wegner1980; Wegner1986; Wegner1987a; Wegner1987b.

We emphasize that the operators of the second order in enter the magnetic part of the action, Eq. (11), precisely in combination . This implies that

 γ2(t)=14e2t,γ11(t)=−18e2t. (36)

### a.2 Operators of the fourth order in Q

The next nontrivial order in involves operators which are of the fourth order in . Their flow is described by the system

 ∂t^CT4 =6(^CT4)−4(^CT31)−2(^CT22)−4α^CT2, (37a) ∂t^CT31 =−6(^CT4)+3(^CT31)−3(^CT211)−6α^CT2−3α^CT11, (37b) ∂t^CT22 =−8(^CT4)+2(^CT22)−2(^CT211), (37c) ∂t^CT211 =−8(^CT31)−2(^CT22)+(^CT211)−(^CT1111)−8α^CT11, (37d) ∂t^CT1111 =−12(^CT211). (37e)

The operators of the forth order in are mixed under the renormalization. In addition, the operators of the second order in are generated. The system of equations (37) can be cast in the matrix form

 ∂t⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝^CT4^CT31^CT2,2^CT211^CT1111^CT2^CT11⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=M4⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝^CT4^CT31^CT2,2^CT211^CT1111^CT2^CT11⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,M4=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝6−4−200−4α0−630−30−6α−3α−802−20000−8−21−10−8α000−12000000001−100000−20⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (38)

Here we used Eq. (34). We emphasize that the matrix is the upper triangular block matrix. This reflects the fact that under renormalization the operators with the same or fewer number of matrices are generated only. The matrix has the following eigenvalues: 12, 5, 2, 2, , , and . The largest eigenvalue corresponds to the operator . It is known that this operator is the pure scaling operator from arguments based on the group representation theory Wegner1980; Wegner1986; Wegner1987a; Wegner1987b. It is worth emphasizing that the operators of the forth order in enter the magnetic part of the action, Eq. (11), precisely in the combination . This implies that the coefficients in the action (13) are simply

 γ4(t)=18e12t,γ31(t)=−112e12t,γ22(t)=−132e12t,γ211(t)=132e12t,γ1111(t)=−1384e12t. (39)

### a.3 Renormalization of operators of arbitrary order in Q

In general, one can derive the following set of renormalization group equations:

 ∂t^CTn=n(n−1)2^CTn−n2n−1∑k=1(^CTk,n−k−(−α)min(k,n−k)^CT|n−2k|), (40)
 (41)
 ∂t^CTm,n,p=−2[mn(^CTm+n,p−(−α)min(m,n)^CT|m−n|,p)+mp(^CTm+p,n−(−α)min(m,p)^CT|m−p|,n)+np(^CTm,n+p−(−α)min(n,p)^CTm,|n−p|)]+m(m−1)+n(n−1)+p(p−1)2(^CTm,n,p)−[m2m−1∑k=1(^CTk,m−k,n,p−(−α)min(k,m−k)^CT|m−2k|,n,p)+n2n−1∑l=1(^CTm,l,n−l,p−(−α)min(l,n−l)^CTm,|n−2l|,p)+p2p−1∑s=1(^CTm,n,s,p−s−(−α)min(s,p−s)^CTm,n,|p−2s|)]. (42)

and so on. Using these equations we find for the renormalization of the action:

 ∂tSmag=−ns∫d2r{∞∑m=1(−1)m−12m[m(m−1)2^CTm+m2m−1∑k=1(−α)min(k,m−k)^CT|m−2k|] +12!∞∑m,n=1(−1)m+n4mn(−2mn)[^CTm+n−(−α)min(m,n)^CT|m−n|]+∞∑m=1(−1)m2mm2m−1∑k=1^CTk,m−k +12!∞∑m,n=1(−1)m+n4mn[m(m−1)^CTm,n+mm−1∑k=1(−α)min(k,m−k)^CT|m−2k|,n] +13!∞∑m,n,p=1(−1)m+n+p−18mnp(−6mn)[^CTm+n,p−(−α)min(m,n)^CT|m−n|,p]+…} =−ns∫d2r{∞∑m=1(−1)m−12mm(m−1)^CTm+12!∞∑m,n=1(−1)m+n4mn(m+n)(m+n−1)^CTmn+…} (43)

Note that all terms in Eq. (43) which contain cancel each other.

## Appendix B The renormalized action Smag

In this Appendix we present the details of derivation of Eq. (8).

All in all, we find from Eq. (43) that the coefficients , where , behaves in the same way:

 γk1k2…kq(t)=γk1k2…kq(0)en(n−1)t. (44)

In what follows we are interested in the mean-field analysis of the renormalized action (13) for which the singlet sector of the theory is important only. Therefore, one can operate with matrix which is the unit matrix in the spin space, . Then averaging over directions of the impurity magnetization becomes trivial. We find (all indices, , , are even)

 ^CTm=(−α)m/2tr(Qτ3)m,^CTmn=(−α)(m+n)/2tr(Qτ3)mtr(Qτ3)n, (45)

and so on. Then the renormalized action for magnetic impurities becomes

 (46)

Decoupling the Gaussian part with an auxiliary integral over we obtain

 Smag=−¯¯¯ns∫