The non-commutative topology of dirty superconductors: the case of the Spin Hall effect

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The non-commutative topology

of dirty superconductors:

the case of the Spin Hall effect

Giuseppe De Nittis

(Universität Erlangen-Nürnberg)

——————————————————————————–

Université Claude Bernard, Lyon, France. 29 de Mars, 2013

——————————————————————————–

Joint work with:

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Outline

1 BdG Hamiltonians and PH-symmetry Tight-binding BdG Hmiltonians Translational invariant pair potentials Spectral consequences of PH-symmetry 2 Spin Hall conductance as a topological index

SU(2)and TR invariant BdG Hamiltonians Observable algebra of homogeneous operators Real structure associated to PH-symmetry Spin derivation and spin Hall current Kubo formula for the spin Hall conductance 3 Spin Hall conductance for the d+ıd model

Spectral properties of the d+ıd model The Bloch bundle for the d+ıd model

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Overview on physics

•Superconductivity: zero electrical resistance and expulsion of magnetic fields for certain materials whenT <Tcrit. Observed

in1911(Onnes);

•BCS-theory: microscopic theory proposed in1957(Bardeen, Cooper & Schrieffer). Key notion:Cooper pairs(particle-hole), i.e.pairs of electrons interacting through the exchange of phonons. Nobel Prize in1972;

•BdG Hamiltonian: derivation of the BCS wavefunction from a canonical transformation (Bogoliubov transformations) of the electronic Hamiltonian (Bogoliubov & de Gennes1958);

•Topological insulators: BdG Hamiltonians provide models for D-typeandC-typetopological insulators: AZ classification (Altland & Zirnbaur,1997);

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The spin quantum Hall conductance

B3(y) ≡Zeeman field, jxz:=−σxys

dB3 dy

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Tight-binding model

h =

_∑

n,n0_∈_Γ

r

∑

l,l0₌₁

h(n,n0)l,l0|n,lihn0,l0| =

_∑

n,n0_∈Γ

|nih(n,n0)hn0|

•the “one-particle” Hilbert space isH :=`2(Γ)⊗_Cr_;

•Γis some lattice. Mainly interested onΓ'_Z2_; •r ∈_Ninternal degrees of freedom (spin+isospin);

•h(n,n0)is ar×r complex matrix for alln,n0∈Γ; Extra assumptions:

•self-adjointnessh=h†_{: this implies}_h₍_n_,_n0_{) =}_h₍_n0_,_n₎†_; •finite range(of sizeR): h(n,n0) =0 if|n−n0|>R;

•space homogeneity:hcan depend on “disorder”ω∈Ωin a “covariant” way

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Fock space representation and pairing potential

Thesecond quantization functoris defined by

|n,lihn0,l0| 7−→ c†_n_,_lcn0_,_l0 .

Herec†_n_,_l andc_n_,_l are thecreationandannihilationoperators of a “spin state”l on the sitenacting on the Fock space

F(H) :=

∞ M

N=0

AntisymH ⊗. . .⊗H

| {z }

N−times

.

Under this map one has h 7−→ h :=

_∑

n,n0_∈Γ

c†_nh(n,n0)c_n0 ≡ c†hc

withc†_n:= (c†_n_,₁, . . . ,c†_n_,_r)andc†:={c†_n}n∈Γ.

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The (fermionic)anticommutationrelations

{c†_n_,_l;cn0_,_l0} = _δ_n_,_n0_δ_l_,_l0 , {c†

n,l;c

†

n0_,_l0} = 0 = {cn,l;cn0_,_l0} provide

∑

n,n0_∈Γ

c†_nh(n,n0)c_n0 = −

_∑

n,n0_∈Γ

cnh(n0,n)t c†_n0 +C1F

whereC:=Tr_H[h] =∑n∈ΓTrCrh(n,n).

The self-adjoint conditionh(n0,n)t =h(n,n0₎_{allows to write} h = 1

2 c

†_h_c₋ 1 2chc

†

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BdG Hamiltonians and pair potentials

H := 1

2c

†_h_c₋1 2 chc

†

| {z }

=h

+ 1

2 c

†_∆_c†₋1 2c∆c

| {z }

pairing therm

.

More precisely

c†∆c† ≡

_∑

n,n0_∈Γ

c†_n∆(n,n0)c†_n0

where∆(n,n0)is ar×r complex matrix for alln,n0∈Γ. Assumptions:

•self-adjointnessH=H†_{: this implies}_∆(_n_,_n0_{) =}_−∆(_n0_,_n₎†_; •finite range(of sizeR): ∆(n,n0) =0 if|n−n0|>R;

•space homogeneity:∆can depend on “disorder”ω∈Ωin a “covariant” way

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Bogoliubov representation and PH-symmetry

Since[H;N]6=0 the BdG HamiltonianHdoes not preserves the particle number. HoweverHisquadraticand can be written in the Bogoliubov representation

H = 1

2(c †_c₎

h ∆

−∆ −h

| {z }

=:H

c c†

.

Thefirst quantizedBdG HamiltonianHacts on theparticle-hole Hilbert space

Hph := H ⊗C2ph (particle-hole fiber).

ThePH-symmetryis

K HK† = −H, K :=

0 1H

1H 0

.

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Outline

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Without disorderhand∆are chosentranslational invariant. SetΓ =Z2and consider onH =`2(Z2)⊗_Cr _the_{shift operator}

(Sjψ)(n) := ψ(n−ej) j=1,2.

along the canonical vectorej∈R2.

Thedynamical termh(without magnetic field) is given by h(µ) := S1 +S₁† + S2 + S₂† − µ1H (discrete laplacian) whereµ is thechemical potential. With magnetic field one can consider theHarper operator.

Thepair potential∆dictates the creation ofCooper pairs:

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d+ıd pair potential (spin quantum Hall effect)

∆dxy := (S1−S †

1)(S2−S † 2) ⊗ ıσ

2 _(singlet_d

xy-wave)

∆d_x2₋_y2 := (S1+S †

1−S2−S † 2) ⊗ ıσ

2 _(singlet_d

x2₋_y2-wave)

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Outline

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As a consequence ofK H K†₌₋_H _{one has:} Spectrum symmetry

H+z1Hph

−1

= −K†H−z1Hph

−1 K

implies

E∈Spec(H) ⇐⇒ −E∈Spec(H).

Symmetry of Fermi projection

Letf±:R→Rsuch thatf±(−x) =±f±(x), then

K f±(H)K† = ±f± H

.

Letf_β(x) := 1+e−βx−1

be theFermi-Dirac function, then K f_β(H)K† = 1Hph − fβ H

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Density of states and pseudo-gap

LetN be theintegrated density of states(IDOS) forH with normalization conditionN(0) =0, i.e. forE≥0

NH(E) = T χ[0,E](H)

, NH(−E) = −T χ[−E,0](H)

withT thetrace per unit volume(usual normalizationN(−∞) =0).

PH-symmetry =⇒ NH(−E) = −NH(E)

=⇒ NH(E) =

1 2NH2(E

2₎_.

IfNH andNH2 are absolutely continuous withDOSρandρh2i respectively, then one deduces

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Generic behavior for the DOS of BdG Hamiltonians

E ΡX2\

E Ρ

Eg

,_E g

E ΡX2\

E Ρ

0

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Non-generic behavior for the DOS of BdG Hamiltonians

E ΡX2\

E Ρ

Eg

,Eg

E ΡX2\

E Ρ

0

E ΡX2\

E Ρ

0

Schematic representation of the DOSρh2iand correspondingρ for various non-generic cases.

InD=2 theVan Hove singularitiesofρh2iare at most of order E−(1−ε) _with

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Outline

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U(1)andSU(2)invariance

Letσ1,σ2,σ3be anirreduciblerepresentation (hermitianand traceless) ofsu(2)onCr (spins= r−₂1).σ1,σ3=real;σ2= purely imaginary. The BdG second quantized is

σ σ σj := 1

2(c †_c₎

1`2⊗σj 0

0 −₁_`2⊗σj

c c†

.

LetHbe a second quantized BdG Hamiltonian associated to the the first quantized operatorH, then

[H;σσσj] = 0 ⇐⇒

H;

1`2⊗σj 0

0 −₁_`2⊗σj

= 0.

•U(1)- invariance:[H;σσσ3] = 0;

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THEOREM

LetHbe aU(1)- invariant(first quantized) BdG Hamiltonian, then H =          

h1 ∆1

. .. .·˙

hr ∆r

−∆1 −h1

.·˙ . ..

−∆r −hr

         

withhj†=hj and∆j†=−∆r−j+1bounded on`2(Γ).

LetHbeSU(2)- invariant, then

h1 = . . . = hr =: hred, ∆j = (−1)j+1∆red, withh_red†₌_h

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In the case ofSU(2)- invariance

H =



  

hred⊗1r ∆red⊗Yr

−∆_red⊗_Yr −hred⊗1r 

  

with₁r ther×r identity matrix and

Yr := 

  

1

−1

.·˙ (−1)r+1



  

.

•_Yr†=Yrt = (−1)r+1Yr;

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H ≡ H_red+ ⊕. . .⊕H_red+

| {z }

h_r 2

i

−times(+1)

⊕H_red− ⊕. . .⊕H_red−

| {z }

h_r 2 i

−times

whereH_red± are bounded operators on`2(Γ)⊗_C2 ph

H_red+ =

hred ∆red (−1)r ∆red −hred

, H_red− =

hred −∆red (−1)r+1∆red −hred

•H_red+ andH_red− areunitarily equivalent JH_red+ J† = H_red− , J :=

1`2 0

0 −1`2

.

•∆red†= (−1)r ∆redimplies

H_red± =

 



D-type if r odd

C-type if r even −→IHI†=−H, I2=−₁

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Time-reversal symmetry

The (first quantized) BdG HamiltonianHisTR-invariantif RhR† = h, R∆R† = ∆ R := 1`2⊗eıπ σ

2

.

Physical phenomena associated to BdG classes

TRS SU(2) U(1) AZ PP Effect Invariant 0 0 0 D ∆p±ıp TQHE Z

±1 √ √ CI ∆dxy,∆d_x2−y2 0

0 √ √ C ∆d±ıd SQHE Z Spatial dimensionD=2; spinhalf-integer(r even);

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Outline

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•Disorder: probability space(Ω,dP,τ)with:Ωcompact;τ a

Γ-action anddPτ-ergodic.

•Algebraic operations: onCc Ω×Γ

⊗Mat(2r,C)one defines

(A∗B)(ω,n) :=

_∑

m∈Γ

A(ω,m)B τ−m(ω),m−n

A∗(ω,n) := A τ−n(ω),−n†

n,m∈Γ, ω∈Ω.

•Representations: for allω∈Ωandψ∈Hph πω(A)ψ

(n) :=

_∑

m∈Γ

A τ−n(ω),m−nψ(m).

•C∗-norm:kAk:=sup_ω∈Ωkπω(A)kB(Hph).

•Observable algebra:k·k-closure ofCc Ω×Γ

⊗Mat(2r,C)

A = C(Ω)oτΓ

| {z } crossed product

⊗Mat(r,C)

| {z } spin & isospin

⊗Mat(2,C)

| {z } particle-hole

.

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Differential calculus forA

•Spatial derivations (gradient):∇:= (∇1, . . . ,∇d)defined by

(∇jA)(ω,n) := ı(n·ej)A(ω,n)

withe1, . . . ,ed the generators ofΓ. LetXj be the unbounded operator

(Xjψ)(n) := (n·ej)ψ(n), ψ∈H =`2(Γ)⊗Cr, then, onHph=H ⊗C2_ph

πω ∇jA

= ıπω(A);Xj⊗1ph

.

•Trace per unit volume (integration): it is the trace state onA defined by

T(A) :=

Z

Ω

dP(ω)Tr_C2r

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Outline

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Real structure

κ:A →A is ananti-linear automorphismdefined by κ(A) := K†AK , K :=

0 1H

1H 0

.

Observe thatκ2=Id (involution).

•Real algebra:Aℜ _:= _{_A_∈A _|

κ(A) =A}. In the particle-hole grading

A∈Aℜ _⇐⇒ _A ₌

a b b a

a,b∈ C(Ω)oτΓ⊗Mat(r,C).

•Imaginary part: the observable algebraA splits as

A = Aℜ _⊕Aℑ_, Aℑ _:= _ıAℜ_.

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•RealK-theory:KR0(A)is the set of homotopy classes of

P=P2=P†_in_A _⊗_Mat₍_∞_,_C₎_{such that}_κ₍_P_{) =}_P.

•Fermi projections: for a BdG Hamiltonians one has

κ(P) =1−P, κ(P) =1−P

which impliesT(P) =1₂ (half-dimensionality);

•Grothendieck group structure: inK0(A)one verifies

κ [1−P],[P]' [P],[1−P]≡”−” [1−P],[P].

•ImmaginaryK-theory !?: The set of([Q1],[Q2])∈K0(A)s.t. κ [Q1],[Q2]

= [Q2],[Q1]

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Outline

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A = C(Ω)oτΓ | {z }

crossed product

⊗Mat(r,C)

| {z }

spin & isospin

⊗Mat(2,C)

| {z }

particle-hole

.

•U(1)andSU(2)-invariant subalgebras:

AU(1) :=

n

A∈A |[A;1⊗σ3⊗J] =0 o

ASU(2) :=

n

A∈A |[A;1⊗σi⊗J] = [A;1⊗σ2⊗1ph] =0, i=1,3 o

whereJ= 1 0 0 −1

. One hasASU(2)⊂AU(1)⊂A. Moreover

Aℜ,ℑ

U(1) := AU(1)∩Aℜ,ℑ, Aℜ

,ℑ

SU(2) := ASU(2)∩Aℜ,ℑ.

•Spin derivation: onA_U₍₁₎∩A1_{one defines}

∇s_jA := ∇jA

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•Spin current: ifA∈A_U₍₁₎∩A1then πω ∇

s

jA

= ı[πω(A);Xj⊗1ph] (1⊗σ3⊗J).

IfH∈A_U₍₁₎∩A1_{is a BdG Hamiltonian, then} Jspin

j (ω) := πω ∇sjH

= ı[πω(H);Xj⊗J]

| {z }

ph-velocity

(1⊗σ3⊗1ph)

| {z }

spin component

.

~ Jspin

∝~v σ3is a current of spin for physicists.

•Equilibrium condition: letA∈A_SU₍₂₎andB∈A_SU₍₂₎∩A1_:

T A∗(∇s_jB) = 0. IfH∈A_SU₍₂₎∩A1_{is a BdG Hamiltonian and}

ρ:=f_β(H)is the Fermi-Dirac (equilibrium) state, then

hJ~spiniρ := T ρ∗(∇sH)

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Outline

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Linear response theory

We need to perturb the BdG HamiltonianH∈A_SU₍₂₎∩A1_with aZeeman fieldλ Pwith

P := X2 ⊗ σ3⊗J, λ ∝ ~B·zˆ.

•TheLiouville operatorsassociated toH andPare LH(A) := ı[A;H], LP(A) := ∇s₂A.

•The operatorLH+λL_P defines atime evolutiononAU(1). •We consider theT=0equilibrium stateP=χ(−∞,0](H). •The spin Hall conductance in the direction 1≡x for a Zeeman field depending on the direction 2≡y is given by

σ_xs_,_y := lim

δ→0 T

(∇s₂P) [δ+LH]−1(∇s1H)

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THEOREM (D. & Schulz-Baldes, 2013)

LetH∈A_SU₍₂₎∩A1_and_P₌

χ(−∞,0](H). Assume that:

(a) T |∇sP|2<+∞, (localization condition); (b) TheDOSofH has no atom in E=0.

Then theKubo-Chern formulaholds:

σ₁s_,₂ = ıT

P

∇s₁P;∇s₂P

= 1

8π 2πıT

P

∇1P;∇2P

| {z }

Chern number ofP

∈ 1

8πZ

Proof (sketch of).

H∈A_SU₍₂₎ ⇒P,∇_jP∈ASU(2)then

(∇s_iP)∗(∇s_jP) = (∇iP)∗(∇jP)∗(1⊗σ3⊗J)2=

1

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Outline

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d+ıd wave model in Fourier variables

Hµ,α(k1,k2) = 2

∑

i=1,2,3

fi(k1,k2)⊗Σi

with(k1,k2)∈[0,1)2'T2(Brillouin torus), f1(k1,k2) := cos(2πk1)−cos(2πk2) f2(k1,k2) := 2α sin(2πk1) sin(2πk2) f3(k1,k2) := cos(2πk1) +cos(2πk2) +µ2 and, introducingγ= 0ı −0ı

,

Σ1 := ı

0 γ

−γ 0

, Σ2 :=

0 γ γ 0

, Σ3 :=

12 0

0 −₁2

.

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Spectral analysis

•Dispersion relation: two energy bandsE± given by

E±(k1,k2) := ±r(k1,k2), r(k1,k2) := 3

∑

i=1

fi(k1,k2)2 !1₂

THEOREM

The d+ıd wave BdG HamiltonianHµ,αas a gap around zero if

and only ifµ6=0,±4and for allα∈R\ {0}.

•Polar variables: together withrone introduces two angles       

ϕ(k1,k2) := arctan

f2(k1,k2) f1(k1,k2)

06ϕ(k1,k2)<2π

θ(k1,k2) := arccos

f3(k1,k2)

r(k1,k2)

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•Eigenvector: thegap conditionr(k1,k2)6=0 allows to rewrite

Hµ,α(k) =r



  

cos[θ] 0 0 sin[θ]e−ıϕ 0 cos[θ] −sin[θ]e−ıϕ 0 0 −sin[θ]eıϕ −cos[θ] 0 sin[θ]eıϕ 0 0 −cos[θ]



  

.

The eigenvectors for thenegativeenergy band are

Ψ1(k) :=



  

sin₁ 2θ

e−ıϕ

0 0

−cos1₂θ 

  

, Ψ2(k) :=



  

0 sin₁

2θ

e−ıϕ

cos₁ 2θ 0     .

•Critical Points: the polar representation is singular in

N := nk ∈_T2|f1(k) =0=f2(k), f3(k)>0 o

(north pole)

S :=

n

k ∈_T2|f1(k) =0=f2(k), f3(k)<0 o

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Outline

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•The vector bundle: hermitian and rank-twoπ:E−→T2

E− :=

G

k∈T2

Ran[P−(k)] = G

k∈T2 n

v∈_C4| P−(k)v=v

o

whereP−(k)is the (globaly defined)negative energyprojection P−(k) := |Ψ₁(k)ihΨ₁(k)|+ |Ψ₂(k)ihΨ₂(k)|

•Local trivialization: inT2\S is given by the mapsΨ1(k)and

Ψ2(k). InT2\N we can use the “twisted” eigenvectors ˜

Ψ1(k) := eıϕ(k)Ψ1(k), Ψ˜2(k) := eıϕ(k)Ψ2(k).

•Transition functions: if bothN 6= /0andS 6= /0thenE− has

(non triviali) transition functions g_{N S}(k) :=

eıϕ(k) ₀ 0 eıϕ(k)

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Outline

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Chern invariant as a winding number

•Singular points for various regimes: for allα6=0 

 

 

N = k∗,k∗∗ , S = /0 if µ>4 (I)

N = /0, S = k∗,k∗∗ if µ<−4 (II)

N =

k∗ , S = k∗∗ if −4<µ<4 (III) withk∗= (0,0)andk∗∗= (1₂,1₂). The topology isnon-trivialonly

in the case III.

•Chern-Euler class: we need the followng facts: -[c1]is thetopChern class onT2;

- the top Chern class equals theEuler class, i.e.[c1] = [e];

-[c1](E−) =[c1](det(E−)), thedeterminate line-bundle;

- the Euler class for the line-bundledet(E−)is given by [e] = d

− 1 2πd 1 ılog h

det(g_{N S})

i

=d

−1

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•The Chern-number: the Stokes’ theorem provides

C1(E−) :=

Z

T2[e](E−) =

1 π

Z

Γ_N

dϕ

withΓ_N aregular closed pathswhich wrap in counterclockwise direction the setN. Then we can parametrizedΓ_N by

[0,2π]3t 7−→ ξε(t) :=

ε

2π cos(t),sin(t)

, ε>0. Sincedϕ=F1(k)dk1+F2(k)dk2one has

C1(E−) = 1

π Z 2π

0

F ξε(t)

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•Residue integral: one computesC1in the limitε→0, F ξε(t)

·ξ˙_ε(t) = 2α

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Phase diagram forC1in the d+ıd modelHµ,α

C1= 0

C₁= -4 C1= +4

C1= 0

C1= -4 C1= +4

Α

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