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Space group symmetry, spin-orbit coupling and the low energy effective Hamiltonian for iron based superconductors (arXiv: ) 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 , NSF Cooperative Agreement No. DMR , 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, (R) (2010); O. Vafek, Phys. Rev. B 82, (2010); R.E. Throckmorton and O. Vafek, Phys Rev B 86, (2012); VC, R.E. Throckmorton, and O. Vafek, Phys Rev 86, (2012); VC and O. Vafek, arXiv: 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, (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, (2009) K. Kuroki, et al., Phys. Rev. Lett. 101, (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, (2009) K. Kuroki, et al., Phys. Rev. Lett. 101, (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, R (2008) J. Hu and N. Hao, Phys. Rev. X 2, (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, (2008) at and M: correct symmetry properties of the bands spurious Fermi surface M. Daghofer, et al., Phys. Rev. B 81, (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, (2006). = 80meV (Fe clusters) Kane-Mele like term = 70meV (bcc Fe) Y. Yao, et al., Phys. Rev. Lett. 92, (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, (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: ) 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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