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Sept.20,2011@Cornell
Functional Renormalization Group Studies of Iron-based Superconductors
Fa WangDept. of Physics, UC Berkeley
Dept. of Physics, MIT
Ref.:arXiv:0805.3343PRL 102, 047005 (2009); EPL 85, 37005 (2009);PRB 80, 064517 (2009); PRB 81, 184512 (2010).
S±
Sept.20,2011@Cornell
Acknowledgment
● UC Berkeley:Dung-Hai Lee, Hui Zhai(current: Tsinghua U.), Ashvin Vishwanath, Ying Ran(current: Boston College).
● Renmin University:Zhong-Yi Lu, Fengjie Ma
● U. of Tokyo:Arita Ryotaro
Sept.20,2011@Cornell
Outline
● Brief introduction to iron-based superconductors
● The functional renormalization group(FRG) method
● FRG results
● Remaining challenges
Sept.20,2011@Cornell
Iron-based HTSCs
LaFeAsO
SmFeAsO
T = 4.2 K
Onnes 1911
Bednorz & Muller1986 Hosono et al.
2008
T = 77 K
Sept.20,2011@Cornell
A brief history of the “iron age”
● 2006, Tc~5K, LaFePO [1111], Hosono et al.
● Feb.'08, Tc=26K, LaFeAsO
1-xF
x, Hosono et al.
● Apr.'08, Tc=55K, SmFeAsO
1-xF
x, ZX Zhao et al.;
Gd1-x
ThxFeAsO, ZA Xu et al.
● May'08, Tc=38K, Ba
1-xK
xFe
2As
2 [122], Rotter et al.
● July'08, Tc=18K, Li
1-δFeAs [111], CQ Jin et al.
● July'08, Tc=8K, Fe
1+δSe [11], MK Wu et al.
● ... c.f.: early review by Ishida&Nakai&Hosono, JPSJ'09.
Sept.20,2011@Cornell
Properties of iron-based superconductors
● Phase diagram:Fe-based vs. Cu-based.SC close to AFM SC close to AFMmetallic AFM Mott insulating AFM
J Zhao et al.Nature Mat.'08
Sept.20,2011@Cornell
Properties of iron-based superconductors
● Phase diagram
Sept.20,2011@Cornell
Properties of iron-based superconductors
● Lattice structureFe-based vs. Cu-basedFe square lattice Cu square latticeFe
2X
2 tri-layer CuO
2 layer
(X=As, P, Se, Te...)FeX above Fe layerFeX below Fe layer
Sept.20,2011@Cornell
Properties of iron-based superconductors
● Lattice structure
Sept.20,2011@Cornell
Properties of iron-based superconductors
● AntiferromagnetismFe-based vs. Cu-basedstripe AFM staggered AFM
strong AFM exchange J2
strong AFM J1.
LaFeAsOCruz et al.'08
J1
J1
J2
Sept.20,2011@Cornell
Properties of iron-based superconductors
● Electronic structure (DFT)Fe-based vs. Cu-basedquasi-2D quasi-2Dmultiple Fermi surfaces one Fermi surface3d6: all d-orbitals active 3d9: one d-orbital active
(quasi-)nesting
Γ(0,0)Singh&Du,PRL'08
Sept.20,2011@Cornell
Iron-based vs. cuprates: summary
● Simpler:metallic parent compound→weaker correlation→perturbative methods maybe ok
● More complicated:multiple orbitals/bands/Fermi surfaces.→need to use realistic multi-band
electronic structures→details may vary between
different materials
Rotter et al.PRL'08
Qazilbash et al. Nat.Phys.'09
Sept.20,2011@Cornell
Big question for theorists:the pairing problem
● Mechanism for pairing.
● Electron-phonon coupling is too weak.● Can repulsive electron-electron interaction cause pairing?
● Symmetry(shape) of pairing order parameter.
Sept.20,2011@Cornell
Early theories
● An incomplete listMazin et al. PRL'08 DFT s
±.
Kuroki et al. PRL'08 RPA, 5-band s± or d-wave
Dai et al. PRL'08 2-band tripletLee&Wen PRL'08 3-band tripletSi&Abraham PRL'08 t-J d
xy.
ZJ Yao et al. NJP'09 FLEX, 2-band d-wave or s±.
Qi et al. arXiv'08 RPA, 2-band d- or p-waveSeo et al. PRL'08 t-J, 2-band s
± or d-wave.
● Need less biased method.
Sept.20,2011@Cornell
Outline
● Brief introduction to iron-based superconductors
● The functional renormalization group(FRG) method
● FRG results
● Remaining challenges
Sept.20,2011@Cornell
Effective Theory and Renormalization Group
● “Solve” the interacting electron problem
● Effective theory and renormalization group:Polchinski, NuclPhysB'84; Shankar, RMP'94
● Integrate out high-energy modes (decrease cutoff Λ), generate corrections to low-energy theory.
● In principle unbiased, practically difficult: too many terms
Sept.20,2011@Cornell
Functional Renormalization Group
● Approximations
● One-loop: justified for weak coupling, keep only two-particle scattering, recursive 2nd order perturbation theory
● Technical: discretize frequency-momentum space
● One-loop FRG equation: Zanchi&Schulz PRB'96, Honerkamp PRB'01
All one-loop corrections
=d
d ln(cutoff)
Sept.20,2011@Cornell
FRG details
● Free fermion action: ab initio band structure fit by d-orbitalsLaFeAsO: Kuroki et al. PRL'08; LaFePO: Arita Ryotaro; FeSe: Fengjie Ma, ZY Lu
● Bare interaction: onsite Kanamori interactions for d-orbitalsU=4, U'=2, J
H=0.7eV, from LDA+DMFT,
Craco et al. PRB'08, Haule et al. PRL'08
Sept.20,2011@Cornell
FRG details
● K-space discretization:16 patches*5 FSs
● RG flow of 803*3=1,536,000 couplings:V(k
1,k
2;k
3,k
4) depends
on three momenta
Sept.20,2011@Cornell
Outline
● Brief introduction to iron-based superconductors
● The functional renormalization group(FRG) method
● FRG results
● Remaining challenges
Sept.20,2011@Cornell
Result: pairing symmetry
● LaFeAsO: FW et al. PRL'09
● for all reasonable input, s± is the leading pairing instability;
● gap around electron pockets usually have large variation: ~consistent with NMR T
1 vs. T, Nakai et al. JPSJ'08; Grafe
et al. PRL'08.
Grafe et al. PRL'08
Sept.20,2011@Cornell
What is s±
● Proposed by Mazin et al. PRL'08
● “s-wave”: trivial representation(A1g
) of point group(D4h
).
● ±: unconventional, Δ changes sign in k-space
● s±
vs. d-wave
~cos(kx)cos(k
y) ~ cos(k
x)-cos(k
y)
~2nd neighbor pairing ~1st neighbor pairing
+
─
Sept.20,2011@Cornell
Result: pairing symmetry
● LaFePO: FW et al.PRB'10
● “s±” with accidental nodes:consistent with penetration depth, Fletcher et al.PRL'09, Hicks et al.PRL'09
Hicks et al. PRL'09
Sept.20,2011@Cornell
Result: pairing symmetry
● FeSe: FW et al.PRB'10
● s±, like 1111,
consistent with ARPES, Nakayama et al. PRL'10
Sept.20,2011@Cornell
Result: gap anisotropy
● Four-fold anisotropy generically exists.Sign/magnitude depends on detail
● Inconsistent w/ ARPES: uniform gap
● Consistent w/ bulk measurements.
H.Ding et al.EPL'08
Zeng et al. Nat.Comm.'10
Sept.20,2011@Cornell
Result: the mechanism
● AFM correlation is driving force: FW et al. EPL'09
● A special class of scattering processes (P1), shared by magnetic and pairing channel, dominates RG flow (and >0)
● Δk (hole Fermi surface) and Δ
-k+Q (electron Fermi surface)
must have opposite sign (s±).
Sept.20,2011@Cornell
Result: the effective interaction
● Low energy effective interaction is well captured by J1-J
2
exchange: Zhai&FW&Lee, PRB'09
● AFM and pairing order parameter obtained from J
1-J
2 exchange is
very similar to FRG result
● Consistent with other “spin fluctuation” theories:Mazin et al.; Kuroki et al.; JP Hu et al.; Scalapino et al.
J1
J1
J2
Sept.20,2011@Cornell
Result: competing orders
● Competing orders driven by AFM:Zhai&FW&Lee PRB'09
● Fermi surface distortion: relative shift of hole&electron bandsARPES evidence M.Yi et al. PNAS'11
breaking of 4-fold sym. (nematic, Pomeranchuk)STM evidence Chuang et al. Science'10
SDW
SC
FS Distortion
...
Sept.20,2011@Cornell
Result: comparison to “cuprates”
● Similar pairing mechanism and competing orders are found in FRG of one band Hubbard model: Zhai&FW&Lee, PRB'09
Sept.20,2011@Cornell
Check by a strong coupling method
● Partially Gutzwiller projected wavefunction:Yang&Zhai&FW&Lee.PRB'11 (VMC)
● Results are qualitatively consistent
● unconventional s-wave pairing● Fermi surface distortion
Sept.20,2011@Cornell
Outline
● Brief introduction to iron-based superconductors
● The functional renormalization group(FRG) method
● FRG results
● Remaining challenges
Sept.20,2011@Cornell
How to detect the sign change of s±
● Indirect method
● neutron resonance mode below 2Δ:proposed: Maier&Scalapino, Korshunov&Eremin PRB'08;measured: Lumsden et al. PRL'09, etc.
● STM quasi-particle interference:proposed: FW et al.EPL'09measured: Hanaguri et al. Science'10
● Direct method: no definitive result yet
Hanaguri et al. Science 328, 474 (2010)
Sept.20,2011@Cornell
How can s± be stable against impurities?
● Non-magnetic impurity can be pair breaking in unconventional superconductors: “violation” of Anderson's theorem.
● Zn substitution of Cu quickly suppresses Tc in cuprates
● Zn substitution of Fe in iron-based: still controversialYuke Li et al. NJP'09, NJP'10S. Kitagawa et al. PRB'10, PRB'11
Sept.20,2011@Cornell
KxFe
2-ySe
2: same or different
● Recently synthesized KxFe
2-ySe
2: J Guo et al. PRB'10
● NO hole-like Fermi surfaces around Γ,but high T
c (>30K).
● some experiments suggest coexistence of large moment AFM (3.3μ
B/Fe) and
superconductivity.
Qian et al.'11
W Bao et al. arXiv:1102.0830
Sept.20,2011@Cornell
Summary and outlook
● FRG method to interacting electrons:
● Less biased for weak/intermediate electron correlations● For most iron-based superconductors:
s± pairing driven by AFM correlation.
● S± gap still needs direct experimental confirmation.
● Higher Tc?
● Not likely in iron-based:Fe-based, J~600K, T
c~55K; Cu-based, J~1700K, T
c~160K.
● Dope a stronger AFM: Sr2IrO
4? FW&Senthil, PRL'11.
Sept.20,2011@Cornell
Other applications of FRG
● 1/4-doped graphene: Fermi surfaces cross von Hove.Levitov et al. ArXiv:1107.1903:“three patch” RG,d+id topological superconductor
● Our FRG (arXiv:1109.3884): chiral SDW(Chern insulator, AQHE) at 1/4-doping, d+id away from 1/4-doping
¼ doping 0.194 doping
Sept.20,2011@Cornell