Complete Localization for Disordered Chiral Chains · Jacob Shapiro based on joint work with Gian Michele Graf Spectral Theory of Quasi-Periodic and Random Operators CRM Montreal

Complete Localization for Disordered Chiral Chains

Jacob Shapirobased on joint work with Gian Michele Graf

Spectral Theory of Quasi-Periodic and Random OperatorsCRM Montreal

November 14, 2018

J. Shapiro (Columbia U.) Complete Loc. for Disordered Chiral Chains November 14, 2018

Chiral Hamiltonians

Before explaining what is ”chiral”, reminder about the usual setting:

Usual Anderson model

Let µ be some probability distribution on N × N Hermitian matrices.Define an i.i.d. sequence of matrices {Vn}n∈Z where each term isdistributed with µ. Define a random Hamiltonian H on H := `2(Z)⊗ CN

by(Hψ)n := ψn+1 + ψn−1 + Vnψn


Chiral Hamiltonians




Kunz, Souillard ’80; Klein, Lacroix, Speis ’90; Aizenman & Warzel ’15:Localization for the entire spectrum.


Chiral Hamiltonians




Chiral model

Let α0, α1 be two probability distributions on GLN(C). Define anindependent alternatingly distributed sequence of matrices {Tn}n∈Z whereeach even / odd element is distributed with α0 / α1 respectively. Define arandom Hamiltonian H on H by

(Hψ)n := T ∗n+1ψn+1 + Tnψn−1


Chiral Hamiltonians

Chiral model


(Hψ)n := T ∗n+1ψn+1 + Tnψn−1

This is the disordered version of the Su-Schrieffer-Heeger model forpolyacetylene. Belongs to the 1D class AIII of the Kitaev table oftopological insulators. Multiscale: Chapman, Stolz (2015).

ψ1

ψ2

ψ3

ψ4

ψ5

ψ6

α0α1

T3

α0

T∗4

α1 α0


Chiral Hamiltonians

Chiral model


(Hψ)n := T ∗n+1ψn+1 + Tnψn−1

Consequences of chirality

Since {H,Π} = 0 with Π := (−1)X the chirality symmetry operator, X theposition operator, σ(H) is symmetric about zero.


Chiral Hamiltonians

Chiral model


(Hψ)n := T ∗n+1ψn+1 + Tnψn−1


Since {H,Π} = 0 with Π := (−1)X the chirality symmetry operator, X theposition operator, σ(H) is symmetric about zero.If Hψ = λψ then HΠψ = −ΠHψ = −Πλψ = −λΠψ.


Chiral Hamiltonians

Chiral model


(Hψ)n := T ∗n+1ψn+1 + Tnψn−1


Since {H,Π} = 0 with Π := (−1)X the chirality symmetry operator, X theposition operator, σ(H) is symmetric about zero.Zero will turn out to be a special point for localization: completelocalization on entire spectrum except possibly at zero energy. Cf. randompolymer model of Jitomirskaya, Schulz-Baldes and Stolz (2004).


Assumptions on αi , i = 0, 1

Assumption 1

suppαi contains an open subset of GLN(C).

Assumption 2

Fixing all hopping matrix elements except the real or imaginary part of anyone entry, the induced conditional probability measure µ on R is τ -Holdercontinuous: ∃τ ∈ (0, 1] : ∃C <∞ : ∀ interval J ⊆ R with |J| ≤ 1,µ(J) ≤ C |J|τ .

Assumtion 3

The second moments exist:∫T∈GLN(C)

‖T±1‖2 dαi (T ) <∞ .

Note: not assuming suppαi is bounded.



Assumption 1


Assumption 2


Assumtion 3


‖T±1‖2 dαi (T ) <∞ .




Assumption 1


Assumption 2


Assumtion 3


‖T±1‖2 dαi (T ) <∞ .




Assumption 1


Assumption 2


Assumtion 3


‖T±1‖2 dαi (T ) <∞ .

Note: not assuming suppαi is bounded.J. Shapiro (Columbia U.) Complete Loc. for Disordered Chiral Chains November 14, 2018

First result

Theorem (Graf, S.)

Under above assumptions on αi we have localization (via FMM) at allnon-zero energies: ∀E 6= 0, we have some s ∈ (0, 1),C <∞, µ > 0 :

supη>0

E[‖Gx ,y (E + i η)‖s ] ≤ C e−µ|x−y | (x , y ∈ Z)


Transfer matrices

Define the transfer matrix An(z) :=

[zT ◦n −Tn

T ◦n 0

]with T ◦ ≡ (T−1)∗. One

verifies that

(Hψ)n = T ∗n+1ψn+1 + Tnψn−1 = zψn ∀n ∈ Z

iff [T ∗n+1ψn+1

ψn

]=: Ψn = An(z)Ψn−1 .


Transfer matrices

Define the transfer matrix An(z) :=

[zT ◦n −Tn

T ◦n 0

]with T ◦ ≡ (T−1)∗. One

verifies that

(Hψ)n = T ∗n+1ψn+1 + Tnψn−1 = zψn ∀n ∈ Z

iff [T ∗n+1ψn+1

ψn

]=: Ψn = An(z)Ψn−1 .

Conclusion: understanding the behavior of singular values ofBn(z) := An(z) . . .A1(z) as n→∞ is important.


Transfer matrices

One further verifies that

An(z)∗JAn(z) = J ≡[

0 −1N

1N 0

](z ∈ C, n ∈ Z) .


Transfer matrices



0 −1N

1N 0

](z ∈ C, n ∈ Z) .

Hence for real z , An(z) belongs to the Hermitian symplectic groupSp∗2N(C) ≡ { A ∈ Mat2N(C) | A∗JA = J }, not to be confused with thecomplex symplectic group defined however with the transpose.


Transfer matrices



0 −1N

1N 0

](z ∈ C, n ∈ Z) .

Hence for real z , An(z) belongs to the Hermitian symplectic groupSp∗2N(C) ≡ { A ∈ Mat2N(C) | A∗JA = J }, not to be confused with thecomplex symplectic group defined however with the transpose.Conclusion: singular values of Bn(z) ≡ An(z) . . .A1(z) are symmetricabout one for z ∈ R.


Transfer matrices

Specialty at zero energy:

An+1(0)An(0) =

[0 −Tn+1

T ◦n+1 0

] [0 −Tn

T ◦n 0

]=

[−Tn+1T

◦n 0

0 −T ◦n+1Tn

].

That is, the even and odd sites are uncoupled, and one must understandnow sequences of invertible complex N × N matrices (rather thanHermitian symplectic 2N × 2N matrices) in each sub-sector separately.Within each sub-sector, there is no symmetry between the singular values,though of course the complete matrix is still Hermitian symplectic.


The Lyapunov exponents

Define the Lyapunov exponents as

γj(z) := limn→∞

1

nE[log(σj(An(z) . . .A1(z)))] ∀z ∈ C, j = 1, . . . , 2N

where σ1, . . . , σ2N are the 2N singular values of a matrix in descendingorder.





1


where σ1, . . . , σ2N are the 2N singular values of a matrix in descendingorder.By Assumption 3 these limits exist.





1


When z ∈ R the Hermitian symplectic property of An(z) implies that theLyapunov exponents become symmetric about zero, so only need toconsider the first N. γN(z) is called the localization length at energy z .





1


λ

γj(λ)

Red: even subsector, yellow: odd subsector

Previous resultNumber of negative red dots = topological index.





1


λ

γj(λ)







1


λ

γj(λ)




Main result

Definition

The system ”exhibits localization at zero energy” iff α0, α1 are such that

0 /∈ {γj(0)}2Nj=1 . (1)

Theorem (Graf, S.)

Under above assumptions on αi and (1), FMC localization extends to zeroenergy: we have some s ∈ (0, 1),C <∞, µ > 0 :

supη>0

E[‖Gx ,y (i η)‖s ] ≤ C e−µ|x−y | (x , y ∈ Z) (2)


Main result

Definition

The system ”exhibits localization at zero energy” iff α0, α1 are such that

0 /∈ {γj(0)}2Nj=1 . (1)

Theorem (Graf, S.)

Under above assumptions on αi and (1), FMC localization extends to zeroenergy: we have some s ∈ (0, 1),C <∞, µ > 0 :

supη>0

E[‖Gx ,y (i η)‖s ] ≤ C e−µ|x−y | (x , y ∈ Z) (2)


Important corollary

Corollary (Aizenman, Graf)

If the FMC at zero energy holds, then the Fermi projection (Fermi energymust be at zero to match the symmetry of the Hamiltonian) obeysalmost-surely: There is some µ (deterministic) such that ∀ε > 0 there issome (random) Cε <∞ with

‖χ(−∞,0)(H)x ,y‖ ≤ Cε e−µ|x−y |+ε|x | (x , y ∈ Z)

and zero is almost-surely not an eigenvalue of H.

This corollary is the (deterministic) input one needs in order to study thetopology of these chiral 1D systems, and has been used in our proof of thebulk-edge correspondence of such systems. It shows how the possiblefailure of localization at the special energy value zero leads to a richtopology; see Graf-Shapiro (2018).


Important corollary





This corollary is the (deterministic) input one needs in order to study thetopology of these chiral 1D systems, and has been used in our proof of thebulk-edge correspondence of such systems.

It shows how the possiblefailure of localization at the special energy value zero leads to a richtopology; see Graf-Shapiro (2018).


Important corollary





This corollary is the (deterministic) input one needs in order to study thetopology of these chiral 1D systems, and has been used in our proof of thebulk-edge correspondence of such systems. It shows how the possiblefailure of localization at the special energy value zero leads to a richtopology; see Graf-Shapiro (2018).


A Green’s function formula

Define the enlarged 2N × N Green’s function

Gx ,y (z) :=

[T ∗x+1Gx+1,y (z)

Gx ,y (z)

].

It also has a recursion relation Gx ,y (z) = Bx(z)G0,y (z) as long asy /∈ [1, x ].

In particular, Gx−1,x(z) = Bx−1(z)G0,x(z).Since we need Sp∗2N(C)-valued transfer matrices, z must be real, so we areforced to study the system at finite volume. Hence consider the Dirichlet

restriction H [1,x].By B.C. G[1,x]0,y (z) = 0 so that G[1,x]0,y (z) =

[T ∗1G

[1,x]1,y (z)

0N

]and

|G[1,x]x−1,x(z)|2 = |Bx−1(z)

[1N

0N

]T ∗1G

[1,x]1,x (z)|2

=⇒ |G [1,x]1,x (z)|2 ≤ ‖T−11 ‖

2tr(|Bx−1(z)

[1N

0N

]|−2)|G[1,x]x−1,x(z)|2




Gx ,y (z) :=

[T ∗x+1Gx+1,y (z)

Gx ,y (z)

].

It also has a recursion relation Gx ,y (z) = Bx(z)G0,y (z) as long asy /∈ [1, x ].In particular, Gx−1,x(z) = Bx−1(z)G0,x(z).

Since we need Sp∗2N(C)-valued transfer matrices, z must be real, so we areforced to study the system at finite volume. Hence consider the Dirichlet


[T ∗1G

[1,x]1,y (z)

0N

]and

|G[1,x]x−1,x(z)|2 = |Bx−1(z)

[1N

0N

]T ∗1G

[1,x]1,x (z)|2

=⇒ |G [1,x]1,x (z)|2 ≤ ‖T−11 ‖

2tr(|Bx−1(z)

[1N

0N

]|−2)|G[1,x]x−1,x(z)|2




Gx ,y (z) :=

[T ∗x+1Gx+1,y (z)

Gx ,y (z)

].

It also has a recursion relation Gx ,y (z) = Bx(z)G0,y (z) as long asy /∈ [1, x ].In particular, Gx−1,x(z) = Bx−1(z)G0,x(z).Since we need Sp∗2N(C)-valued transfer matrices, z must be real, so we areforced to study the system at finite volume. Hence consider the Dirichlet

restriction H [1,x].

By B.C. G[1,x]0,y (z) = 0 so that G[1,x]0,y (z) =

[T ∗1G

[1,x]1,y (z)

0N

]and

|G[1,x]x−1,x(z)|2 = |Bx−1(z)

[1N

0N

]T ∗1G

[1,x]1,x (z)|2

=⇒ |G [1,x]1,x (z)|2 ≤ ‖T−11 ‖

2tr(|Bx−1(z)

[1N

0N

]|−2)|G[1,x]x−1,x(z)|2




Gx ,y (z) :=

[T ∗x+1Gx+1,y (z)

Gx ,y (z)

].



[T ∗1G

[1,x]1,y (z)

0N

]

and

|G[1,x]x−1,x(z)|2 = |Bx−1(z)

[1N

0N

]T ∗1G

[1,x]1,x (z)|2

=⇒ |G [1,x]1,x (z)|2 ≤ ‖T−11 ‖

2tr(|Bx−1(z)

[1N

0N

]|−2)|G[1,x]x−1,x(z)|2




Gx ,y (z) :=

[T ∗x+1Gx+1,y (z)

Gx ,y (z)

].



[T ∗1G

[1,x]1,y (z)

0N

]and

|G[1,x]x−1,x(z)|2 = |Bx−1(z)

[1N

0N

]T ∗1G

[1,x]1,x (z)|2

=⇒ |G [1,x]1,x (z)|2 ≤ ‖T−11 ‖

2tr(|Bx−1(z)

[1N

0N

]|−2)|G[1,x]x−1,x(z)|2


A Green’s function formula (cont.)

Using a lemma in Lacroix 1985 we find that

tr(|Bx−1(z)

[1N

0N

]|−2) ≤

N∑j=1

‖ ∧N−1 Bx−1(z)uj‖2

‖ ∧N Bx−1(z)u‖2

with { ej }2Nj=1 the std. basis of R2N , u := e1 ∧ · · · ∧ eN ,uj = e1 ∧ · · · ∧ ej ∧ · · · ∧ eN . Conclusion:

E[‖G [1,x]1,x (z)‖s ] ≤∼ E[‖T−11 ‖

3s ]1/3E[‖G[1,x]x−1,x(z)‖3s ]1/3 e−γN(z)|x |


Roadmap

1 Prove an a-priori bound for all z : supη>0 E[‖Gx,x+1(E + i η)‖s ] ≤ C <∞. Impliesthe same for x , x matrix elements for all E 6= 0.

2 Prove that the the semigroup generated by the support of the distribution oftransfer matrices at non-zero real energies contains an open subset of Sp∗

2N(C).Furstenberg then says that 0 /∈ {γj(z)}2Nj=1 for all z ∈ R \ {0}.

3 Conclude localization at finite-volume non-zero energy.

4 Prove a Combes-Thomas estimate for unbounded hopping (this is our setting here,will only hold in expectation).

5 Extend off the real axis via the subharmonicity of z 7→ E[‖Gx,y (z)‖s ] to getpolynomial decay off the real axis.

6 For this model, a decoupling estimate and a Fekete-type theorem says that anyoff-diagonal decay implies exponential decay.

7 Get exp. off-diagonal decay off the real axis, and hence for infinite volume.

8 Solve the model explicitly at zero energy (recall it is uncoupled) to get that ana-priori bound for x , x matrix element at zero energy too, assuming 0 /∈ {γj(0)}2Nj=1.This then implies FMC at zero energy using the above.


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Furstenberg

Lemma (Graf, S.)

If the support of α0, α1 contains an open subset of GLN(C) then thesemigroup generated by the support of the transfer matrices atE ∈ R \ {0} contains an open subset of Sp∗2N(C).


Furstenberg

Lemma (Graf, S.)


Proof

Idea: Show that the following map has a continuous local inverse

suppα1 × suppα0 × suppα1 3 (T1,T2,T3) 7→

7→[ET ◦1 −T1

T ◦1 0

] [ET ◦2 −T2

T ◦2 0

] [ET ◦3 −T3

T ◦3 0

]∈ Sp∗2N(C) .


Furstenberg

Lemma (Graf, S.)


Proof

Use the following chart of Sp∗2N(C):

GLN(C)× HermN(C)2 3 (D,R, S) 7→[? D◦R

SD◦ D

]∈ Sp∗2N(C)

Restrict to open subset defined by: |D|2 > 1zR and

(z2 − zD◦RD−1)−1 > 1z S .

Only makes sense when z 6= 0!


Furstenberg

Lemma (Graf, S.)


Corollary: By Furstenberg’s theorem (Bougerol ’85, Goldscheid, Margulis’89) it follows that the Lyapunov spectrum is simple, and henceγN(E ) > 0!


A-priori bound

It turns out that because the randomness is only in the hopping, one canonly hope to get an a-priori bound on E[‖Gx ,x−1(z)‖s ].

By finite-rankperturbation theory, we find that for i , j = 1, . . .N,

Gx ,x−1(z)ij = ((M +

[0 ¯(Tx)ij

(Tx)ij 0

]︸︷︷︸

=:Q

)−1)21

where M is some 2× 2 matrix that does not depend on the randomvariable (Tx)ij . Expanding any 2× 2 matrix in the Pauli basis {σj}3j=0 tocontrol the 2, 1 entry it is sufficient to control only the σ1 and σ2components. If we wiggle only Re (Tx)ij =: λ then the σ1 component ofQ−1 is of the form

−q1 − λq20 − q22 − q23 − (q1 + λ)2

for some q ∈ C4 which is independent of λ.


A-priori bound

It turns out that because the randomness is only in the hopping, one canonly hope to get an a-priori bound on E[‖Gx ,x−1(z)‖s ]. By finite-rankperturbation theory, we find that for i , j = 1, . . .N,

Gx ,x−1(z)ij = ((M +

[0 ¯(Tx)ij

(Tx)ij 0

]︸︷︷︸

=:Q

)−1)21

where M is some 2× 2 matrix that does not depend on the randomvariable (Tx)ij . Expanding any 2× 2 matrix in the Pauli basis {σj}3j=0 tocontrol the 2, 1 entry it is sufficient to control only the σ1 and σ2components.

If we wiggle only Re (Tx)ij =: λ then the σ1 component ofQ−1 is of the form

−q1 − λq20 − q22 − q23 − (q1 + λ)2



A-priori bound

It turns out that because the randomness is only in the hopping, one canonly hope to get an a-priori bound on E[‖Gx ,x−1(z)‖s ]. By finite-rankperturbation theory, we find that for i , j = 1, . . .N,

Gx ,x−1(z)ij = ((M +

[0 ¯(Tx)ij

(Tx)ij 0

]︸︷︷︸

=:Q

)−1)21

where M is some 2× 2 matrix that does not depend on the randomvariable (Tx)ij . Expanding any 2× 2 matrix in the Pauli basis {σj}3j=0 tocontrol the 2, 1 entry it is sufficient to control only the σ1 and σ2components. If we wiggle only Re (Tx)ij =: λ then the σ1 component ofQ−1 is of the form

−q1 − λq20 − q22 − q23 − (q1 + λ)2



A-priori bound (cont.)

Proposition (Graf, S.)

Consider the subset of the complex plane

Dc,t := {z ∈ C|| z

z2 − c2| > t}

for c ∈ C and t > 0 and the line R 3 λ 7→ z := αλ+ β (α, β ∈ C,|α| = 1) parametrized by arclength. Then its intersection with Dc,t isbounded in Lebesgue measure as

|{λ ∈ R|z ∈ Dc,t}| ≤4

t.


Open questions

1 Calculate almost-sure spectrum, not as simple asσ(H) = σ(−∆) + suppµ because the randomness is not additive!

2 Prove converse of main theorem: If χ(−∞,0)(H) is exp. decaying

off-diagonal matrix elements then 0 /∈ {γj(0)}2Nj=1.


Open questions

1 Calculate almost-sure spectrum, not as simple asσ(H) = σ(−∆) + suppµ because the randomness is not additive!

2 Prove converse of main theorem: If χ(−∞,0)(H) is exp. decaying

off-diagonal matrix elements then 0 /∈ {γj(0)}2Nj=1.


Documents

Complete Localization for Disordered Chiral Chains · Jacob Shapiro based on joint work with Gian Michele Graf Spectral Theory of Quasi-Periodic and Random Operators CRM Montreal