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Unconventional Cooper pairing results in a pseudogap-like phase in s-wave superconductors

The impact of disorder on the superconducting (SC) pairing mechanism is the centre of much debate. Some evidence suggests a loss of phase coherence of pairs while others point towards the formation of a competing phase. In our work we show that the two perspectives may be different sides of the same coin. Using an extension of the perturbative renormalization group approach we compare the impact of different disorder-induced interactions on a SC ground state. We find that in the strongly disordered regime an interaction between paired fermions and their respective disordered environment replaces conventional Cooper pairing. For these unconventional Cooper pairs the phase coherence condition, required for the formation of a SC condensate, is not satisfied.


SC fixed point
The conventional perturbative RG method is used to understand the influence of perturbations on the noninteracting Fermi liquid fixed point [30]. In our work we extend this technique to directly study the SC fixed point of the spinless BCS-like Hamiltonian where c (c † ) is the annihilation (creation) operator, k ( )  denotes the kinetic energy measured relative to the Fermi surface and U represents the attractive s-wave coupling between fermions with antiparallel momenta. The summation in the interaction term is restricted to positive k i to prevent double counting of the equivalent pair configurations (k k , For a fixed-point theory the Hamiltonian and hence the quantum partition function must remain invariant under the RG transformation. This requires a rescaling of the quantum fields. To make (7) invariant under the RG transformation the field rescaling relation must be which is the well known result for the Fermi liquid fixed point [30]. With this choice, the second part of the action (S SC ) (8) grows with increasing s and hence the relation (11) does not determine the SC fixed point. For the field rescaling relation the interaction term S SC remains invariant while the free part S 0 decreases with increasing s. Hence, in the limit of large length scales only S SC is important while S 0 is irrelevant. This describes the reasonable scenario that formation of fermion pairs dominates over unpaired charge transfer in the SC state. The SC fixed point condition at tree-level (12) is independent of the coupling strength U and hence applies equally to the weak and strong coupling limits. This is in contrast to strong coupling expansions where the kinetic term is treated perturbatively [31,32]. In our RG treatment, the kinetic term can be ignored at the SC fixed point due to its irrelevant long-range scaling, rather than its negligible magnitude in comparison to the pairing term U. of the averaged term in the effective action (6) produces quantum corrections that may modify the field rescaling relation (12) found at tree-level. On the right-hand side, the first term in the exponent with two external legs adds a correction to the free hopping of unpaired charge carriers. This will be negligible at the SC fixed point. At second order there are two types of quantum corrections that resemble the same external leg structure as the original vertex: (1) the particle-hole channel (figure 1) and (2) the particle-particle channel (figure 2). The particle-hole channel may be ignored as it is restricted to forward scattering and therefore only affects a small region of the scattering phase space. For the particle-particle diagram the in-and out-going momenta are chosen from the entire phase space. To obtain the contribution from internal lines the Grassmann path integrals  Figure 1. To maintain that the external momenta of the particle-hole diagram are pairwise antiparallel the diagram has to be restricted to forward scattering. This results in a strongly restricted phase space compared to figure 2. The contribution from this loop diagram is therefore negligible. over all internal fast fields (y > ) have to be evaluated. These non-Gaussian integrals (the exponent S [ ] y > contains a quadratic (free hopping) and a quartic (pairing) term) cannot be evaluated directly and in general constitute a significant challenge. However, due to the special properties of Grassmann numbers we show in appendix B that these integrals rescale to zero at the SC fixed point. Hence the rescaling relation (12) is exact for the SC fixed point.

Disorder-induced interactions
We assume that the introduction of disorder gives rise to two types of perturbations that might cause a transition out of the SC ground state. One of them breaks Cooper pairs to induce a transition towards an unpaired non-SC state. The other conserves pairs and either generates a transition into an unconventional SC state or into a non-SC paired state.

Pair-breaking perturbations
The simplest pair-breaking vertex consists of a single Cooper pair and an unpaired fermion or hole (1CP1e). The energy required to break the pair is provided by the unpaired particle, while momentum must also be conserved. For such a simple vertex we discuss the momentum conservation condition graphically (figure 3). The grey region represents the narrow band of slow modes around the Fermi momentum (K F ). Modes outside the grey band are not important for the long-range behaviour and are integrated out. The incoming Cooper pair (dashed green) carries zero momentum and interacts with a single fermion or hole (dashed black) with a momentum within the grey region. The outgoing lines (red) have to conserve the incoming momentum and also lie within  . Illustration of the momentum conservation condition for a three-particle interaction. Only momentum configurations within the grey region close to K F survive the RG transformation. The momentum conservation for these slow modes forces two outgoing lines to remain antiparallel. the narrow grey band. This restriction is only satisfied if two of the outgoing lines are antiparallel again. Therefore the only quantum process that survives the RG rescaling of the 1CP1e vertex describes conventional Cooper pairing in the presence of an unpaired fermion or hole.
The next higher-order pair-breaking vertex consists of a single Cooper pair and two unpaired fermions Unlike the 1CP1e vertex, there are no kinematic restrictions for the 1CP2e vertex. When we perform the usual RG transformations and insert (12) into (15) we find that the 1CP2e vertex scales as s −3 at tree-level and hence is irrelevant with respect to the SC fixed point. The same is true for all higher-order pair-breaking vertices that consist of a single Cooper pair and two or more unpaired charge carriers. This resilience of BCS and hightemperature SC towards impurities has also been shown in other theoretical works [33,34]. Apart from the tree-level term a relevant contribution may come from higher-order quantum corrections. However, as discussed in (appendix B) quantum corrections renormalize to zero in the SC fixed point. Using the same reasoning as for the SC fixed point we find that quantum corrections of pair-breaking interactions with U   can be neglected. However, for U  > we cannot evaluate the Grassmann path integrals by induction anymore. In particular for U   higher-order quantum corrections will not vanish. For this regime of strongly disordered weakly coupled BCS-like superconductors, it is expected that during the SIT the SC ground state vanishes due to the breaking of Cooper pairs [1].

Pair-conserving perturbations
The disorder-induced direct localization of Cooper pairs during the SIT [21,22] can be described as a pairconserving interaction between paired fermions and impurities. In the limit of high disorder the free propagation of the fermion pair will be suppressed which eventually leads to localization. For a pair to survive an interaction with impurities, equal amounts of energy and momentum must be transfered for both fermions. This interaction can be described by the action with and q g as the transferred energy and momentum respectively, and To obtain the scaling of S 1CP2ẽ with respect to the SC fixed point we perform the usual RG transformation. By inserting (12) into (17), the scaling of the 1CP2e vertex at tree level is Hence, this pair-conserving Cooper pair-impurity interaction dominates at large length scales. This induces a transition away from the SC state towards a new fixed point. The ground state of this novel type of pairing interaction may either be an unconventional SC condensate or a non-SC gapped phase. To distinguish between the two scenarios it is sufficient to determine whether the ground state of the 1CP2e fixed point is accompanied by the formation of a macroscopically coherent quantum phase (CQP).

Quantum phase coherence
It is known from the BCS theory that the condensation into the SC ground state is accompanied by the formation of a macroscopically CQP. The for all k C where is a constant [35]. Since the BCS trial state (20) cannot describe systems containing unpaired charge carriers, which we must necessarily include to check the CQP condition subject to 1CP2e pair-conserving interactions, we chose instead of (20) the most general linear combination of states with paired and unpaired charge carriers Here the amplitudes u K and v K (u K and v K ) describe the occupation probability of unpaired fermionic (paired bosonic) states. Although we use a fermionic notation for clarity, the paired operators vc c vb = and that each configuration appears only once. Equation (22) then describes product states of a single Cooper pair and its local environment consisting of two unpaired fermions.
For ordinary Cooper pair scattering this trial state has to reproduce the CQP. Indeed, when we apply the variational scheme on the energy expectation value the matrix element of the first half of the pair scattering reads We note that only the paired states (P P , -) give a non-zero contribution. Since double occupancy is not allowed and each configuration must appear only once all terms where the states (P P , -) are occupied can be pulled out of summation Here KRS  accounts for all remaining states that do not appear in the first bracket. When we apply c c  Here we can factor out the phase for all u Q to combine them to an overall constant phase. This is done by setting 0 Q a = and substituting Q To find the ground state the minimization has to be performed with respect to both, the phase as well as the amplitude. Variation with respect to the amplitude was performed in [36] for conventional BCS pairing. Here we are only interested in the phase coherence condition. Therefore we drop the amplitudes u and v | | | | and keep the phase factors only to get e .
The Rayleigh-Ritz approach then yields To simplify this expression we group δ-functions with the same sign and relabel the summations over S, T, W and P, Q, R according to After further relabeling we combine the summations and set it equal to zero 3 ie ie The CQP condition is obtained if each individual summand in (28) becomes zero [35]. It is then easy to see that the CQP condition for conventional Cooper pairing (21) is one of infinite configurations to satisfy 0. 29 Since the ground state is not exclusively realized for CQP condition but for an infinite number of non-CQP configurations, the ground state of the 1CP2e interaction is not SC.

Conclusion
In the first part of our work we present an extension of the well known RG method that allows us to determine the SC fixed point of a BCS-like Hamiltonian. From the comparison of pair-breaking and pair-conserving perturbations at the SC fixed point we find, that pair-breaking interactions are irrelevant on large length scales. While the SC state is resilient towards disorder induced pair-breaking [33], we identify a particular pairconserving interaction (1CP2e) that describes the coupling of Cooper pairs in the presence of a disordered environment. For this unconventional pairing we find a relevant RG flow away from the conventional BCS ground state. In the ground state of this unconventional pairing, Cooper pairs are not coherent and therefore do not form a SC condensate. This behaviour is reminiscent of a PG-like state with pre-formed or disordermodified Cooper pairs. We conclude that the observation of a second energy gap [13] may be related to the 1CP2e pairing. Moreover, because disorder-modified Cooper pairing is restricted to strongly disordered regions, we expect the formation of non-SC islands, separated by regions with a high SC order [29].
While the existence of pre-formed Cooper pairs can be anticipated, we are able to identify the 1CP2e mechanism as the actual quantum process that describes non-coherent Cooper pairs in a disordered BCS type system. Although we analyze a simplified Hamiltonian, the scalings of disorder induced pair-breaking and pairconserving perturbations with respect to a s-wave fixed point, are universal. Therefore our results suggest that the formation of a PG-like phase is a general phenomenon of a disordered SC state instead of merely a peculiar property of HTSC materials.

Appendix A. Effective action
The interaction part of the action S = S 0 + S SC can be decomposed into We substitute the partition function  > and take the average over fast fields to find  After averaging the full action (instead of S 0 only) over the fast modes, we now obtain an effective action defined over the slow modes. For the RG operations the term  > only adds a constant factor which can be ignored. We therefore find for the modified effective action e e e . 3 3 where the action terms contain momentum and energy integrals themselves. We can discretize these integrals as i j k l m n i k i l j m j n SC , , , , , According to the Grassmann integration rules (36) the only non-vanishing terms in (37) must contain each field configuration exactly once. The implication of this is best understood by restricting the momentum and frequency phase space to a few allowed configurations. We demonstrate the evaluation of (34) for 2, 4 and 6 allowed momentum values for a one-dimensional system first and then extrapolate to infinitely many allowed momenta. In this way we can understand how the RG transformation changes the path integral over fast modes. To begin, we consider only summation over the momenta (K k K k ,    The result for a single allowed momentum (43) is almost reproduced except for the additional U 2 in the denominator. The trend of a growing denominator with respect to the numerator continues for the case (k k k k k k , , , ( )¯( ) ( ) ( ) y y y y á --ñ .
In summary, the convergence of fractions of Grassmann path integrals remains trivial only at the fixed point where the multinomials in the numerator and denominator contain the same potential U 0 > . Moreover, it is interesting to note that for U 0 < the additional terms in  come with alternating signs and therefore the average k k k k