Unparticle mediated superconductivity

Unparticle mediated superconductivity

James P. F. LeBlanc1,2 and Adolfo G. Grushin1

1Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

E-mail: [email protected]

Abstract. In this work we introduce the possibility of unparticle mediatedsuperconductivity. We discuss a theoretical scenario where it can emerge and show thata superconducting state is allowed by deriving and solving the gap equation for s-wavepairing of electrons interacting through the unparticle generalization of the Coulombinteraction. The dependence of the gap equation on the unparticle energy scale ΛU

and the unparticle scaling dimension dU enables us to find a richer set of solutionscompared to those of the conventional BCS paradigm. We discuss unconventionalfeatures within this construction, including the resulting insensitivity of pairing to thedensity of states at the Fermi energy for dU = 3/2 of the superconducting gap andsuggest possible experimental scenarios for this mechanism.

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Unparticle mediated superconductivity 2

1. Introduction

One of the most remarkable consequences of the discovery of unconventionalsuperconductivity (SC) in both cuprates and pnictides [1, 2] is, arguably, the richness oftheoretical ideas that have been put forward to understand this physical phenomenon[3, 4]. Searching for novel mechanisms that lead to superconducting states differingfrom the conventional Bardeen-Cooper-Schrieffer (BCS) paradigm [5, 6, 7], as wellas a complete understanding of the pseudogap state [8, 9] has driven the emergenceof elaborate theoretical concepts that well transcend the goal of understanding anyparticular material.There is strong evidence that suggests that interactions are mainly responsible for, orplay a major role in generating the rich phase diagrams of these systems [10, 11, 12,13, 14, 15]. Under the umbrella of such an observation, many novel interesting ideashave emerged that were later found to be relevant also in other, completely unrelatedsystems. The marginal Fermi liquid [16, 17, 18] introduced phenomenologically toaccount for experimental observations [19] is a prime example. It was argued thatstrong interactions could cause the imaginary part of the self-energy, usually associatedwith the quasiparticle life-time, to behave as ImΣ(ω) ∼ ω instead of the quadraticFermi-liquid like behaviour [20]. Remarkably, this particular idea was also shown to berelevant for the physics of graphene [21, 22].On the other hand, high-energy physics has also profited from similar phenomenologicalapproaches predicting a plethora of verifiable consequences for experiments at the LargeHadron Collider (LHC) [23]. A particularly appealing idea, proposed recently by Georgi[24, 25], is the existence of a conformally invariant sector that couples to ordinarystandard model particles. The former sector does not behave as ordinary mattersince it is described by propagators without poles, and hence has no a priori particleinterpretation. Such unparticle "stuff" (as dubbed by Georgi) could in principle leavea very particular signature in LHC scattering cross-sections as missing spectral weightcorresponding to a non-integer number of ordinary particles [24, 26]. The parallelismwith non-Fermi liquid properties [18] recently motivated the idea that unparticlesresulting from strong interactions could be responsible for the missing spectral weightin the pseudogap phase [27], accounting also for a possible breakdown of Luttinger’stheorem [28]. These works complement the numerous scenarios that can lead to exoticsuperconductivity proposed in the literature, which have explored the dependence ofthe superconducting transition temperature with model parameters [29, 30, 31, 32]. Inthe case of unparticles, within a BCS framework it was shown [27] that if electronsparticipating in superconductivity were promoted to unparticles that pair with astandard BCS interaction, unusual phenomenology followed.Inspired by these ideas, we aim to discuss a mechanism which is distinct from otherexotic forms of superconductivity and which can in principle exist in strongly correlatedmaterials. In this work we explore how superconductivity of normal electrons can arisefrom mediating unparticles. In essence, we discuss how the unparticle construction


E

⇤U

{A, O}

{�}

{ }

{ }

Sector SectorUnparticle

L �⇥ , †,�

⇤

L A

⇥ , †, A

⇤ {

† †VU

Particle

Figure 1. (Color online) Schematic picture showing the emergence of unparticles.At high energies ordinary particles (particle sector) couple to a second sector, theunparticle sector, that has a non-trivial infrared fixed point below a characteristic scaleΛU where the fields turn effectively into unparticles. Below ΛU this sector couples tothe particle sector that is unaffected by the RG flow by construction. This couplingresults in an effective four point interacting potential VU , the unparticle analogue ofthe Coulomb interaction, defined by (7) [see main text].

provides a path from a high-energy theory with repulsive interactions between fermionsto an effective low energy theory where particles attract. The price to pay is thatthe mediating glue is composed of unparticles, ultimately resulting in unconventional(i.e. away from standard BCS) behaviour. Being a generic theoretical scenario, thismechanism could potentially emerge in strongly correlated matter. In this context, thiswork aims to clarify two important issues (i) the simplest theoretical construction wherethis effective unparticle mediating interaction emerges (ii) how this state departs fromtypical BCS phenomenology.To address these points, we will first discuss a minimal theoretical framework for themechanism to emerge, partially reviewing known properties of unparticles, but focusingon those relevant for our work. We will write down a generic model containing theunparticle analogue of the Coulomb interaction for which we write and solve the gapequation at the mean-field level and s-wave pairing. We will find that even at the meanfield level, the unparticle nature of the mediators manifests itself through non-BCSphenomenology. We finish with a discussion of our results and a summary of our mainconclusions.

2. Emergence of unparticles

Although the concept of unparticles may seem exotic at first, especially in thecontext of strongly correlated electrons, it is not entirely novel to condensed mattersystems. Within the renormalization group approach and due to electron interactions,quasiparticle propagators can acquire anomalous dimensions near non-trivial fixed


points [33, 24], turning them into unparticles. As discussed in [27] the Tomonoga-Luttinger liquid exhibits such behaviour [34, 35, 36]. A second particularly clean, andto some extent unexplored example in 2 + 1 dimensions that illustrates how unparticlesemerge is that of the low energy interacting electrons in graphene. In this systeminteractions renormalize the Fermi velocity vF [37, 38] which increases at low energies,a prediction consistent with recent experiments [39]. However, as vF increases, the ratiowith the speed of light vF/c ceases to be small and thus one has to consider the fullrelativistic Coulomb interaction to have a fully consistent theory [37]. In this case, thetheory can be shown to have a non trivial infra-red (IR) fixed point at vF = c. Close tothis point, the electron propagator acquires an anomalous dimension γ and satisfies thescaling law

G(λω, λk) = λγG(ω, k), (1)

where λ is the scaling parameter and γ is the anomalous dimension, which is in generalnot an integer and depends on the coupling constant at the non-trivial critical point. Forgraphene γ = α2/(12π2)− 1 where α = e2/~c is the fine structure constant [37]. Such apropagator corresponds by definition to an unparticle since it is defined around a fixedpoint (i.e. it is conformally invariant) and has no simple quasiparticle poles in general.Thus electrons in graphene can be ultimately reinterpreted as genuine unparticles belowan energy scale ΛU ‡.Motivated by this observation, we now reinterpret Georgi’s original construction,[24]where an unparticle sector is coupled to ordinary particles, to obtain a theory that willlead us to unparticle mediated superconductivity. The process, summarized in Fig. 1starts by considering a theory with Lagrangian

L[ψ†, ψ, A,O

]= L1

[ψ†, ψ, A

]+ L2 [A,O] , (2a)

L1

[ψ†, ψ, A

]= Lψ

[ψ†, ψ

]+ LψA

[ψ†, ψ, A

], (2b)

L2 [A,O] = LA [A] + LAO [A,O] + LO [O] , (2c)

defined for energies E � ΛU . The theory is composed of two sectors which we illustrateschematically in Fig. 1. The first sector, L1

[ψ†, ψ, A

], represents ordinary particles ψ

(e.g. electrons) coupled through a generic interaction A, (e.g. the Coulomb interaction).In the original unparticle set up, this sector would correspond to the standard model ofparticle physics. On the other hand L2 [A,O] represents a different interacting sectorwhere other degrees of freedom, collectively denoted by the field O, interact also throughA. At this point we impose that L2 [A,O] has a non trivial IR fixed point below theenergy scale ΛU . Above this energy scale L2 [A,O] has an ordinary interpretation asa particle theory. Below ΛU unparticles emerge as exemplified at the beginning of thesection (see Fig. 1). We emphasize that the scale ΛU must exist in order to constrainthe unparticles to exist only below this energy scale. Remarkably, there are explicitexamples where it is possible to estimate this energy scale (see for instance eq. (3) in[41] or eq. (23) in [40]).

‡ The energy scale at which this happens is extremely small, albeit not zero, see [40].


In particular, if L1 is unaffected by the IR renormalization group (RG) flow, at energieslower than ΛU , ordinary particles ψ will effectively only couple to unparticles [24], whichwe label φ with an action of the form

LIR = Lψ + Lψφ + Lφ. (3)

So far, we have just reviewed how unparticles generically emerge. We now assume thatLIR is quadratic in the fields and integrate out φ to obtain an effective interactionbetween electrons, mediated by an unparticle propagator of the form (1). After thisstep the Lagrangean reads

LIR = Lψ +g

2

∑

ij

ψ†iψiGφ;ijψ†jψj, (4)

where g is a coupling constant and Gφ;ij is the unparticle propagator in real space,which is in general a matrix depending on the physical degrees of freedom encoded inψ. Generically, the functional form of the propagator Gφ;ij is fixed by dimensionalityand conformal invariance of the unparticle sector at the IR fixed point. In particularfor a scalar unparticle the propagator is of the form [24, 42]

Gφ(q) =AdU (q2)dU−2

2(Λ2U)dU−1 sin(πdU)

, (5)

AdU =16π5/2

(2π)2dU

Γ(dU + 12)

Γ(dU − 1)Γ(2dU), (6)

Comparing to Eq. (1), γ = 2(dU − 2). The quantity dU is referred to as the unparticlescaling dimension. For a scalar particle it satisfies dU ≥ 1 to preserve unitarity ofthe theory [43]. We are interested in the unparticle generalization of the Coulombinteraction and thus we focus on the effective static potential VU associated to thisscalar propagator which was derived in Refs. [42, 44]. This is a good approximationwhen retardation effects associated to field φ (i.e. current-current interactions) can beneglected, which implies that the typical Fermi velocity should satisfy vF/c � 1 [45].In this approximation the effective static potential reads

VU(q) = 4πgAdU

2(Λ2U)dU−1 sin(πdU)

(q2)dU−2

≡ CdU (q2)dU−2. (7)

Note that mathematically, and for 1 ≤ dU < 2, it is possible to rewrite this poten-tial to be proportional to the integral

´∞0dM2ρ(M2, dU)/(q2 + M2) with ρ(M2, dU) =

(M2)dU−2. This form explicitly reveals that unparticles can be interpreted as a towerof infinitely many massive particles distributed according to ρ(M2, dU) [46, 47, 48]. In-deed, when examined in real space, this potential is nothing but a Yukawa potentialintegrated over the screening momentum scaleM2. Physically, it is possible to interpretsuch an interaction as critical in the sense that it has the form of a screened interaction,but the screening occurs at all length scales, effectively resulting in a potential of theform VU(r) ∝ 1/ |x|2dU−1 [44, 42], the unparticle counterpart of the Coulomb interaction


§. Interactions of this form have also been discussed in the context of the even denomi-nator fractional quantum Hall states [49, 50, 32]. From Eq. (7) the long range Coulombinteraction is recovered for dU = 1 if we identify g = e2.A crucial observation here is that the sign of g, and thus the attractive or repulsive char-acter of the interaction is not fixed by the renormalization that generates the unparticles,described above [51]. A repulsive interaction between fermions ψ at high energies in (2b),represented by the term LψA

[ψ†, ψ, A

]can turn into an effective attractive interaction

mediated by unparticles at low energies. The actual sign will depend on the nature ofthe IR fixed point itself and the particular value of dU trough the prefactor CdU . Weuse this freedom to fix the low energy theory to have an attractive character.We finish this section with a general remark on the validity of the present approach.In a general scenario there can exist an additional energy scale, Λe, below which thefield ψ, acquires anomalous dimensions, and is described by an unparticle propagator.In this work we assume that there exists an energy window such that Λe < E < ΛU

where the field ψ is still defined by quasiparticle propagators that interact via the un-particle potential (7). In this notation, Ref. [27] can be interpreted as the case whereΛU < E < Λe and the scenario where E < (ΛU ,Λe) is yet to be explored.

3. Mean field approximation

Given the above discussion we now ask if the attractive unparticle Coulomb potential(7) can lead to unparticle mediated superconductivity, and if so, what is its particularsignature. In order to accomplish this, we consider the case of coherent quasiparticlesin the presence of scale invariant bosonic unparticles. We then proceed in the spirit ofa modified BCS-theory, which now contains a potential due to mediating unparticleswhich has a natural cutoff, gΛU , the highest energy an unparticle might have. Weproceed without explicit interactions, but keep in mind that we are free to considerany renormalized quasiparticle weight by including Zk factors in the effective mass.Switching to the equivalent Hamiltonian formalism the effective low energy Hamiltonianfollows from (4)

H =∑

k

ξkψ†kψk +

∑

kk′q

VU(q)ψ†kψk−qψ†k′+qψk′ , (8)

where the sums are a short hand notation for three-dimensional momentum spaceintegrals. The first term corresponds to the diagonalized Hamiltonian stemming fromLψ in (4) with ξk = εk − µ with µ being the chemical potential and εk a dispersionrelation (fixed below). The second part contains the unparticle mediated interaction,which we set to be attractive by choosing the sign of CdU or the appropriate dU regime.For concreteness we choose ψ to describe fermions in three spatial dimensions with

§ Technically, the integral over M2 only converges for 1 < dU < 2. For the same interpretation tohold for dU > 2 a ultraviolet cut-off must be provided for the the integral, which is on the other handnatural for condensed matter systems


quadratic dispersion εk = k2/2m. Following standard techniques [52] we can writedown the self consistent mean field equation for the superconducting gap ∆k as

∆k = −∑

k′

VU(k − k′)∆k′

2√ξ2k′ + ∆2

k′

[1− 2nF (Ek′)] , (9)

where nF (Ek) is the Fermi-Dirac distribution function and Ek =√ξ2k + ∆2

k. Unparticlephysics enters through the potential VU(k−k′). We look for an isotropic s-wave solutionfor the superconducting gap and thus we impose that ∆k = ∆0. It is now possible tointegrate the angular variables in (9), resulting in a gap equation of the form (9) butwith the effective potential (see Appendix A)

Veff(k, k′) = − |CdU |4π(dU − 1)kk′

∑

s=±

s(k + sk′)2dU−2, (10)

that depends only on the absolute values of both k = |k| and k′ = |k′|. In order tosolve the gap equation with this potential, we make use of the standard Fermi surfacerestricted (FSR) approximation [53] by setting k = kF to obtain, at T = 0

1 = −ˆk2dk

2π

Veff (kF , k)

2√ξ2k + ∆2

0

. (11)

At this point we find it convenient to integrate over the energy variable εk and write theequation in a more transparent dimensionless form by defining k = k/kF , ∆0 = ∆0/εF ,ε = k2 − 1. After a few lines of algebra we obtain

1 =2m(k2

F )dU−3/2|CdU |16π2(dU − 1)

ˆdε

∑s=±

s[(

1 + s√ε+ 1

)2]dU−1

2√ε2 + ∆2

0

. (12)

The integral is evaluated in an energy shell around the Fermi surface defined by theinequality |ε| < gΛU/εF , where gΛU is the typical energy scale under which themediating interaction due to unparticles is non-zero‖.Note that the density of states (DOS) at the Fermi level of a 3D electron gas isρ(εF )2 ∝ k2

F and therefore the multiplicative prefactor in front of the integral isproportional to [ρ(εF )2]

dU−3/2. This is in sharp contrast with conventional BCS wherethe equivalent prefactor is proportional to ρ(εF ) [52]. The latter is recovered for dU → 2

since the interaction (7) approaches the short range BCS like interaction. We emphasizein particular that for dU = 3/2 the dependence on the DOS at the Fermi level drops outcompletely from the gap equation. This case is special in that it corresponds to a 1/x2

interaction which in 3+1 dimensions scales exactly as the kinetic term [54].We continue the comparison with the BCS scenario by employing an approximationwhere we only allow particles to exchange momentum in a infinitesimal shell close to

‖ We use units where g has dimensions of energy×(momentum)−1 and ΛU has units of momentum.Thus, gΛU is the natural energy scale for unparticles.


the Fermi surface. In this approximation we are able to set k = 1 (ε = 0) in thenumerator of (12). By solving the remaining integral we arrive at the gap formula

∆0 = 2gΛUexp[−1/V ρ(εF )2dU−3

], (13)

with an effective interaction V =|CdU

|dU−1

[4π2

2m

]2dU−4

. In this format it is tempting tointerpret gΛU as the analogue of the Debye frequency in BCS superconductivity [52]since Eq. (13) approaches the standard BCS result as dU → 2. However, the BCScharacter is lost in general because the exponential factor in Eq. (13) depends both ondU and ΛU , the latter entering through V . Indeed, CdU ∝ 1/Λ2dU−2

U that results in agap dependence ∆0 ∝ ΛUexp

[−Λ2dU−2

U

]that has no BCS analogue.

The fact that both the unparticle energy scale and the effective interaction strength arefunctions of ΛU allows for distinctly non-BCS solutions of (12). To prove that thesesolutions exist we have solved (12) numerically for ∆0 with no implicit assumption onk′ or the size of ΛU . In order to proceed, note that the prefactors in Eq. (12) can berewritten as AdU

8π(dU−1) sin(πdU )gkFεF

1

(Λ2U)

dU−1 where ΛU = ΛU/kF . If we assume that g is of

the order of the bare Coulomb interaction then the dimensionless ratio gkF/εF is oforder one for typical values of kF and εF . Thus we are able to focus on the effect ofincreasing the shell around the Fermi surface defined by ΛU as well as the effect of dU .Furthermore, since the superconducting gap should be smaller than εF , it should satisfy∆0 � 1.Our results are summarized in Fig. 2(a) where we plot the numerical solution of ∆0 asa function of ΛU and dU . It is possible to distinguish two regions where ∆0 vanisheslabelled A and B. We find that regions A and B are separated by robust physicallyacceptable solutions where ∆0 � 1 over a wide range of dU and ΛU . In Fig. 2(b) weshow an example of such an acceptable region for ΛU = 0.2. The two curves comparesolutions obtained with (12) and (13) showing that the latter is an increasingly betterapproximation as dU → 2.In contrast, we find that, as ΛU → 0 and dU > 3/2 [see Fig. 2(c)]. the approximationsthat lead to (13) break down (for instance V ceases to be small) and (12) leads tosolutions beyond the upper bound ∆0 = 1, which diverge when ΛU → 0 and dU → 2.As dU → 1 the gap decreases leading once more to physically valid solutions.All the discussed solutions are only possible because both the effective interactionstrength and the upper limit for the unparticle energies are set by ΛU and the twoeffects can level out at a given value of dU to generate a physically acceptable solutionfor the superconducting gap. Together with their dependence on physical parametersdiscussed above, their emergence is in sharp contrast to how standard BCS solutionsoccur. Thus, their existence and scaling with ΛU and dU represents one of the mainresults of this work.


1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

0.02

0.04

0.06

0.08

0.1

1 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.02

0.04

0.06

0.08

0.1

∆0

_

Λ

_

U=0.2

(b)

(c)

Region B

Region A

(a)

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

Eq. (13)

Eq. (12)

Figure 2. (Color online)(a) The gap in units of the Fermi energy ∆0 as a functionof ΛU = ΛU/kF and dU calculated from Eq. (12) with gkF /εF = 1 (see main text).Solutions below the solid black line can be disregarded as unphysical since they satisfy∆0 > ∆c = 0.2, where ∆c � 1 is chosen for illustrative purposes. (b) Comparisonof the analytic result of Eq. (13) and the numerical result for ΛU = 0.2 as a functionof the unparticle dimension dU . (c) Numerical plot of small ΛU region of numericalcomputation shown in (a). The shaded region represents unphysical ∆0 > ∆c solutions.

4. Discussion and conclusions

In this work we have described how it is possible for unparticles in strongly correlatedmatter to mediate superconductivity. We have shown that this scenario can departstrongly from the conventional BCS paradigm. In particular, we have found that fors−wave superconductivity, the superconducting gap acquires an unusual dependenceon the DOS at the Fermi level that can even vanish when the scaling dimension ofthe mediating unparticle is dU = 3/2. Interestingly, this vanishing dependence on thedensity of states occurs at the value of dU where kinetic and potential energy termsscale equally in energy[54]. This may be important in the high-Tc materials which arethought to switch from typical potential energy driven pairing in the overdoped side ofthe phase diagram, to kinetic energy driven in the underdoped[55, 56]. Such a crossovermay correspond to a similar crossover of the unparticle scaling determined by the valueof dU .As further exemplified above with electrons in graphene, obtaining dU for a particularcondensed matter system is sometimes possible. However, this is a hard task in generalsince it implies characterizing the crossover from the high-energy particle descriptionto unparticles, which could be dominated by a strong coupling IR fixed point hard toaccess analytically [24, 25]. The particular value of dU will generically depend on thecoupling constants in the unparticle sector (that differ from g in general) [37, 24] and


can therefore scale with an external experimentally tunable parameter that modifies theinteraction strength. This idea resembles the situation in hole-doped high-Tc materialsthat can be doped from strongly correlated Mott insulators towards Fermi liquids forlarge hole doping [57]. The presented dependence on dU can significantly affect the sizeof the gap as shown in Fig. 2(b).To conclude, our work also leaves many interesting open extensions such as unparticlemediated d-wave superconductivity and the effects of reduced dimensionality or com-peting orders. Furthermore, the presented construction may be useful to generalizeprevious studies in the context of even denominator fractional quantum Hall state theo-ries [49, 50, 32]. The exact connection to the latter and its consequences will be exploredin future work.

Appendix A. Effective Potential

In this section we present the derivation of the gap formula and effective potential usedin the main text. We start from the standard gap equation (9) given in the main text

∆k = −∑

k′

VU(k − k′)∆k′

2√ξ2k′ + ∆2

k′

[1− 2nF (Ek′)] , (A.1)

we are most interested in isotropic s−wave pairing and thus set ∆k = ∆0. This enablesus to perform the angular integral first that defines the effective potential

Veff(k,k′) =

ˆdθk′dφk′

(2π)2sin(θk′)VU(k − k′). (A.2)

In the isotropic s−wave pairing case we can fix the direction of k to conveniently solvethe integral

Veff(k, k′) =CdU2π

πˆ

0

(k2 + k′

2 − 2kk′ cos(θ′))du−2

sin(θ′)dθ′, (A.3)

which results in

Veff(k, k′) =CdU2π

[(k + k′)2dU−2 − (k − k′)2dU−2

2kk′(dU − 1)

], (A.4)

=CdU

4π(dU − 1)kk′

∑

s=±

s(k + sk′)2dU−2, (A.5)

valid for 1 < dU < 2 presented in the main text. To solve the remaining integral wemake use of the Fermi surface restricted (FSR) approximation (see for instance [53])and set k = kF . For this case, and T = 0, the gap equation reads

1 =

ˆk2dk

2π

Veff (kF , k)

2√ξ2k + ∆2

0

, (A.6)


where we have dropped the primes. The effective potential, used in the main text isgiven therefore by

Veff(kF , k) =CdU (k2

F )dU−2

4π(dU − 1)k

∑

s=±

s(1 + sk)2dU−2, (A.7)

with k = k/kF .

Acknowledgments

A.G.G. acknowledges F. de Juan and M. A. H. Vozmediano for related collaborationsfrom which this work originated as well as critical reading of the manuscript. We areindebted to Jens H. Bardarson, E. V. Castro, P. Fulde and B. Valenzuela for usefulcomments and M. Peiro for discussions at the early stages of this work.

[1] J. G. Bednorz and K. A. Müller, 1986 Z. Physik B 64 189[2] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, 2008 J. Am. Chem. Soc. 130 3296[3] M. R. Norman, 2011 J. Supercond. Nov. Magn. 25 2131[4] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, 2011 Reports on Progress in Physics 74 12

124508[5] L. N. Cooper, 1956 Phys. Rev. 104 1189. doi:10.1103/PhysRev.104.1189[6] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, 1957 Phys. Rev. 106 162. doi:10.1103/PhysRev.

106.162[7] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, 1957 Phys. Rev. 108 1175. doi:10.1103/PhysRev.

108.1175[8] T. Timusk and B. Statt, 1999 Reports on Progress in Physics 62 1 61[9] M. R. Norman, D. Pines, and C. Kallin, 2005 Advances in Physics 54 8 715. doi:10.1080/

00018730500459906[10] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M.

De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo,E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, 2012 Science337 6096 821. doi:10.1126/science.1223532

[11] I. M. Vishik, M. Hashimoto, R.-H. He, W.-S. Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang,W. Meevasana, T. Sasagawa, S. Uchida, K. Fujita, S. Ishida, M. Ishikado, Y. Yoshida, H. Eisaki,Z. Hussain, T. P. Devereaux, and Z.-X. Shen, 2012 PNAS 109 18332

[12] E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin, E. Schierle, E. Weschke, A. Gyenis, J. Wen,J. Schneeloch, Z. Xu, S. Ono, G. Gu, M. Le Tacon, and A. Yazdani, 2014 Science 343 6169 393.doi:10.1126/science.1243479

[13] R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle, E. Weschke, R. Sutarto,F. He, A. Soumyanarayanan, Y. He, M. Le Tacon, I. S. Elfimov, J. E. Hoffman, G. A. Sawatzky,B. Keimer, and A. Damascelli, 2014 Science 343 6169 390. doi:10.1126/science.1242996

[14] M. Hashimoto, E. A. Nowadnick, R.-H. He, I. M. Vishik, B. Moritz, Y. He, K. Tanaka, R. G.Moore, D. Lu, Y. Yoshida, M. Ishikado, T. Sasagawa, K. Fujita, S. Ishida, S. Uchida, H. Eisaki,Z. Hussain, T. P. Devereaux, and Z.-X. Shen, 2014 arXiv:cond-mat 1405.5199

[15] A. Kaminski, T. Kondo, T. Takeuchi, and G. Gu, 2014 arXiv:cond-mat 1403.0492[16] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, 1989 Phys.

Rev. Lett. 63 1996. doi:10.1103/PhysRevLett.63.1996[17] C. M. Varma, 1997 Phys. Rev. B 55 14554. doi:10.1103/PhysRevB.55.14554[18] C. Varma, Z. Nussinov, and W. van Saarloos, 2002 Physics Reports 361 5 267 . ISSN 0370-1573.

doi:http://dx.doi.org/10.1016/S0370-1573(01)00060-6


[19] G. A. Thomas, J. Orenstein, D. H. Rapkine, M. Capizzi, A. J. Millis, R. N. Bhatt, L. F.Schneemeyer, and J. V. Waszczak, 1988 Phys. Rev. Lett. 61 1313. doi:10.1103/PhysRevLett.61.1313

[20] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, 1975 Methods of quantum field theory instatistical physics. Dover, New York

[21] J. González, F. Guinea, and M. A. H. Vozmediano, 1999 Phys. Rev. B 59 R2474. doi:10.1103/PhysRevB.59.R2474

[22] A. G. Grushin, B. Valenzuela, and M. A. H. Vozmediano, 2009 Phys. Rev. B 80 155417. doi:10.1103/PhysRevB.80.155417

[23] K. M. Zurek, 2010 ArXiv e-prints[24] H. Georgi, 2007 Phys. Rev. Lett. 98 221601. doi:10.1103/PhysRevLett.98.221601[25] H. Georgi, 2007 Physics Letters B 650 4 275 . ISSN 0370-2693. doi:http://dx.doi.org/10.1016/j.

physletb.2007.05.037[26] K. Cheung, W.-Y. Keung, and T.-C. Yuan, 2007 Phys. Rev. Lett. 99 051803. doi:10.1103/

PhysRevLett.99.051803[27] P. W. Phillips, B. W. Langley, and J. A. Hutasoit, 2013 Phys. Rev. B 88 115129[28] K. B. Dave, P. W. Phillips, and C. L. Kane, 2013 Phys. Rev. Lett. 110 090403[29] V. Khodel and V. Shaginyan, 1990 JETP Lett. 51 553[30] N. B. Kopnin, T. T. Heikkilä, and G. E. Volovik, 2011 Phys. Rev. B 83 220503. doi:

10.1103/PhysRevB.83.220503[31] D. Yudin, D. Hirschmeier, H. Hafermann, O. Eriksson, A. I. Lichtenstein, and M. I. Katsnelson,

2014 Phys. Rev. Lett. 112 070403. doi:10.1103/PhysRevLett.112.070403[32] M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, 2014 arXiv: 1403.3694[33] M. L. Bellac, 1991 Quantum and Statistical Field Theory. Oxford Graduate Texts, New York[34] S. Tomonaga, 1950 Prog. in Theo. Phys. 5 544[35] J. M. Luttinger, 1963 Jour. of Math. Phys. 4 1154[36] S. Sachdev, 2011 Quantum Phase Transitions. Cambridge University press[37] J. Gonzalez, F. Guinea, and M. Vozmediano, 1994 Nuclear Physics B 424 3 595 . ISSN 0550-3213.

doi:http://dx.doi.org/10.1016/0550-3213(94)90410-3[38] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, 2012 Rev. Mod. Phys.

84 1067. doi:10.1103/RevModPhys.84.1067[39] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A.

Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, 2011 Nat. Phys.7 701

[40] F. de Juan, A. G. Grushin, and M. A. H. Vozmediano, 2010 Phys. Rev. B 82 125409. doi:10.1103/PhysRevB.82.125409

[41] H. Georgi and Y. Kats, 2008 Phys. Rev. Lett. 101 131603[42] P. Gaete and E. Spallucci, 2008 Physics Letters B 661 4 319 . ISSN 0370-2693. doi:http:

//dx.doi.org/10.1016/j.physletb.2008.02.036[43] G. Mack, 1977 Communications in Mathematical Physics 55 1 1[44] H. Goldberg and P. Nath, 2008 Phys. Rev. Lett. 100 031803. doi:10.1103/PhysRevLett.100.031803[45] T. Holstein, R. E. Norton, and P. Pincus, 1973 Phys. Rev. B 8 2649. doi:10.1103/PhysRevB.8.2649[46] N. V. Krasnikov, 2007 International Journal of Modern Physics A 22 28 5117. doi:10.1142/

S0217751X07037342[47] M. A. Stephanov, 2007 Phys. Rev. D 76 035008. doi:10.1103/PhysRevD.76.035008[48] N. G. Deshpande and X.-G. He, 2008 Phys. Rev. D 78 055006. doi:10.1103/PhysRevD.78.055006[49] B. I. Halperin, P. A. Lee, and N. Read, 1993 Phys. Rev. B 47 7312. doi:10.1103/PhysRevB.47.7312[50] C. Nayak and F. Wilczek, 1994 Nuc. Phys. B 417 359[51] W. Kohn and J. M. Luttinger, 1965 Phys. Rev. Lett. 15 524. doi:10.1103/PhysRevLett.15.524[52] H. Bruus and K. Flensberg, 2004Many-Body Quantum Theory in Condesed matter physics. Oxford

Graduate Texts, New York


[53] V. Mishra, U. Chatterjee, J. C. Campuzano, and M. R. Norman, 2014 Nat. Phys. 10 4357[54] L. D. Landau and E. M. Lifshitz, 1991 Quantum Mechanics. Pergamon press, Oxford[55] D. N. Basov, S. I. Woods, A. S. Katz, E. J. Singley, R. C. Dynes, M. Xu, D. G. Hinks, C. C.

Homes, and M. Strongin, 1999 Science 283 49[56] E. Gull and A. J. Millis, 2012 Phys. Rev. B 86 241106(R)[57] M. R. Norman and C. Pépin, 2003 Rep. Prog. Phys. 66 1547

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