1. Zlatko Tesanovic, Johns Hopkins University zbt@pha.jhu.edu http://www.pha.jhu.edu/~zbtzbt@pha.jhu.eduhttp://www.pha.jhu.edu/~zbt o Strongly correlated normal state exhibiting a “BCS-ish” pairing instability (?pnictides, 3 He, heavy fermions, organics, e-doped cuprates, …) o Intrinsic strongly correlated SC, exerting major influence on surrounding “normal” state(s) (?pnictides, h-doped cuprates, …) Correlated superconductivity comes in (at least) two varieties: Electron Correlations and HTS in Fe-Pn Electron Correlations and HTS in Fe-Pn

2. 122 Correlated Superconductors: Cu-oxides vs Fe-pnictides However, there are also many differences! This may add up to new and interesting physics Fe As O RE 1111 11 111

3. Key Difference: 9 versus 6 d-electrons In CuO 2 a single hole in a filled 3d orbital shell  A suitable single band model might work In FeAs large and even number of d-holes  A multiband model is likely necessary ZT, Physics 2, 60 (2009)

4. Phase diagram of Cu-oxides Cu-oxides: Mott Insulators  Superconductors ?? How Mott insulators turn into superconductors, particularly in the pseudogap region, remains one of great intellectual challenges of condensed matter physics U Only when doped with holes (or electrons) do cuprates turn into superconductors

5. All superconductors have thermal fluctuations No such ground state in BCS theory (at weak coupling) !! Correlated superconductors have quantum (anti)vortex fluctuations Ground state with enhanced pairing (and other!) correlations but no SC !! (gauge theories, QED 3, chiral SB, AdS/CMT,… PW Anderson, Balents, Burkov, MPA Fisher, Nayak, Nikolic, Sachdev, Senthil, PA Lee, Franz, Vafek,,ZT ) How Correlated Superconductors turn into Mott Insulators Near T c these are always phase fluctuations ZT, Nature Physics 4, 408 (2008) P. Nikolic and ZT, PRB 83, 064501 (2011) BCS-Eliashberg-Migdal Optimal T c in HTS is determined by quantum fluctuations

6. Giant Nernst Effect and Quantum Diamagnetism in HTS Vortex fluctuations and enhanced quantum diamagnetism in pseudogap state !! Z. A. Xu et al., Nature 406, 486 (2000) L. Li et al., Nature-Phys (2007 ) Vortex excitations in conductivity !! N. P. Armitage et al, Nature Physics

7. A sequence of thermal KT transitions as T c drops from 40K to 2K However, there is a clear break below ~ 30K indicating T c  -2 (0)   s (0)   magnitude of T c (x) is dictated by Quantum critical SC fluctuations !! Quantum Superconducting Fluctuations in Underdoped Cuprates “Thermal metal” in non-SC YBCO M. Sutherland et al., PRL 94, 147004 (2005) I. Hetel et al., Nature Physics 3, 700 (2007) d-wave nodal liquid in highly underdoped BSCCO U. Chatterjee et al., Nature Physics 6, 99 (2009) dHvA  Small Fermi surfaces (pockets) in underdoped YBCO? C. Jaudet et al., PRL 100, 187005 (2008); S. E. Sebastian et al., PNAS 107, 6175 (2010) d-wave pseudogap in YBCO in very high fields S. C. Riggs. et al., NHMFL preprint (2010) Pockets & Nodes? Millis & Norman Chubukov Johnson et al.

8. Fermi sea + + - - Cooper pairing driven by repulsion (No BEC limit) !! HTS, other correlated SCs  Cooper pairing is a way of reducing repulsion !!  Competition with other states !!

9. P. W. Anderson, KITP Conference on Higher Tc, June 2009 Projection eliminates doubly occupied sites

10. Phase diagram of underdoped cuprates from a wave-function? This is just d-wave BCS SC. However, inside © there is:  ij  ij must start fluctuating as x  0 and system becomes Mott insulator.  ij Including fluctuations in  ij leads to improved ground state wavefunction.  N  & 1 PW Anderson PA Lee, Senthil, Wen, Hermele Balents, MPA Fisher, Nayak Franz, Vafek, Melikyan, ZT Senthil & MPA Fisher, Balents, Burkov, Sachdev, … Salk et al, … Pockets !? Nodes !? Pockets & Nodes !?

11. Correlated s-wave superconductor Quantum s-wave/bosonic duality: Automatic conservation of vorticity  Cooper pairs, bound by strong attraction !! view  i -  j as a gauge field a ij coupled to electrical charge P. Nikolic and ZT, PRB 83, 064501 (2011)

12. Quantum Phase Fluctuations on SITES: Formation of Local Spin Singlets (Strong Attraction, U < 0) Real space Cooper pairs (center-of-mass motion only) CDW SC Strong local attraction favors formation of spin singlet Cooper “molecules” (BEC limit):

13. Bond Phases (dSC) versus Site Phases (sSC) Automatic conservation of vorticity  Cooper pairs !! Vafek, Melikyan, ZT

14. d-wave SC Duality is More Complex Quantum d-wave/fermionic duality (U > 0): Monopole-antimonopole configurations  Non-conservation of vorticity  Cooper pairs !! Quantum s-wave/bosonic duality (U < 0): Automatic conservation of vorticity  Cooper pairs !! Translation: Cooper pairs can fall apart !! They are held together by kinetic energy and cannot survive without motion in a correlated medium (No BEC) ! ZT, Nature Physics 4, 408 (2008) P. Nikolic and ZT, PRB 83, 064501 (2011)

15. What controls fluctuations in µ ij ? Kinetic energy. view  ij as gauge field a ij coupled to staggered charge ZT, Nature Physics 4, 408 (2008)

16. Formation of Local Singlets on BONDS (Strong Repulsion, U >> t ) !! For CuO 2 plane this translates into a d-wave superconductor Or an antiferromagnet !! dSC or AF as the ground state depends on kinetic energy and dynamics of center-of-mass versus relative motion of bond Cooper pairs If center-of-mass moves around freely while relative motion is suppressed CuO 2 plane  dSC CuO 2 plane is Neel AF !! If relative motion becomes too strong Cooper pairs break up !! As relative fluctuations become too strong and motion ceases, bond Cooper pairs fade away !!  charge e insulator !!  Dual of dSC is AF Strong local repulsion favors formation of spin singlets on bonds: Cooper pairs are held together by electronic motion  repulsion is minimized

17. ZT, Nature Physics 4, 408 (2008) Wave-function © BCS is not enough  Quantum disorder in µ ij !! Must include quantum fluctuations of: dSC nodal fermions AF+x Lu Li & Ong Phase diagram of Cu-oxides

18. Fe-pnictides: Semimetals  Superconductors In contrast to CuO 2, all d- bands in FeAs are either nearly empty (electrons) or nearly full (holes) and far from being half-filled. This makes it easier for electrons (holes) to avoid each other.  FeAs are less correlated than CuO 2 (correlations are still important !! )

19. C. de la Cruz, et al., Nature 453, 899 (2008) Phase diagram of Fe-pnictides Like CuO 2, phase diagram of FeAs has SDW (AF) in proximity to the SC state. SC coexists with SDW (AF) in 122 compounds  H. Chen, et al., arXiv/0807.3950 However, unlike CuO 2, all regions of FeAs phase diagram are (bad) metals !! SmFeAsO 1-x F x parent (SDW) SC x = 0.0 x = 0.18 T. Y. Chen, et al..

20. Important: Near E F e and h bands contain significant admixture of all five Wannier d- orbitals, d xz and d yz of odd parity (in FeAs plane) and the remaining three d-orbitals of even parity in FeAs plane  Minimal Model of FeAs Layers V. Cvetkovic and ZT, EPL 85, 37002 (2009) C. Cao, P. J. Hirschfeld, and H.-P. Cheng, PRB 77, 220506 (2008) K. Kuroki et al, PRL 101, 087004 (2008) d xz odd parity even parity d yz d xy d xx-yy d 2zz-xx-yy

21. Nesting in Fe-pnictides Turning on moderate interactions  VDW = itinerant multiband CDW (structural), SDW (AF) and orbital orders at q = M = ( ¼, ¼ ) Cvetkovic & ZT, Korshunov & Eremin SemiconductorSemimetal  d c dd cc SDW, CDW, ODW or combinations thereof  VDW

22. Interactions in FeAs I V. Cvetkovic and ZT, PRB 80, 024512 (2009); J. Kang and ZT, PRB 83, 020505 (2011)

23. Interactions in FeAs II Typically, we find W s is dominant  Valley density-wave(s) (VDW) in FeAs h1h1 h2h2 e1e1 e-h These “Josephson” terms are not crucial for SDW  Could they be the cause of SC ? Hirschfeld, Kuroki, Bernevig, Thomale, Chubukov, Eremin,

24. Interband pairing acts like Josephson coupling in k-space. If G 2 is repulsive  antibound Cooper pairs (s’SC) M Two Kinds of Interband Superconductivity Type-A interband SC: c FS cd d Type-B (intrinsic) interband SC: cd FS sSC s’SC G2G2 sSC s’SC G2G2 ZT, Physics 2, 60 (2009)

25. If G 1, G 2 << U, W  relevant vertices: U, W, & G 2 Interactions in FeAs III V. Cvetkovic & ZT (RG) ; A. V. Chubukov, I. Eremin et al (parquet); F. Wang, H. Zhai, Y. Ran, A. Vishwanath & DH Lee (fRG) R. Thomale, C. Platt, J. Hu, C. Honerkamp & A. Bernevig (fRG) The condition for interband SC is actually milder: suffices to have G 2 * > U* even if G 2 << U

26. RG flows (near SDW): RG Theory of Interband Mechanism of SC in FeAs V. Cvetkovic and ZT, PRB 80, 024512 (2009) In Fe-pnictides interband superconductivity (s’ or s+- state) is a strong possibility but there is some fine tuning with SDW/CDW/ODW

27. What is a (THE) Model for Iron-Pnictides ?  U(4) £ U(4) Theory of Valley-Density Wave (VDW) Key assumption I: Eremin, Knolle Hirschfeld, Kuroki, Bernevig, Thomale, Chubukov, Eremin, Key assumption II:

28. U(4) £ U(4) Theory of Valley-Density Wave (VDW)  U(4) £ U(4) symmetry  unified spin and pocket/orbital flavors V. Cvetkovic and ZT, PRB 80, 024512 (2009); J. Kang and ZT, PRB 83, 020505 (2011)

29. Hierarchy of RG Energy Scales U, W >> G 1, G 2  U(4) £ U(4) Theory of Valley-Density Wave (VDW) VDW in Fe-pnictides is a (nearly) U(4) £ U(4) symmetric combination: SDW/CDW/ODW (...)  D. K. Pratt, et al., arxiv/0903.2833 T S (K)T N (K)m ord (μ B ) LaFeAsO1551370.36 CeFeAsO1551400.83 PrFeAsO1531270.48 NdFeAsO1501410.9 CaFeAsF1341140.49 SrFeAsF175120 CaFe 2 As 2 173 0.8 SrFe 2 As 2 220 0.94-1.0 BaFe 2 As 2 140 0.9 V. Cvetkovic and ZT, PRB 80, 024512 (2009); J. Kang and ZT, PRB 83, 020505 (2011)

30. Hierarchy of RG Energy Scales U, W >> G 1, G 2  U(4) £ U(4) Theory of Valley-Density Wave (VDW) U(4) £ U(4) symmetry at high energies  Spin and pocket/orbital flavors all mixed  VDW ground state (any combination of SDW/CDW/ODW) D. K. Pratt, et al., arxiv/0903.2833 At low energies, numerous terms break this U(4) £ U(4) symmetry Key assumptions: V. Cvetkovic and ZT, PRB 80, 024512 (2009); J. Kang and ZT, PRB 83, 020505 (2011)

31. U(4) £ U(4) Symmetry vs Reality Since ¸ 1 » 0 el-ph interaction or dynamical polarization from Pn bands could easily lead to ¸ 1 < 0  Pnictides are near ¸ 1 = 0 QCP !! J. Kang and ZT, PRB 83, 020505 (2011)

32. “Near” U(4) £ U(4) Symmetry and Experiments Orbital “AF”  Can this modulated current pattern be observed by neutrons? ¹ SR? J. Kang and ZT, PRB 83, 020505 (2011)