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Space group symmetry, spin-orbit coupling and the low energy effective Hamiltonian for iron based superconductors (arXiv:1304.3723) Vladimir Cvetkovic National High Magnetic Field Laboratory Tallahassee, FL Superconductivity: the Second Century Nordita, Stockholm, Sweden, August 29, 2013

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Together with… Dr. Oskar Vafek (NHMFL, FSU) NSF Career award (Vafek): Grant No. DMR-0955561, NSF Cooperative Agreement No. DMR-0654118, and the State of Florida National High Magnetic Field Laboratory Florida State University

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Motivation: Electronic multicriticality in iron-pnictide superconductors quasi 2D system parent state is a compensated semi-metal low carrier density competing instabilities

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Solution: Electronic multicriticality in bilayer graphene We know how to do it in bilayer and trilayer graphene! O. Vafek and K. Yang, Phys. Rev. B 81, 041401(R) (2010); O. Vafek, Phys. Rev. B 82, 205106 (2010); R.E. Throckmorton and O. Vafek, Phys Rev B 86, 115447 (2012); VC, R.E. Throckmorton, and O. Vafek, Phys Rev 86, 075467 (2012); VC and O. Vafek, arXiv:1210.4923 The first step is to build the low energy effective theory based on the symmetry. J.M. Luttinger, Phys. Rev. 102, 1030 (1956). G. Bir and G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (John Wiley, New York, 1974).

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Lattice structure of iron-pnictides Pnictide families: 1111: REOFeAs, LaOFeP, REFFeAs 122: BaFeAs 11: FeTe, FeSe 111: LiFeAs Space group: 1111: P4/nmm (129) 122: I4/mmm (139) 11: P4/nmm (129) 111: P4/nmm (129) Literature: C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin Heidelberg, 1990)

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Space group P4/nmm P4/nmm is non- symmorphic Generators: Operations: Integer lattice translations `Point group’, i.e., symmetries of the unit cell: The gap structure different in materials with a non-symmorphic space group (T. Micklitz and M. R. Norman, Phys. Rev. B 80, 100506(R) (2009))

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Irreducible representations of the space group Bloch states, order parameters at wave-vector k characterized by an irreducible representation of D 4h C 2v CsCs ?? Literature: C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin Heidelberg, 1990) ? ?

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Irreducible representations of the space group at the M-point The group of the wave-vector, P M, is a factor group of P4/nmm w.r.t. ``even’’ translations (C. Herring, 1942) 32 elements (16 from D 4h and 16 with an odd translation added) Only 2D irreducible representations are physical! At M-point: D 4h is not closed due to fractional translations

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Symmetry adapted functions at M-point The lowest harmonics EM2XEM2X EM2YEM2Y EM4YEM4Y EM4XEM4X Next harmonics EM2XEM2X EM4XEM4X EM3XEM3X E M1 X

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Full tight banding band structure Range: ±2eV from the Fermi level (3d-iron orbitals) V. Cvetkovic, Z. Tesanovic, Europhys. Lett. 85, 37005 (2009) K. Kuroki, et al., Phys. Rev. Lett. 101, 087004 (2008) Fermi surface states’ symmetries:

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Low-energy effective theory Low-energy spinor ( : E g states; M: E M1 and E M3 states):

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Low-energy effective theory The individual blocks: Fitting to the full models for iron-pnictides

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Comparison of the low-energy effective theory to the full models V. Cvetkovic, Z. Tesanovic, Europhys. Lett. 85, 37005 (2009) K. Kuroki, et al., Phys. Rev. Lett. 101, 087004 (2008)

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Comparison of the low energy effective theory to 2-orbital models Only d xz and d yz iron orbitals: at : E g and E u states at M: E M1 and E M2 states S. Raghu, et al., Phys. Rev. B 77, 220503R (2008) J. Hu and N. Hao, Phys. Rev. X 2, 021009 (2012)Misidentified symmetry:

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Comparison of the low energy effective theory to 3-orbital models Only d xz, d yz, and d XY iron orbitals P. A. Lee and X.-G. Wen, Phys. Rev. B 78, 144517 (2008) at and M: correct symmetry properties of the bands spurious Fermi surface M. Daghofer, et al., Phys. Rev. B 81, 014511 (2010) no spurious Fermi surfaces at and M wrong band ordering

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Spin-orbit interaction in the low-energy effective theory On-site spin-orbit interaction for iron 3d orbitals comparable to other energy scales M. L. Tiago, et al., Phys. Rev. Lett. 97, 147201 (2006). = 80meV (Fe clusters) Kane-Mele like term = 70meV (bcc Fe) Y. Yao, et al., Phys. Rev. Lett. 92, 037204 (2004).

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Spin-orbit interaction in the low-energy effective theory The effect on the spectrum All states doubly degenerate (Kramers degeneracy) The only symmetry allowed 4-fold degeneracy is at the M-point center of inversion

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Spin-density wave order parameters Collinear SDW order parameter – one of the E M components condenses E M1 Y = E M4 X S X = E M2 Y S z E M2 Y = E M4 X S Y E M3 X = E M4 X S z = E M2 Y S X E M4 X = E M2 Y S Y Spin-orbit interaction: Magnetic moment locking Magnetic moment on iron the orbital part is E M4 Experiments (e.g., 1111 – C. de la Cruz et al., Nature 453, 899 (2008); 122 – J. Zhao et al., Nat. Mater. 7, 953 (2008)): the total order parameter is E M4 X S X = E M1 Y Induced magnetic moment on pnictogen atoms

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Nodal Dirac fermions in the collinear SDW phase E M4 SDW order parameter – symmetry protected Dirac nodes Y. Ran, et al., Phys. Rev. B 79, 014505 (2009) Intermediate-coupling regime ( ~ 0.7eV): another band admixes; Dirac nodes not protected anymore. Spin-orbit coupling: All the Dirac nodes lifted (gaps ~ 0.25meV and higher The degeneracies at the M-point lifted by the SDW The Kramers degeneracy still present

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Spin-density wave order parameters Ba 0.76 Na 0.24 Fe 2 As 2 (S. Avci et.al. arXiv:1303.2647) C 4 -symmetric phase

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The spectrum in the coplanar SDW phase No Kramers degeneracy Fermi surfaces split + = Coplanar SDW order parameter – both of the E M components condense

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Superconductivity SC order parameters classified according to the space group Zero momentum pairingLarge (M) momentum pairing - PDW Spin-singlet pairing terms:

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Superconductivity A 1g spin-singlet SC specified by three k-independent parameters Hole FS’s – the gap is isotropic Electron FS’s – the gap anisotropy determined by M1 and M3 Bogolyubov-de Gennes Hamiltonian

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Superconductivity (spin-singlet) The gap on the electron Fermi surfaces given by This is also applicable to B 2g -superconductivity (d-wave)

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Superconductivity in the presence of spin-orbit coupling Spin orbit interaction: spin-triplet SC admixture A 1g spin-triplet SC: two more gap parameters The gap on the hole FS’s is t hole FS’s gap anisotropy ``Near nodes’’ in the gap on one FS The other FS relatively isotropic Bogolyubov-de Gennes Hamiltonian at

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Superconductivity in the presence of spin-orbit coupling At the M-point: The gap on the electron FS’s is Fourfold gap symmetry

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Conclusions Used space group symmetry to build the low energy effective model - degeneracy at M-point - spin-orbit interaction is readily included Order parameters classified according to the symmetry breaking - collinear SDW – a single E M -component (Kramers present) - coplanar SDW – both E M -components (Kramers broken) - spin-orbit: spin direction locking and induced pnictogen magnetic moment A 1g -superconductivity (s-wave): - spin-singlet: 3 parameters; gap isotropic at , anisotropic at M A 1g -superconductivity (s-wave) with spin-orbit: - spin-triplet admixture; 2 parameters; anisotropy and near nodes at , 4-fold gap dependence at M

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Future directions We wish to study how e-e interaction drives the system toward a symmetry breaking phase The interaction Hamiltonian Where i,j (m) ’s are 6x6 Hermitian matrices 30 independent couplings

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Theory Winter School National High Magnetic Field Laboratory, Tallahassee, FL, USA T (F)

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