Correlation and disorder-enhanced nematic spin response in superconductors with weakly broken rotational symmetry
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov epl draft Correlation and disorder-enhanced nematic spin response in su-perconductors with weakly broken rotational symmetry
Brian M. Andersen , Siegfried Graser and P. J. Hirschfeld Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augs-burg, D-86135 Augsburg, Germany Department of Physics, University of Florida, Gainesville, Florida 32611, USA
PACS – Broken symmetry phases
PACS – Theories and models of superconducting state
PACS – Cuprate superconductors
Abstract. - Recent experimental and theoretical studies have highlighted the possible role of aelectronic nematic liquid in underdoped cuprate superconductors. We calculate, within a modelof d -wave superconductor with Hubbard correlations, the spin susceptibility in the case of a smallexplicitly broken rotational symmetry of the underlying lattice. We then exhibit how the inducedspin response asymmetry is strongly enhanced by correlations as one approaches the instabilityto stripe order. In the disorder-induced stripe phase, impurities become spin nematogens witha C symmetric impurity resonance state, and the disorder-averaged spin susceptibility remainsonly C symmetric at low energies, similar to recent data from neutron scattering experiments onunderdoped YBCO. Introduction. –
Incommensurate one dimensionalcomposite spin and charge density waves, often called“stripes” [1–6], have been observed and play an impor-tant role in discussions of the underdoped cuprates andother systems. Such states were predicted theoreticallyin the context of mean-field studies of Hubbard mod-els [7–10], and later observed in neutron scattering ex-periments in La − x Ba x CuO and La − x − y Nd y Sr x CuO [11,12]. Stripes break the discrete translation and rotationsymmetries of the CuO planes. Rotational symmetry isalso broken in so-called liquid crystal analogs called “elec-tronic nematic” states, but these preserve translationalsymmetry and may occur as the initial instability of aparamagnetic state before an ordered state of charge, spin,or combined spin and charge order is reached [3, 13–16].While stripe-like ground states were originally thoughtto be a very special feature of the 214 compounds, thisview has changed in recent years, in particular with thediscovery of broken C symmetry in the spin response ofhighly underdoped samples of YBa Cu O δ [17], and,more recently, a subtle charge order in the same sys-tem [18]. Signatures of nematic order have also beenrecently reported in transport and tunneling measure-ments [19–21]. In all these cases, however, the ques- tion of C → C symmetry breaking is muddied by thefact that the crystal is not tetragonal, since for exam-ple in YBa Cu O δ the CuO chains give a well-definedanisotropy in the untwinned samples on which the ex-periments were performed, such that the system is for-mally orthorhombic. Nevertheless, the evolution from op-timally to highly underdoped samples, which is accom-panied by a dramatic enhancement of the anisotropy ofthe responses, is quite striking, and leads to the com-mon assumption that these highly correlated underdopedmaterials display a strongly enhanced “nematic suscep-tibility”, i.e. a tendency to create nematic order whichis driven by the very small symmetry-breaking field pro-vided by the x − y anisotropy in the band-structure. How-ever, these ideas have rarely been cast in a concrete mi-croscopic model allowing a direct study of how disorderand local electronic correlations which drive the Mott in-sulating state influence nematicity. Previous studies havemainly focussed on the high-energy spin fluctuations ofthe RPA susceptibility of a homogeneous d -wave super-conductor with an anisotropic band-structure [22–26], orutilized phenomenological Ginzburg-Landau approaches[16, 27, 28]. More recently, the nematic response has alsobeen studied within the two-dimensional Hubbard modelp-1. M. Andersen, S. Graser, and P. J. Hirschfeldwith slight x − y hopping asymmetry using strong-couplingcluster methods, and found to be significantly enhanced byinteractions at low temperatures in the underdoped pseu-dogap regime [29, 30].The salient features of the inelastic neutron scatteringexperiments on strongly underdoped untwinned YBCOsamples which should be reproduced by a reasonably com-plete theoretical analysis are as follows:1. The neutron intensity near ( π, π ) evolves from a pat-tern of four symmetrically placed incommensuratepeaks at high energies which merge at ( π, π ) at thespin resonance energy Ω , to a pattern with C sym-metry with two incommensurate peaks at quite lowenergies ω ≪ Ω .2. The details of this pattern appear to be important:the intensity in the nematic case has a saddle pointform in q space, with a maximum at ( π, π ) for cutsalong b and a minimum there for cuts along a .3. The strength of the C symmetry breaking increasesas one underdopes.Some phenomenologies have been rather successful in ac-counting for some of these features, but to our knowl-edge there is no microscopic approach which has thus farsuccessfully accounted for all of them. In this work wepresent a simple version of such a theory, based on ourearlier work exploring the effect of Hubbard correlationson the d -wave superconducting state, to which we nowadd a small symmetry-breaking field to simulate e.g. theeffect of the chains in YBa Cu O δ . We show that thenematicity in the spin response is enhanced by correlationsand decreasing temperature; what is perhaps more surpris-ing is our finding that the phenomenon can be enhancedfurther by pairing and disorder. We find that disorder iscrucial to explain the saddle point structure of the inelas-tic scattering intensity in q -space. In addition, we presentlocal investigations which exhibit the explicit formationof nematogens, nematically driven impurity states, whichhave not to our knowledge been observed in the cuprates,although recently reported in the stripe-ordered phase ofFe-based systems [31–33]. Indications of an enhanced ne-matic susceptibility above the magnetic transition havealso been observed in these systems [34]. Model. –
The Hamiltonian is given byˆ H = − X ijσ t ij ˆ c † iσ ˆ c jσ + X iσ ( V i − µ )ˆ n iσ + U X iσ n i − σm i n iσ + X iδ (cid:16) ∆ δi ˆ c † i ↑ ˆ c † i + δ ↓ + ∆ ∗ δi ˆ c i + δ ↓ ˆ c i ↑ (cid:17) , (1)where ˆ c † iσ creates an electron on site i with spin σ , and t ij = { t x , t y , t ′ } denote the hopping integrals to the twonearest neighbors. In Eq.(1), n i and m i refer to the chargedensity and magnetization, respectively, V i is an impurity potential from a set of N point-like scatterers, µ is thechemical potential and ∆ ij is the d -wave pairing potentialbetween sites i and j . The amplitude of ∆ ij is set bythe superconducting coupling constant g and will exhibita slight anisotropy inherited from a finite δ = ( t y − t x ).Below we fix the parameters t ′ = − . t , adjust µ to givea hole doping x = 1 − n ≃ g = 0 . . t . The hoppingasymmetry δ = 0 .
05 is fixed for all results presented here,and we use units where t = t y = 1 .
0. We have solvedEq.(1) self-consistently on unrestricted N × N lattices bydiagonalizing the associated Bogoliubov-de Gennes (BdG)equations at T = 0 . t . [35]The model given by Eq.(1) has been used extensivelyin the literature to study the competition between bulksuperconducting and magnetic phases, and field-inducedmagnetization [36]. It has also been used to study momentformation around nonmagnetic impurities in correlated d -wave superconductors [37]. In the case of many impu-rities, Eq.(1) was used to model static disorder-inducedmagnetic droplets [38–42], and explain how these mayincrease in volume fraction when moving to lower dop-ing levels and eventually form a quasi-long-range orderedmagnetic stripe phase. More recently, Eq.(1) extended tothe vortex state was used to obtain a semi-quantitativedescription of the temperature dependence of the elasticneutron response in underdoped LSCO [43, 44].The transverse bare spin susceptibility χ xx ( ~r i , ~r j , ω ) = − i R ∞ dt e iωt (cid:10)(cid:2) σ xi ( t ) , σ xj (0) (cid:3)(cid:11) , can be expressed in termsof the BdG eigenvalues E n and eigenvectors u n , v n as χ xx ( ~r i , ~r j , ω ) = X m,n f ( u, v ) f ( E m ) + f ( E n ) − ω + E m + E n + i Γ+ X m,n g ( u, v ) f ( E m ) + f ( E n ) − ω − E m − E n + i Γ , (2) f ( u, v ) = u ∗ m,i v ∗ n,i ( u m,j v n,j − u n,j v m,j ) , (3) g ( u, v ) = v m,i u n,i (cid:0) u ∗ m,j v ∗ n,j − u ∗ n,j v ∗ m,j (cid:1) . (4)Including the electronic interactions within RPA we findfor the full susceptibility χ xx ( ~r i , ~r j , ω ) = X ~r l [1 − U χ xx ( ω )] − ~r i ,~r l χ xx ( ~r l , ~r j , ω ) . (5)Fourier transforming with respect to the relative coordi-nate ~r = ~r i − ~r j defines the spatially resolved momentum-dependent susceptibility χ ( ~q, ~R, ω ) = P ~r e i~q · ~r χ ( ~R, ~r, ω ).Averaging over the center of mass coordinate ~R = ( ~r i + ~r j ) /
2, this expression gives the susceptibility χ ( q , ω ) rele-vant for comparison with neutron measurements. Results. –
Figure 1 shows the susceptibility χ ′′ ( q, ω ) = Im χ ( q , ω ) at low energy ω/t = 0 . q x and q y for different U in the homogeneous superconduct-ing phase. As seen, the apparent asymmetry in the neu-tron response is clearly enhanced as the correlations arep-2orrelation and disorder-enhanced nematic spin response in superconductors q x / π q y / π (a) q x / π q y / π (b) q x / π q y / π (c) q x / π q y / π (d) Fig. 1: (Color online) Constant energy cuts of the susceptibility χ ′′ ( q, ω ) at low energy ω/t = 0 . δ = 0 . N = 80, and U/t = 2 . U/t = 2 . U/t = 2 . U/t = 2 . U/t η Fig. 2: Nematic spin response η versus U for δ = 0 .
05 in thelow- U homogeneous d -wave superconducting phase. enhanced. Quantitatively, we can define a nematic spin-response ”order parameter” as η = max { χ ′′ ( q x , π, ω ) } − max { χ ′′ ( π, q y , ω ) } max { χ ′′ ( q x , π, ω ) } + max { χ ′′ ( π, q y , ω ) } . (6)Figure 2 shows the dramatic increase of η as U approachesthe stripe instability from below for fixed hopping asym-metry δ = 0 .
05. In cuprates with weak orthorhombicity,it is therefore natural to expect a significant enhancementof the nematic response upon lowering the doping levelfrom the optimally doped regime [16, 29, 30].Experimentally, it is known that the observed x − y asymmetry disappears as one increases the energy ω whichseems natural in light of the presumably very small energyscale associated with the nematic aligning field. Here, the q x / π q y / π (a) q x / π q y / π (b) q x / π q y / π (c) q x / π q y / π (d) Fig. 3: (Color online) Constant energy cuts of the susceptibility χ ′′ ( q, ω ) with U/t = 2 . δ = 0 . N = 80, and ω/t = 0 . t (a), ω/t = 0 . t (b), ω/t = 0 . t (c), and ω/t = 0 . t (d). same result is obtained as seen from Fig. 3 where thepanels show constant-energy cuts of χ ′′ ( q, ω ) at varying ω with fixed U = 2 . t and δ = 0 . d -wave superconductor. Within thepresent mean-field formalism this is the relevant regimefor clean systems with U < U c ≃ . t . In the oppositeregime, where U > U c , the clean phase is an orderedsmectic stripe phase which disorder may turn into a ne-matic as discussed e.g. in Refs. [45–47]. We do not ex-pect a small explicit symmetry breaking term to qualita-tively alter this scenario. Instead, we focus on the weak- U limit where disorder may induce a stripe magnetic phaseas shown in Refs. [38, 39, 41, 42], and study the evolutionof the nematic spin response to disorder. For this purposewe need to investigate how impurities respond to a finite δ .Figure 4(a) shows the C symmetric pattern of localmagnetic order induced around such an impurity, whichwe will refer to as a “spin nematogen”, while in Figure4(b) we see that a significant anisotropy is absent in thecharge sector, as may be expected in weak-coupling the-ories of this type. Similar to the bulk results in Fig. 1,the impurity-induced magnetization becomes increasinglyasymmetric as one approaches U c from below. The ques-tion arises how to detect these spin nematogens in thecuprates. NMR is clearly sensitive to the local distributionof spins, and the lineshape in other disordered, correlatedmaterials has been analyzed in terms of postulated struc-tures of this type [32]. In addition, one might expect asignature in the local density of states around an impurity;the well known fourfold-symmetric impurity resonance as-sociated with a potential scatterer in a d -wave supercon-ductors has been detected by STM [48] in near-optimallyp-3. M. Andersen, S. Graser, and P. J. Hirschfeld
14 16 18 20 22 24141618202224 x y −0.2−0.100.10.2 (a)
14 16 18 20 22 24141618202224 x y (b)
14 16 18 20 22 241520 x y (c) q x / π q y / π (d) q/ π I m χ ( q , ) (e) q/ π I m χ ( q , ) (f) Fig. 4: (Color online) Magnetization (a), charge density (b),and local density of states (c) plotted in real-space near a singlestrong scatterer ( V imp = 100 t ) with hopping asymmetry δ =0 .
05 and
U/t = 2 .
5. In (a-c) we have used N = 35, but showonly the center 11 ×
11 sites near the impurity for clarity. In (c)the LDOS is shown at ω = − . t which is the energy of theimpurity resonance for the present choice of parameters. (d)Constant energy cut of the susceptibility χ ′′ ( q, ω ) with U/t =2 . ω/t = 0 . t for the case of disorder-induced nematogenscorresponding to (a). (e) shows q x ( q y ) cuts of χ ′′ ( q, ω ) along q y = π [black line] ( q x = π [red line]) for the same case asshown in panel (d). (f) same as (e) but in the homogeneouscase similar to Fig. 3(a). doped samples. In the presence of x − y anisotropy, thisresonance acquires within our theory a similar anisotropyas shown in Fig. 4(c) which should be observable nearstrong scatterers in the underdoped regime.The spin nematogens around an impurity arise becauseof a ”freezing” of gapped incommensurate spin fluctua-tions present in the clean system [44, 45, 49]. In agree-ment with the x − y anisotropy shown in Fig. 4(a),the low-energy disorder-induced spin excitations also be-come highly anisotropic, as shown in Fig. 4(d). Becauseof numerical limitations the spin-susceptibility from in-homogeneous real-space configurations is obtained fromsystems with N = 24. Surprisingly, the disorder signif-icantly enhances the nematic response as seen by com-paring Figs. 4(e,f); (e) displays two distinct cuts alongeither q x = π or q y = π from Fig. 4(d), and reveals amuch larger anisotropy in the spin response for the dis- x y −0.2−0.100.10.2 (a) q x / π q y / π (b) q x / π q y / π (c) q/ π χ ’’ ( q , ) (d) Fig. 5: (Color online) (a) Static magnetization shown in real-space for one of the disordered systems used in (b-d). Asseen, the frozen magnetization roughly consists of anti-phase-coupled antiferromagnetic domains. (b) Constant energy cutsof the susceptibility χ ′′ ( q, ω ) with U/t = 2 . ω/t = 0 . t averaged over ten different 24 ×
24 systems each containing4% impurities of strength V i = 5 t . (c) same as (b) but at ω/t = 0 . t . (d) shows q x ( q y ) cuts of χ ′′ ( q, ω ) from (b) along q y = π [black line] ( q x = π [red line]). ordered case (e) compared to the clean case (f). Since agapped spectrum near the Fermi level (caused by super-conductivity or pseudo-gap physics) is a prerequisite fordisorder-induced magnetization, we obtain the interestingsituation where a disordered superconductor in the pres-ence of sub-dominant electronic correlations and a small x − y symmetry breaking field, work as a catalyst for ob-serving a nematic response in neutron scattering.We end this section with a brief discussion of the real-istic many-impurity situation where the impurity-inducedmagnetization form a glassy pattern shown in Fig. 5(a).Panels (b) and (c) in Fig. 5 displays respectively the low-and high-energy spin susceptibility, χ ′′ ( q, ω ), in the pres-ence of a few percent added disorder similar to the exper-imental study in e.g. Ref. [49]. The particular concen-tration of disorder (and their strength) is not importantfor the present discussion. In agreement with the neutronmeasurements, the low-energy spin response from a collec-tion of overlapping spin nematogens peaks at the incom-mensurate (commensurate) position along q x ( q y ). Againthis distinct asymmetry is only present in the low-energysector as seen from Fig. 5(c). In Fig. 5(d) we show cutsthrough Fig. 5(b) along q y = π [black line] and q x = π [redline], revealing a semi-quantitative fit to the experimentallow-energy spin fluctuations obtained in Ref. [17]. Conclusions. –
We have calculated the spin suscep-tibility of a d -wave superconductor with Hubbard correla-tions in the presence of small explicit symmetry breakingof the underlying lattice. Correlations were found to sig-p-4orrelation and disorder-enhanced nematic spin response in superconductorsnificantly enhance the low-energy anisotropy of the spinresponse as the stripe instability is approached. Compar-isons with experiment [17, 49], as well as with work basedon the same model and recent analysis of the doping de-pendence of such effects at strong coupling [50] suggestthat this effect is responsible for the strongly enhancednematic tendency in the spin response observed in theYBCO system as it is underdoped. In addition, we haveshown here that disorder significantly enhances the ne-maticity, via generation of local spin nematogens whichexhibit an anisotropic low-energy spin response that canbe significantly enhanced compared to the clean case, andis crucial to understand the q -space form of the neutronresponse near ( π, π ) observed in experiments on stronglyunderdoped, untwinned YBCO samples. Our predictionscan be tested by studying the effect of Zn substitutionon the anisotropic spin response, and by analysis of NMRand STM spectroscopy of impurity bound states in under-doped cuprates. ∗ ∗ ∗ B.M.A. acknowledges support from The Danish Coun-cil for Independent Research | Natural Sciences. P.J.Hacknowledges support from NSF-DMR-1005625.
REFERENCES[1] V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Proc.Natl. Acad. Sci. USA , (1999) 8814.[2] J. Zaanen, Physica C , (1999) 217.[3] S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J.M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod.Phys. , (2003) 1201 .[4] H. B. Brom and J. Zaanen, in: Handbook of MagneticMaterials, Vol. xx, K. H. J. Buschow, ed. (Elsevier, 2003).[5] A. H. Castro Neto and C. Morais Smith, in: Strong Inter-actions in Low Dimensions, D. Baeriswyl and L. Degiorgi,eds. (Kluwer, 2004), pg. 277.[6] M. Raczkowski, A. M. Ole´s, and R. Fr´esard, Low. Temp.Phys. , (2006) 305.[7] J. Zaanen and O. Gunnarsson, Phys. Rev. B , (1989)7391.[8] D. Poilblanc and T. M. Rice, Phys. Rev. B , (1989) 9749.[9] H. J. Schulz, J. de Physique , (1989) 2833.[10] K. Machida, Physica C , (1989) 192.[11] J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura,and S. Uchida, Nature (London) , (1995) 561.[12] J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura,S. Uchida, and B. Nachumi, Phys. Rev. B , (1996) 7489.[13] S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature (Lon-don) , (1998) 550.[14] M. Vojta, Adv. Phys. , (2009) 699.[15] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein,and A. P. Mackenzie, Annu. Rev. Condens. Matter Phys. , (2010) 153.[16] M. Vojta, Eur. Phys. J. Special Topics , (2010) 49. [17] V. Hinkov, D. Haug, B. Fauqu´e, P. Bourges, Y. Sidis, A.Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science , (2008) 597.[18] T. Wu, H. Mayaffre, S. Kr¨amer, M. Horvati´c, C. Berthier,W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien,Nature (London) , (2011) 191.[19] Y. Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys.Rev. Lett. , (2002) 137005.[20] R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choini´ere, F. Lal-ibert´e, N. Doiron-Leyraud, B. J. Ramshaw, R. Liang, D.A. Bonn, W. N. Hardy, and L. Taillefer, Nature (London) , (2010) 519.[21] M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt, Y.Kohsaka, C. Koo Kim, H. Eisaki, S. Uchida, J. C. Davis,J. P. Sethna, and E.-A. Kim, Nature (London) , 347(2010).[22] T. Zhou and J.-X. Li, Phys. Rev. B , (2004) 224514;Phys. Rev. B , (2005) 134512.[23] Y.-J. Kao and H.-Y. Kee, Phys. Rev. B , (2005) 024502.[24] I. Eremin and D. Manske, Phys. Rev. Lett. , (2005)067006.[25] A. P. Schnyder, D. Manske, C. Mudry, and M. Sigrist,Phys. Rev. B , (2006) 224523.[26] H. Yamase and W. Metzner, Phys. Rev. B , (2006)052501; H. Yamase, Phys. Rev. B , (2009) 052501.[27] M. Vojta, T. Vojta, and R. K. Kaul, Phys. Rev. Lett. ,(2006) 097001.[28] K. Sun, M. J. Lawler, and E.-A. Kim, Phys. Rev. Lett. , (2010) 106405.[29] S. Okamoto, D. S´en´echal, M. Civelli, and A.-M. S. Trem-blay, Phys. Rev. B , (2010) 180511(R).[30] S.-Q. Su and T. A. Maier, arXiv:1110.5297.[31] T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L.Budko, G. S. Boebinger, P. C. Canfield, and J. C. Davis,Science , (2010) 181.[32] A.P. Dioguardi, N. apRoberts-Warren, A. C. Shockley,S. L. Bud’ko, N. Ni, P. C. Canfield, and N. J. Curro, J.Phys. Chem. Solids (2011); A. P. Dioguardi, N. apRoberts-Warren, A. C. Shockley, S. L. Bud’ko, N. Ni, P. C. Canfield,and N. J. Curro, Phys. Rev. B , (2010) 140411(R).[33] T. Hanaguri, conference presentation .[34] I. R. Fisher, L. Degiorgi, Z. X. Shen, Rep. Prog. Phys. ,(2011) 124506.[35] J. W. Harter, B. M. Andersen, J. Bobroff, M. Gabay, andP. J. Hirschfeld, Phys. Rev. B , (2007) 054520.[36] I. Martin, G. Ortiz, A. V. Balatsky, A. R. Bishop, Int. J.Mod. Phys. , (2000) 3567; M. Ichioka, M. Takigawa,and K. Machida, J. Phys. Soc. Jpn. , (2001) 33; Y.Chen, Z. D. Wang, J.-X. Zhu, and C. S. Ting, Phys. Rev.Lett. , (2002) 217001; J. X. Zhu, I. Martin, and A. R.Bishop, Phys. Rev. Lett. , (2002) 067003; B. M. Ander-sen, P. Hedeg˚ard, and H. Bruus, Phys. Rev B , (2003)134528; H.-Y. Chen and C. S. Ting, Phys. Rev B , (2005)220510(R); B. M. Andersen, I. V. Bobkova, P. J. Hirschfeld,and Yu. S. Barash, Phys. Rev B , (2005) 184510; B. M.Andersen and P. Hedeg˚ard, Phys. Rev. Lett. , (2005)037002.[37] For a review see, H. Alloul, J. Bobroff, M. Gabay, and P.J. Hirschfeld, Rev. Mod. Phys. , (2009) 45.[38] G. Alvarez, M. Mayr, A. Moreo, and E. Dagotto, Phys.Rev. B , (2005) 014514.[39] B. M. Andersen, P. J. Hirschfeld, A. P. Kampf, and M. p-5. M. Andersen, S. Graser, and P. J. Hirschfeld Schmid, Phys. Rev. Lett. , (2007) 147002.[40] B. M. Andersen and P. J. Hirschfeld, Physica C (Amster-dam) , 744 (2007).[41] W. A. Atkinson, Phys. Rev. B , (2007) 024510.[42] B. M. Andersen and P. J. Hirschfeld, Phys. Rev. Lett. , (2008) 257003.[43] M. Schmid, B. M. Andersen, A. P. Kampf, and P. J.Hirschfeld, New J. Phys. , (2010) 053043.[44] B. M. Andersen, S. Graser, M. Schmid, A. P. Kampf, andP. J. Hirschfeld, J. Phys. Chem. Solids , (2011) 358.[45] B. M. Andersen, S. Graser, and P. J. Hirschfeld, Phys.Rev. Lett. , (2010) 147002.[46] J. A. Robertson, S. A. Kivelson, E. Fradkin, A. C. Fang,and A. Kapitulnik, Phys. Rev. B , (2006) 134507.[47] A. Del Maestro, B. Rosenow, and S. Sachdev, Phys. Rev.B , (2006) 024520.[48] S. H. Pan, E. W. Hudson, K. M. Lang, H. Eisaki, S.Uchida, and J. C. Davis, Nature (London) , (2000) 746.[49] A. Suchaneck V. Hinkov, D. Haug, L. Schulz, C. Bernhard,A. Ivanov, K. Hradil, C. T. Lin, P. Bourges, B. Keimer, andY. Sidis, Phys. Rev. Lett. , (2010) 037207.[50] R. B. Christensen, P. J. Hirschfeld, and B. M. Andersen, in press , Phys. Rev. B (2011)., Phys. Rev. B (2011).