1. FFLO and pair density wave superconductors 1- General motivation for studying PDW phases 2- Microscopic PDW mechanism in superconductors without parity symmetry 3- Ginzburg-Landau-Wilson theories of PDW phases: ground states, topological line defects, fluctuations 4- Novel defect driven superfluid/superconudctor phases in 2D. Daniel F. Agterberg, U. of Wisconsin – Milwaukee (UWM) Manfred Sigrist (ETH-Zurich), Hiro Tsunetsugu (ISSP), Raminder Kaur, Shantanu Mukherjee, Zhichao Zheng (UWM) PRL 102, 207004 (2009), PRB 82, 024506 (2010) PRL 100, 107001 (2008) Nature Physics 4, 639 (2008) PRL 94, 137002 (2005), PRB 75 064511 (2007),

2. PDW and FFLO phases Superconductor/superfluids that “ oscillate ” in space periodically are PDW phases. FFLO is a specific example of a PDW, with spatial oscillations on the order of the coherence length. Finite momentum BEC and supersolids.

3. Singlet pairing in a Zeeman field Key Point of FF and LO: Pairing between fermions with different Fermi surfaces leads to “finite momentum” pairing states (FFLO). (FF) (LO) Matusda and Shimahara, JPSJ 76, 051005 (2007). T.K. Koponen, et al, New J. Phys 10, 045014 (2008). 1D, 2D, 3D

4. Relevant Materials: Heavy Fermion Superconductors CeCoIn 5 (d-wave SC +PDW) CeRhSi 3 (non-centrsymmetric SC) N. Kimura, et al (2005) Radovan et al., Nature (2003). Bianchi, Phys. Rev. Lett. (2003).

5. Organic Superconductors Yonezawa et al, PRL 100, 117002 (2008) Lortz et al, PRL 99, 187002 (2007)

6. Zero Field PDW phases: E. Berg, et al PRL (2007) Berg, Fradkin, Kivelson, PRB (2009) Himeda et al, PRL (2002). Voronstov PRL (2009), Voronstov and Sauls PRL (2007): frustrated unconventional SC in confined regions. Vortex-antivortex lattices in 2D 4 He, Zhang PRL/Gabay, Kaputulnik PRL (1993) La 2-x Ba x CuO 4 near x=1/8: P-band in an optical lattice: Wirth et al, arXiv:1006.0509 Finite momentum BEC,W. V.Liu and C. Wu PRA (2006) X.Lee, W.V.Liu, C.Lin arXiv:1005.4027 (2010)

7. PDW phase in superconductors without parity symmetry

8. Non-centrosymmetric: CePt 3 Si, CeRhSi 3, CeIrSi 3 CePt 3 Si: Heavy fermion superconductor: E. Bauer et al PRL 92 027003. CeRhSi 3 and CeIrSi 3 also have H c2 >H P N. Kimura, et al, PRL 95,247004 I.Sugitani, et al. JPSJ 75, 043703 Yasudai et al, JPSJ, 73, 1657 (2004)

9. Interface Superconductors Reyren et al, Science, 317 1196 (2007): Ohtomo,Hwang, Nature 427, 423 (2004): Superconductivity in the 2D electron gas under electric fields applied normal to the interface

10. Theoretical Formulation Broken Inversion exists solely through ASOC: Single Particle excitations become : With spins polarized along

11. Band Structure For CePt 3 Si Anisimov et al. : This implies  is much larger than  Model by Rashba SO interaction (ASOC): Model applies to interface superconductors: SrTiO 3 /LaAlO 3

12. PDW phase in non-centrosymmetric materials No ASOCWith ASOCWith ASOC and Zeeman Field field k+q -k+q The two bands prefer opposite q vectors.

13. Phase Diagram Uniform phase N1=N2 (N1-N2)=.05 (N1+N2) Helical phase Multiple q-phase Helical phase Helical phase: Multiple-q phase: spatially varying gap H H T/Tc DFA, RP Kaur, PRB 75, 0645111 (2007).

14. Helical Phase: Free Energy  induces a helical solution in a uniform magnetic field when |  | is uniform (this state carries no current) Kaur, DFA, Sigrist, PRL 94, 13702 (2005) GL free energy: constrained by symmetry: Broken parity symmetry allows a new term (Lifshitz invariant)

15. Symmetry Based Theories of Pair Density Wave Phases

16. General questions about PDW phases 1- What possible ground states are there? 2 –What topological defects (vortices) exist and how do they shape the physics? Matusda,Shimahara, JPSJ (2007).

17. Ginzburg-Landau-Wilson Theory: Square lattice 1- Four Q-modes related by symmetry 2- Applies to CeCoIn 5 and cuprates 3- Possible PDW orders labeled by Q i and G Q Symmetry properties leads to Ginzburg Landau Wilson free energy for the four-component order parameter Generic mixing of spin-singlet and spin-triplet order parameters

18. PDW Ground States Can find all possible “homogeneous” ground states (the phase factors due to U(1)xU(1)xU(1) symmetry): Vortex-antivortex (v-av) phase breaks time reversal Stripe phase breaks time reversal and parity

19. Theory of  Qx  and  -Qx Gauge invariance Translational invariance symmetry! General feature :

20. FFLO/PDW “ Vortices ” (n,m) vortices. Consider (1,0) vortex in a phase where both components have equal magnitudes (FFLO) case: (1,1) vortex is usual Abrikosov vortex with flux   then Related arguments in different U(1)xU(1), Babaev, PRL 89, 067001 (2002).

21. ½ vortices usual phase field PDW displacement field Uniform SC (charge 4): Edge dislocation in charge density from ½ vortex U(1)xU(1) symmetry implies  -Q and  Q single valued Charge Density : Single valued :

22. 2D superfluid: thermal excitation of vortices Free energy governed by phase fluctuations: The energy of a vortex is: The entropy is: above which vortices proliferate below Tc (Berezinskii, Kosterlitz, Thouless phase transition) R=radius of system

23. Fluctuations of ½ vortices A screens 1/r leading to a finite energy vortex Dislocations still cost log(R/a) energy. CDW order survives while SC order does not. (DFA, Tsunetsugu, Nature Physics 2008) Superconductors Rotationally invariant superfluids  Q screens 1/r leading to a finite energy dislocation, vortices cost log(R/a) energy. Charge 4 superfluid order survives while CDW order does not. (Radzihovsky and Vishwanath PRL 2009) Berg, Fradkin, Kivelson argue charge 4e SC without rotational invariance (Nature Physics 2009)

24. Possible High Field FFLO ½ Vortex Solution Around this circle we have (1,0) vortex properties Field due to screening currents is proportional to The field is that of a lattice of half flux quanta. We have recently shown that this lattice is stable in FFLO superconductors. Zero of Abrikosov vortex can decay: (1,1)=(1,0)+(0,1)

25. Other exotic PDW phases? Possible 6e superfluidity/superconductivity Can prove this phase has stable  0 /3 vortices: This phases exists in materials with hexagonal or cylindrical symmetry. Has SDW and CDW order.

26. Singlet Triplet Mixing in FFLO Spin-singlet Spin-triplet “Nodeless FFLO”: Spin-triplet mixing affects the low- energy states and stability of FFLO Symmetry implies singlet-triplet mixing in all PDW phases: how does this look in the FFLO phase? ( DFA, Z Zheng 2010 )

27. Phase Diagram Find small region of stable FF phase. This implies existence of ½ vortex phase.

28. Andreev mid-gap states Mid gap states are “removed”, that is pushed away from zero energy. Provides microscopic reason for large induced triplet component.

29. Simple way to understand this mixing Lifshitz Invariants If exists in free energy then Ifthenis non-zero Also: then The existence of this Lifshitz invariant guarantees the s-wave/p-wave (singlet-triplet mixing)

30. Conclusions Natural mechanism for field-induced PDW phase when parity symmetry broken. Carried out classification of PDW phases in tetragonal materials (distinguishable by induced CDW and SDW order). Fractional vortex/dislocations exist in PDW and FFLO phases. Vortices/Dislocations play an important role in 2D fluctuation physics (4e SC/CDW phases, more exotic phases). Abrikosov vortices can decay into fractional vortex pairs. ½ flux quantum lattice. Symmetry guaranteed triplet admixture changes properties of FFLO phase.