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)