Space group symmetry, spin-orbit coupling and the low energy effective Hamiltonian for iron based

superconductors (arXiv:1304.3723)

Vladimir CvetkovicNational High Magnetic Field Laboratory

Tallahassee, FL

Superconductivity: the Second CenturyNordita, 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 LaboratoryFlorida State University

Motivation: Electronic multicriticality iniron-pnictide superconductors

•quasi 2D system• parent state is a compensated semi-metal• low carrier density• competing instabilities

Solution: Electronic multicriticality in bilayer grapheneWe 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:1111: REOFeAs, LaOFeP, REFFeAs122: BaFeAs11: FeTe, FeSe111: 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)

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))

Irreducible representations of the space group

Bloch states, order parameters at wave-vector k characterized by an irreducible representation of

D4h

C2v

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

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!

At M-point: D4h is not closed due to fractional translations

Symmetry adapted functions at M-point

The lowest harmonics

EM2X EM2

Y EM4YEM4

X

Next harmonics

EM2X EM4

XEM3XEM1

X

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:

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 effectivetheory 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 effectivetheory 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)

J. Hu and N. Hao, Phys. Rev. X 2, 021009 (2012)Misidentified symmetry:

Comparison of the low energy effectivetheory 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

M. L. Tiago, et al., Phys. Rev. Lett. 97, 147201 (2006).l = 80meV (Fe clusters)

Kane-Mele like term

l = 70meV (bcc Fe) Y. Yao, et al., Phys. Rev. Lett. 92, 037204 (2004).

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

Spin-density wave order parametersCollinear SDW order parameter – one of the EM components condenses

EM1Y = EM4

X SX

= EM2Y Sz

EM2Y = EM4

X SY EM3X = EM4

X Sz

= EM2Y SX

EM4X = EM2

Y SY

Spin-orbit interaction:• Magnetic moment locking

Magnetic moment on iron the orbital part is EM4

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 EM4

X SX = EM1Y

Induced magnetic moment on pnictogen atoms

Nodal Dirac fermions in the collinear SDW phase

EM4 SDW order parameter – symmetry protected Dirac nodesY. 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

• No Kramers degeneracy• Fermi surfaces split

+ =Coplanar SDW order parameter – both of the EM components condense

SuperconductivitySC order parameters classified according to the space group

Zero momentum pairing Large (M) momentum pairing - PDW

Spin-singlet pairing terms:

SuperconductivityA1g spin-singlet SC specified by three k-independent parameters

• Hole FS’s – the gap is isotropic• Electron FS’s – the gap anisotropy determined by DM1 and DM3

Bogolyubov-de Gennes Hamiltonian

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 admixtureA1g spin-triplet SC: two more gap parameters

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

Bogolyubov-de Gennes Hamiltonian at G

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 directionsWe 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 SchoolNational High Magnetic Field Laboratory, Tallahassee, FL, USA

T (F)