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Strongly Disordered Floquet Topological Systems Jacob Shapiro based on joint work with Cl ement Tauber arXiv:1807.03251 ETH Zurich Recent progress in mathematics of topological insulators September 4, 2018 J. Shapiro (ETH Zurich)

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Strongly Disordered Floquet Topological Systems

Jacob Shapiro

based on joint work with Cl´ ement Tauber arXiv:1807.03251 ETH Zurich Recent progress in mathematics of topological insulators

September 4, 2018

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 1 / 14

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Floquet systems

Periodically time-dep. Hamiltonian H : S1 → B(H) induces a unitary map U : [0, 1] → U(H) via the Schr¨

  • dinger equation i ˙

U = HU with U(0) ≡ 1.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 2 / 14

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Floquet systems

Periodically time-dep. Hamiltonian H : S1 → B(H) induces a unitary map U : [0, 1] → U(H) via the Schr¨

  • dinger equation i ˙

U = HU with U(0) ≡ 1. Models non-int. electrons subject to driving beyond adiabatic regime.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 2 / 14

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Floquet systems

Periodically time-dep. Hamiltonian H : S1 → B(H) induces a unitary map U : [0, 1] → U(H) via the Schr¨

  • dinger equation i ˙

U = HU with U(0) ≡ 1. Models non-int. electrons subject to driving beyond adiabatic regime. Long time dynamics of the system determined by U(1) because U(n + t) = U(1)nU(t) for t ∈ (0, 1), n ∈ N.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 2 / 14

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Floquet systems

Periodically time-dep. Hamiltonian H : S1 → B(H) induces a unitary map U : [0, 1] → U(H) via the Schr¨

  • dinger equation i ˙

U = HU with U(0) ≡ 1. Models non-int. electrons subject to driving beyond adiabatic regime. Long time dynamics of the system determined by U(1) because U(n + t) = U(1)nU(t) for t ∈ (0, 1), n ∈ N. Main object however is U, not H, and all the questions (such as existence of a gap) are asked w.r.t. U(1).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 2 / 14

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Simple example in zero dimensions

In zero dimensions, H = CN (atom with N internal levels); get a

  • cont. map U : [0, 1] → U(N).
  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 3 / 14

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Simple example in zero dimensions

In zero dimensions, H = CN (atom with N internal levels); get a

  • cont. map U : [0, 1] → U(N).

Cannot use the winding number of det U since U is not a loop!

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 3 / 14

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Simple example in zero dimensions

In zero dimensions, H = CN (atom with N internal levels); get a

  • cont. map U : [0, 1] → U(N).

Cannot use the winding number of det U since U is not a loop! Relative construction: straight line to next integer value below; get loop on the circle in whose winding may be computed. 1 2π 4π 6π − i log ◦U

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 3 / 14

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Simple example in zero dimensions

In zero dimensions, H = CN (atom with N internal levels); get a

  • cont. map U : [0, 1] → U(N).

Cannot use the winding number of det U since U is not a loop! Relative construction: straight line to next integer value below; get loop on the circle in whose winding may be computed.

1 2

1 2π 4π 6π − i log ◦U

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 3 / 14

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Higher dimensions

In d > 1, H = ℓ2(Zd) ⊗ CN with N the internal levels; We ask that H : S1 → B(H) be piecewise continuous in time and local in the sense that δx, H(t)δy is exp. decaying in x − y (uniformly in t ∈ S1).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 4 / 14

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Higher dimensions

In d > 1, H = ℓ2(Zd) ⊗ CN with N the internal levels; We ask that H : S1 → B(H) be piecewise continuous in time and local in the sense that δx, H(t)δy is exp. decaying in x − y (uniformly in t ∈ S1). This implies the locality of U : [0, 1] → U(H), and also of the loop Urel : S1 → U(H) obtained via the relative construction as before,

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 4 / 14

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Higher dimensions

In d > 1, H = ℓ2(Zd) ⊗ CN with N the internal levels; We ask that H : S1 → B(H) be piecewise continuous in time and local in the sense that δx, H(t)δy is exp. decaying in x − y (uniformly in t ∈ S1). This implies the locality of U : [0, 1] → U(H), and also of the loop Urel : S1 → U(H) obtained via the relative construction as before, if ∃ spectral gap, i.e. S1 \ σ(U(1)) = ∅, where we pick a branch cut for the logarithm, which in turn makes it local (Combes-Thomas). branch cut σ(U(1))

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 4 / 14

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Higher dimensions

In d > 1, H = ℓ2(Zd) ⊗ CN with N the internal levels; We ask that H : S1 → B(H) be piecewise continuous in time and local in the sense that δx, H(t)δy is exp. decaying in x − y (uniformly in t ∈ S1). This implies the locality of U : [0, 1] → U(H), and also of the loop Urel : S1 → U(H) obtained via the relative construction as before, if ∃ spectral gap, i.e. S1 \ σ(U(1)) = ∅, where we pick a branch cut for the logarithm, which in turn makes it local (Combes-Thomas). branch cut σ(U(1)) Topology depends on choice of gap, but not on branch within it! In IQHE Chern ♯ also depends on choice of gap.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 4 / 14

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Higher dimensions

In d > 1, H = ℓ2(Zd) ⊗ CN with N the internal levels; We ask that H : S1 → B(H) be piecewise continuous in time and local in the sense that δx, H(t)δy is exp. decaying in x − y (uniformly in t ∈ S1). This implies the locality of U : [0, 1] → U(H), and also of the loop Urel : S1 → U(H) obtained via the relative construction as before, if ∃ spectral gap, i.e. S1 \ σ(U(1)) = ∅, where we pick a branch cut for the logarithm, which in turn makes it local (Combes-Thomas). branch cut σ(U(1)) Topology depends on choice of gap, but not on branch within it! In IQHE Chern ♯ also depends on choice of gap. Gap condition is not related to insulator property (unlike static case)!

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 4 / 14

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Higher dimensions (cont.)

In transl. invar. case we get a cont. loop Urel : S1 × Td → U(N) based at 1, i.e. an element in suspension of C-star algebra C(Td). Hence such unitary loops are classified by K1(SC(Td)) ∼ = K0(C(Td)); get same classification as static top. insulators of class A in d dim.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 5 / 14

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Higher dimensions (cont.)

In transl. invar. case we get a cont. loop Urel : S1 × Td → U(N) based at 1, i.e. an element in suspension of C-star algebra C(Td). Hence such unitary loops are classified by K1(SC(Td)) ∼ = K0(C(Td)); get same classification as static top. insulators of class A in d dim. Hence get for the strong invariants: Dimension 1 2 3 4 . . . Invariant Z Z . . . which has Bott periodicity of two in d, like class A row in Kitaev table.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 5 / 14

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Higher dimensions (cont.)

In transl. invar. case we get a cont. loop Urel : S1 × Td → U(N) based at 1, i.e. an element in suspension of C-star algebra C(Td). Hence such unitary loops are classified by K1(SC(Td)) ∼ = K0(C(Td)); get same classification as static top. insulators of class A in d dim. Hence get for the strong invariants: Dimension 1 2 3 4 . . . Invariant Z Z . . . which has Bott periodicity of two in d, like class A row in Kitaev table. Can consider also other symmetry classes, but need to decide how symmetry operations should interact with time variable. Can Get analogous periodic table (see Roy, Harper (2017)).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 5 / 14

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Higher dimensions (cont.)

In transl. invar. case we get a cont. loop Urel : S1 × Td → U(N) based at 1, i.e. an element in suspension of C-star algebra C(Td). Hence such unitary loops are classified by K1(SC(Td)) ∼ = K0(C(Td)); get same classification as static top. insulators of class A in d dim. Hence get for the strong invariants: Dimension 1 2 3 4 . . . Invariant Z Z . . . which has Bott periodicity of two in d, like class A row in Kitaev table. Can consider also other symmetry classes, but need to decide how symmetry operations should interact with time variable. Can Get analogous periodic table (see Roy, Harper (2017)). As in static case, ∃ bulk picture (on H ≡ ℓ2(Zd) ⊗ CN) and edge picture on half-space HE := ℓ2(Zd−1 × N) ⊗ CN obtained by truncating a given bulk Hamiltonian with some B.C. (truncation always on H, not U!).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 5 / 14

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What we studied and previous results

We study the 2D no-symmetries case in the bulk and on the edge. The input is a bulk H : S1 → B(H) (piecewise) cont. in time and local in space. It induces a bulk evolution U : [0, 1] → U(H) via Sch¨

  • dinger, an edge Hamiltonian HE : S1 → HE (via truncation

to half-space with Dirichlet) and an edge evolution UE : [0, 1] → U(HE) via Schr¨

  • dinger

from HE.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 6 / 14

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What we studied and previous results

We study the 2D no-symmetries case in the bulk and on the edge. The input is a bulk H : S1 → B(H) (piecewise) cont. in time and local in space. It induces a bulk evolution U : [0, 1] → U(H) via Sch¨

  • dinger, an edge Hamiltonian HE : S1 → HE (via truncation

to half-space with Dirichlet) and an edge evolution UE : [0, 1] → U(HE) via Schr¨

  • dinger

from HE.

Previous studies Physics: Rudner, Lindner, et al (2013) Math: Schulz-Baldes, Sadel (2017) and Graf, Tauber (2018)

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 6 / 14

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What we studied and previous results

We study the 2D no-symmetries case in the bulk and on the edge. The input is a bulk H : S1 → B(H) (piecewise) cont. in time and local in space. It induces a bulk evolution U : [0, 1] → U(H) via Sch¨

  • dinger, an edge Hamiltonian HE : S1 → HE (via truncation

to half-space with Dirichlet) and an edge evolution UE : [0, 1] → U(HE) via Schr¨

  • dinger

from HE.

Previous studies Physics: Rudner, Lindner, et al (2013) Math: Schulz-Baldes, Sadel (2017) and Graf, Tauber (2018)

K-theoretic classification says this case should have a Z strong invariant.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 6 / 14

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What we studied and previous results

We study the 2D no-symmetries case in the bulk and on the edge. The input is a bulk H : S1 → B(H) (piecewise) cont. in time and local in space. It induces a bulk evolution U : [0, 1] → U(H) via Sch¨

  • dinger, an edge Hamiltonian HE : S1 → HE (via truncation

to half-space with Dirichlet) and an edge evolution UE : [0, 1] → U(HE) via Schr¨

  • dinger

from HE.

Previous studies Physics: Rudner, Lindner, et al (2013) Math: Schulz-Baldes, Sadel (2017) and Graf, Tauber (2018)

K-theoretic classification says this case should have a Z strong invariant. Previous studies assume a spectral gap for U(1) which allows one to take a log(U(1)) which is local, then Urel : S1 → B(H) is U concat. with static e· log(U(1)). Bulk invariant is 3D winding of the loop Urel.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 6 / 14

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SLIDE 23

What we studied and previous results

We study the 2D no-symmetries case in the bulk and on the edge. The input is a bulk H : S1 → B(H) (piecewise) cont. in time and local in space. It induces a bulk evolution U : [0, 1] → U(H) via Sch¨

  • dinger, an edge Hamiltonian HE : S1 → HE (via truncation

to half-space with Dirichlet) and an edge evolution UE : [0, 1] → U(HE) via Schr¨

  • dinger

from HE.

Previous studies Physics: Rudner, Lindner, et al (2013) Math: Schulz-Baldes, Sadel (2017) and Graf, Tauber (2018)

K-theoretic classification says this case should have a Z strong invariant. Previous studies assume a spectral gap for U(1) which allows one to take a log(U(1)) which is local, then Urel : S1 → B(H) is U concat. with static e· log(U(1)). Bulk invariant is 3D winding of the loop Urel. Define Hrel

E

as the concatenation of HE and the truncation of − i log(U(1)). Induces evol. Urel

E

: [0, 1] → U(HE) (not a loop). Edge invar. is charge pumped along 1 direction after one period of Urel

E : depends only on endpoint Urel E (1)!

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 6 / 14

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Our results

1st result: mobility gap Relax the set-theoretic spectral gap assumption with an estimate from dynamical localization.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 7 / 14

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Our results

1st result: mobility gap Relax the set-theoretic spectral gap assumption with an estimate from dynamical localization. 2nd result: stretch function New formulation the bulk and edge invariants in a new way that avoids the relative construction.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 7 / 14

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Our results

1st result: mobility gap Relax the set-theoretic spectral gap assumption with an estimate from dynamical localization. 2nd result: stretch function New formulation the bulk and edge invariants in a new way that avoids the relative construction. 3rd result: magnetization and time-averaged charge pumping Investigate the physical meaning of the invariants in completely localized case.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 7 / 14

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Our results

1st result: mobility gap Relax the set-theoretic spectral gap assumption with an estimate from dynamical localization. 2nd result: stretch function New formulation the bulk and edge invariants in a new way that avoids the relative construction. 3rd result: magnetization and time-averaged charge pumping Investigate the physical meaning of the invariants in completely localized case. 4th result: equality All invariants are equal, including bulk-edge correspondence. Uses continuity argument.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 7 / 14

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The mobility gap regime

Via Combes-Thomas, S1 = σ(U(1)) implies that h(U(1))xy decays in x − y for h holomorphic. This off-diagonal decay is apparently all we need for a well-defined topological phase.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 8 / 14

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The mobility gap regime

Via Combes-Thomas, S1 = σ(U(1)) implies that h(U(1))xy decays in x − y for h holomorphic. This off-diagonal decay is apparently all we need for a well-defined topological phase. Hamza, Joye, Stolz (2009) e.g. prove that certain random unitary ops. have dyn. loc. We assume the a.-s. results of loc. deterministically, i.e. we assume that ∃µ > 0 s.t. for any ε > 0 ∃Cε < ∞ with sup

g∈B1(∆)

g(U(1))xy ≤ Cε e−µx−y+εx with B1(∆) the set of Borel bdd. maps |g| ≤ 1 constant outside of ∆ ⊆ S1, which is called the mobility gap. Implies spectral localization in ∆ via RAGE.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 8 / 14

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The mobility gap regime (cont.)

∆

  • Spec. gap

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∆ Mobility gap σ(U(1)) = S1 No gap σ(U(1)) = S1

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 9 / 14

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The mobility gap regime (cont.)

∆

  • Spec. gap

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∆ Mobility gap σ(U(1)) = S1 No gap σ(U(1)) = S1 Theorem If ∆ is a mob. gap for U(1), placing the branch cut of the logarithm in ∆, the relative construction still goes through, as well as its bulk- edge correspondence proof.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 9 / 14

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The mobility gap regime (cont.)

∆

  • Spec. gap

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∆ Mobility gap σ(U(1)) = S1 No gap σ(U(1)) = S1 Theorem If ∆ is a mob. gap for U(1), placing the branch cut of the logarithm in ∆, the relative construction still goes through, as well as its bulk- edge correspondence proof.

Main point over [GT18]: Use loc. instead of Combes-Thomas to get (weak) locality of log(U(1)); then generalize all notions from uniform decay in x − y to allow possible explosion in x simultaneously, which we call weakly-local operators: Axy ≤ Cε e−µx−y+εx.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 9 / 14

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Problems with the relative construction

Not clear what the invariant W (Urel) (3D non-comm. winding) measures in an experiment or how to implement it:

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 10 / 14

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Problems with the relative construction

Not clear what the invariant W (Urel) (3D non-comm. winding) measures in an experiment or how to implement it: W (Urel) ≡ −1 2

  • S1 tr

˙ Urel(Urel)∗[Urel

,1 (Urel)∗, Urel ,2 (Urel)∗]

where A,i ≡ i[Λi, A] with Λi a switch function. We have W (Urel) = W (U) − W (e· logλ(U(1))), so that some winding of e· logλ(U(1)) is removed, but what does it mean physically? (non-top. transport contributions?)

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 10 / 14

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Problems with the relative construction

Not clear what the invariant W (Urel) (3D non-comm. winding) measures in an experiment or how to implement it: W (Urel) ≡ −1 2

  • S1 tr

˙ Urel(Urel)∗[Urel

,1 (Urel)∗, Urel ,2 (Urel)∗]

where A,i ≡ i[Λi, A] with Λi a switch function. We have W (Urel) = W (U) − W (e· logλ(U(1))), so that some winding of e· logλ(U(1)) is removed, but what does it mean physically? (non-top. transport contributions?) Edge invariant contains significant information from the bulk, namely, it depends on Urel

E

which is the evolution of Hrel

E , which is the

concatenation of HE and the truncation of − i log(U(1)). The latter is a bulk object. Want bulk-edge correspondence where bulk and edge invariants depend on H and HE alone, without intertwining their evolutions during the proof.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 10 / 14

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The stretch function construction

The stretch function (used by Sadel, Schulz-Baldes (2017) only for the edge in spec. gap case) smooth map F∆ : C \ { 0 } → C; restricted to S1: constant 1 outside ∆, has winding number 1.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 11 / 14

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The stretch function construction

The stretch function (used by Sadel, Schulz-Baldes (2017) only for the edge in spec. gap case) smooth map F∆ : C \ { 0 } → C; restricted to S1: constant 1 outside ∆, has winding number 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ(U(1)) ∆ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∆ σ(F∆(U(1)))

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 11 / 14

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SLIDE 38

The stretch function construction

The stretch function (used by Sadel, Schulz-Baldes (2017) only for the edge in spec. gap case) smooth map F∆ : C \ { 0 } → C; restricted to S1: constant 1 outside ∆, has winding number 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ(U(1)) ∆ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∆ σ(F∆(U(1))) F∆(U(1)) is dynamically localized on S1 \ { 1 }.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 11 / 14

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SLIDE 39

The stretch function construction

The stretch function (used by Sadel, Schulz-Baldes (2017) only for the edge in spec. gap case) smooth map F∆ : C \ { 0 } → C; restricted to S1: constant 1 outside ∆, has winding number 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ(U(1)) ∆ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∆ σ(F∆(U(1))) F∆(U(1)) is dynamically localized on S1 \ { 1 }. Idea: If we can understand the situation for completely localized

  • perators then we could work with F∆ ◦ U and F∆ ◦ UE for bulk and

edge respectively. The application of F∆ on UE uses no information from the bulk except the position of the chosen gap!

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 11 / 14

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SLIDE 40

The stretch function construction

The stretch function (used by Sadel, Schulz-Baldes (2017) only for the edge in spec. gap case) smooth map F∆ : C \ { 0 } → C; restricted to S1: constant 1 outside ∆, has winding number 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ(U(1)) ∆ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∆ σ(F∆(U(1))) F∆(U(1)) is dynamically localized on S1 \ { 1 }. Idea: If we can understand the situation for completely localized

  • perators then we could work with F∆ ◦ U and F∆ ◦ UE for bulk and

edge respectively. The application of F∆ on UE uses no information from the bulk except the position of the chosen gap! F∆ chooses the gap for Floquet just like χ(−∞,EF ) chooses the gap for the IQHE, so F∆ is like the Floquet’s Fermi projection.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 11 / 14

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The completely localization case

Let V : [0, 1] → U(H) be some bulk evolution s.t. V (1) is completely localized, in the sense that it obeys a det. dyn. loc. estimate on S1 except some finitely many special points; we ask that the Chern ♯

  • assoc. to each such point vanish.

♠

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 12 / 14

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SLIDE 42

The completely localization case

Let V : [0, 1] → U(H) be some bulk evolution s.t. V (1) is completely localized, in the sense that it obeys a det. dyn. loc. estimate on S1 except some finitely many special points; we ask that the Chern ♯

  • assoc. to each such point vanish.

Define the bulk magnetization operator M(V ) :=

  • [0,1] I♠ V ∗Λ1 i ˙

V V ∗Λ2V and the total (orbital) magnetization M(V ) :=

  • z∈S1 tr M(V ) d P(z) with P the proj.

valued spectral measure of V (1). Related to magnetization studied by Rudner, Lindner et al (2017). If Λi ∼ xi then like orbital angular momentum 1

2r(t) × ˙

r(t).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 12 / 14

slide-43
SLIDE 43

The completely localization case

Let V : [0, 1] → U(H) be some bulk evolution s.t. V (1) is completely localized, in the sense that it obeys a det. dyn. loc. estimate on S1 except some finitely many special points; we ask that the Chern ♯

  • assoc. to each such point vanish.

Define the bulk magnetization operator M(V ) :=

  • [0,1] I♠ V ∗Λ1 i ˙

V V ∗Λ2V and the total (orbital) magnetization M(V ) :=

  • z∈S1 tr M(V ) d P(z) with P the proj.

valued spectral measure of V (1). Related to magnetization studied by Rudner, Lindner et al (2017). If Λi ∼ xi then like orbital angular momentum 1

2r(t) × ˙

r(t). Define the edge time-avg. charge pumping assoc. to VE(1), the evolution of the truncated Hamiltonian assoc. to V : PE(VE(1)) := limn→∞ limr→∞ 1

n tr(VE(1)∗)n[Λ1, VE(1)n]Λ⊥ 2,r where

Λ⊥

2,r restricts to a vertical band from zero to r.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 12 / 14

slide-44
SLIDE 44

Connecting everything

Theorem If U : [0, 1] → U(H) is s.t. U(1) is completely loc. as above, then M(U) = W (Urel).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 13 / 14

slide-45
SLIDE 45

Connecting everything

Theorem If U : [0, 1] → U(H) is s.t. U(1) is completely loc. as above, then M(U) = W (Urel). Theorem If U : [0, 1] → U(H) is s.t. U(1) is completely loc. as above, then PE(UE(1)) = M(U).

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 13 / 14

slide-46
SLIDE 46

Connecting everything

Theorem If U : [0, 1] → U(H) is s.t. U(1) is completely loc. as above, then M(U) = W (Urel). Theorem If U : [0, 1] → U(H) is s.t. U(1) is completely loc. as above, then PE(UE(1)) = M(U). Theorem If U : [0, 1] → U(H) has a mobility gap at ∆, and Urel : S1 → U(H) is the rel. construction w.r.t. a cut in ∆ then W (Urel) = W ((F∆ ◦ U)rel) = M(F∆ ◦ U) = PE(F∆(UE(1))) .

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 13 / 14

slide-47
SLIDE 47

Idea for proof

We start with W (Urel) = W (U) − W (e· logλ(U(1))) (δα := − i U∗U,α) = 1 2 tr

  • [0,1]

εαβ(δα ˙ δβ − δλ

α ˙

δλ

β)

(U,α ≡ i[Λα, U] ∧ δα(t) = δλ

α(t)∀t ∈ { 0, 1 })

= tr M(U) − M(e· logλ(U(1))) Now use localization to prove (the regularized) trace of M(e· log(U(1))) is finite and actually zero.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 14 / 14

slide-48
SLIDE 48

Idea for proof

We start with W (Urel) = W (U) − W (e· logλ(U(1))) (δα := − i U∗U,α) = 1 2 tr

  • [0,1]

εαβ(δα ˙ δβ − δλ

α ˙

δλ

β)

(U,α ≡ i[Λα, U] ∧ δα(t) = δλ

α(t)∀t ∈ { 0, 1 })

= tr M(U) − M(e· logλ(U(1))) Now use localization to prove (the regularized) trace of M(e· log(U(1))) is finite and actually zero. For W (Urel) = W ((F∆ ◦ U)rel) we use continuity of W under interpolation from the smooth F∆ to the identity map, in the mobility gap regime.

  • J. Shapiro (ETH Zurich)

Disordered Floquet Topological Systems September 4, 2018 14 / 14