Topological states of matter in correlated electron · PDF file ·...
Transcript of Topological states of matter in correlated electron · PDF file ·...
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Topological states of matter in correlated electron systems
Qiang-Hua Wang National Lab of Solid State Microstructures, Nanjing
University, Nanjing 210093, China
Seminar @ Tsinghua, Dec.5/2012
Collaborators:Dunghai Lee (Berkely), F Wang (MIT), F Yang (Beijing) Students: Wan-Sheng Wang and Yuan-Yuan Xiang
Wang WS, et al, PRB 2012; Xiang YY, et al, PRB 2012
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Outline
• Introduction and motivation • T-breaking topological phases in doped
Graphene and kagome lattices • T-invariant topological superconductors • Conclusions
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Topology in daily life
Zero handle
One handle
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Topological states in 1D: kink and soliton
∫ −+∂= ]cos1)([ 2 φφxKdxH
0
2*pi
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Topological states in 2D: vortex
W=1
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Topological states in 2D: vortex
W=0
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Topology and topological states of matter Hasan and Kane, RMP 2010
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Topology and mapping k à H(k)
.2},{
,])([
ββα δ
ψεψ
a
kkk
k IkBH
=ΓΓ
+Γ⋅=∑ +
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TKNN invariant, Chern number and edge states Thouless, Kohmoto, Nightingale, and den Nijs 1982
Topological number can not change smoothly. Topologically distinct phases are connected by a) gap closing in the quantum phase transition point in parameter space, or b) gapless edge states in real space. There is a 1 to 1 correspondence between the change of Z across the boundary and the number of edge states.
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TKNN invariant, Chern number and edge states
Skippy cyclotron orbits Edge state in the Haldane model
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Spin polarized p+ip superconductor, Z=1
Read and Green 2000
The effective field B(k) cover the Bloch sphere once.
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Majorana fermions in 1d and 2d cases
+E and –E forms a canonical fermion, not protected E=0 comes in pair and sit on opposite edges, protected
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Majorana fermion and non-Abelian statistics
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Topological quantum computing
April 2006 www.sciam.com
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Spin singlet d+id superconductor
)2exp(~)(,),(
,),(
,)()()()(
0*0
k
Tkkk
Tkkk
kk
k
ikdcc
cc
kHkdkdkH
H
θ
ψ
ψ
ψµ
µψ
+
+↓−↑
↓−+↑
+
−
++
=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
−Δ
Δ−=∑
The effective field B(k) cover the Bloch sphere twice, thus Z=2
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Edge states for d+id pairing (Z=2)
−1 0 1
−6
−4
−2
0
2
4
6
q//
Energy
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Z2 number in T-invariant insulators
Enen crossings:Z2=0 Odd crossings: Z2=1
Kramers degeneracy on T-invariant momenta Г
In-gap edge states In-gap edge states
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Quantum spin Hall system
Konig et al, 2007
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T-invariant topological insulator/superconductor
Roy et al 2008; Schnyder et al 2008 Kitaev 2009; Qi et al 2009; Qi et al, RMP 2011
For a topological insulator,ψk
+ = (ak↑+ ,bk↑
+ ,ak↓+ ,bk↓
+ )
ψk = (ak↑,bk↑,ak↓,bk↓)T .
For a superconductor,ψk
+ = (ak↑+ ,a−k↑,ak↓
+ ,a−k↓)
ψk = (ak↑,a−k↑+ ,ak↓,a−k↓
+ )T .
)( yx ippip +=±
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Edge state in a T-invariant topological superconductor
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A convenient criterion for T-invariant topological superconductor
Roy et al 2008; Schnyder et al 2008 Kitaev 2009; Qi et al 2009; Qi et al, RMP 2011
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3d topological insulators
Fang et al Xue et al
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3d topological superconductors
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Ando et al, PRL 2010
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Fu and Berg, PRL 2010
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Periodic table of topology
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Challenges in the search of (intrinsic) topological superconductors
• Energy scale of topological insulators: eV
• Finding a BdG hamiltonian finishes only a half of the job
• Proximity effect generated topological superconductor depends solely on the edge (surface) states of topological insulator (many literatures in this direction)
• Intrinsic topological superconductor relies on the system itself, such as Sr2RuO4 and He-III B-phase.
• In repulsive systems, the energy scale involved in superconducting transition: 1 ~ 40meV. Energy hierarchy requires RG treatment.
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Ideas of RG and FRG Wilson RG Wetterich FRG
Energy S
cale
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Ideas of RG and FRG Wilson RG Wetterich FRG
Energy S
cale
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Ideas of RG and FRG Wilson RG Wetterich FRG
Energy S
cale
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Ideas of RG and FRG Wilson RG Wetterich FRG
Energy S
cale
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Singular-mode FRG
: orthonormal form factors
P
C D
Cf: Husemann and Salmhofer
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FRG flow
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FRG flow
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FRG flow
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A simple view of mode-mode coupling
±ciσ+ ciσcjτ
+ cjτ ⇔ ± ciσ+ cjτ
+ cjτciσ
Si ⋅Sj ⇔ −12(ci↑
+cj↓+ − ci↓
+cj↑+ )(cj↓ci↑ − cj↑ci↓)+ ... ⇒ ↑↓−↓↑
−Si ⋅Sj ⇔ −14ci↑+cj↑
+ cj↑ci↑+ ... ⇒ ↑↑, ↓↓, ↑↓+↓↑
Singlet pair
Triplet pair
+ more general bond-type density wave interactions
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Instabilities
• Q=0 p-p susceptibility always logarithmically divergent à universal Cooper instability wrt infinitesimal attraction
• p-h susecptibility usually finite (unless in case of perfect nesting or van Hove singularity) àStoner instability wrt finite interaction
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• Introduction and motivation • T-breaking topological phases in doped
Graphene and kagome lattices • T-invariant topological superconductors • Conclusions
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Band structure of graphene
The band structure of graphene with t1=2.8ev, t2=0.1ev , t3=0.07ev at ¼ doping
Physics near van Hove singularity?
Physics near the Dirac point
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Semenoff, PRL53,2449,1984
Zhang et al., Nature 438,201(2005)
Relativistic quantum mechanics near the Dirac point
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C. L. Kane and E.J. Mele , QSHE in Graphene , PRL95,226801(2005)
C. L. Kane and E. J. Mele, Z2 topological Order and the QSHE, PRL95, 146802 (2005)
Spin quantum Hall effect?
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X.Du et al., Nature 462,192; Bolotin et al., Nature 462,196
Correlations revealed by fractional QH
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Doping graphene…
Eli Rotenberg
Extended van Hove singularity
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What’s so special of graphene • van Hove singularity and correlation effect
• Under C6v point group,( ,xy) and (x,y) are doublets. Candidates for the gap function.
• T-breaking mixing of degenerate pairing gaps very likely, leading to a full gap
• Possible pairing symmetries: s, d+id, p+ip, f
22 yx −
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x=1/4, U=3.6t, V=0 Van Hove singularity and perfect nesting
V_sdw V_sc
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Chiral SDW
LiTao, arxiv 1103.2420, honeycomb lattice Martin and Batista, PRL101,156402, triangle lattice
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Chern number and quantized anomalous Hall conductivity
1
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Non-perturbative quantum Monte Carlo
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x=0.211, U=3.6t, V=0
V_sdw V_sc
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T-breaking d+id Two degenerate d-wave pairing:
d+id
MF or GL theory predict that the d+id pairing is energetically more favorable
R Nandkishore et al, Nature Physics 8, 158 (2012)
Full gap
Nodal gap
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Edge states for d+id pairing (Z=2)
−1 0 1
−6
−4
−2
0
2
4
6
q//
Energy
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Phase diagram
Both chiral SDW and chiral d+id are topological.
Cf: Keisel et al, arxiv 1109.2953
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Kagome lattice
Upper van Hove filling Lower van Hove filling
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Upper van Hove filling
D-wave Permoranchuk
FM
Intra-cell AFM
sSC
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Lower van Hove filling
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• Topological states of matter and challenges of the search of intrinsic topological superconductors
• T-breaking topological phases in doped Graphene
• T-invariant topological superconductors, a road map
• Conclusions
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Gap function of a T-invariant superconductor
Qi et al, Kitaev et al: 1) Even number of spin-split pockets, each encircles an odd number of T-invariant momenta. 2) Number of pockets with + and - signs: Even = Odd + Odd
singlet triplet
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Gap-function of a T-invariant superconductor
H0 =Ψ k+(εkσ 0 +λγ k ⋅σ )Ψ k → ψk
+(εk ±λ |γ k |)ψk,
−k,a = iσ 2K k,a
If dk ~ γ k, T − invar iant, and
HP =Ψ k+(φkσ 0 + dk ⋅σ )iσ 2 (Ψ−k
+ )T
→−ψk,a+ ( φk ± | dk | )(ψ−k,b
+ )Tδa,b
γ k = (−sinky, sinkx, 0)
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i(px + ipy )↓↓+i(px − ipy )↑↑
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Our road map
• Seek ferromagnetic spin fluctuations to favor triplet pairing
• Seek point group with odd parity degenerate irreducible representation (such as C4v and C6v)
• Seek a system with 2(2n+1) spin-split pockets
• Rashba coupling causes degenerate triplets to recombine into a T-invariant gap, plus small induced singlet component.
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Non-centrosymmetric systems
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Spin-resolved fully anti-symmetrized SM-FRG
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Topological pairing near van Hove singularity
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Topological pairing triggered by small-q inter-pocket scattering
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Possible candidates with ferromagnetic spin fluctuations
Aoki and Flouquet, JPSJ 81, 011003 (2012).
Shimizu et al, NATURE 412, 316 (2001).
Reyren, etal, Science 317, 1196 (2007).
Iron under high pressure
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• Motivation • T-breaking topological phases in doped
Graphene • T-invariant topological superconductors, a
road map • Conclusions
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Conclusions • Graphene near ¼ doping is either a Chern
insulator or a chiral d+id superconductor.
• Ferromagnetic instability is the key to T-invariant topological insulator, plus Rashba coupling and 2(2n+1) spin split fermi pockets (encircling T-invarint momenta).
• T-invariant topo-SC can be triggered by 1) proximity to van Hove singularity and 2) by small-q inter-pocket scattering
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Thank you for your attention