Report

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 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 Motivation: Electronic multicriticality in iron-pnictide superconductors •quasi 2D system • parent state is a compensated semi-metal • low carrier density • competing instabilities 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). Lattice structure of iron-pnictides Pnictide families: Space group: 1111: REOFeAs, LaOFeP, REFFeAs 122: BaFeAs 11: FeTe, FeSe 111: LiFeAs 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) Space group P4/nmm Operations: Integer lattice translations `Point group’, i.e., symmetries of the unit cell: Generators: P4/nmm is nonsymmorphic 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)) Irreducible representations of the space group Bloch states, order parameters at wave-vector k characterized by an irreducible representation of C2v D4h ?? ? ? Cs 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) Irreducible representations of the space group at the M-point At M-point: D4h is not closed due to fractional translations The group of the wave-vector, PM, is a factor group of P4/nmm w.r.t. ``even’’ translations (C. Herring, 1942) 32 elements (16 from D4h and 16 with an odd translation added) Only 2D irreducible representations are physical! Symmetry adapted functions at M-point The lowest harmonics EM2X EM2Y EM4X EM4Y Next harmonics EM1X EM2X EM3X EM4X Full tight banding band structure Range: ±2eV from the Fermi level (3d-iron orbitals) V. Cvetkovic, Z. Tesanovic, Europhys. Lett. 85, 37005 (2009) Fermi surface states’ symmetries: K. Kuroki, et al., Phys. Rev. Lett. 101, 087004 (2008) Low-energy effective theory Low-energy spinor (G: Eg states; M: EM1 and EM3 states): Low-energy effective theory The individual blocks: Fitting to the full models for iron-pnictides 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) Comparison of the low energy effective theory to 2-orbital models Only dxz and dyz iron orbitals: • at G: Eg and Eu states • at M: EM1 and EM2 states S. Raghu, et al., Phys. Rev. B 77, 220503R (2008) Misidentified symmetry: J. Hu and N. Hao, Phys. Rev. X 2, 021009 (2012) Comparison of the low energy effective theory to 3-orbital models Only dxz, dyz, and dXY iron orbitals P. A. Lee and X.-G. Wen, Phys. Rev. B 78, 144517 (2008) • at G 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 G and M wrong band ordering Spin-orbit interaction in the low-energy effective theory On-site spin-orbit interaction for iron 3d orbitals comparable to other energy scales l = 80meV (Fe clusters) M. L. Tiago, et al., Phys. Rev. Lett. 97, 147201 (2006). l = 70meV (bcc Fe) Y. Yao, et al., Phys. Rev. Lett. 92, 037204 (2004). Kane-Mele like term Spin-orbit interaction in the low-energy effective theory The effect on the spectrum center of inversion • All states doubly degenerate (Kramers degeneracy) • The only symmetry allowed 4-fold degeneracy is at the M-point Spin-density wave order parameters Collinear SDW order parameter – one of the EM components condenses Magnetic moment on iron the orbital part is EM4 EM1Y = EM4X SX = EM2Y Sz EM2Y = EM4X SY EM3X = EM4X Sz = EM2Y SX EM4X = EM2Y SY Spin-orbit interaction: • Magnetic moment locking 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 EM4X SX = EM1Y Induced magnetic moment on pnictogen atoms Nodal Dirac fermions in the collinear SDW phase EM4 SDW order parameter – symmetry protected Dirac nodes Y. Ran, et al., Phys. Rev. B 79, 014505 (2009) Intermediate-coupling regime (D ~ 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 Spin-density wave order parameters Ba0.76Na0.24Fe2As2 (S. Avci et.al. arXiv:1303.2647) C4-symmetric phase The spectrum in the coplanar SDW phase Coplanar SDW order parameter – both of the EM components condense + = • No Kramers degeneracy • Fermi surfaces split Superconductivity SC order parameters classified according to the space group Zero momentum pairing Spin-singlet pairing terms: Large (M) momentum pairing - PDW Superconductivity A1g spin-singlet SC specified by three k-independent parameters Bogolyubov-de Gennes Hamiltonian • Hole FS’s – the gap is isotropic • Electron FS’s – the gap anisotropy determined by DM1 and DM3 Superconductivity (spin-singlet) The gap on the electron Fermi surfaces given by This is also applicable to B2g-superconductivity (d-wave) Superconductivity in the presence of spin-orbit coupling Spin orbit interaction: spin-triplet SC admixture A1g spin-triplet SC: two more gap parameters Bogolyubov-de Gennes Hamiltonian at G The gap on the hole FS’s is • DGt hole FS’s gap anisotropy • ``Near nodes’’ in the gap on one FS • The other FS relatively isotropic Superconductivity in the presence of spin-orbit coupling At the M-point: The gap on the electron FS’s is Fourfold gap symmetry 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 EM-component (Kramers present) - coplanar SDW – both EM-components (Kramers broken) - spin-orbit: spin direction locking and induced pnictogen magnetic moment • A1g-superconductivity (s-wave): - spin-singlet: 3 parameters; gap isotropic at G, anisotropic at M • A1g-superconductivity (s-wave) with spin-orbit: - spin-triplet admixture; 2 parameters; anisotropy and near nodes at G, 4-fold gap dependence at M Future directions We wish to study how e-e interaction drives the system toward a symmetry breaking phase The interaction Hamiltonian Where Gi,j(m)’s are 6x6 Hermitian matrices 30 independent couplings Theory Winter School National High Magnetic Field Laboratory, Tallahassee, FL, USA T (F)