Competition between spin density wave order and superconductivity in the underdoped cuprates
aa r X i v : . [ c ond - m a t . s t r- e l ] J un arXiv:0905.2608 Competition between spin density wave order andsuperconductivity in the underdoped cuprates
Eun Gook Moon and Subir Sachdev
Department of Physics, Harvard University, Cambridge MA 02138 (Dated: May 14, 2009)
Abstract
We describe the interplay between d -wave superconductivity and spin density wave (SDW) orderin a theory of the hole-doped cuprates at hole densities below optimal doping. The theory assumeslocal SDW order, and associated electron and hole pocket Fermi surfaces of charge carriers inthe normal state. We describe quantum and thermal fluctuations in the orientation of the localSDW order, which lead to d -wave superconductivity: we compute the superconducting criticaltemperature and magnetic field in a ‘minimal’ universal theory. We also describe the back-actionof the superconductivity on the SDW order, showing that SDW order is more stable in the metal.Our results capture key aspects of the phase diagram of Demler et al. (Phys. Rev. Lett. ,067202 (2001)) obtained in a phenomenological quantum theory of competing orders. Finally, wepropose a finite temperature crossover phase diagram for the cuprates. In the metallic state, theseare controlled by a ‘hidden’ quantum critical point near optimal doping involving the onset of SDWorder in a metal. However, the onset of superconductivity results in a decrease in stability of theSDW order, and consequently the actual SDW quantum critical point appears at a significantlylower doping. All our analysis is placed in the context of recent experimental results. . INTRODUCTION A number of recent experimental observations have the potential to dramatically advanceour understanding of the enigmatic underdoped regime of the cuprates. In the present paper,we will focus in particular on two classes of experiments (although our results will also haveimplications for a number of other experiments): • The observation of quantum oscillations in the underdoped region of YBCO.
The period of the oscillations implies a carrier density of order the density of dopants.LeBoeuf et al. have claimed that the oscillations are actually due to electron-likecarriers of charge − e . We will accept this claim here, and show following earlierwork , that it helps resolve a number of other theoretical puzzles in the underdopedregime. • Application of a magnetic field to the superconductor induces a quantum phase tran-sition at a non-zero critical field, H sdw , involving the onset of spin density wave(SDW) order. This transition was first observed in La − x Sr x CuO with x = 0 . et al. . Chang et al. have provided detailed studies of the spin dy-namics in the vicinity of H sdw , including observation of a gapped spin collective modefor H < H sdw whose gap vanishes as H ր H sdw . Most recently, such observationshave been extended to YBa Cu O . by Haug et al. , who obtained evidence for theonset of SDW order at H ≈
15 T. These observations were all on systems which donot have SDW order at H = 0; they build on the earlier work of Lake et al. whoobserved enhancement of prexisting SDW order at H = 0 by an applied field in La − x Sr x CuO with x = 0 . The phase diagramin the work of Demler et al. is reproduced in Fig. 1. The parameter t appears in a Landautheory of SDW order and tunes the propensity to SDW order, with SDW order being favoredwith decreasing t . We highlight a number of notable features of this phase diagram:A. The upper-critical field above which superconductivity is lost, H c , decreases withdecreasing t . This is consistent with the picture of competing orders, as decreasing t enhances the SDW order, which in turn weakens the superconductivity.B. The SDW order is more stable in the non-superconducting ‘normal’ state than in thesuperconductor. In other words, the line CM, indicating the onset of SDW orderin the normal state, is to the right of the point A where SDW order appears inthe superconductor at zero field; i.e. t c (0) > t c . Thus inducing superconductivitydestabilizes the SDW order, again as expected in a model of competing orders.2 c tH SC MAB DC
SC+SDWSDW "Normal" H c2 H sdw t c(0) FIG. 1: From Ref. 14: Phase diagram of the competition between superconductivity (SC) andspin density wave (SDW) order tuned by an applied magnetic field H , and a Landau parameter t controlling the SDW order (the effective action has a term t~ϕ , where ~ϕ is the SDW order). Thelabels identifying H c , H sdw , and t c (0) have been added to the original figure, but the figure isotherwise unchanged. The dashed line does not indicate any transition or crossover; it is just thecontinuation of the line CM to identify t c (0). A key feature of this phase diagram is that SDWorder is more stable in the metal than in the superconductor i.e. t c (0) > t c . C. An immediate consequence of the feature B is the existence of the line AM of quantumphase transitions within the superconductor, representing H sdw , where SDW orderappears with increasing H . As we have discussed above, this prediction of Demler etal. has been verified in a number of experiments.A related prediction by Demler et al. that an applied current should enhance the SDWorder, also appears to have been observed in a recent muon spin relaxation experiment. A glance at Fig. 1 shows that it is natural to place the quantum oscillationexperiments in the non-superconducting phase labeled “SDW”. Feature B aboveis crucial in this identification: the normal state reached by suppressing superconductivitywith a field is a regime where SDW order is more stable. The structure of the Fermi surfacein this normal state can be deduced in the framework of conventional spin-density-wave the-ory, and we recall the early results of Refs. 20,21 in Fig. 2. Recent studies have extendedthese results to incommensurate ordering wavevectors Q , and find that the electron pockets(needed to explain the quantum oscillation experiments) remain robust under deviationsfrom the commensurate ordering at ( π, π ). The present paper will consider only the case ofcommensurate ordering with Q = ( π, π ), as this avoids considerable additional complexity.The above phenomenological theory appears to provide a satisfactory framework for inter-preting the experiments highlighted in this paper. However, such a theory cannot ultimatelybe correct. A sign of this is that within its parameter space is a non-superconducting, non-3 ncreasing SDW order(a) (b) (c) (d) FIG. 2: (Color online) Fermi surface evolution in the SDW theory . Panel (d) is the “largeFermi surface” state appropriate for the overdoped superconductor. The SDW order parameter, ~ϕ ,desribes ordering at the wavevector Q = ( π, π ), and mixes fermion states whose wavevectors differby Q . This leads to the SDW metal state with electron (red) and hole (blue) pockets in panel (b),which is the state used here to explain the quantum oscillation experiments. SDW normal state at H = 0 and T = 0 (not shown in Fig. 1). Indeed, such a state isthe point of departure for describing the onset of the superconducting and SDW order inRef. 14. There is no such physically plausible state, and the parameters were chosen so thatthis state does not appear in Fig. 1. Furthermore, we would like to extend the theory tospectral properties of the electronic excitations probed in numerous other experiments. Thisrequires a more microscopic formulation of the theory of competing orders in terms of theunderlying electrons. We shall provide such a theory here, building upon the proposals ofRefs. 7,8,24,25. Our theory will not have the problematic H = 0, T = 0 “normal” state ofthe phenomenological theory, and so cannot be mapped precisely onto it. Nevertheless, wewill see that our theory does reproduce the key aspects of Fig. 1. We will also use our theoryto propose a finite temperature phase diagram for the hole-doped cuprates; in particular,we will argue that it helps resolve a central puzzle on the location of the quantum criticalpoint important for the finite temperature crossovers into the ‘strange metal’ phase. Theseresults appear in Section IV and Fig. 10.The theory of superconductivity mediated by exchange of quanta of the SDW orderparameter, ~ϕ , has been successful above optimal doping. However, it does not appear tobe compatible with the physics of competing orders in the underdoped regime, at least inits simplest version. This theory begins with the “large Fermi surface” state in panel (d) ofFig. 2, and examines its instability in a BCS/Eliashberg theory due to attraction mediatedby exchange of ~ϕ quanta. An increase in the fluctuations of ~ϕ is therefore connected to anincrease in the effective attraction, and consequently a strengthening of the superconductingorder. This is evident from the increase in the critical temperature for superconductivityas the SDW ordering transition is approached from the overdoped side (see e.g. Fig. 4 in4ef. 27). Thus rather than a competition, this theory yields an effective attraction betweenthe SDW and superconducting order parameters. This was also demonstrated in Ref. 14 by amicroscopic computation in this framework of the coupling between these order parameters.It is possible that these difficulties may be circumvented in more complex strong-couplingversions of this theory , but a simple physical picture of these is lacking.As was already discussed in Ref. 14, the missing ingredient in the SDW theory of the or-dering of the metal is the knowledge of the proximity to the Mott insulator in the underdopedcompounds. Numerical studies of models in which the strong local repulsion associated withMott insulator is implemented in a mean-field manner do appear to restore aspects of thepicture of competing orders. Here, we shall provide a detailed study of the model of theunderdoped cuprates proposed in Refs. 7,8,24,25, and show that it is consistent with thefeatures A, B, and C of the theory of competing orders noted above, which are essential inthe interpretation of the experiments.As discussed at some length in Ref. 8, the driving force of the superconductivity in theunderdoped regime is argued to be the pairing of the electron pockets visible in panel (b) ofFig. 2. Experimental evidence for this proposal also appeared in the recent photoemissionexperiments of Yang et al. . In the interests of simplicity, this paper will focus exclusively onthe electron pockets, and neglect the effects of the hole pockets in Fig. 2. Further discussionon the hole pockets, and the reason for their secondary role in superconductivity may befound in Refs. 8,24,25.The degrees of freedom of the theory are the bosonic spinons z α ( α = ↑ , ↓ ), and spinlessfermions g ± . The spinons determine the local orientation of the SDW order via ~ϕ = z ∗ α ~σ αβ z β (1.1)where ~σ are the Pauli matrices. The electrons are assumed to form electron and hole pocketsas indicated in Fig. 2b, but with their components determined in a ‘rotating reference frame’set by the local orientation of ~ϕ . This idea of Fermi surfaces correlated with the local orderis supported by the recent STM observations of Wise et al. . Focussing only on the electronpocket components, we can write the physical electron operators c α as c ↑ = e i G · r h z ↑ g + − z ∗↓ g − i + e i G · r h z ↑ g + + z ∗↓ g − i c ↓ = e i G · r h z ↓ g + + z ∗↑ g − i + e i G · r h z ↓ g + − z ∗↑ g − i (1.2)where G = (0 , π ) and G = ( π,
0) are the anti-nodal points about which the electronpockets are centered. We present an alternative derivation of this fundamental relation fromspin-density-wave theory in Appendix A.Note that when z α = (1 , z direction with ~ϕ = (0 , , c ↑ = g + ( e i G · r + e i G · r ) and c ↓ = g − ( e i G · r − e i G · r ). Thus, for this SDW state, the ± labels on the g ± are equivalent5o the z spin projection, and the spatial dependence is the consequence of the potentialcreated by the SDW order, which has opposite signs for the two spin components (as shownin Appendix A). The expression in Eq. (1.2) for general ~ϕ is then obtained by performinga spacetime-dependent spin rotation, determined by z α , on this reference state.Another crucial feature of Eqs. (1.1) and (1.2) is that the physical observables ~ϕ and c α are invariant under the following U(1) gauge transformation of the dynamical variables z α and g ± : z α → e iφ z α ; g + → e − iφ g + ; g − → e iφ g − . (1.3)Thus the ± label on the g ± can also be interpreted as the charge under this gauge transfor-mation. This gauge invariance implies that the low energy effective theory will also includean emergent U(1) gauge field A µ .We will carry out most of the computations in this paper using a “minimal model” for z α and g ± with the imaginary time ( τ ) Lagrangian L = L z + L g , (1.4)where the fermion action is L g = g † + (cid:20) ( ∂ τ − iA τ ) − m ∗ ( ∇ − i A ) − µ (cid:21) g + + g †− (cid:20) ( ∂ τ + iA τ ) − m ∗ ( ∇ + i A ) − µ (cid:21) g − , (1.5)and the spinon action is L z = 1 t " N X α =1 (cid:18) | ( ∂ τ − iA τ ) z α | + v | ( ∇ − i A ) z α | (cid:19) + i̺ N X α =1 | z α | − N ! . (1.6)Here the emergent gauge field is A µ = ( A τ , A ), and, for future convenience, we have general-ized to a theory with N spin components (the physical case is N = 2). The field ̺ imposes afixed length constraint on the z α , and accounts for the self-interactions between the spinons.This effective theory omits numerous other couplings involving higher powers or gradi-ents of the fields, which have been discussed in some detail in previous work. It alsoomits the 1 /r Coulomb repulsion between the g ± fermions–this will be screened by the Fermisurface excitations, and is expected to reduce the critical temperature as in the traditionalstrong-coupling theory of superconductivity. For simplicity, we will neglect such effects here,as they are not expected to modify our main conclusions on the theory of competing orders.Non-perturbative effects of Berry phases are expected to be important in the superconduct-ing phase, and were discussed earlier; they should not be important for the instabilitiestowards superconductivity discussed here. 6s has been discussed earlier, the theory in Eq. (1.4) has a superconducting ground statewith a a simple momentum-independent pairing of the g ± fermions h g + g − i 6 = 0. Combiningthis pairing amplitude with Eq. (1.2), it is then easy to see that the physical c α fermionshave the needed d -wave pairing signature (see Appendix A).The primary purpose of this paper is to demonstrate that the simple field theory inEq. (1.4) satisfies the constraints imposed by the framework of the picture of competingorders. In particular, we will show that it displays the features A, B, and C listed above.Thus, we believe, it offers an attractive and unified framework for understanding a largevariety of experiments in the underdoped cuprates. We also note that the competing orderinterpretation of Eq. (1.4) only relies on the general gauge structure of theory, and notspecifically on the interpretation of g ± as electron pockets in the anti-nodal region; thus itcould also apply in other physical contexts.Initially, it might seem that the simplest route to understanding the phase diagram ofour theory Eq. (1.4) is to use it to compute the effective coupling constants in the phe-nomenological theory of Ref. 14. However, such a literal mapping is not possible, because,as we discussed earlier, the phenomenological theory does have additional unphysical phases.Rather, we will show that our theory does satisfy the key requirements of the experimentallyrelevant phase diagram in Fig. 1.A notable feature of the theory in Eq. (1.4) is that it is characterized by only 2 dimen-sionless couplings. We assume the chemical potential µ is adjusted to obtain the requiredfermion density, which we determine by the value of the Fermi wavevector k F . The effectivefermion mass m ∗ and the spin-wave velocity then determine our first dimensionless ratio α ≡ ~ k F m ∗ v . (1.7)Although we have inserted an explicit factor of ~ above, we will set ~ = k B = 1 in mostof our analysis. Note that we can also convert this ratio to that of the Fermi energy, E F = ~ k F / (2 m ∗ ) and the energy scale m ∗ v : E F m ∗ v = α , m ∗ = 1 . m e and πk F =5 . − , and the spin-wave velocity in the insulator v ≈
670 meV ˚A, we obtain the estimate α ≈ .
76. We will also use m ∗ v ≈
112 meV (1.9)as a reference energy scale.The second dimensionless coupling controls the strength of the fluctuations of the SDWorder, which are controlled by the parameter t in Eq. (1.6). Tuning this coupling leads to atransition from a phase with h z α i 6 = 0 to one where the spin rotation symmetry is preserved.7e assume that this transition occurs at the value t = t c (0) in the metallic phase (thesignificance of the argument of t c will become clear below): this corresponds to the line CMin Fig. 1. Then we can charaterize the deviation from this quantum phase transition by thecoupling α ≡ (cid:18) t c (0) − t (cid:19) m ∗ . (1.10)Note that α < α > α >
0, we can also characterize this coupling by the valueof the spinon energy gap ∆ z in the N = ∞ theory, which is (as will become clear below)∆ z m ∗ v = 4 πα . (1.11)It is worth noting here that our “minimal model” (Eq. (1.4)) in two spatial dimensions hasaspects of the universal physics of the Fermi gas at unitarity in three spatial dimensions. Thelatter model has a ‘detuning’ parameter which tunes the system away from the Feshbachresonance; this is the analog of our parameter α . The overall energy scale is set in theunitary Fermi gas by the Fermi energy; here, instead, we have 2 energy scales, E F and m ∗ v .The outline of the remainder of the paper is as follows. In Section II, we will consider thepairing problem of the g ± fermions, induced by exchange of the gauge boson A µ . We willdo this within a conventional Eliashberg framework. Our main result will be a computationof the critical field H c , which will be shown to be suppressed as SDW order is enhancedwith decreasing t . Section III will consider the feedback of the superconductivity on theSDW ordering, where we will find enhanced stability of the SDW order in the metal overthe superconductor. Section IV will summarize our results, and propose a crossover phasediagram at non-zero temperatures. II. ELIASHBERG THEORY OF PAIRING
In our mininal model, the charge and spin excitations interact with each other throughthe A µ gauge boson. So the gauge fluctuation is one of the key ingredients in our analysis.We begin by computing the gauge propagator, and then we will determine the criticaltemperature and magnetic field within the Eliashberg theory in the following subsections.We use the framework of the large N expansion. In the limit N = ∞ , the gauge field issuppressed, and the constraint field ̺ takes a saddle point value ( i̺ = m ) that makes thespinon action extremum in Eq. (1.6). At leading order, the spinon propagator has the form tv k + ω n + m (2.1)8 - Α mT FIG. 3: The parameter m in Eq. (2.1) for T / ( m ∗ v ) = 0 . where k is spatial momentum, ω n is the Matsubara frequency. The saddle point equationfor m is T X ω n Z d k π (cid:20) v k + ω n + m (cid:21) = − m ∗ α + Z dω π Z d k π v k + ω . (2.2)The solution of this is m = 2 T ln " e +2 πm ∗ v α /T + √ e +4 πm ∗ v α /T + 42 (2.3)which holds for −∞ < α < ∞ . This result is plotted in Fig. 3. Clearly, m is a monotonicallyincreasing function of α . Recall that the positive α region has no SDW order, and m islarge here. As we will see below, the value of m plays a significant role in the photonpropagators.The photon propagator is determined from the effective action obtained by integratingout the spinons and non-relativistic fermions. Using gauge invariance, we can write downthe effective action of the gauge field as follows: S A = N T X ǫ n Z d k π (cid:20) ( k i A τ − ǫ n A i ) D ( k, ǫ n ) k + A i A j (cid:18) δ ij − k i k j k (cid:19) D ( k, ǫ n ) (cid:21) . (2.4)As in analogous computation with relativistic fermions in Ref. 30, we separate the photonpolarizations into their bosonic and fermionic components: D = N D b + D f D = N D b + D f . (2.5)9e use the Coulomb gauge, k · A = 0 in the computation. After imposing the gaugecondition, the propagator of A τ from the above action is 1 /D , while that of A i is (cid:18) δ ij − k i k j k (cid:19) D + ( ǫ n /k ) D . (2.6)We will approximate D b and D b by their zero frequency limits. Computation of thespinon polarization in this limit, as in Ref. 30 yields D b ( k ) = − Tπv ln (cid:16) (cid:16) m T (cid:17)(cid:17) + 12 πv Z dx p m + v k x (1 − x ) coth p m + v k x (1 − x )2 T ! (2.7)and D b ( k ) = v k π Z dx p m + v k x (1 − x ) coth p m + v k x (1 − x )2 T ! (2.8)For the fermionic contributions, we include the contribution of the g ± fermions with effectivemass m ∗ and Fermi wavevector k F . Calculation of the fermion compressibility yields D f ( k, ǫ n ) = 2 Z d q π ( n F ( ε q − k / ) − n F ( ε q + k / ))( iǫ n + k · q /m ∗ ) ≈ m ∗ π , (2.9)where n F is the Fermi function. For the transverse propagator, we obtain from the compu-tation of the fermion current correlations D f ( k, ǫ n ) + ǫ n k D f ( k, ǫ n )= k F πm ∗ − m ∗ Z d q π (cid:18) q − ( q · k ) k (cid:19) ( n F ( ε q − k / ) − n F ( ε q + k / ))( iǫ n + k · q /m ∗ ) ≈ k F | ǫ n | πk (2.10)Putting all this together, we have the final form of the propagators. The propagator of A τ is 1 N D b ( k ) + m ∗ /π (2.11)while that of A i is (cid:18) δ ij − k i k j k (cid:19) N D b ( k ) + k F | ǫ n | / ( πk ) . (2.12)10 . Eliashberg equations We now address the pairing instability of the g ± fermions. Both the longitudinal andtransverse photons contribute an attractive interaction between the oppositely chargedfermions, which prefers a simple s -wave pairing. However, we also know that the trans-verse photons destroy the fermionic quasiparticles near the Fermi surface, and so havea depairing effect. The competition between these effects can be addressed in the usualEliashberg framework. Based upon arguments made in Refs. 34,35, we can anticipate thatthe depairing and pairing effects of the transverse photons exactly cancel each other in thelow-frequency limits, because of the s -wave pairing. The higher frequency photons yield anet pairing contribution below a critical temperature T c which we compute below.Closely related computations have been carried out by Chubukov and Schmalian on ageneralized model of pairing due to the exchange of a gapless bosonic collective mode; ournumerical results for T c below agree well with theirs, where the two computations can bemapped onto each other. The Eliashberg approximation starts from writing the fermion Green function usingNambu spinor notation. ˆΣ( ω n ) = iω n (1 − Z ( ω n ))ˆ τ + ǫ ˆ τ + φ. ( ω n )ˆ τ (2.13) G − ( ǫ, ω n ) = iω n Z ( ω n )ˆ τ − ǫ ˆ τ − φ ( ω n )ˆ τ where ˆ τ are the Pauli matrices in the particle-hole space. Then self-consistency equation isconstructed by evaluating the self-energy with the above Green function, which yields thefollowing equation: ˆΣ( iω n ) = T X ω m Z d k ′ π ˆ G ( k ′ , ω m ) e D( ~q, ~k, ω m − ω n ) (2.14)= T X ω m λ tot ( ω m − ω n ) Z dǫ ′ ˆ G ( ǫ ′ , ω m )Note that the first line is a formal expression, with e D( ~q, ~k, iω m ) being a combination ofthe photon propagator and the matrix elements of the vertex. The equations are thereforecharacterized by the coupling λ tot ( ω n ); computation of the photon contribution yields theexplicit expression λ tot ( ω n ) = λ T ( ω n ) + λ L (2.15) λ T ( ω n ) = k F π m ∗ Z k F dk p − ( k/ k F ) N D b ( k ) + k F | ω n | / ( πk ) λ L = m ∗ π k F Z k F dk p − ( k/ k F ) (cid:20) N D b ( k ) + m ∗ /π (cid:21) (2.16)11 - Α Λ L - - Α Λ T FIG. 4: (Color online) The pairing coupling constants associated with the longitudinal ( λ L )and transverse ( λ T ( ω n )) gauge interactions. The parameter α measures the distance fromthe SDW ordering transition in the metal, as defined in Eq. (1.10). The dotted (red), dot-dashed (green), dashed (blue), and continuous (black) lines correspond to α / E F / ( m ∗ v ) =0 . , . , . , .
29. We show λ T ( ω n = 8 πT ) with T / ( m ∗ v ) = 0 .
016 for the transverse interaction.Note that λ T ( ω n ) function is analytic near α ∼ We have divided the total coupling into two pieces based on the different frequency depen-dence of the longitudinal and transversal gauge boson propagators. The frequency indepen-dent term will need a cutoff for the actual calculation as we will see below. The typicalbehaviors of the dimensionless couplings λ T ( ω n ) , λ L are shown in Fig 4.The longitudinal coupling λ L is around 0 .
35, and has a significant dependence upon α ,which is a measure of the distance from the SDW ordering transition. Note that λ L is larger in the SDW-disordered phase ( α > λ T ( ω n ). Thisis divergent at low frequencies with λ T ( ω n ) ∼ | ω n | − / . As we noted earlier, this diver-gent piece cancels out between the normal and anomalous contributions to the fermion selfenergy. We plot the dependence of λ T ( ω n ) on the coupling α for a fixed ω n in Fig. 4.As was for the case of the longitudinal coupling, the transverse contribution is larger in theSDW-disordered phase.The full self-consistent Eliashberg equations are obtained by matching the coefficients ofthe Pauli matrices term by term. iω n (1 − Z ( ω n )) = − πT X ω m λ tot ( ω m − ω n ) iω m p ω m + ∆ ( ω m ) (2.17)∆( ω n ) = πT X ω m λ tot ( ω m − ω n ) ∆( iω m ) p ω m + ∆ ( ω m ) (2.18)where ∆( ω n ) is the frequency-dependent pairing amplitude.Now we can solve the self-consistent equations to determine the boundary of the super-conducting phase. Our goal is to look for the critical temperature and magnetic field, andwe can linearize the equations in ∆( ω n ) in these cases; in other words we would neglect thegap functions in the denominator. Z ( ω n ) = 1 + πTω n X ǫ n sgn( ǫ n ) λ T ( ω n − ǫ n )= 1 + πT | ω n | X | ǫ n | < | ω n | λ T ( ǫ n ) (2.19)Then the solution of the critical temperature of linearized Eliashberg equation is equivalentto the condition that the matrix K ( ω n , ω m ) first has a positive eigenvalue, where K ( ω n , ω m ) = λ T ( ω n − ω m ) + λ L Λ( ω n − ω m ) − δ n,m | ω n | Z ( ω n ) πT (2.20)with the soft cutoff function with cutoff E F Λ( ω n ) ≡
11 + c ( ω n /E F ) (2.21)where c is a constant of order unity. The cutoff E F is the highest energy scale of theelectronic structure, so it is not unnatural to set the cutoff with the scale. With this, thenumerics is well-defined and we plot the resulting critical temperature in Fig. 5.For comparison, we show in Fig. 6 the results for T c obtained in a model with only thetransverse interaction associated with λ T ( ω n ). We can use this T c to define an effective13 - Α T c m * v - - Α T c E F FIG. 5: (Color online) The critical temperature for superconductivity obtained by solution of theEliashberg equations. The lines are for the same parameter values as in Fig. 4. The top plot hascritical temperature scaled with m ∗ v , and the bottom is one scaled with E F . transverse coupling, λ T , by T c /E F = exp( − /λ T ). Using T c /E F ≈ .
008 for α ≈ λ T ≈ .
2. This is of the same order as the longitudinal coupling λ L for α ≈ λ T ( ω n ) and λ L clearly induces a higher critical temperaturein the SDW-disordered region. Note that this behavior is different from the one of previousSDW-mediated superconductivity. (See the results of Ref . 27 Fig. [4]; near the criticalregion, T c shows the opposite behavior there.) We have also compared the plots obtainedby scaling T c by m ∗ v and E F . The dependencies on the parameter α are reversed in twoplots in the SDW-disordered region. To interpret α as the doping related parameter, weshould choose the scaling by m ∗ v because the mass m ∗ and spin wave velocity v are notaffected much by doping. With this scaling (the first plot in Fig. 5), the critical temperaturerises with increase doping at fixed α ; of course, in reality, α is also an increasing function14 - Α T c E F FIG. 6: (Color online) As in the top panel of Fig. 5, but with only the transverse pairing interaction, λ T ( ω n ), included. of doping. B. Critical field
This subsection will extend the above analysis to compute the upper-critical magneticfield, H c at T = 0. We will neglect the weak Zeeman coupling of the applied field, andassume that it couples only to the orbital motion of the g ± fermions. This means that L g in Eq. (1.5) is modified to L g = g † + (cid:20) ( ∂ τ − iA τ ) − m ∗ ( ∇ − i A − i ( e/c ) a ) − µ (cid:21) g + + g †− (cid:20) ( ∂ τ + iA τ ) − m ∗ ( ∇ + i A − i ( e/c ) a ) − µ (cid:21) g − , (2.22)where ∇ × a = H is the applied magnetic field.Generally, the magnetic field induces non-local properties in the Green’s function. How-ever, in the vanishing gap limit, Helfand and Werthamer proved the non-locality only ap-pears as a phase factor (see Ref. 41). The formalism has been developed by Shossmann andSchachinger , and we will follow their method. As they showed, in the resulting equationfor H c , the magnetic field only appears in the modification of the frequency renormalization Z ( ω n ).The Eliashberg equations in zero magnetic contain a term | ω n Z ( ω n ) | , which comes from15he inverse of the Cooperon propagator type at momentum q = 0, C ( ω n , C ( ω n , q ) = Z d p π − iω n Z ( ω n ) + ε p + q )( iω n Z ( ω n ) + ε − p ) ≈ N (0) Z π dθ π Z ∞−∞ dε − iω n Z ( ω n ) + ε + v F q cos θ )( iω n Z ( ω n ) + ε )= 2 πN (0) p ω n Z ( ω n ) + v F q (2.23)where N (0) is the density of states at the Fermi level per spin.Now we discuss the extension of this to H = 0, as described in Refs. 40,41,42. For this,we need to replace C ( ω n ,
0) by the smallest eigenvalue of the operatorˆ L ( ω n ) = Z d ρ Z d q π C ( ω n , q ) e i q · ρ e − i ρ · ˆ π (2.24)where ˆ π = ˆ p − (2 e/ ~ c ) A (ˆ r ). Using Eq. (22) from Ref. 40, we find the smallest eigenvalueof ˆ L ( ω n ) is L ( ω n ) = Z ∞ ρdρ Z ∞ qdqJ ( qρ ) C ( ω n , q ) e − ρ / (2 r H ) = r H Z ∞ qdqe − q r H / C ( ω n , q ) (2.25)where r H = p ~ c/ eH is the magnetic length.So the only change in the presence of a field is that the wavefunction renormalization Z ( ω n ) is replaced by Z H ( ω n ), where1 Z H ( ω n ) = 2 | ω n | r H Z ∞ qdq e − q r H / p ω n Z ( ω n ) + v F q (2.26)= 2 | ω n | Z ∞ xdx e − x / q ω n Z ( ω n ) + v F r − H x (2.27)We can now insert the modified Z ( ω n ) into Eq. (2.27) into Eq. (2.20), and so compute H c as a function of both α and the SDW tuning parameter α . The natural scale for themagnetic field is H m ≡ (cid:18) ~ c e (cid:19) k F ≈
534 Tesla , (2.28)where in the last step we have used values from the quantum oscillation experiment quotedin Section I. Our results for H c /H m are shown in Fig. 7. We can see that the criticalfield dependence on α is similar to the critical temperature dependence: it is clear thatSDW competes with superconductivity, and that H c decreases as the SDW ordering is16 - Α H c2 H m - - Α Α H c2 H m FIG. 7: (Color online) The upper critical field H c as a function of α and α using the sameconventions as in Fig. 4. The magnetic field is measured with the units induced by the fermionmass via H m defined in Eq. (2.28). enhanced by decreasing α . Also, we can compare this with the phenomenological phasediagram of Fig 1; the critical field line in Fig. 7 determines the line B-M-D within Eliashbergapproximation. Finally, the values of H c in Fig. 7 are quite compatible with the quantumoscillation experiments. III. SHIFT OF SDW ORDERING BY SUPERCONDUCTIVITY
We are interested in the feedback on the strength of magnetic order due to the onsetof superconductivity. Rather than using a self-consistent approach, we will address thequestion here systematically in a 1 /N expansion.We will replace the fermion action in Eq. (1.5) by a theory which has N/ L g = N/ X a =1 ( g † + a (cid:20) ( ∂ τ − iA τ ) − m ∗ ( ∇ − i A ) − µ (cid:21) g + a + g †− a (cid:20) ( ∂ τ + iA τ ) − m ∗ ( ∇ + i A ) − µ (cid:21) g − a − ∆ g + a g − a − ∆ g †− a g † + a ) (3.1)Here we consider the gauge boson fluctuation more rigorously in the sense of accountingfor full fermion and boson polarization functions. But we will treat the fermion pairingamplitude ∆ as externally given: the previous section described how it could be determinedin the Eliashberg theory with approximated polarization.The large N expansion proceeds by integrating out the z α and the g ± a , and then expandingthe effective action for ̺ and A µ – formally this has the same structure as the computationin Ref. 30, generalized here to non-relativistic fermions. At N = ∞ , the g ± a and z α remaindecoupled because the gauge propagator is suppressed by a prefactor of 1 /N . So at thislevel, the magnetic critical point is not affected by the presence of the fermions, and appearsat t = t c where 1 t c = Z dωd k π ω + v k . (3.2)We are interested in determining the 1 /N correction to the magnetic quantum criticalpoint, which we write as 1 t c (∆) = 1 t c + 1 N F (∆); (3.3)note that in the notation of Fig. 1, t c ≡ t c (∆). The effect of superconductivity on themagnetic order will therefore be determined by F (∆) − F (0), which is the quantity to becomputed. The shift of the critical point at this order will be determined by the graphs inFig. 3 of Ref. 30, which are reproduced here in Fig. 8. Evaluating these graphs we find F (∆) = Z d qdω π Z d pdǫ π ǫ + v p ) " D ( q, ω ) (cid:18) (2 ǫ + ω ) ( ǫ + ω ) + v ( p + q ) − ω ω + v q (cid:19) + 1[ D ( q, ω ) + ( ω /q ) D ( q, ω )] (cid:18) v ( p − ( p.q ) /q )( ǫ + ω ) + v ( p + q ) (cid:19) + 1Π ̺ ( q, ω ) (cid:18) ω + v q − ω + ǫ ) + v ( p + q ) (cid:19) , (3.4)where 1 / Π ̺ ( q, ω ) = 8 p ω + v q is the propagator of the Lagrange multiplier field ̺ . Thelast term involving Π ̺ is independent of ∆, and so will drop out of our final expressions18 a) (f)(e) (b)(c) (d) FIG. 8: Feynman diagrams for the self energy of z α from Ref. 30. The full line represents z α , thewavy line is the A µ propagator, and the dashed line is the ̺ propagator which imposes the lengthconstraint on z α . measuring the influence of superconductivity: we will therefore omit this term in subsequentexpressions for F (∆).It is now possible to evaluate the integrals over p and ǫ analytically. This is done by usinga relativistic method in 3 spacetime dimensions. Using a 3-momentum notation in which P µ ≡ ( vp i , ǫ ) and Q µ ≡ ( vq i , ω ) and R P ≡ v R dǫd p/ (8 π ), some useful integrals obtainedby dimensional regularization are: Z P P = 0 Z P P ( P + Q ) = 0 Z P P µ P ( P + Q ) = − Q µ Q Z P P µ P ν P ( P + Q ) = δ µν Q + Q µ Q ν Q . (3.5)While some of the integrals above appear infrared divergent, there are no infrared divergen-cies in the complete original expression in Eq. (3.4), and we have verified that dimensionalregularization does indeed lead to the correct answer obtained from a more explicit subtrac-19ion of the infrared singularities. Using these integrals, we obtain from Eq. (3.4) F (∆) = Z d qdω π q ω + v q ) / " ω + v q ) D ( q, ω ) + 1 q D ( q, ω ) + ω D ( q, ω ) . (3.6)The above expression was obtained in the Coulomb gauge, but we have verified that it isindeed gauge invariant.We can now characterize the shift of the critical point in the superconductor by determin-ing the spinon gap, ∆ z , at the coupling t = t c (0) where there is onset of magnetic order inthe metal i.e. the spinon gap in the superconductor at H = 0 at the value of t correspondingto the line CM in Fig. 1. To leading order in 1 /N , this is given by∆ z m ∗ v = 4 πm ∗ (cid:18) t c (∆) − t c (0) (cid:19) = 4 πm ∗ N ( F (∆) − F (0)) . (3.7)This expression encapsulates our main result on the backaction of the superconductivity ofthe g ± fermions, with pairing gap ∆, on the position of the SDW ordering transition.Before we can evaluate Eq. (3.7), we need the gauge field propagators D , . For com-pleteness, we give explicit expressions for the boson and fermionic contributions by writing D ( q, ω ) + ω q D ( q, ω ) = D bT ( q, ω ) + D fT ( q, ω ) (3.8) D ( q, ω ) = D bL ( q, ω ) + D fL ( q, ω ) . (3.9)We can read off the bosonic polarization functions D bL,T ( q, ω ) from the exact relativisticresult of Ref. 30, and the Eq. (2.4). D bT ( q, ω ) = p v q + ω
16 (3.10) D bL ( q, ω ) = 116 q p v q + ω (3.11)For the fermion contribution, let us introduce the Nambu spinor Green’s function¯ g ( q, ω ) = 1( iω ) − E q iω + ξ q − ∆ − ∆ iω − ξ q ! (3.12)= Z d Ω π − iω Im [¯ g ( q, Ω)] (3.13)Im [¯ g ( q, Ω)] = ( − π )2 E q ( δ (Ω − E q ) − δ (Ω + E q )) Ω + ξ q − ∆ − ∆ Ω − ξ q ! , (3.14)20here ξ q = q / (2 m ∗ ) − µ and E q = p ξ q + ∆ . With the matrix elements of longitudinaland transverse parts, the polarizations of the fermions are as: D fL ( q, ω ) = − Z d k (2 π ) dǫ (2 π ) tr [¯ g ( k, ǫ )¯ g ( q + k, ω + ǫ )]= Z d k (2 π ) Z d Ω ′ π d Ω π n F (Ω ′ ) − n F (Ω) iω + Ω − Ω ′ tr [Im¯ g ( k, Ω)Im¯ g ( q + k, Ω ′ )]= Z d k (2 π ) (cid:18) − ξ k ξ k + q + ∆ E k E k + q (cid:19) (cid:18) E k + E k + q ) ω + ( E k + E k + q ) (cid:19) . (3.15) D fT ( q, ω ) = D fT,dia + D fT,para D fT,para = − Z d k (2 π ) m ∗ ) (cid:18) k − ( k · q ) q (cid:19) Z dǫ (2 π ) tr [¯ g ( k, ǫ )ˆ τ ¯ g ( q + k, ω + ǫ )ˆ τ ]= − Z d k (2 π ) k sin θ ( m ∗ ) Z d Ω π d Ω ′ π n F (Ω ′ ) − n F (Ω) iω + Ω − Ω ′ × tr [Im¯ g ( k, Ω)ˆ τ Im¯ g ( q + k, Ω ′ )ˆ τ ]= − Z d k (2 π ) k sin θ ( m ∗ ) (cid:18) − ξ k ξ k + q − ∆ E k E k + q (cid:19) (cid:18) E k + E k + q ) ω + ( E k + E k + q ) (cid:19) (3.16) D fT,dia = ρ f m ∗ , (3.17)where ρ f is the density of the fermions and ˆ τ is a Pauli matrix in the Nambu particle-holespace. With these results we are now ready to evaluate Eq. (3.7).One of the key features of the theory of competing orders was the enhanced stability ofSDW ordering in the metallic phase. This corresponds to feature B discussed in Section I: t c (0) > t c in Fig. 1. In the notation of our key result in Eq. (3.7), where t c (∆) ≡ t c , thisrequires ∆ z >
0. We show numerical evaluations of Eq. (3.7) in Fig 9 and find this indeedthe case. (The values of ∆ used in Fig. 9 are similar to those obtained in Section II near theSDW ordering critical point.) Indeed, the sign of ∆ z is easily understood. In the metallicphase, the gauge fluctuations are quenched by excitations of the Fermi surface. On the otherhand, in the superconducting state, this effect is no longer present: gauge fluctuations areenhanced and hence SDW ordering is suppressed. Note that the fact that the g ± fermionshave opposite gauge charges is crucial to this conclusion. The ordinary Coulomb interaction,under which the g ± have the same charge, continues to be screened in the superconductor.In contrast, a gauge force which couples with opposite charges has its polarizability stronglysuppressed in the superconductor, much like the response of a BCS superconductor to aZeeman field. 21 .2 0.25 0.3 0.35 0.40.0.0050.010.0150.020.025 10 D m * v D z m * v FIG. 9: The energy ∆ z in Eq. (3.7) determining the value of the shift in the SDW ordering criticalpoint, t c (0) − t c (∆). The horizontal axis is the externally given superconducting gap. For numericswe fix the parameter α / E F /m ∗ v = 0 . IV. CONCLUSIONS
This paper has described the phase diagram of a simple ’minimal model’ of the under-doped, hole-doped cuprates contained in Eqs. (1.4), (1.5), and (1.6). This theory describesbosonic neutral spinons z α and spinless charge − e fermion g ± coupled via a U(1) gauge field A µ . We have shown that the theory reproduces key aspects of a phenomenological phasediagram of the competition between SDW order and superconductivity in Fig. 1 in anapplied magnetic field, H . This phase diagram has successfully predicted a number of recentexperiments, as was discussed in Section I.In particular, in Section II, we showed that the minimal model had a H c which decreasedas the SDW ordering was enhanced by decreasing the coupling t in Eq. (1.6).Next, in Section III, we showed that the onset of SDW ordering in the normal state with H > H c occurred at a value t = t c (0) which was distinct from the value t = t c (∆) inthe superconducting state with H = 0. As expected from the competing order picture inFig. 1, we found t c (0) > t c (∆). The enhanced stability of SDW ordering in the metal wasa consequence of the suppression of A µ gauge fluctuations by the g ± Fermi surfaces. TheseFermi surfaces are absent in the superconductor, and as a result the gauge fluctuations arestronger in the superconductor.We conclude this paper by discussing implications of our results for the phase diagramat
T >
0, and in particular for the pseudogap regime above T c . In our application of themain result in Section III, t c (0) > t c (∆) we have assumed that the ∆ = 0 state was reachedby application of a magnetic field. However, this result also applies if ∆ is suppressed bythermal fluctuations above T c . Unlike H , thermal fluctuations will also directly affect the22 SCSC+SDW
Small Fermi pocketswith pairing fluctuations (RC)
Large Fermi surface (QD)
Strange metal (QC)RC QCQD t c tt c(0) FIG. 10: Proposed finite temperature crossover phase diagram for the cuprates. The labels at T = 0 are as in Fig. 1: the onset of SDW order in the superconductor is at t c ≡ t c (∆), while t c (0)is a ‘hidden’ critical point which can be observed only at H > H c as in Fig. 1. The computationsin Section III show that t c (0) > t c (∆). The full line is the phase transition at T c representing lossof superconductivity. The dashed lines are crossovers in the fluctuations of the SDW order. Thedotted lines are guides to the eye and do not represent any crossovers. Thus, in the pseudogapregime at T > T c the SDW fluctuations are in the ‘renormalized classical’ (RC) regime; uponlowering temperature, they crossover to the ‘quantum critical’ (QC) and ‘quantum disordered’(QD) regime in the superconductor. SDW order, in addition to the indirect effect through suppression of superconductivity. Inparticular in two spatial dimensions there can be no long-range SDW order at any
T > T , t plane. We anticipate that t c (0) is near optimal doping. Thus in the underdoped regime above T c , there is local SDW order which is disrupted by classical thermal fluctuations: this is theso-called ‘renormalized classical’ regime of the hidden metallic quantum critical point at t c (0). Going below T c in the underdoped regime, we eventually reach the regime controlledby the quantum critical point associated with SDW ordering in the superconductor, whichis at t c (∆). Because t c (∆) < t c (0), the SDW order can now be ‘quantum disordered’ (QD).Thus neutron scattering in the superconductor will not display long-range SDW order as T →
0, even though there is a RC regime of SDW order above T c . This QD region will haveenhanced charge order correlations ; this charge order can survive as true long-rangeorder below T c , even though the SDW order does not. Thus we see that in our theory theunderlying competition is between superconductivity and SDW order, while there can besubstantial charge order in the superconducting phase.23urther study of the nature of the quantum critical point at t c (0) in the metal is animportant direction for further research. In our present formulation in Eq. (1.4), this pointis a transition from a conventional metallic SDW state to an ‘algebraic charge liquid’ inthe O(4) universality class. However, an interesting alternative possibility is a transitiondirectly to the large Fermi surface state. Finally, we note that a number of experimental studies have discussed ascenario for crossover in the cuprates which is generally consistent with our Fig. 10.
Acknowledgments
We thank A. Chubukov, V. Galitski, P. D. Johnson, R. Kaul, A. Keren, Yang Qi, L. Taille-fer, Cenke Xu, and A. Yazdani for valuable discussions. We are especially grateful toA. Chubukov for pointing out numerical errors in an earlier version of this paper. Thisresearch was supported by the NSF under grant DMR-0757145, by the FQXi foundation,and by a MURI grant from AFOSR. E.G.M. is also supported in part by a Samsung schol-arship.
APPENDIX A: FIELD RELATIONS FROM SPIN DENSITY WAVE THEORY
This appendix will give a derivation of the relation (1.2) between the physical electronoperators c α and the fields g ± and z α using spin density wave theory. This will complementthe derivation obtained from the doped Mott insulator approach in previous works. We begin the quasiparticle Hamitonian which determines the ‘large’ Fermi surface in theoverdoped regime H = − X i
0, the24amiltonian for these electrons is H + H sdw = ε (cid:16) c † α c α + c † α c α (cid:17) − ϕ (cid:16) c † ↑ c ↑ − c † ↓ c ↓ + c † ↑ c ↑ − c † ↓ c ↓ (cid:17) (A4)We diagonalize this by writing H + H sdw = ( ε − ϕ ) (cid:16) g † + g + + g †− g − (cid:17) + ( ε + ϕ ) (cid:16) h † + h + + h †− h − (cid:17) (A5)where c ↑ = ( g + + h + ) / √ c ↑ = ( g + − h + ) / √ c ↓ = ( g − + h − ) / √ c ↓ = ( − g − + h − ) / √ h ± fermions.We obtain the electron operators for a general polarization of the N´eel order as in Eq. (1.1)by performing an SU(2) rotation defined by the z α (and dropping the unimportant factor of1 / √ c ↑ c ↓ ! = R z g + g − ! ; c ↑ c ↓ ! = R z g + − g − ! (A7)where the SU(2) rotation is R z = z ↑ − z ∗↓ z ↓ z ∗↑ ! . (A8)These results lead immediately to Eq. (1.2). In the superconducting state, where h g + g − i 6 = 0,they yield h c ↑ c ↓ i = (cid:10)(cid:0) | z ↑ | + | z ↓ | (cid:1) g + g − (cid:11) h c ↑ c ↓ i = − (cid:10)(cid:0) | z ↑ | + | z ↓ | (cid:1) g + g − (cid:11) , (A9)which implies a d -wave pairing signature for the electrons. N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang,D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature , 565 (2007). E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, andJ. R. Cooper, Phys. Rev. Lett. , 047003 (2008). A. F. Bangura, J. D. Fletcher, A. Carrington, J. Levallois, M. Nardone, B. Vignolle, P. J. Heard,N. Doiron-Leyraud, D. LeBoeuf, L. Taillefer, S. Adachi, C. Proust, and N. E. Hussey, Phys.Rev. Lett. , 047004 (2008). C. Jaudet, D. Vignolles, A. Audouard, J. Levallois, D. LeBoeuf, N. Doiron-Leyraud, B. Vignolle,M. Nardone, A. Zitouni, Ruixing Liang, D. A. Bonn, W. N. Hardy, L. Taillefer, and C. Proust,Phys. Rev. Lett. , 187005 (2008). S. E. Sebastian, N. Harrison, E. Palm, T. P. Murphy, C. H. Mielke, Ruixing Liang, D. A. Bonn,W. N. Hardy, and G. G. Lonzarich, Nature , 200 (2008). D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J.-B. Bonnemaison, N. E. Hussey, L. Bal-icas, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Adachi, C. Proust, and L. Taillefer,Nature , 533 (2007). R. K. Kaul, M. A. Metlitski, S. Sachdev and C. Xu, Phys. Rev. B , 045110 (2008). V. Galitski and S. Sachdev, Phys. Rev. B , 134512 (2009). B. Khaykovich, S. Wakimoto, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, P. Smeibidl, P. Vorder-wisch, and K. Yamada, Phys. Rev. B , 220508 (2005). J. Chang, Ch. Niedermayer, R. Gilardi, N. B. Christensen, H. M. Rønnow, D. F. McMorrow,M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono, M. Oda, M. Ido, andJ. Mesot, Phys. Rev. B , 104525 (2008). J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann, H. M. Rønnow, D. F. McMorrow,A. Schneidewind, P. Link, A. Hiess, M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda,M. Ido, and J. Mesot, Phys. Rev. Lett. , 177006 (2009). D. Haug, V. Hinkov, A. Suchaneck, D. S. Inosov, N. B. Christensen, Ch. Niedermayer,P. Bourges, Y. Sidis, J. T. Park, A. Ivanov, C. T. Lin, J. Mesot, and B. Keimer, arXiv:0902.3335. B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorder-wisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T. E. Mason,Nature , 299 (2002). E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. , 067202 (2001); Y. Zhang, E. Demlerand S. Sachdev, Phys. Rev. B , 094501 (2002). S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B , 144516 (2002). S. Sachdev, Rev. Mod. Phys. , 913 (2003). S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, andC. Howald, Rev. Mod. Phys. , 1201 (2003). M. Shay, A. Keren, G. Koren, A. Kanigel, O. Shafir, L. Marcipar, G. Nieuwenhuys, E. Moren-zoni, M. Dubman, A. Suter, T. Prokscha, and D. Podolsky, arXiv:0906.2047. Wei-Qiang Chen, Kai-Yu Yang, T. M. Rice, and F. C. Zhang, EuroPhys. Lett. , 17004 (2008)[arXiv:0706.3556]. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B , 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports , 355 (1997). A. J. Millis and M. R. Norman, Phys. Rev. B , 220503(R) (2007) [arXiv:0709.0106]. N. Harrison, arXiv:0902.2741. R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, Phys. Rev. B , 235122(2007). R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil, Nature Physics , 28 (2008). D. J. Scalapino, Physics Reports , 328 (1995). Ar. Abanov, A. V. Chubukov and J. Schmalian, Adv. Phys. , 119 (2003) and referencestherein. K. Park and S. Sachdev, Phys. Rev. B , 184510 (2001). S. Pathak, V. B. Shenoy, M. Randeria, and N. Trivedi, Phys. Rev. Lett. , 027002 (2009). R. K. Kaul and S. Sachdev, Phys. Rev. B , 155105 (2008). H.-B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Nature , 77(2008). W. D. Wise, Kamalesh Chatterjee, M. C. Boyer, Takeshi Kondo, T. Takeuchi, H. Ikuta, ZhijunXu, Jinsheng Wen, G. D. Gu, Yayu Wang, and E. W. Hudson, Nature Physics , 213 (2009). D. J. Scalapino in