Himeji Conference on PDEs 2023
04/03/2023 (Sat)A topological index for one-dimensional quantum walks with asymptotically periodic
parameters
Yohei Tanaka (Shinshu University, JSPS Research Fellow PD)△ Please feel free to download
my slides with...............................
!

Motivation§1

Motivationp.1△ What exactly is a ...............................................
(discrete-time) quantum walk??
!

▶ Let ℓ 2 (Z, C2 ) be the Hilbert space of square-summable C2 -valued sequences.
▶ A (discrete-time) 2-state quantum walk on the integer lattice Z is characterised by aunitary time-evolution operator 𝑈 on ℓ 2 (Z, C2).
▶▶

Suppose that a (quantum) walker has an initial state Ψ0 = (Ψ0 (𝑥))𝑥∈Z in ℓ 2 (Z, C2),
where we assume ∥Ψ0 ∥ ℓ2 (Z,C2) = 1.

▶▶

Then the state of the walker at time 𝑡 ∈ {1, 2, . . . } is given by 𝑈 𝑡 Ψ0 .

▶▶

The probability of ﬁnding the walker at position 𝑥 ∈ Z and at time 𝑡 ∈ {1, 2, . . . } isgiven by 𝑃𝑡 (𝑥) := ∥(𝑈 𝑡 Ψ0) (𝑥) ∥ 2C2 .

Conservation of total probability:∑∑
𝑃𝑡 (𝑥) =∥(𝑈 𝑡 Ψ0)(𝑥) ∥ 2C2 = ∥𝑈 𝑡 Ψ0 ∥ ℓ22 (Z,C2) = ∥Ψ0 ∥ ℓ22 (Z,C2) = 1,▶▶

𝑥∈Z𝑥∈Z𝑡 ∈ {1, 2, . . . }.

§1Motivationp.2▶ We consider index theory for unitary time-evolution operators of the form;

𝑈 = 𝛤 × 𝛤 ′,where 𝛤, 𝛤 ′ are unitary self-adjoint operators on a state Hilbert space H .
▶ Given 𝑈 = 𝛤 × 𝛤 ′, we get the following chiralsymmetry condition:..........................................

𝑈∗ =Aim of the talkLet 𝜆 be a ﬁxed number in the unit-circle T. We wish to introduce a certain well-deﬁned
index, say ind 𝜆 (𝛤, 𝑈), satisfying:(i) ind 𝜆 (𝛤, 𝑈) is stable against a wide range of perturbations.
(ii) |ind 𝜆 (𝛤, 𝑈)| ≤ dim ker(𝑈 − 𝜆) (known as symmetry protection of eigenstates).

△ We are interested in the case 𝜆 = ±1.
!

Preliminaries

§2

Preliminariesp.3▶ Let 𝛤 be a (bounded) operator on an abstract Hilbert space H :▶▶

𝛤 is self-adjoint, if 𝛤 ∗ = 𝛤.

▶▶

𝛤 is unitary, if 𝛤 ∗ = 𝛤 −1 .

▶▶

𝛤 is involutory, if 𝛤 2 = 1.

▶ If 𝛤 has any two of the above properties, then it automatically has the third property.
▶ For example, the following 2 × 2 matrices are unitary self-adjoint;(𝑎𝛤𝑎 := √1 − 𝑎2√

1 − 𝑎2−𝑎),𝑎 ∈ [−1, 1].

§2Preliminaries

p.4▶ If 𝛤 : H → H is unitary self-adjoint, then:▶▶

▶▶

We have H = ker(𝛤 − 1) ⊕ ker(𝛤 + 1). Indeed, for any Ψ ∈ H)()
(1−𝛤1+𝛤Ψ=Ψ+Ψ ∈ ker(𝛤 − 1) ⊕ ker(𝛤 + 1).
22The operator 𝛤 has the following block-operator matrix representation;()
1 0
𝛤==0 −1ker(𝛤−1)⊕ker(𝛤+1)△ Indeed, if Ψ± ∈ ker(𝛤 ∓ 1), then 𝛤Ψ± = ±Ψ± .
!

▶ A decomposition of this kind is often used, for example, in supersymmetric quantum

mechanics (SUSYQM).

Index Theory for Chiral Unitaries§3Index Theory for Chiral Unitaries

p.5▶ Let 𝑈 = 𝛤 × 𝛤 ′, where 𝛤, 𝛤 ′ are unitary self-adjoint operators on H .
▶▶

The operator 𝑅 := (𝑈 + 𝑈 ∗)/2, which is the realpart of 𝑈, satisﬁes..............
ker(𝑈 ∓ 1) = ker(𝑅 ∓ 1).

▶▶

(1)

We can also easily prove that 𝑅 can be written as
𝑅 = 𝑅1 ⊕ 𝑅2

w.r.t. ker(𝛤 − 1) ⊕ ker(𝛤 + 1).

▶ It immediately follows from (1) and (2) that ker(𝑈 ∓ 1) = ker(𝑅1 ∓ 1) ⊕ ker(𝑅2 ∓ 1).
▶ This motivates us to introduce the following two formal indices:

ind ± (𝛤, 𝑈) := dim ker(𝑅1 ∓ 1) − dim ker(𝑅2 ∓ 1),
where |ind ± (𝛤, 𝑈)| ≤ dim ker(𝑈 ∓ 1).
▶ The formal index ind ± (𝛤, 𝑈) is well-deﬁned, if dim ker(𝑈 ∓ 1) < ∞.

(2)

§3

Index Theory for Chiral Unitaries▶ Proof of ker(𝑈 ∓ 1) = ker(𝑅 ∓ 1) (equality (1) on the previous slide):▶ Proof of 𝑅 = 𝑅1 ⊕ 𝑅2 (equality (2) on the previous slide):△ Let us ﬁrst show that 𝑈 ∗ = 𝛤𝑈𝛤 implies 𝑅 = 𝛤 𝑅𝛤 (that is, [𝑅, 𝛤] = 0).
!

p.6

§3

Index Theory for Chiral Unitaries

p.7▶ The operator 𝑄 := (𝑈 − 𝑈 ∗ )/2𝑖, which is the ........................
imaginary part of 𝑈, satisﬁes(𝑄=0𝑄 ∗0𝑄0 0),ker(𝛤−1)⊕ker(𝛤+1)

because of the anti-commutation relation 𝛤𝑄 + 𝑄𝛤 = 0.
▶ We deﬁne the following formal index

ind (𝛤, 𝑈) := dim ker 𝑄 0 − dim ker 𝑄 ∗0,which is the Fredholm index of 𝑄 0, provided that 𝑄 0 is Fredholm. Moreover,ind (𝛤, 𝑈) = ind + (𝛤, 𝑈) + ind − (𝛤, 𝑈),(3)|ind (𝛤, 𝑈)| ≤ dim ker(𝑈 − 1) + dim ker(𝑈 + 1),(4)

where (4) is a weaker version of |ind ± (𝛤, 𝑈)| ≤ dim ker(𝑈 ∓ 1) previously mentioned.

A Concrete Model§4

A Concrete Model: Split-step Quantum Walkp.8▶ We consider the following block-operator matrices on ℓ 2 (Z, C2 ) = ℓ 2 (Z, C) ⊕ ℓ 2 (Z, C) :(𝛤 :=(1 00 𝐿∗)(× √𝑎𝛤 ′ := √1 − 𝑎2√√𝑝
1 − 𝑝2

1 − 𝑎2−𝑎1 − 𝑝2−𝑝))(×1 00 𝐿)

,.

(5)

(6)

where:▶▶

𝐿 is the left-shift operator on ℓ 2 (Z, C).

▶▶

𝑝 = ( 𝑝(𝑥))𝑥∈Z and 𝑎 = (𝑎(𝑥))𝑥∈Z are two arbitrary sequences taking values in [−1, 1].

▶ Note that 𝛤, 𝛤 ′ are unitary self-adjoint.
▶ Let 𝑈 := 𝛤 × 𝛤 ′ (time-evolution of the so-called split-stepquantum walk).
......................................
▶ Let 𝜎ess (𝑈) be the essential
spectrum of 𝑈. If ±1 ∉ 𝜎ess (𝑈), then ker(𝑈 ∓ 1) is ﬁnite.............................

dimensional, and so ind ± (𝛤, 𝑈) is well-deﬁned.

§4

A Concrete Model: Split-step Quantum Walkp.9

Theorem 1. Symmetry Protection of Eigenstates for the SSQW
Let us assume the existence of the following limits for each ★ = −∞, +∞ :𝑝(★) := lim 𝑝(𝑥),𝑥→★

𝑎(★) := lim 𝑎(𝑥).
𝑥→★

(7)

Then ±1 ∉ 𝜎ess (𝑈 ) if and only if 𝑝(★) ∓ 𝑎(★) ≠ 0 for each ★ = −∞, +∞. In this case,(i) Index Formula. We have ind ± (𝛤, 𝑈 ) ∈ {−1, 0, 1}. More precisely,ind ± (𝛤, 𝑈 ) =sign ( 𝑝(+∞) ∓ 𝑎(+∞)) − sign ( 𝑝(−∞) ∓ 𝑎(−∞)),2(8)

where sign denotes the sign function.
(ii) Symmetry Protection. If sup𝑥∈Z |𝜁 (𝑥)| < 1 for each 𝜁 = 𝑝, 𝑎, then|ind ± (𝛤, 𝑈 )| = dim ker(𝑈 ∓ 1).

△ The equality (9) can be viewed as an analogue of the bulk-edgecorrespondence.
.........................................
!

(9)§4A Concrete Model: Split-step Quantum Walk

p.10

△ Let T be the unit-circle.
!

▶ It can be shown that for each ★ = −∞, +∞ there exist continuous T ∋ 𝑧 ↦−→ 𝛾± (𝑧, ★) ∈ Cwith the following properties:▶▶

The continuous curve 𝛾± (·, ★) depends only on 𝑝(★), 𝑎(★).

▶▶

The integer ind ± (𝛤, 𝑈 ) is a ...........................
topological index in the sense thatind ± (𝛤, 𝑈 ) = wn(𝛾± (·, +∞)) − wn(𝛾± (·, −∞)),(10)where wn denotes the winding
number.
...........................
▶ Therefore, the equality |ind ± (𝛤, 𝑈 )| = dim ker(𝑈 ∓ 1) previously mentioned can beunderstood as the bulk-edgecorrespondence for the one-dimensional SSQW.
.........................................
▶ The non-trivial equality (10) can be proved by the well-known index theorem for Toeplitz

operators.

§4A Concrete Model: Split-step Quantum Walk

p.11

▶ The Hilbert space 𝐿 2 (T) admits the standard complete orthonormal basis (𝑒 𝑥 )𝑥∈Z deﬁnedby T ∋ 𝑧 ↦−→ 𝑧 𝑥 ∈ C.
▶ The Hardy-Hilbert space H2 is the closure of the linear span of {𝑒 𝑥 | 𝑥 ≥ 0}.
▶ Let 𝜄 : H2 ↩→ 𝐿 2 (T) be the inclusion mapping, and let 𝑓∈ 𝐶 (T). Then the Toeplitz

operator 𝑇 𝑓 with symbol 𝑓 is deﬁned by𝑇 𝑓 := 𝜄∗ × 𝑀 𝑓 × 𝜄,

(11)

where 𝑀 𝑓 : 𝐿 2 (T) → 𝐿 2 (T) is the bounded multiplication operator by 𝑓 .
▶▶

The Toeplitz operator 𝑇 𝑓 is Fredholm if and only if the curve T ∋ 𝑧 ↦−→ 𝑓 (𝑧) ∈ C does
NOT pass through the origin. In this case, ind 𝑇 𝑓 = −wn( 𝑓 ).

▶▶

The essential spectrum of 𝑇 𝑓 is given by 𝜎ess (𝑇 𝑓 ) = { 𝑓 (𝑧) | 𝑧 ∈ T} = ran 𝐹.

Two Generalisations of Theorem 1

§5

Two Generalisations of Theorem 1

p.12▶ What follows is the main result of arXiv:2111.04108. This is a joint work with▶▶

Y. Matsuzawa (Shinshu University), A. Suzuki (Shinshu University),
N. Teranishi (Hokkaido University), K. Wada (Hachinohe Kosen).

▶ In Theorem 1, we have

±1 ∉ 𝜎ess (𝑈 ) ⇐⇒ 𝑝(★) ∓ 𝑎(★) ≠ 0 for each ★ = −∞, +∞.
▶ We say that 𝑈 is .............
gap-less, if the above condition fails to hold.

In this case, scattering

theory for supersymmetric quantum mechanics allows us to show
ind ± (𝛤, 𝑈 ) =sign ( 𝑝(+∞) ∓ 𝑎(+∞)) − sign ( 𝑝(−∞) ∓ 𝑎(−∞)),2

where we now agree to set sign (0) := 0. Therefore, ind ± can take half-integer
values:..............................
}{−1 1
ind ± (𝛤, 𝑈 ) ∈ −1, , 0, , 1 .
22§5

Two Generalisations of Theorem 1p.13▶ What follows is the main result of arXiv:2111.12652. This is a joint work with▶▶

Y. Matsuzawa (Shinshu University) and K. Wada (Hachinohe Kosen).

▶ In Theorem 1, we assume the existence of the following two-sided limits:

𝜁 (±∞) := lim 𝜁 (𝑥),𝑥→±∞𝜁 = 𝑝, 𝑎.

(12)

We can obtain a generalisation of Theorem 1, if we replace (12) by the so-called “asymptotically periodic assumption”.
▶ For example, in the “asymptotically 2-periodic case”, we assume the existence of𝜁 (★, 0) := lim 𝜁 (2𝑥 + 0),𝑥→★

𝜁 (★, 1) := lim 𝜁 (2𝑥 + 1).
𝑥→★

▶ If you are interested, I am happy to give you more details about this.

§6References

p.14△ This talk is based on the following existing literature:!

▶ J. K. Asbóth, H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B 88(12), 121406(2013).
▶ C. Cedzich et al., Bulk-edge correspondence of one-dimensional quantum walks, J. Phys. A 49(21), 21LT01 (2016).
▶ C. Cedzich et al., The Topological Classiﬁcation of One-Dimensional Symmetric Quantum Walks, Ann. Henri Poincaré 19(2),325–383 (2018).
▶ C. Cedzich et al., Complete homotopy invariants for translation invariant symmetric quantum walks on a chain, Quantum 2, 95(2018).
▶ T. Fuda, D. Funakawa, A. Suzuki, Localization for a one-dimensional split-step quantum walk with bound states robust against
perturbations, J. Math. Phys. 59(8), 082201 (2018).
▶ A. Suzuki, Supersymmetry for chiral symmetric quantum walks, Quantum Inf. Process. 18(12), 363 (2019).
▶ Y. Matsuzawa, An index theorem for split-step quantum walks, Quantum Inf. Process. 19(8), 227 (2020).

My slides can be found in