A Background field renormalization of the nonlinear sigma model action

Superconductor-insulator transitions: Phase diagram and magnetoresistance

Abstract

Influence of disorder-induced Anderson localization and of electron-electron interaction on superconductivity in two-dimensional systems is explored. We determine the superconducting transition temperature , the temperature dependence of the resistivity, the phase diagram, as well as the magnetoresistance. The analysis is based on the renormalization group (RG) for a nonlinear sigma model. Derived RG equations are valid to the lowest order in disorder but for arbitrary electron-electron interaction strength in particle-hole and Cooper channels. Systems with preserved and broken spin-rotational symmetry are considered, both with short-range and with long-range (Coulomb) interaction. In the cases of short-range interaction, we identify parameter regions where the superconductivity is enhanced by localization effects. Our RG analysis indicates that the superconductor-insulator transition is controlled by a fixed point with a resistivity of the order of the quantum resistance . When a transverse magnetic field is applied, we find a strong nonmonotonous magnetoresistance for temperatures below .

pacs:
72.15.Rn ,   71.30.+h ,   74.78.-w,   74.62.-c

I Introduction

Superconductivity [(1); (2)] and Anderson localization [(3)] are among most important and fundamental quantum phenomena in condensed matter physics. These two phenomena are in a sense antagonists: in the case of superconductivity the Cooper interaction creates a collective state with vanishing resistivity, while the Anderson localization resulting from disorder-induced quantum interference drives the system into a state with zero conductivity. Therefore, when both interaction and disorder are present, a competition between the superconductivity and localization naturally arises. This competition is of particular interest in two-dimensional (2D) geometry, where even a weak disorder makes the system an Anderson insulator. Thus, a 2D system may be expected to undergo a direct quantum phase transition (QPT) between the insulating and superconducting states, the superconductor-insulator transition (SIT).

Experimentally SIT has been studied in a variety of 2D structures, including amorphous Bi and Pb [(4); (5)], MoC [(6)], MoGe [(7)], Ta [(8)], InO [(9); (11)], NbN [(12)] and TiN films [(13); (14)], see also the reviews [(15)]. In recent years, there have been also a growing experimental activity on SIT in novel 2D materials and nanostructures, such as LaAlO/SrTiO interfaces [(16); (17)], SrTiO surfaces [(18); (19)], MoS flakes [(20); (21)], FeSe thin films [(22)], LaSrCuO surfaces [(23)], and LiZrNCl layered materials [(24)]. Characteristic for many of the novel structures is a strong screening of the Coulomb interaction due to a large dielectric constant of the substrate (such as SrTiO). In addition, strong spin-orbit coupling is present in many of the novel materials (MoS, LaAlO/SrTiO, SrTiO).

To drive the system through SIT, one changes a parameter (film thickness, gate voltage, doping) controlling the high-temperature sheet resistivity. With lowering temperature, systems with lower resistivity become superconducting (resistivity drops to zero), while those with higher resistivity get insulating (resistivity becomes exponentially large). The most salient observations common to the majority of the above experiments are as follows:

  1. Most of the experiments are interpreted as supporting a direct transition between the superconducting and insulating phases, although some of them suggest a possibility of existence of an intermediate metallic phase. The critical resistivity (the low-temperature limit of the separatrix curve separating the temperature dependence of resistivity in the insulating and superconducting phases) is of the order of the quantum resistance . However, the precise value of varies from one experiment to another, roughly in the range between and .

  2. For those systems that are superconducting (at low temperature and magnetic field ), a non-monotonous dependence of resistivity on and is observed. In particular, a giant non-monotonous magnetoresistance is found in such systems at very low temperatures, .

  3. The temperature dependence of resistivity on the insulating side is very fast (activation or even stronger).

Theoretical investigation of the interplay of interaction and disorder in systems with Cooper attraction has a long history. Soon after the development of the microscopic theory of superconductivity by Bardeen, Cooper, and Schrieffer (BCS) [(2)], the question of influence of disorder on superconductivity attracted a great deal of attention. It was found [(25); (26)] that the diffusive motion of electrons does not affect essentially the temperature of superconducting transition, i.e the mean free path does not enter the expression for . This statement is conventionally called “Anderson theorem”.

Effects of disorder-induced Anderson localization [(3)] on superconductivity were considered in Refs. [(27); (28)]. It was found that, within the BCS approach, the superconductivity in a disordered system persists up to the localization threshold and even in the localized regime near the Anderson transition. Furthermore, Refs. [(27); (28)] came to the conclusion that the mean-field transition temperature in these regimes remains unaffected by disorder (i.e. the Anderson theorem holds). In a parallel line of research, it was discovered [(29); (30); (31)] that an interplay of long-range () Coulomb interaction and disorder leads to suppression of . These ideas were put on the solid basis by Finkelstein [(32)] who developed the nonlinear sigma model (NLSM) renormalization-group (RG) formalism.

An alternative approach to the SIT known as “bosonic mechanism” was proposed in Refs. [(33)]. It takes into account the superconducting phase fluctuations and discards completely all other degrees of freedom, in particular, the localization effects. It was also proposed that an intermediate “Bose metal” phase may separate the superconductor and insulator [(34)]. A relation between the bosonic and fermionic mechanisms as a well as a status of the Bose metal conjecture remain quite obscure.

Recently, Feigelman et al. [(35); (36)] found that the eigenfunction multifractality near the localization threshold in three dimensions strongly affects properties of a superconductor. Their remarkable finding is that is dramatically enhanced: its dependence on the coupling constant is no longer exponential (as in the conventional BCS solution) but rather of a power-law type. This result was obtained on the basis of the BCS-type self-consistency equation, with Cooper attraction being the only interaction included.

In a preceding work by the present authors [(37)] the influence of disorder-induced Anderson localization on the temperature of superconducting transition was studied within the field-theoretical framework. Electron-electron interaction in particle-hole and Cooper channels was taken into account. The focus was put on the case of a weak short-range interaction (which is relevant to materials with large dielectric constant, as well to cold atom systems). Two-dimensional systems in the weak localization and antilocalization regime, as well as systems near mobility edge were investigated. A systematic analytical approach to the problem was developed in the framework of the interacting NLSM and its RG treatment. The approach took into account the mutual renormalization of disorder and all interaction constants (that, in particular, leads to mixing of different interaction channels). This methodology allows us to explore both the cases of a long-range (Coulomb) interaction previously studied by Finkelstein [(32)] and of a weak short-range interaction within a unified formalism. More specifically, in the case of short-range interactions a system of coupled RG equations for the problem was derived in the lowest order in disorder and three interaction couplings (singlet, triplet, and Cooper channels).

The analysis of RG equations for the weak short-range interaction showed the behavior which is exactly opposite to that predicted by Ref. [(32)] for Coulomb interaction. It was found that the interplay of such interactions and Anderson localization leads to strong enhancement of superconductivity in a broad range of parameters in dirty 2D systems, as well as in three dimensional (3D) systems near the Anderson transition (in contrast to the suppression in the Coulomb case). In the latter case (vicinity of the Anderson transition), the microscopic theory of Ref. [(37)] justified previous theoretical results obtained from the self-consistency equation [(35); (36)].

This result of Ref. [(37)] is of fundamental importance and represents an unexpected physics (enhancement of superconductivity by localization, which is naively its exact antagonist). Indeed, remarkably, the localization physics, responsible for the increase of resistivity and thus driving the system towards an insulating state, favors at the same time the superconductivity. The key condition is a suppression of the long-range component of the Coulomb interaction (see also Ref. [(38)]). This opens a new way for searching novel materials exhibiting high-temperature superconductivity: one needs the combination of a large dielectric background constant and disorder in layered structures.

In this paper, we extend the formalism of Ref. [(37)] by deriving the RG equations to the lowest order in disorder but, formally, for arbitrary interaction couplings. We use this framework to explore systematically the interplay of superconductivity, interaction, and localization in 2D systems, with a focus on the SIT in thin films. More specifically:

  1. We evaluate the temperature dependence of the resistivity for given bare (high-temperature) couplings down to the temperature at which the finite expectation value of the superconducting order parameter emerges, or else, down to the temperature where the system enters the insulating regime.

  2. We use the RG equations to determine the structure of the phase diagram. In particular, we identify parameter regions where the superconductivity is enhanced by localization. Our results also indicate that in some cases the phase diagram may include a critical-metal phase.

  3. We study the magnetoresistance near the SIT within two-step RG approach. Since the magnetic field suppresses both superconductivity and localization, a non-monotonous magnetoresistance arises, as observed experimentally. Furthermore, this magnetoresistance becomes very strong at low temperatures, again in agreement with experiments. Both orbital and Zeeman effects of the magnetic field are incorporated in the unifying RG scheme.

All the above analysis is performed for the cases of short-ranged and long-ranged Coulomb interaction, both with and without spin-orbit interaction.

The structure of the article is as follows. In Sec. II we introduce the NLSM formalism. The corresponding RG equations (valid to the lowest order in disorder and for arbitrary interaction strength) are presented in Sec. III. The RG equations are used in Sec. IV to analyze the phase diagram in zero magnetic field. The temperature dependence of resistivity in zero magnetic field is discussed in Sec. V. In Sec.  VI this analysis is extended to calculate the magnetoresistance in a transverse and in a parallel magnetic field. Section VII contains a discussion of obtained results, their implications, limitations, possible extensions, comparison with numerical and experimental results. Finally, our results and conclusions are summarized in Sec. VIII. Several Appendices contain technical details of the derivation of RG equations and of their analysis.

Ii Formalism

ii.1 NLSM action

The action of the NLSM is given as a sum of the non-interacting part, , and contributions arising from the interactions in the particle-hole singlet, , particle-hole triplet, , and particle-particle (Cooper), , channels (see Refs. [(32); (39)] for review):

(1)

where

Here is the total Drude conductivity (in units and including spin), , and we use the following matrices

(2)

with standing for replica indices and corresponding to the Matsubara fermionic energies . The sixteen matrices,

(3)

operate in the particle-hole (subscript ) and spin (subsrcipt ) spaces with the corresponding Pauli matrices denoted by

(4)
(5)

Matrices and stand for the unit matrices. The matrix field (as well as the trace ) acts in the replica, Matsubara, spin, and particle-hole spaces. It obeys the following constraints:

(6)

The charge conjugation matrix satisfies the following relation . Matrix can be parameterized as where the matrices obey (symbol denotes the complex conjugation)

(7)

In order to avoid notational confusion, it is instructive to compare our notation with that of the reviews [(32)] and [(39)]. In both references, a different definition of Pauli matrices in the particle-hole space has been used, namely, instead of for . In Ref.  [(32)] Pauli matrices in the spin space coincide with our definition (5). In Ref. [(39)] the spin-space Pauli matrices (for ) were used instead of our definition (5). The interaction terms , and coincide with terms in Eqs. (3.9a), (3.9b), and (3.9b) of Ref. [(32)] provided the following relations between the couplings , and in , and and , and in Ref. [(32)] hold: , , and . Here the thermodynamic density of states includes the spin-degeneracy factor. Note that Ref. [(32)] focuses on the case of unscreened (long-ranged) Coulomb interaction. Hence the interaction amplitude in the singlet particle-hole channel is expressed through the frequency renormalization factor there. We consider both long-ranged (Coulomb) and short-ranged interactions. In the latter case the quantities and are independent variables. The interaction terms , and coincide with the terms in Eqs. (3.92d), (3.92e), and (3.92f) of Ref. [(39)] provided , , and . The parameters and in are related to the corresponding quantities of Ref. [(32)] as and and to the parameters and in Ref. [(39)] as and .

ii.2 Interaction in the Cooper channel

The Cooper-channel interaction term can be rewritten as

(8)

Here the matrix is defined as

(9)

However, for we find

(10)

Therefore, the term describing the interaction in the Cooper channel is fully determined by the Cooper-singlet channel:

(11)

ii.3 Relation with the BCS hamiltonian

In general, bare values of the interaction parameters and can be estimated for a given electron-electron interaction in a microscopic hamiltonian. It is convenient to introduce the dimensionless parameters . Then their bare values can be written as

(12)

where ,

(13)

Here stands for the statically screened interaction and denotes averaging over the Fermi surface. In the BCS case (for example, for a weak short-range attraction mediated by phonons), the interaction can be written as where . Neglecting screening in this case we find

(14)

Thus, for the BCS case (i.e. when neither screened nor unscreened Coulomb repulsion is taken into account), we get the following interaction parameters at the ultraviolet scale (which is given by Debye frequency in the case of phonon-induced superconductivity):

(15)

If disorder is strong, , the relations (15) determine initial values of the interaction parameters for the action (1). In what follows, we will refer to the line determined by relations as the “BCS line”. When disorder is weak, , the relations (15) hold at the scale corresponding to the Debye frequency . Then the Cooper interaction constant is renormalized at ballistic scales (between and ) such that

(16)

where .

ii.4 algebra and invariance

The NLSM action (1) involves the matrices which are formally defined in the infinite Matsubara frequency space. To perform calculations with these matrices, it is convenient to introduce an ultraviolet cutoff for the Matsubara frequencies. In addition, it is useful to introduce another cutoff indicating the size of a non-trivial part of the matrix (beyond which the matrix equals ). At the end of calculations both cutoffs should be sent to infinity.

Global rotations of the matrix with any matrix of the type , where , play an important role [(40); (41)]. In the limit and , the set of rules known as algebra [(40)] allows one to establish the following relations (for and ):

(17)

Using Eqs. (17), one can check that, provided , the action (1) is invariant under global rotations of the matrix with the matrix (so called invariance). The constraint corresponds to the case of Coulomb interaction [(32)]. Since the relation is dictated by the symmetry of the action (1) it should remain fulfilled under the RG flow.

Iii One-loop renormalization-group equations

iii.1 Preserved spin-rotational symmetry

To derive RG equations in the one-loop approximation (i.e., to the lowest order in disorder strength), we employ the background-field method and apply it to renormalization of the NLSM action (1). Details of the derivation can be found in Appendix A. In dimensions the one-loop RG equations read []:

(18)
(19)
(20)
(21)
(22)

where ( denotes the mean free path) and . These RG equations describe the evolution of the system with spin-rotational and time-reversal symmetries upon changing the characteristic length scale . We stress that RG equations (18) - (22) satisfy the particle number conservation since . Further, it is worth emphasising that the right-hand-sides of the equations are nonsingular in the limit of Coulomb interaction, .

The ultraviolet value of the NLSM coupling that describes the disorder strength is given by the dimensionless Drude resistivity. The renormalization of at larger scales involves the contributions to the resistivity induced by interference effects and by virtual (elastic) processes due to interactions in particle-hole singlet () and triplet (), as well as in Cooper channel ().

We emphasize that Eqs. (18) - (22) are obtained in the lowest order in but they are formally exact in interactions . It is worth noting that the Cooper-interaction coupling enters all the RG equations only in a polynomial way. Interestingly, the contribution of Cooper channel to the renormalization of is fully described by the linear term only, thus rendering Eq. (18) for arbitrary the same as in the weak-coupling limit [(42)], .

The first term in Eq. (21) describes the standard BCS instability; in accordance with the “Anderson theorem” this term is not affected by disorder. Moreover, the “Anderson theorem” manifests itself in Eq. (21) through the absence of the terms on the right hand side. To the lowest order in interaction couplings, the effect of disorder on the renormalization of is solely due to the presence of the interaction in the particle-hole channels.

Somewhat counter-intuitively, Eq. (18) suggests an insulating behavior (an increase of the resistivity with increasing ) for . We note, however, that the (dimensionless) physical resistivity is not exactly equal to the NLSM coupling because of the inelastic contribution to the conductivity governed by superconducting fluctuations, see Sec. V below for details. Near the superconducting instability (for large ), this antilocalizing inelastic contribution to the conductivity becomes large.

Furthermore, towards the superconducting instability, , the disorder-induced renormalization of in Eq. (21) is dominated by the term which tends to impede a development of the superconducting instability. Thus, if Eqs. (18) - (22) would constitute the ultimate truth, the superconducting instability would not, strictly speaking, develop. An explanation for this apparent paradox is as follows. It turns out that the one-loop RG equations become insufficient in a vicinity of the superconducting instability, namely, on scales larger than where reaches a value . In other words, the weak-disorder condition of validity of the one-loop RG, , should in fact be supplemented by the condition .

The emergence of the latter condition (and thus of the scale ) becomes evident from a comparison of the terms of the zeroth and the first order in in Eq. (21). This scale arises also in the calculation of the conductivity (see Sec. V): at this scale the inelastic contribution to the conductivity reaches in magnitude the elastic one. We expect that in the vicinity of the superconducting instability higher-loop terms of the type in the beta-function for and in the equation governing renormalization of should emerge. Upon resummation, they are expected to restore the divergence of at a scale slightly larger than . At the same time, since the second-loop () terms are similar to those describing mesoscopic fluctuations of the superconducting order parameter [(44); (45)], we expect for (i.e., for temperatures slightly above the transition) strong spatial fluctuations of observables (in particular, of the local tunneling density of states [(46); (45)], as observed in experiments, see, e.g., Ref. [(11)]).

To the lowest order in , Eqs. (18) - (22) coincide with results obtained by Finkelstein long ago [(42)]. Recently, one-loop RG equations beyond the lowest order in interactions were reported in Ref. [(43)] for the case of preserved spin-rotational and time-reversal symmetries. It should be stressed, however, that our RG equations (18) - (22) differ from those of Ref. [(43)]. It is instructive to highlight the difference. First of all, the right hand side of a RG equation for in Ref. [(43)] [see Eq. (A12) there] contains a term proportional to rather than to as in our Eq. (19). Since the quantity should have no renormalization by virtue of the particle number conservation, this would imply the presence of a term proportional to in the RG equation for . Being divergent for the case of Coulomb interaction, , such a term would, however, violate the -invariance of the NLSM action (1) and is thus not allowed. Second, the RG equation for reported in Ref. [(43)] does not contain the term proportional to , in contrast to our Eq. (20). Finally, the RG equation for reported in Ref. [(43)] contains an additional term proportional to as compared to our Eq. (21). We note that a similar term was reported by Belitz and Kirkpatrick in Ref. [(39)] [see Eq. (6.8g) there]. In our opinion, such terms, divergent for the case of Coulomb interaction, , cannot appear in the course of renormalization of -invariant operators, including . In Ref. [(47)], the appearance of a term proportional to in the RG equation for of Ref. [(39)] was attributed to an improper treatment of the gauge invariance. In our background-field RG calculations, terms proportional to do appear in the course of renormalization of at intermediate steps but cancel each other in the final results, in agreement with the -invariance, see Appendix A.

iii.2 General case

The RG equations (18) - (22) have been derived for the case of preserved spin-rotational symmetry. We are now going to generalize them to systems with spin-rotational symmetry broken (partly or fully) due to spin-orbit coupling and/or spin-orbit impurity scattering. Both these symmetry-breaking mechanisms induce finite relaxation rates (, , ) for corresponding components of the electron spin. The relaxation rates determine the mass of the corresponding triplet modes (diffusons and cooperons). As an example, the mode corresponding to the spin component acquires a mass proportional to . This mode thus become effectively frozen and drops out of RG equations at length scales .

In the presence of spin-orbit coupling, there is the spin relaxation due to D’yakonov-Perel’ mechanism. The corresponding relaxation rates are given by , where denotes the spin-orbit splitting [(48)]. Therefore, all triplet modes (both for diffusons and cooperons) are suppressed at the length scales , i.e., the number of triplet modes contributing to the RG equations is . In the case of a 2D electron system with the spin-orbit impurity scattering but without spin-orbit coupling, the spin relaxation is anisotropic: , , where denotes the skew scattering rate [(49)]. Thus, for the triplet modes corresponding to the total spin component remain massless. Therefore, in this case triplet mode still contributes to the RG equations.

If the spin-orbit coupling and spin-orbit scattering are both present, then different regimes with , and can be realized depending on the relations between , and . For all three cases the one-loop RG equations can be written as

(23)
(24)
(25)
(26)
(27)

In the case , Eq. (25) should be omitted. The RG equations (23) - (26) constitute one of the main results of the paper. In the rest of the paper, we will analyze these equations to investigate phase diagrams and observables for the cases of preserved and broken spin-rotational symmetry.

The system of RG equations (23) - (26) has the fixed plane corresponding to the case of long-ranged Coulomb interaction. In fact, this statement is not restricted to the one-loop RG equations. The existence of such a fixed plane is a consequence of the particle-number conservation and of the -invariance of the NLSM action (1). Due to the charge conservation, RG equations for and are related to all orders in :

(28)

The value of the anomalous dimension at a fixed point determines the dynamical critical exponent . The latter controls the temperature behavior of the specific heat, [(50)]. Typically, one expects that () which implies the instability of the fixed point in the plane with respect to the increase of .

It is worth reminding the reader that RG equations (23) - (27) are of one-loop order with respect to diffusive modes (i.e., are derived by expansion of the right hand side to the lowest nontrivial order in ) but are exact in interaction. Typically, one expects that one-loop RG equations are valid until entering the insulating (strong-disorder) phase, i.e. for . This requires a tacit assumption that in the expansion of the right hand side of RG equations in powers of all coefficients (which are functions of interaction amplitudes) are of the order of unity. In the case of superconducting instability, diverges at some scale , so that coefficients of the expansion in powers of become much larger than unity. As discussed in Sec. III.1, near the superconducting instability (i.e., at ) the general condition of validity of the one-loop approximation becomes more restrictive: . Similarly, near the Stoner instability (which corresponds to the divergence of ) the two-loop analysis [(51); (52)] demonstrates that expansion in is justified for .

Up to now we have discussed the renormalization as a flow of couplings with the length scale. In practice, one usually has a sufficiently large system and the infrared cutoff is controlled not by the system size but rather by the temperature . In this situation, the renormalization due to the contributions to RG equations (23) - (27) induced by interactions should be stopped at the length scale which is determined as follows [see Eq. (75)]:

(29)

where and . This transformation of temperature into the length scale [(53)] allows us to investigate the temperature dependence of observables. In particular, the electrical resistivity in the absence of magnetic field is addressed in Sec. V. The inclusion of magnetic field induces two additional length scales, and , related to the orbital and Zeeman effect of magnetic field and leading to the magnetoresistivity, Sec. VI.

Iv Phase diagram at zero magnetic field

iv.1 Preserved spin rotational symmetry

We start our analysis of RG equations (23) - (27) from the case in which spin rotational and time reversal symmetries are preserved, i.e., there are triplet modes. We note that in notations of Ref. [(39)] this case is termed as G(LR) for Coulomb interaction and G(SR) for short-ranged interaction.

Coulomb interaction

For the case of Coulomb interaction, , which is the fixed plane of Eqs. (23) - (26), the RG equations can be simplified as (we set )

(30)
(31)
(32)

Let us now analyze fixed points of Eqs. (30) - (32). It turns out that the structure of the set of fixed points and of the three-dimensional phase diagram is very rich. Specifically:

  • There is a marginally unstable line of fixed points at (with arbitrary ). These fixed points describe a conventional clean Fermi liquid without Cooper-channel attraction.

  • There is a line of fixed points at and (with arbitrary ) corresponding to the superconducting (SC) phase.

  • Further, Eqs. (30) - (32) contain also the attractive line of fixed points at and . The divergence of corresponds to a ferromagnetic instability.

  • Formally, in Eqs. (30) - (32), there exists also a fixed point at , and . While the range of is beyond the accuracy of the one-loop RG, it is expected on general grounds that full RG equations should contain an attractive fixed point (or a family of fixed points) with describing the insulating phase.

  • Within Eqs. (30) - (32) there is a possibility at some length scale to enter the phase with . At this length scale there are finite values and . We note that corresponds to the infinitely strong attraction in the triplet particle-hole channel indicating a possibility of exciton condensation. Since the value is reached at a length scale close to , full RG equations are needed to study a competition of exciton condensation in the spin channel and superconductivity in the Cooper channel. We leave this as a prospect for future research and do not discuss a possibility of exciton condensation in the rest of the paper.

  • Going beyond the one-loop RG equations (30) - (32), we expect a fixed point at , , and a certain value of governing the transition between the superconductor and insulator phases. The corresponding phase boundary is a critical surface with a flow towards this SIT fixed point originating at the trivial fixed point with . We will discuss the SIT fixed point in more detail in Sec. VII below.

  • Similarly, we expect strong-coupling fixed points that control the ferromagnet-insulator and the ferromagnet-superconductor transitions. We will not discuss these fixed points in the present paper [(55)].

Let us now discuss properties of the emerging phases (see Figs. 1 and 2) and corresponding fixed points, in more detail.

Superconducting phase. We first note that within the RG equations (30) - (32) the superconducting line of fixed points at and is unstable, which makes the superconducting phase formally unreachable. As we have already discussed, this indicates a failure of the one-loop (lowest order in ) RG equations near the superconducting instability. In the absence of disorder (i.e., at ), Eq. (32) describes the usual BCS-type scenario. The Cooper-channel interaction diverges at some finite length scale as . To estimate the length scale in the case of finite disorder, we shall use the scale defined by the condition . Assuming that the divergence of is of the BCS type, we get an estimate . Thus, while the one-loop RG is not sufficient to follow the flow up to the singularity scale , it works up to a scale which is only slightly smaller than .

Figure 1: (Color online) The case of preserved spin rotational symmetry with Coulomb interaction, : the RG flow obtained from numerical solution of Eqs. (30) - (32). The initial condition fixes . The arrows indicate the flow towards the infrared. The gray region indicates the part of the phase diagram which is not accessible within one-loop RG equations. The lines describing the flow to the superconducting (SC), insulating (I), and ferromagnetic (F) phases are shown in red, green, and blue, correspondingly. Gray flow lines correspond to the region of superconductor-insulator transition (SIT).

Figure 2: (Color online) The case of preserved spin rotational symmetry with Coulomb interaction, : a projection of the phase diagram on the plane. The color indicates the value of the Drude resistivity at which the quantum phase transition from SC to I occurs. Above the dashed line, the FM phase appears in addition to SC and I phases. The figure is obtained from numerical solutions of RG Eqs. (30) - (32).

Insulating phase and superconductor-insulator transition. On general grounds, we assume that once the RG flow reaches , the system is in the insulating phase, i.e., it flows into the insulating (I) fixed point with . On the other hand, as discussed above, if remains small when reaches a value , the system flows into a superconducting fixed point. There should be thus a fixed point at (i.e., with resistivity of order of quantum resistance ) and certain values of and that controls the quantum phase transition between superconductor and insulator, see Sec. VII for a further discussion. At small values of and , , the separatrix surface between the two phases is parametrized by the following equation: .

“Ferromagnetic” phase. For the attractive line of fixed points at and , the value is fixed by a cancelation of terms in the right-hand side of Eq. (32) which are proportional to . The divergence of occurs at some finite length scale . Due to a delocalizing effect of the interaction (Altshuler-Aronov) contribution to renormalization of the resistance at large , the fixed point value remains finite and is non-universal (i.e., determined by the initial conditions). Therefore, Eqs. (30) - (32) predict ferromagnetic metallic phase with a non-universal resistivity. Strictly speaking, one-loop equations are insufficient to describe accurately the regime (see Refs. [(51); (52)]) but this is not expected to modify essentially the emergence of instability.

However, since the emergent fixed points are characterized by a finite value of dimensionless resistivity , the diffusive RG continues at larger scales. Specifically, to describe properly the system at scales larger than , one needs to take into account breaking of spin rotational symmetry and derive a new set of RG equations. In this case all triplet diffusive modes in the particle-hole channel and singlet and triplet modes in the Cooper channel are suppressed. One can thus assume that the system at is described by RG equations(23) with , and that results in insulating behavior at large length scales. Moreover, due to enhanced spin fluctuations near the Stoner instability, the system at can demonstrate a spin-glass behavior [(62)]. In what follows, we shall term this phase ferromagnetic (FM) for simplicity.

Figure 3: (Color online) The case of preserved spin rotational symmetry with Coulomb interaction, : dependence of (“renormalized Drude resistivity”) on the length scale across the quantum phase transition between superconducting (SC, red curves) and insulating (I, blue curves) phases. The curves are obtained from numerical solutions of RG equations (30) - (32) for , and (from the bottom to the top).

Figure 4: (Color online) The case of preserved spin rotational symmetry with Coulomb interaction, : dependence of on the length scale across the quantum phase transition between superconducting (SC, red curves), insulating (I, green curve), and ferromagnetic (FM, blue curves) phases. The curves are obtained from numerical solutions of RG equations (30) - (32) for , and (from the bottom to the top). For higher values of , another insulating phase (not shown on this scale) emerges. Inset: dependence of and on .

Overall RG flow and phase diagram. A part of the RG flow for Eqs. (30) - (32) is shown in Fig. 1. In general, a projection of the flow in a three-dimensional parameter space onto a 2D plane, as in Fig. 1 depends on initial conditions for the couplings. For the plot shown in Fig. 1, we have assumed a realistic relation between the triplet (third axis) and Cooper amplitudes, which has allowed us to avoid intersections in the projected flows. Furthermore, the RG flow is shown only in the region of validity of the one-loop approximation: . The flows towards the superconducting, insulating, and ferromagnetic phases are plotted in red, green, and blue, correspondingly. The grey part of the flow describes the vicinity of the SIT. One of the grey curves is the separatrix between the superconducting and insulator phases. However, the one-loop precision is insufficient to determine the separatrix in the region . At small values of the separatrix is parametrized by .

The phase diagram expected on the basis of the RG equations (30) - (32) is shown in Fig. 2 in the plane of bare interaction couplings and . For , the superconducting phase exists at small values of . For given and the quantum phase transition from superconductor to insulator occurs with increase of . In addition, for a sufficiently large (above the dashed line) a ferromagnetic phase emerges. In this part of the plane, a sequence of transitions S – I – FM – I takes place with increasing bare resistivity . For , there is no superconducting phase; changing drives a transition from the ferromagnetic to the insulator phase.

The dependence of the NLSM coupling on the length scale across the quantum phase transition from the superconducting to insulating phase (in the part of the phase diagram in Fig. 2 where FM phase does not occur) is shown in Fig. 3. This dependence dominates the corresponding evolution of the total electrical resistivity (apart from a narrow region close to the superconducting instability, where the inelastic contributions due to fluctuating Cooper pairs becomes dominant, see Sec. V for details).

In Fig. 4 we choose the values of and such that the FM phase exists in addition to the SC and I ones. We thus show the length dependence of across the quantum phase transitions from SC to I and from I to FM phases. We note that within RG Eqs. (30) - (32) the insulating phase (between SC and FM phases) exists in a very narrow interval of , see Fig. 4. As one can see, the scale (at which red curves in Fig. 4 are stopped), which yields approximately the superconducting coherence length, is larger than the BCS coherence length . In the ferromagnetic phase, the corresponding length scale (where blue curves end) is still larger than .

At finite temperature, the interaction contributions to the RG equations (30) - (32) are stopped at the length scale . Neglecting the difference between and the temperature-induced dephasing length (which cuts off the localization corrections), we can stop the whole RG at . Then the transition temperatures to superconducting () and ferromagnetic phases () is estimated as follows (see also a discussion in the end of Sec. III): and . A typical dependence of and on is shown in the insets to Figs. 3 and 4. The effect of disorder on depends on the sign of the term in the square brackets in the right hand side of Eq. (32). It occurs that for and this term is always positive, except for a small region at small negative values of and . Therefore, as was first found by Finkelstein [(32)], disorder in the presence of Coulomb interaction suppresses the superconducting phase (i.e., lowers ). At the same time, disorder induces the ferromagnetic phase which exists in an intermediate range of disorder. This implies a nonmonotonous dependence of on .

We note that evaluated from the RG equations (30) - (32) is in fact somewhat larger than the true superconducting (Berezinskii-Kosterlitz-Thouless) transition temperature due to the presence of phase fluctuations of the order parameter at temperatures below , see Sec. V for more detail. The relative difference between and is, however, small for weak disorder, and thus does not essentially affect a much stronger variation of with disorder explored in this paper.

Short-ranged interaction

In the case of short-ranged interaction, RG equations (23) - (26) with read

(33)
(34)
(35)
(36)

Contrary to the Coulomb-interaction case (where we had ), the singlet particle-hole amplitude is not fixed now, so that the RG flow occurs in the four-dimensional parameter space. However, the structure of the set of attractive fixed points (quantum phases) and of fixed points describing quantum phase transitions between them remains qualitatively the same as in the Coulomb case. Specifically, the fixed points of the RG flow for the short-ranged interaction are as follows [(55)]:

  • There is a surface of clean-Fermi-liquid fixed points at (with arbitrary and ).

  • The fixed-point surface at and corresponds to the superconducting phase.

  • The line of fixed points with , , , and arbitrary , is attractive in the direction. Therefore, the RG equations (33)-(36) lead to the same ferromagnetic phase that exists in the case of Coulomb interaction.

  • Exactly as in the Coulomb case, there should be a fixed point (or a family of fixed points) with describing the insulating phase.

  • For the same token as in the Coulomb case, a SIT fixed point with should separate the superconducting and insulating phases.

The phase diagram for a given is similar to that for the case of Coulomb interaction, (shown in Fig. 2). With increase of , the destruction of the superconducting phase gets shifted towards larger values of . The crucial difference between the cases of short-ranged and Coulomb interactions is the existence of large region of the phase diagram with (and thus ). In the case of a bare repulsion in the particle-hole channel, and , the superconducting transition temperature is typically lower than the clean BCS result, (see Fig. 5). However, the situation changes if the bare interaction in the triplet particle-hole channel is attractive, . As illustrated in Fig. 5, a significant part of the phase diagram is occupied by superconductor with . It should be emphasized that the superconducting phase with enhanced exists also for . However, it occurs only in a small region of (see Fig. 5). Typical RG evolution of the resistance in this region of initial values of interactions is shown in Fig. 6. Being initially suppressed by disorder, can be significantly (several orders of magnitude) enhanced with respect to near the superconductor-insulator quantum phase transition, as illustrated in the inset to Fig. 6. This is in agreement with the conclusion of our work [(37)] where RG equations (18)-(21) with the right-hand sides expanded to the lowest nontrivial order in , and were analyzed.

The mechanism of enhancement of the transition temperature is as follows. For small initial values of interaction parameters