Functional Renormalization Group Studies of Iron … · Functional Renormalization Group Studies of...

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±


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


Outline

● Brief introduction to iron-based superconductors

● The functional renormalization group(FRG) method

● FRG results

● Remaining challenges


Iron-based HTSCs

LaFeAsO

SmFeAsO

T = 4.2 K

Onnes 1911

Bednorz & Muller1986 Hosono et al.

2008

T = 77 K


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.


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



● Phase diagram



● 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



● Lattice structure



● AntiferromagnetismFe-based vs. Cu-basedstripe AFM staggered AFM

strong AFM exchange J2

strong AFM J1.

LaFeAsOCruz et al.'08

J1

J1

J2



● 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


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


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.


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.


Outline



● FRG results



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


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)


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


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


Outline



● FRG results



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


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

+

─



● 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



● FeSe: FW et al.PRB'10

● s±, like 1111,

consistent with ARPES, Nakayama et al. PRL'10


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


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±).


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


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

...


Result: comparison to “cuprates”

● Similar pairing mechanism and competing orders are found in FRG of one band Hubbard model: Zhai&FW&Lee, PRB'09


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


Outline



● FRG results



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)


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


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


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.


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


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