The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins

TL;DR

This paper classifies the Witten index for 1D supersymmetric quantum walks with anisotropic coins, revealing its topological significance and similarities to supersymmetric Dirac particles.

Contribution

It provides a complete classification of the Witten index for one-dimensional split-step quantum walks, linking it to topological and supersymmetric properties.

Findings

Witten index classifies topologically protected states

Similarity to supersymmetric Dirac particles

Provides a complete mathematical framework for the index

Abstract

Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index associated with a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits striking similarity to the one associated with a Dirac particle in supersymmetric quantum mechanics.

Equations

$$\mathrm{ind}(\Gamma ,C)={\Delta }_{\Gamma }(H):=\dim \ker H_{+}-\dim \ker H_{-}.$$

$$\Gamma =\left(\begin{array}{cc}p& qL\\ q^{\ast }{L}^{\ast }& -p\end{array}\right),$$

$$C=\left(\begin{array}{cc}{a}_1& b^{\ast }\\ b& a_2\end{array}\right),$$

$$C(♯)=\left(\begin{array}{cc}a(♯)& b^{\ast }(♯)\\ b(♯)& -a(♯)\end{array}\right)$$

$$(\Gamma ,C)$$

$$\mathrm{ind}(\Gamma ,C)=\{\begin{array}{ll}+\mathrm{sgn}p,& |a(\mathrm{R})|<|p|<|a(\mathrm{L})|,\\ -\mathrm{sgn}p,& |a(\mathrm{L})|<|p|<|a(\mathrm{R})|,\\ 0,& \text{mbox}otherwise.\end{array}$$

$$\mathrm{ind}(\Gamma ,C)=\mathrm{ind}{Q}_{{ϵ}_{+}}=\dim \ker Q_{{ϵ}_{+}}-\dim \ker Q_{{ϵ}_{+}}^{\ast }.$$

$$\alpha (x)\Psi (x+1)+{\alpha }^{\prime }(x)\Psi (x-1)+\beta (x)\Psi (x)=0,\Psi =(\Psi (x){)}_{x\in \mathbb{Z}}\in {\ell }^2(\mathbb{Z}),$$

$$C=\left(\begin{array}{cc}{a}_1& b^{\ast }\\ b& a_2\end{array}\right),$$

$$b(a_1+a_2)=0\text{mbox}and{a}_j^2+|b{|}^2=1,j=1,2.$$

$$C=\left(\begin{array}{cc}a& b^{\ast }\\ b& -a\end{array}\right)\text{mbox}and{a}^2+|b{|}^2=1.$$

$$Q$$

$$H$$

$${\Delta }_{\Gamma }(H):=\dim \ker H_{+}-\dim \ker H_{-},$$

$$\ker A^{\ast }A=\ker A,$$

$$\inf \sigma (A^{\ast }A)\setminus \{0\}>0$$

$$\mathrm{ind}(\Gamma ,C)={\Delta }_{\Gamma }(H),$$

$${\ell }^2(\mathbb{Z},\mathrm{X}):=\{\Psi :\mathbb{Z}\to \mathrm{X}\mid \sum _{x\in \mathbb{Z}}\Vert \Psi (x){\Vert }_{\mathrm{X}}^2<\infty \},$$

$$(L\Psi )(x)=\Psi (x+1)\text{mbox}and(L^{\ast }\Psi )(x)=\Psi (x-1),x\in \mathbb{Z}.$$

$$\Gamma ={\left(\begin{array}{cc}p& qL\\ q^{\ast }{L}^{\ast }& -p\end{array}\right)}_{{\ell }^2(\mathbb{Z})\oplus {\ell }^2(\mathbb{Z})}\text{mbox}andC={\left(\begin{array}{cc}{a}_1& b^{\ast }\\ b& a_2\end{array}\right)}_{{\ell }^2(\mathbb{Z})\oplus {\ell }^2(\mathbb{Z})},$$

$$\theta =\mathrm{Arg}q,$$

$$p^2+|q{|}^2=1,$$

$$a_j(x{)}^2+|b(x){|}^2=1,j=1,2,$$

$$b(x)(a_1(x)+a_2(x))=0,$$

$$\underset{x\to ♯}{\lim }C(x)=C(♯)={\left(\begin{array}{cc}{a}_1(♯)& b(♯{)}^{\ast }\\ b(♯)& a_2(♯)\end{array}\right)}_{{\ell }^2(\mathbb{Z})\oplus {\ell }^2(\mathbb{Z})},♯=\mathrm{L},\mathrm{R},$$

$$a(♯):=a_1(♯)=-a_2(♯).$$

$${ϵ}^{\ast }Qϵ={\left(\begin{array}{cc}0& Q_{{ϵ}_{-}}\\ Q_{{ϵ}_{+}}& 0\end{array}\right)}_{{\ell }^2(\mathbb{Z})\oplus {\ell }^2(\mathbb{Z})},$$

$$ϵ$$

$$-2i{Q}_{{ϵ}_{\pm }}$$

$$\mathrm{ind}(\Gamma ,C)=\mathrm{ind}{Q}_{{ϵ}_{+}}=\dim \ker Q_{{ϵ}_{+}}-\dim \ker Q_{{ϵ}_{-}}.$$

$${ϵ}^{\ast }\Gamma ϵ=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

$$2{ϵ}^{\ast }\left(\begin{array}{cc}X& 0\\ 0& X^{\prime }\end{array}\right)ϵ$$

$$2{ϵ}^{\ast }\left(\begin{array}{cc}0& Y^{\prime }\\ Y& 0\end{array}\right)ϵ$$

$$2{ϵ}^{\ast }\Gamma ϵ$$

Full text

1 Introduction

Discrete-time quantum walks are versatile platforms realising topological phenomena Gross ; AsOb13 ; OA ; OK ; MKO ; XZB ; Ced1 ; Ced2 . Kitagawa et al. KRBD proposed a split-step quantum walk with chiral symmetry and experimentally observed topologically protected bound states KBFRBKAD (see Kit for a comprehensive review). For such bound states, Fuda et al. FFSd ; FFS1 proved the robustness against compact perturbations and the spatial exponential decay property with mathematical rigour. Barkhofen et al. BLNSS implemented a chiral symmetric discrete-time quantum walk with supersymmetry. Recently, the first author of the present paper Suzuki18 proved that all the chiral symmetric quantum walks possess supersymmetry and that a discrete-time quantum walk has chiral symmetry if and only if the product of two unitary involutions represents its evolution operator. From these facts, we know that a discrete-time quantum walk can possess supersymmetry even if it does not have apparent chiral symmetry. Indeed, all homogeneous one-dimensional two-state quantum walks ABNVW01 ; Ko02 , multi-dimensional quantum walks FFSd , various types of quantum walks on graphs MNRoS07 ; MNRS09 ; Se13 ; HKSS14 ; Po16 ; HiSe1 ; HiSe2 ; HSS ; KPSS18 , and several quantum-walk based algorithms Gr ; Sz have evolution operators that can be represented by the product of two unitary involutions, and therefore they exhibit supersymmetry. See Suzuki18 for more details, and Oh for many examples of inhomogeneous one-dimensional quantum walks Ko09 ; Ko10 ; ShiKa10 ; KoLuSe13 ; EEKST1 whose evolutions are written by the product of two unitary evolutions.

As shown in Suzuki18 , a supersymmetric quantum walk (SUSYQW) assigns the Witten index, which provides a lower bound for the number of topologically protected bound sates. In this paper, we classify the Witten index for the split-step quantum walk entirely.

1.1 Witten index for SUSYQWs

To give a precise definition of the Witten index for SUSYQWs introduced in Suzuki18 , we briefly review here the supersymmetric structure of chiral symmetric quantum walks. We say that a unitary operator $U$ on a Hilbert space $\mathcal{H}$ has chiral symmetry if there exists a unitary involution $\varGamma$ on $\mathcal{H}$ (i.e., $\varGamma^{-1}=\varGamma^{*}=\varGamma$) such that $\varGamma U\varGamma=U^{-1}$. $U$ has chiral symmetry if and only if it can be represented as a product of two unitary involutions $\varGamma$ and $C:=\varGamma U$.

Suppose that $U$ is the evolution of a chiral symmetric quantum walk. Namely, there are two unitary involutions $\varGamma$ and $C$ such that $U=\varGamma C$. We call $Q:=[\varGamma,C]/2i$ a supercharge and $H=Q^{2}$ the superhamiltonian, where $[X,Y]:=XY-YX$ is the commutator. A direct calculation proves that $\varGamma$ and $H$ commute and hence $H$ can be decomposed into $H_{+}\oplus H_{-}$ with respect to the decomposition $\ker(\varGamma-1)\oplus\ker(\varGamma+1)$. We now define a topological index ${\rm ind}(\varGamma,C)$ so that it coincides with the Witten index $\Delta_{\varGamma}(H)$ of $H$ with respective to $\varGamma$, i.e.,

[TABLE]

In this sense, we call a pair $(\varGamma,C)$ of two unitary involutions a SUSYQW with the evolution $U=\varGamma C$ and call the index ${\rm ind}(\varGamma,C)$ the Witten index for the SUSYQW. We say that a SUSYQW $(\varGamma,C)$ is Fredholm if $H$ is Fredholm. As shown in Suzuki18 , the Fredholmness of $(\varGamma,C)$ depends only on $U$ (or equivalently $H$), and it is independent of the choice of $(\varGamma,C)$. However, the Witten index ${\rm ind}(\varGamma,C)$ depends on the choice of $(\varGamma,C)$. If $(\varGamma,C)$ is Fredholm, then the index ${\rm ind}(\varGamma,C)$ is robust against compact perturbations (see Appendix I).

1.2 Main result

In this paper, we study a split-step quantum walk FFS1 ; FFSd ; FFS1w , which unifies Kitagawa’s split-step quantum walk Kit and a usual one-dimensional quantum walk ABNVW01 ; Ko02 ; Suzuki16 . Let $\mathcal{H}:=\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$ be the state space of the split-step quantum walk. Idetifying $\mathcal{H}$ with $\ell^{2}(\mathbb{Z})\oplus\ell^{2}(\mathbb{Z})$, we define a shift operator $\varGamma$ on $\mathcal{H}$ as

[TABLE]

where $L$ is the left shift operator on $\ell^{2}(\mathbb{Z})$. We suppose that $(p,q)\in\mathbb{R}\times\mathbb{C}$ satisfies $p^{2}+|q|^{2}=1$, which ensures that $\varGamma$ is a unitary involution. We define a coin operator on $\mathcal{H}$ as

[TABLE]

where $a_{1}$, $a_{2}$ and $b$ are the multiplication operator on $\ell^{2}(\mathbb{Z})$ by functions $a_{j}:\mathbb{Z}\to\mathbb{R}$ ($j=1,2$) and $b:\mathbb{Z}\to\mathbb{C}$. We assume $a_{j}(x)^{2}+|b(x)|^{2}=1$ ($x\in\mathbb{Z}$, $j=1,2$), which garantees that $C$ is a unitary involution. The evolution operator of the split-step quantum walk is defined as the product of $\varGamma$ and $C$, i.e., $U=\varGamma C$. Since $\varGamma$ and $C$ are unitary involutions, the evolution $U$ has chiral symmetry, and $(\varGamma,C)$ defines a SUSYQW as explained in Subsection 1.1.

As shown in Suzuki18 , the modulus of ${\rm ind}(\varGamma,C)$ provides the lower bound for the number of topological bound states. Therefore, if ${\rm ind}(\varGamma,C)$ is nonzero, the corresponding quantum walk with the evolution $U=\varGamma C$ has a topological bound state. Motivated by this fact, we give a complete classification of the Witten index for the split-step quantum walk $(\varGamma,C)$ defined by (2) and (3).

To state our main result, we suppose that the coin $C$ is anisotropic Richard-Suzuki-Aldecoa18 ; Richard-Suzuki-Aldecoa19 , i.e., $C(x):=\begin{pmatrix}a_{1}(x)&b^{*}(x)\\ b(x)&a_{2}(x)\end{pmatrix}$ has limits as $x\to\pm\infty$. We denote these limits by $C(\mathrm{L})=\lim_{x\to-\infty}C(x)$ and $C(\mathrm{R})=\lim_{x\to+\infty}C(x)$. Clearly, the limit coins are unitary and hermitian. We say that a unitary and hermitian matrix is trivial if it equals $+1$ or $-1$. If the limit coins $C(\sharp)$ ($\sharp=\mathrm{L},\mathrm{R}$) are nontrivial, they can be assumed to be a unitary involution of the form

[TABLE]

without loss of generality (see Section 2.2 for more details). We are now in a position to state our main result.

Theorem A**.**

Let $\varGamma$ and $C$ be defined by (2) and (3). Suppose that $C$ is anisotropic and the limit coins $C(\sharp)$ ($\sharp=\mathrm{L},\mathrm{R}$) are nontrivial. Then

[TABLE]

In this case, we have

[TABLE]

The case where at least one of the two limits $C(\mathrm{L})$ and $C(\mathrm{R})$ is a trivial unitary involution is excluded here, since the pair $(\varGamma,C)$ with this property automatically fails to be Fredholm (Lemma 6). Theorem A provides a necessary and sufficient condition for the Fredholmness of 1D split-step SUSYQWs endowed with anisotropic coins, together with complete classification of the associated Witten index.

Note also that the model takes its simplest form when $p=0,$ but the associated Witten index is [math] in this case by the formula A2. It is therefore important to consider non-zero $p$ as well as the trivial case $p=0.$

There are close links between quantum walks and Dirac particles. In a continuous limit, quantum walks converge to Dirac particles BES ; Str06 (see MS for a mathematically rigorous and general proof). Klein’s paradox and Zitterbewegung in quantum walks were found in Mey97 ; Str07 ; Ku08 . Theorem A inspires a new relation between quantum walks and Dirac particles in comparison with the rusult of Bole et al. BGGSS87 : for the Dirac operator $Q=-i\sigma_{2}d/dx+\sigma_{1}\phi(x)$ on $L^{2}(\mathbb{R})\oplus L^{2}(\mathbb{R})$ with an anisotropic scalar potential $\phi(x)$ satisfying $\lim_{x\to\pm\infty}\phi(x)=\phi_{\pm}\in\mathbb{R}$, the Witten index equals $\pm 1$ if $\pm\phi_{-}<0<\pm\phi_{+}$ and it equals 0 otherwise.

1.3 Organisation and strategy of the paper

The present paper is organised as follows. In Section 2 we go through some preliminary results including the precise definition of the one-dimensional split-step SUSYQW $(\varGamma,C)$. It is shown in Section 3 that the Witten index of $(\varGamma,C)$ is given by the Fredholm index of a certain well-defined operator $Q_{\epsilon_{+}}$ on $\ell^{2}(\mathbb{Z})$ (see Theorem 1 for details);

[TABLE]

We show that the operator $Q_{\epsilon_{+}}$ is of the form $Q_{\epsilon_{+}}=\alpha L+\alpha^{\prime}L^{*}+\beta,$ where $\alpha,\alpha^{\prime},\beta$ are $\mathbb{C}$-valued sequences indexed by $\mathbb{Z}$ and $L,L^{*}$ are the left and right shift operators on $\ell^{2}(\mathbb{Z})$ respectively. In Section 4 we separately compute the two dimensions on the right hand side 4. With the explicit form of $Q_{\epsilon_{+}}$ mentioned above in mind, we shall end up solving second-order linear difference equations of the form

[TABLE]

which is known to have two linearly independent algebraic solutions. Here, we need not only to algebraically solve Equation 5, but also to ensure the solutions $\Psi$ to be square summable. This is precisely why the difference on the right-hand side of 4 can still be non-zero.

In Section 5 we prove Theorem A by making use of the index formula 4. The present paper concludes with Section 6, the main focus of which is a possible generalisation of the Witten index associated with SUSYQWs which fail to be Fredholm. Finally, Appendix I contains a brief summary of the several invariance principles of the Witten index, each of which plays a supplementary role in this paper.

2 Preliminaries

The primary focus on the present paper is discrete-time quantum walks, and so we shall henceforth assume that all (linear) operators in this paper are everywhere-defined bounded operators.

2.1 A brief overview of supersymmetry

Here, we give a brief overview of supersymmetry by going through some preliminary results in a somewhat rapid manner. What follows can be found in any standard textbook on the subject (see, for example, (Book:Bernd92:TheDiracEquation, , §5) or (Book:Arai17:AnalysisOnFockSpacesAndMathematicalTheoryOfQuantumFields, , §7.13)), and so proofs are omitted. An abstract operator $\varGamma$ on a Hilbert space $\mathcal{H}$ is called an involution, if $\varGamma^{2}=1.$ Note that if an operator possesses any two of the properties “involutory”, “unitary” and “self-adjoint”, then it possesses the third. We shall make use of the following finite-dimensional example throughout this paper;

Example 1** ($2\times 2$ case).**

A $2\times 2$ matrix $C$ is a unitary involution if and only if it is of the following form:

[TABLE]

where the triple $(a_{1},a_{2},b)\in\mathbb{R}\times\mathbb{R}\times\mathbb{C}$ satisfies

[TABLE]

In particular, $C=-1$ or $C=+1,$ which will be referred to as trivial unitary involutions, satisfies all of the above equalities. It is then easy to observe that a $2\times 2$ matrix $C$ is a non-trivial unitary involution if and only if it is of the following form:

[TABLE]

A self-adjoint operator $Q$ on $\mathcal{H}$ is called a supercharge with respect to a unitary involution $\varGamma,$ if it satisfies the anti-commutation relation $Q\varGamma+\varGamma Q=0,$ where the left hand side is commonly denoted by the symbol $\{Q,\varGamma\}.$ With a canonical decomposition $\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$ by $\mathcal{H}_{\pm}:=\ker(\varGamma\mp 1)$ in mind, a supercharge $Q$ and the superhamiltonian $H:=Q^{2}$ admit the following block-operator representations respectively;

[TABLE]

The superhamiltonian $H$ simultaneously represents two non-negative hamiltonians $H_{+},H_{-}$ whose spectra are identical except possibly for $0.$ The Witten index of the superhamiltonian $H$ with respect to $\varGamma$ is given by

[TABLE]

which measure the difference in the number of zero-energy ground states of $H_{+},H_{-}$, whenever the right-hand side is well-defined. Recall that $Q_{+}$ is a Fredholm operator if and only if both $\dim\ker Q_{\pm}$ are finite-dimensional and the range of $Q_{+}$ is closed. In this case, the Fredholm index of $Q_{+}$ is defined by $\mathrm{ind}\,Q_{+}:=\dim\ker Q_{+}-\dim\ker Q_{-}.$ The following two results about a general bounded operator $A:\mathcal{H}\to\mathrm{K}$ are useful:

[TABLE]

where the proof of 12 can be found, for example, in (Book:Arai17:AnalysisOnFockSpacesAndMathematicalTheoryOfQuantumFields, , Lemma 7.27)). With 11 in mind, the Witten index has a precise interpretation as $\Delta_{\varGamma}(H)=\mathrm{ind}\,Q_{+},$ provided that $Q_{+}$ is a Fredholm operator.

Following Suzuki18 we introduce the supercharge associated with a supersymmetric quantum walk (SUSYQW);

Definition 2**.**

We call a pair $(\varGamma,C)$ of two unitary involutions on a Hilbert space $\mathcal{H}$ a SUSYQW with the evolution operator $U=\varGamma C$.

For a SUSYQW $(\varGamma,C)$, $Q:=[\varGamma,C]/2i$ is a supercharge with respect to $\varGamma$, i.e., $\{Q,\varGamma\}=0$. We define the Witten inex of the SUSYQW as

[TABLE]

where the right-hand side is defined by 10 with the superhamiltonian $H=Q^{2}$ for the supercharge $Q$ of the SUSYQW.

Definition 3** (Fredholmness).**

A SUSYQW $(\varGamma,C)$ is said to be Fredholm, if $Q_{+}$ as in 8 is a Fredholm operator.

2.2 Definition of the model

Given $\mathrm{X}=\mathbb{C}$ or $\mathrm{X}=\mathbb{C}^{2},$ we shall consider the Hilbert space of square-summable $\mathrm{X}$-valued sequences:

[TABLE]

where $\|\cdot\|_{\mathrm{X}}$ is the standard norm defined on $\mathrm{X}.$ We shall agree to write elements of $\mathbb{C}^{2}$ as $2\times 1$ column vectors. With this convention in mind, an element $\Psi$ of $\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$ is written by $\Psi=(\Psi_{1},\Psi_{2})^{\mathrm{T}},$ where $\Psi_{1},\Psi_{2}\in\ell^{2}(\mathbb{Z},\mathbb{C}^{2}).$ On the Hilbert space $\ell^{2}(\mathbb{Z}):=\ell^{2}(\mathbb{Z},\mathbb{C}),$ the left-shift operator $L$ and the right-shift operator $L^{*}$ are given respectively by

[TABLE]

Evidently, we have $LL^{*}=L^{*}L=1.$ Let $\mathcal{H}=\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$ be the state space of a quantum walker throughout the present paper. With the canonical identification $\mathcal{H}=\ell^{2}(\mathbb{Z})\oplus\ell^{2}(\mathbb{Z})$ in mind, we are now in a position to introduce the precise definition of the model we shall consider throughout this paper;

Definition 4**.**

A (one-dimensional) split-step SUSYQW is a pair $(\varGamma,C)$ of two unitary involutions on $\mathcal{H}$ that are of the following forms;

[TABLE]

where the pair $(p,q)\in\mathbb{R}\times(\mathbb{C}\setminus\{0\})$ and the triple $(a_{1},a_{2},b)$ of $\mathbb{C}$-valued sequences satisfy all of the following conditions:

[TABLE]

where we assume that both 16 and 17 hold true for each $x\in\mathbb{Z}.$ Note that the sequences $a_{j}=(a_{j}(x))_{x\in\mathbb{Z}}$ and $b=(b(x))_{x\in\mathbb{Z}}$ here are canonically identified with their associated multiplication operators on $\ell^{2}(\mathbb{Z}).$

Definition 5** (anisotropic coins).**

Let $\mathrm{L}=-\infty,$ and let $\mathrm{R}=+\infty.$ Let $(\varGamma,C)$ be a split-step SUSYQW. The coin operator $C$ is called an anisotropic coin, if it admits the following two-sided limits:

[TABLE]

where we assume that 17 and 16 both hold true for each $x=\mathrm{L},\mathrm{R}.$ Note that if $C(\sharp)$ is a non-trivial unitary involution, then we shall assume without loss of generality (see Example 1 for details) that

[TABLE]

As in Definition 5 we shall always let $\mathrm{L}=-\infty$ and $\mathrm{R}=+\infty$ throughout this paper. This commonly used convention is, for example, in accordance with Richard-Suzuki-Aldecoa18 .

3 Diagonalisation

3.1 The main result

The ultimate purpose of the current section is to prove the following index formula for the Witten index;

Theorem 1**.**

Let $(\varGamma,C)$ be a split-step SUSYQW, where $C$ may or may not be anisotropic. Then there exists a unitary operator $\epsilon$ on $\mathcal{H}$ such that the supercharge $2iQ:=[\varGamma,C]$ admits off-diagonalisation of the following form with respect to the orthogonal decomposition $\mathcal{H}=\ell^{2}(\mathbb{Z})\oplus\ell^{2}(\mathbb{Z}):$

[TABLE]

where the three operators $\epsilon,Q_{\epsilon_{+}},Q_{\epsilon_{-}}$ are given respectively by

[TABLE]

Furthermore, the split-step quantum walk $(\varGamma,C)$ is Fredholm if and only if $Q_{\epsilon_{+}}$ is a Fredholm operator. In this case, we have

[TABLE]

Remark 2**.**

A direct computation shows that the supercharge $Q$ itself is not representable as an off-diagonal matrix with respect to the $\ell^{2}(\mathbb{Z})$-decomposition $\mathcal{H}=\ell^{2}(\mathbb{Z})\oplus\ell^{2}(\mathbb{Z})$, unlike the standard representation 8 which makes use of the canonical decomposition $\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}.$ To avoid confusion, we shall henceforth adhere to the convention that the round parentheses are used in the former representations, whereas the square parentheses are used in the latter representations.

3.2 The significance of diagonalisation

The main result of the current section, Theorem 1, might look rather technical at first glance, but as we shall see shortly the basic idea behind the proof is nothing but simple diagonalisation of the shift operator as in the following lemma;

Lemma 3**.**

Let $(\varGamma,C)$ be a split-step SUSYQW. The operator $\epsilon$ given by 21 is a unitary operator which diagonalises the shift operator $\varGamma$ as follows:

[TABLE]

Proof.

It is left as an easy exercise for the reader to verify that $\epsilon$ is unitary, and that the following two equalities hold true:

[TABLE]

With these two equalities in mind, we obtain 24 as follows:

[TABLE]

∎

Remark 4**.**

The diagonalisation of the form 24 is not unique. Indeed, as the experienced reader might immediately notice, one can introduce the discrete Fourier transform $\mathscr{F}$ following GJS and consider following unitary transform;

[TABLE]

where the right-hand side is diagonalisable in infinitely many different ways. Since the transform 27 is reversible, we can then obtain diagonalisation of the form 24. The unitary operator $\epsilon$ given explicitly by 21 is constructed in this precise manner.

In what follows, we shall make use of the unitary invariance of the Witten index as in Theorem 1. Let us fix an arbitrary unitary operator $\epsilon$ which gives the diagonalisation 24. We can then consider a new unitarily equivalent SUSYQW given by $(\varGamma_{\epsilon},C_{\epsilon}):=(\epsilon^{*}\varGamma\epsilon,\epsilon^{*}C\epsilon).$ Since the new shift operator $\varGamma_{\epsilon}$ is given by 24, we see immediately that the two subspaces $\mathcal{H}_{\epsilon_{\pm}}:=\ker(\varGamma_{\epsilon}\mp 1)$ are given respectively by

[TABLE]

Since the two subspaces $\mathcal{H}_{\epsilon_{\pm}}$ can be canonically identified with $\ell^{2}(\mathbb{Z}),$ the following abstract version of Theorem 1 holds true;

Lemma 5**.**

Let $(\varGamma,C)$ be a split-step SUSYQW, and let $\epsilon$ be any unitary operator which gives diagonalisation 24. Then the new supercharge $Q_{\epsilon}:=\epsilon^{*}Q\epsilon$ admits the following off-diagonal block matrix representation with respect to the decomposition $\mathcal{H}=\ell^{2}(\mathbb{Z})\oplus\ell^{2}(\mathbb{Z}):$

[TABLE]

Furthermore, $(\varGamma,C)$ is a Fredholm operator if and only if $Q_{\epsilon_{+}}:\ell^{2}(\mathbb{Z})\to\ell^{2}(\mathbb{Z})$ is a Fredholm operator. In this case, we have

[TABLE]

Proof.

As in Section 2.1 the new supercharge $Q_{\epsilon}=\epsilon^{*}Q\epsilon$ admits

[TABLE]

Observe first that $\ell^{2}(\mathbb{Z})$ can be canonically identified with $\mathcal{H}_{\pm}$ by the unitary operators $\gamma_{\pm}:\ell^{2}(\mathbb{Z})\to\mathcal{H}_{\pm}$ defined respectively by the following formulas:

[TABLE]

If we let $Q_{\epsilon_{\pm}}=\gamma_{\mp}Q^{\prime}_{\epsilon_{\pm}}\gamma_{\pm},$ then $\gamma_{\mp}^{*}Q_{\epsilon_{\pm}}=Q^{\prime}_{\epsilon_{\pm}}\gamma_{\pm}.$ More explicitly,

[TABLE]

With these two equalities in mind, we obtain

[TABLE]

Therefore, 28 holds true. Here, the following easy computation shows that the operators $Q^{\prime}_{\epsilon_{\mp}}Q^{\prime}_{\epsilon_{\pm}}$ and $Q_{\epsilon_{\mp}}Q_{\epsilon_{\pm}}$ are unitarily equivalent:

[TABLE]

With this fact in mind, we obtain the following two equalities:

[TABLE]

That is, $(\varGamma_{\epsilon},C_{\epsilon})$ is Fredholm if and only if $Q_{\epsilon_{+}}$ is a Fredholm operator and

[TABLE]

The claim now follows from Theorem 1. ∎

3.3 Proof of Theorem 1

By virtue of Lemma 5 we may choose to work with any unitary $\epsilon$ which gives diagonalisation 24 in order to compute the Witten index. In particular, as in Theorem 1, we shall henceforth work with the one given explicitly by 21 in this paper. In order to prove Theorem 1, it remains to show that 22 holds true:

Proof of Equality 22.

Let $\epsilon$ be the unitary operator given by 21. We shall first find the matrix representation of the time evolution $U_{\epsilon}:=\epsilon^{*}U\epsilon.$ We obtain

[TABLE]

where 25 to 26 allow us to prove:

[TABLE]

where $La_{2}=a_{2}(\cdot+1)L$ is used in the first equality. Note that 29 becomes

[TABLE]

It can then be shown that the following equalities hold true:

[TABLE]

We are now in a position to prove 22. We get

[TABLE]

where the last equality follows from the fact that $U_{\pm}$ given by 30 are both self-adjoint. On the other hand, $U_{\pm}$ given by 31 admit $(U^{\prime}_{\mp})^{*}=-U^{\prime}_{\pm},$ and so

[TABLE]

The claim follows. ∎

3.4 Coin operators with trivial limits

We shall conclude the current section with one simple corollary of Theorem 1. Recall that in Theorem A the case where at least one of the two limits $C(\mathrm{L})$ and $C(\mathrm{R})$ is a trivial unitary involution is excluded. The following result explains why.

Lemma 6**.**

If $(\varGamma,C)$ is a split-step SUSYQW endowed with an anisotropic coin $C$ with the property that at least one of the two limits $C(\mathrm{L})$ and $C(\mathrm{R})$ is trivial, then $\dim\ker Q_{\epsilon_{\pm}}=\infty.$ That is, $(\varGamma,C)$ automatically fails to be Fredholm in this case.

Proof.

We may assume without loss of generality that $C(\mathrm{L})$ is trivial, and that $C(x)=C(L)$ for each $x\leq 0$ due to the topological invariance 84. If $\Psi_{\pm}\in\ker Q_{\epsilon_{\pm}},$ then it follows from 22 that

[TABLE]

where $a_{2}(x+1)-a_{1}(x)=a_{2}(\mathrm{L})-a_{1}(\mathrm{L})=0.$ Thus, for each $x\leq-1$ the vectors $\Psi_{\pm}(x)\in\mathbb{C}^{2}$ can be freely chosen regardless of the other required conditions $(Q_{\epsilon_{\pm}}\Psi_{\pm})(x)=0$ for each $x\geq 0.$ This implies $\dim\ker Q_{\epsilon_{\pm}}=\infty.$ ∎

4 Classification of $\dim\ker Q_{\epsilon_{\pm}}$

4.1 The main result

In order to state the main theorem of the current section, we introduce the following definition;

Definition 1**.**

Let $(\varGamma,C)$ be a split-step SUSYQW with an anisotropic coin $C$. We shall consider the following mutually exclusive cases:

[TABLE]

We say that the coin operator $C$ is of Type I, if the two unitary involutions $C(\mathrm{L})$ and $C(\mathrm{R})$ are both non-trivial and if (I) holds true. Type II, II’, III coins are defined likewise. That is, we shall always assume that 19 holds true for each $\sharp=\mathrm{L},\mathrm{R},$ whenever the four types of the isotropic coin thus defined.

With this definition in mind, the ultimate aim of the current section is to prove the following classification result:

Theorem 2**.**

Let $(\varGamma,C)$ be a split-step SUSYQW, and let $C$ be an anisotropic coin of the following specific form;

[TABLE]

where $C(\sharp)$ is assumed to be non-trivial for each $\sharp=\mathrm{L},\mathrm{R}.$ Let $d_{\pm}=\dim\ker Q_{\epsilon_{\pm}}.$

If $C$ is of Type I, then $d_{\pm}$ are uniquely determined by the pair $(a(\mathrm{L}),a(\mathrm{R})):$

[TABLE] 2. 2.

If $C$ is of Type II, then $d_{\pm}$ are uniquely determined by the triple $(p,a(\mathrm{L}),a(\mathrm{R})):$

[TABLE] 3. 3.

If $C$ is of Type II’, then $d_{\pm}$ are uniquely determined by the triple $(p,a(\mathrm{L}),a(\mathrm{R})):$

[TABLE] 4. 4.

If $C$ is of Type III, then $d_{\pm}$ are uniquely determined by the triple $(p,a(\mathrm{L}),a(\mathrm{R})):$

[TABLE]

Remark 3**.**

The following comments about Theorem 2 are worth mentioning:

Note that the ultimate purpose of the present paper is not the computation of each individual $d_{\pm},$ but rather the difference $d_{+}-d_{-}.$ Since the latter quantity is invariant under compact perturbations, we may impose 32 without loss of generality. 2. 2.

If $p=0,$ then $d_{+}=d_{-}$ regardless of the coin type:

[TABLE]

That is, the Witten index of $(\varGamma,C)$ is always zero in this case.

4.2 Preliminaries

4.2.1 Notation

We shall always adhere to the notation introduced here throughout the remaining part of the current section. Let $(\varGamma,C)$ be a split-step SUSYQW endowed with an anisotropic coin $C,$ and let $C(\sharp)$ be non-trivial for each $\sharp=\mathrm{L},\mathrm{R}.$ Recall that $Q_{\epsilon_{\pm}}$ introduced in Theorem 1 are operators of the following forms:

[TABLE]

where the unnecessary constant $-2i$ is removed for notational simplicity. We have

[TABLE]

where the two-sided limits of the last two sequences will be denoted respectively by

[TABLE]

where the last equality follows from 19. We shall also make use of the simplification assumption 32 throughout this subsection, so that

[TABLE]

4.2.2 A sketch for the proof of Theorem 2

The main theorem of the current section, Theorem 2, does require a lengthy argument as we need to separately consider the four types of the coin operator. However, the basic idea behind the proof is in fact elementary. Note first that the equation $(Q_{\epsilon_{\pm}}\Psi)(x)=0$ is equivalent to

[TABLE]

This equation, known as the second-order linear difference equation, can then be put into the following first-order matrix equation;

[TABLE]

whenever $\alpha_{\pm}(x+1)\neq 0.$ This idea of transforming a difference equation to the associated matrix equation of less order is well-known (see, for example, Book:Elaydi05:AnIntroductionToDifferenceEquations ), and this is precisely the approach we are going to take. Note that we need not only to algebraically solve Equation 42, but also to ensure the solutions to be square summable. The following coefficient matrix shall be used throughout this section;

[TABLE]

where $A_{\pm}(\mathrm{L})$ are well-defined if $C$ is of either Type II or III, and $A_{\pm}(\mathrm{R})$ are well-defined if $C$ is of either Type II’ or III.

4.2.3 Abstract matrix difference equations

We are interested in solving a first-order linear matrix difference equation which is an equation of the following form:

[TABLE]

where $\mathbb{N}=\{1,2,\dots\}$ and $A_{0}$ is a fixed invertible $2\times 2$ matrix. An easy inductive argument shows that 46 is equivalent to the following equation:

[TABLE]

We call any $\mathbb{C}^{2}$-valued sequence $\Phi$ satisfying 46 an algebraic solution with in mind that it may fail to be square summable. It is easy to see from 47 that any algebraic solution $\Phi$ is uniquely determined by the initial value $\Phi(1).$ Given a $\mathbb{C}^{2}$-valued sequence $\Phi,$ we have that $\Phi$ is an algebraic solution to 46 and $\sum_{x\in\mathbb{N}}\|\Phi(x)\|^{2}<\infty$ if and only if $\Phi\in\ker(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{0}).$ The following well-known result is included merely for the sake of completeness (See, for example, the proof of (Book:Elaydi05:AnIntroductionToDifferenceEquations, , Theorem 2.15) which makes use of the discrete analogue of the Wronskian);

Lemma 4**.**

Let $A_{0}$ be a fixed invertible $2\times 2$ matrix, and let $\Phi,\Phi^{\prime}$ be two algebraic solutions to the difference equation 46. Then $\Phi,\Phi^{\prime}$ are linearly independent if and only if $\Phi(x_{0}),\Phi^{\prime}(x_{0})$ are linearly independent for any $x_{0}\in\mathbb{N}.$

Proof.

If $\Phi(x_{0}),\Phi^{\prime}(x_{0})$ are linearly independent for each $x_{0}\in\mathbb{Z},$ then $\Phi,\Phi^{\prime}$ are obviously linearly independent. To prove the converse, suppose that $\Phi,\Phi^{\prime}$ are linearly independent, and that $c\Phi(x_{0})+c^{\prime}\Phi^{\prime}(x_{0})=0$ for some fixed $x_{0}\in\mathbb{N}$ and some $c,c^{\prime}\in\mathbb{C}.$ Since $\Phi,\Phi^{\prime}$ are both solutions to the difference equation 46, we have $c\Phi(x)+c^{\prime}\Phi^{\prime}(x)=0$ for each $x\in\mathbb{N},$ and so the linear independence of $\Phi,\Phi^{\prime}$ gives $c=c^{\prime}=0.$ It follows that $\Phi(x_{0}),\Phi^{\prime}(x_{0})$ are linearly independent for each $x_{0}\in\mathbb{N}.$ ∎

To put it another way, Lemma 4 states that two algebraic solutions $\Phi,\Phi^{\prime}$ to 46 are either identically linearly independent or identically linearly dependent. It is then easy to observe that $\dim\ker(L\oplus L-\bigoplus_{x\in\mathbb{N}}A_{0})\leq 2.$ Here, the equality may not hold, since an algebraic solution to 47 may fail to be square summable. To check the square summability of solutions, the following lemma is useful:

Lemma 5**.**

Let $A_{0}$ be a fixed invertible $2\times 2$ matrix with two distinct eigenvalues $z_{1},z_{2},$ so that $A_{0}$ admits diagonalisation of the following form for some invertible matrix $P:$

[TABLE]

Suppose that $\Phi$ is an algebraic solution to the difference equation 46, and that

[TABLE]

Then we have $\Phi\in\ker(L\oplus L-\bigoplus_{x\in\mathbb{N}}A_{0})$ if and only if the following sum is finite;

[TABLE]

Proof.

It follows from 47 that

[TABLE]

Then there exist constants $C_{1},C_{2}>0,$ such that

[TABLE]

where

[TABLE]

The claim follows. ∎

In fact, we shall end up solving Equation 46 with a constraint on the initial condition, and so we introduce the following notation:

Lemma 6**.**

Given an invertible $2\times 2$ matrix $A_{0}$ and two complex numbers $a_{0},b_{0}\in\mathbb{C},$ we introduce the following subspace of $\ell^{2}(\mathbb{N},\mathbb{C}^{2}):$

[TABLE]

If $a_{0},b_{0}\in\mathbb{C}$ are both non-zero, then $\dim\mathcal{S}(A_{0},a_{0},b_{0})\leq 1.$

Proof.

If $\Phi,\Phi^{\prime}\in\mathcal{S}(A_{0},a_{0},b_{0}),$ then $\Phi(1),\Phi^{\prime}(1)$ are linearly dependent vectors in $\mathbb{C}^{2}:$

[TABLE]

Thus, $\Phi,\Phi^{\prime}$ are also linearly dependent by Lemma 4. The claim follows. ∎

4.2.4 Concrete matrix difference equations

With 49 in mind, we are now in a position to state the explicit forms of the matrix difference equations we need to solve;

Lemma 7**.**

If the type of the coin $C$ is one of II,II’,III, then we have the following associated linear isomorphisms respectively:

[TABLE]

Proof.

If $C$ is of Type II, then $\alpha_{\pm}(\mathrm{L})=0$ and $\beta(\mathrm{L})\neq 0.$ Thus $\Psi\in\ker Q_{\epsilon_{\pm}}$ if and only if 42 holds true for each $x\geq 1$ together with the following two conditions:

[TABLE]

As in Section 4.2.2, Equation 42 is equivalent to the following:

[TABLE]

and so 50 is a well-defined operator. It remains to show that 50 is surjective, since the injectivity is obvious. It is easy to verify that any vector in $\mathcal{S}(A_{\pm}(\mathrm{R}),\alpha_{\pm}(\mathrm{R}),\pm\beta(0))$ must be of the form $(L\Psi_{0},\Psi_{0})$ for some $\Psi_{0}\in\ell^{2}(\mathbb{N})$ satisfying

[TABLE]

We define $\Psi\in\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$ by

[TABLE]

Then it is easy to show that $\Psi\in\ker Q_{\epsilon_{\pm}},$ and that it gets mapped to $(L\Psi_{0},\Psi_{0})$ under 50. Therefore, 50 is a well-defined linear isomorphism.

Similarly, if $C$ is of Type II’, then $\alpha_{\pm}(\mathrm{R})=0$ and $\beta(\mathrm{R})\neq 0.$ Thus $\Psi\in\ker Q_{\epsilon_{\pm}}$ if and only if 42 holds true for each $x\leq-1$ together with the following two conditions:

[TABLE]

As before 42 is equivalent to the following:

[TABLE]

If we introduce the change of variable $x\leftrightarrow-x,$ then the above equation becomes

[TABLE]

and so 51 is a well-defined operator. It remains to show that 51 is surjective, since the injectivity is obvious. It is easy to verify that any vector in $\mathcal{S}(A_{\pm}(\mathrm{L})^{-1},\mp\beta(0),\alpha_{\mp}(\mathrm{L})^{*})$ must be of the form $(\Psi_{0},L\Psi_{0})$ for some $\Psi_{0}\in\ell^{2}(\mathbb{N})$ with

[TABLE]

We define $\Psi\in\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$ by

[TABLE]

Then it is easy to show that $\Psi\in\ker Q_{\epsilon_{\pm}},$ and that it gets mapped to $(\Psi_{0},L\Psi_{0})$ under 51. Therefore, 51 is a well-defined linear isomorphism. The fact that 52 is a linear isomorphism if $C$ is of Type III is left as an easy exercise. ∎

As in Lemma 5, diagonalisation of the coefficient matrices $A_{\pm}(\sharp)$ is important.

Lemma 8**.**

If $A_{\pm}(\sharp)$ are well-defined for $\sharp=\mathrm{L},\mathrm{R},$ then the two matrices $A_{\pm}(\sharp)$ have two non-zero distinct eigenvalues $z_{\pm,1}(\sharp),z_{\pm,2}(\sharp)$ of the following forms:

[TABLE]

Moreover, the two matrices $A_{\pm}(\sharp)$ admit diagonalisation of the following form;

[TABLE]

Proof.

It is left as an easy exercise to show that a $2\times 2$ matrix of the form

[TABLE]

has two eigenvalues $2z_{j}=s+(-1)^{j}\sqrt{s^{2}+4t},$ where $j=1,2,$ together with the following eigenvalue equations:

[TABLE]

With this result in mind, the matrices $A_{\pm}(\sharp)$ given by 45 are as follows;

[TABLE]

where

[TABLE]

It follows that

[TABLE]

The claim follows. ∎

Definition 9**.**

We define the increasing function $f:[-1,1]\to[0,+\infty]$ by

[TABLE]

The following figure shows the graphs of $y=f(\pm\kappa):$

[TABLE]

We shall also make use of the following obvious identities:

[TABLE]

where $\kappa,\kappa^{\prime}\in(-1,1).$

Corollary 10**.**

With the notation introduced in Lemma 8 in mind, we have

[TABLE]

Proof.

Since $qq^{*}=(1-p)(1+p)$ and $b(\sharp)b(\sharp)^{*}=((-1)^{j}-a(\sharp))((-1)^{j}+a(\sharp)),$

[TABLE]

where

[TABLE]

On the other hand,

[TABLE]

We get

[TABLE]

The claim follows. ∎

4.3 Proof of Theorem 2

It remains to prove 33 to 36.

4.3.1 Type I coin

Proof of Equality 33.

If $C$ is an anisotropic coin of Type I, then $\beta(\sharp)\neq 0$ for each $\sharp=\mathrm{L},\mathrm{R}.$ It follows that $\Psi\in\ker Q_{\epsilon_{\pm}}$ if and only if $\Psi(x)=0$ whenever $x\neq 0.$ We get

[TABLE]

The claim follows. ∎

4.3.2 Type II coin

If $C$ is an anisotropic coin of Type II, then we shall make use of the isomorphism 50;

[TABLE]

We shall compute $d_{\pm}$ by making use of Lemma 5. As in Lemma 8, the matrices $A_{\pm}(\mathrm{R})$ admit diagonalisation of the following form:

[TABLE]

Given $\mathbb{C}$-valued sequences $\Phi_{\pm}=(\Phi_{\pm}(x))_{x\in\mathbb{N}},$ we have $\Phi_{\pm}\in\mathcal{S}(A_{\pm}(\mathrm{R}),\alpha_{\pm}(\mathrm{R}),\pm\beta(0))$ if and only if $\Phi_{\pm}$ are square-summable and the following equalities hold true:

[TABLE]

Lemma 11**.**

If the sequences $\Phi_{\pm}$ satisfy 61 and 62, then

[TABLE]

Proof.

If we let $\sharp=\mathrm{R},$ then

[TABLE]

where Lemma 8 implies

[TABLE]

With this equality in mind we obtain

[TABLE]

where

[TABLE]

Therefore

[TABLE]

∎

Proof of Equality 34.

We shall first assume $a(\mathrm{L})=1.$ If the sequences $\Phi_{\pm}$ satisfy 61 and 62, then

[TABLE]

Thus $\Phi_{+}$ is square summable (resp. $\Phi_{-}$ is square summable) if and only if

[TABLE]

where 59 gives

[TABLE]

Therefore, we obtain

[TABLE]

An analogous argument gives that if $a(\mathrm{L})=-1,$ then

[TABLE]

Thus, 34 is proved. ∎

4.3.3 Type II’ coin

This case is nothing but a repetition of the previous argument, but we include the proof for completeness. If $C$ is an anisotropic coin of Type II’, then we shall make use of the isomorphism 51;

[TABLE]

We shall compute $d_{\pm}$ by making use of Lemma 5. As in Lemma 8, the matrices $A_{\pm}(\mathrm{L})^{-1}$ admit diagonalisation of the following form:

[TABLE]

Given $\mathbb{C}$-valued sequences $\Phi_{\pm}=(\Phi_{\pm}(x))_{x\in\mathbb{N}},$ we get $\Phi_{\pm}\in\mathcal{S}(A_{\pm}(\mathrm{L})^{-1},\mp\beta(0),\alpha_{\mp}(\mathrm{L})^{*})$ if and only if $\Phi_{\pm}$ are square-summable and the following algebraic conditions hold:

[TABLE]

Lemma 12**.**

If the sequences $\Phi_{\pm}$ satisfy 65 and 66, then

[TABLE]

Proof.

It follows from 64 that

[TABLE]

With 60 in mind, 53 becomes

[TABLE]

We get

[TABLE]

Thus we obtain

[TABLE]

∎

Proof of Equality 35.

We shall first assume $a(\mathrm{R})=1.$ If the sequences $\Phi_{\pm}$ satisfy 65 and 66, then as before $\Phi_{+}$ is square summable (resp. $\Phi_{-}$ is square summable) if and only if

[TABLE]

where 59 together with 56 gives

[TABLE]

Therefore, we obtain

[TABLE]

An analogous argument gives that if $a(\mathrm{R})=-1,$ then

[TABLE]

The claim follows. ∎

4.3.4 Type III coin

Let $C$ be of Type III. This case turns out to be the hardest case. Here, we shall make use of the isomorphism 52:

[TABLE]

Lemma 13**.**

Given arbitrary $\mathbb{C}^{2}$-valued sequences $\Phi_{\pm}=(\Phi_{\pm}(x))_{x\in\mathbb{Z}},$ we define two sequences $\Phi_{\pm,\mathrm{L}},\Phi_{\pm,\mathrm{R}}$ by

[TABLE]

Then $\Phi_{\pm}\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ if and only if the following three conditions are simultaneously satisfied:

[TABLE]

Proof.

Evidently, we have $\Phi_{\pm}\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ if and only if the following three conditions are simultaneously satisfied:

[TABLE]

where the last condition is obviously 69 and the first two conditions are equivalent to the following two equations respectively:

[TABLE]

The claim follows. ∎

Lemma 14**.**

We have

[TABLE]

Proof.

As in 60, we can let

[TABLE]

We have

[TABLE]

On one hand,

[TABLE]

On the other hand,

[TABLE]

where for each $j=1,2,$ we have

[TABLE]

We obtain 70 as follows;

[TABLE]

∎

Corollary 15**.**

Suppose that $\mathbb{C}^{2}$-valued sequences $\Phi_{\pm}$ satisfy algebraic equations $\Phi_{\pm}(x+1)=A_{\pm}(x)\Phi_{\pm}(x)$ for each $x\in\mathbb{Z},$ and that

[TABLE]

Then $\Phi_{\pm}\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ if and only if the following sum is finite for each $j=1,2:$

[TABLE]

Moreover,

[TABLE]

Furthermore, we have $d_{\pm}\leq 1.$

Note that $k_{\pm,1}$ and $k_{\pm,2}$ cannot be simultaneously both non-zero.

Proof.

Recall that $\Phi\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ if and only if 67, 68 and 69 hold true. With the notation introduced in Lemma 13 in mind, we have

[TABLE]

where the second last equality follows from 69 and the last equality follows from 70. It follows from Lemma 5 that $\Phi\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ if and only if

[TABLE]

and so we get the criterion 71. It follows from 59 that

[TABLE]

where

[TABLE]

Thus 72 to 75 hold true. Finally, we may assume without loss of generality that $a(\mathrm{L})<a(\mathrm{R}),$ so that 73 to 74 both fail to hold. That is, $\Phi_{\pm}\in\ker\left(L\oplus L-\bigoplus_{x\in\mathbb{Z}}A_{\pm}(x)\right)$ are always of the following forms:

[TABLE]

and so $d_{\pm}\neq 2.$ This is because the conclusion of Lemma 4 still holds true, if the indexing set $\mathbb{N}$ is replaced by $\mathbb{Z}.$ ∎

Proof of Equality 36.

The claim immediately follows from 72 to 75. ∎

5 Proof of the main theorem

We are finally in a position to prove Theorem A, the main theorem of the present paper. This will be done in two separate steps: proof of the Fredholmness characterisation as in A1 and proof of the index formula A2.

5.1 The Fredholmness

In order to prove the Fredholmness characterisation A1, let us first discuss the following simple characterisation of the closedness of the range of $Q_{\epsilon_{+}}$;

Lemma 1**.**

*With the notation introduced in Lemma 5 in mind, the operator $Q_{\epsilon_{+}}$ has a closed range if and only if the time-evolution $U$ has spectral gaps111 That is to say, $\pm 1$ are not accumulation points of the spectrum of $U.$

at $\pm 1.$*

Proof.

It is a well-known fact that $Q_{\epsilon_{+}}$ has a closed range if and only if $\inf\sigma(Q_{\epsilon_{+}}^{*}Q_{\epsilon_{+}})\setminus\{0\}>0$ (see, for example, (Book:Arai17:AnalysisOnFockSpacesAndMathematicalTheoryOfQuantumFields, , Lemma 7.27)). Since $\sigma(H_{\epsilon})=\sigma(H)$ and $H=(\textrm{Im\,}U)^{2},$ the claim follows from the spectral theorem. ∎

To put it another way, Lemma 1 states that the closedness of the operator $Q_{\epsilon_{+}}$ is nothing but a spectral property of the time-evolution $U=\varGamma C.$ The following result is therefore useful;

Theorem 2**.**

Let $(\varGamma,C)$ be a split-step SUSYQW with an anisotropic coin $C,$ and let

[TABLE]

Then the essential spectrum of the time-evolution $U=\varGamma C$ is given by

[TABLE]

More explicitly, we have $\sigma(U_{\sharp})=\{z\in\mathbb{T}\mid\mathrm{Re}\,z\in I_{\sharp}\}$ for each $\sharp=\mathrm{L},\mathrm{R},$ where

[TABLE]

Proof of Theorem 2.

This result is standard, and so we will only give a brief sketch of the proof. Firstly, the well-known equality 76 can be proved by either using Weyl’s criterion for the essential spectrum or an elegant $C^{*}$-algebraic approach (see, for example, (Richard-Suzuki-Aldecoa18, , Theorem 2.2)). Secondly, the fact that each $\sigma(U_{\sharp})$ is characterised by 77 is an easy consequence of the standard approach which makes use of the discrete Fourier transform. ∎

We are now in a position to show that A1 is also a characterisation of the closedness of the range of $Q_{\epsilon_{+}}$;

Theorem 3**.**

Let $(\varGamma,C)$ be a split-step SUSYQW endowed with an isotropic coin $C.$ With the notation introduced in Theorem 1 in mind, the operator $Q_{\epsilon_{+}}$ has a closed range if and only if $|p|\neq|a(\sharp)|$ whenever $C(\sharp)$ is non-diagonal, where $\sharp=\mathrm{L},\mathrm{R}.$

Proof.

It immediately follows from Theorem 2 that for each $\sharp=\mathrm{L},\mathrm{R},$ the set $\sigma(U_{\sharp})$ is a discrete subset of $\mathbb{T}$ if and only if $C(\sharp)$ is a diagonal matrix (i.e. $b(\sharp)=0$). With Lemma 1 in mind, we have that the time-evolution $U$ has spectral gaps at $\pm 1$ if and only if the set $I_{\sharp}$ does not contain both $-1$ and $+1,$ whenever the limit $C(\sharp)$ is not a diagonal matrix. For such $\sharp,$ we introduce the following parametrisation:

[TABLE]

With this parametrisation in mind, we obtain

[TABLE]

The addition formula for the cosine gives:

[TABLE]

so that the set $I_{\sharp}$ becomes the following closed interval:

[TABLE]

where $-\pi<\Theta\pm\Theta_{\sharp}<\pi.$ Thus, we have $\pm 1\in I_{\sharp}$ if and only if $\Theta\mp\Theta_{\sharp}=0.$ That is, $I_{\sharp}$ does not contain both $-1$ and $+1$ if and only if $|p|\neq|a(\sharp)|.$ The claim follows. ∎

Proof of the characterisation A1.

Let $(\varGamma,C)$ be a split-step SUSYQW endowed with an anisotropic coin $C,$ and let $C(\sharp)$ be a non-trivial unitary involution for each $\sharp=\mathrm{L},\mathrm{R}.$ Recall the Fredholmness is invariant under compact perturbations (see 84 for details). Therefore, we may assume without loss of generality that $C$ is of the form 32, so that Theorem 2 implies $d_{\pm}\leq 1.$ That is, Theorem 3 implies that $(\varGamma,C)$ is Fredholm if and only if $|p|\neq|a(\sharp)|$ for each $\sharp=\mathrm{L},\mathrm{R}.$ Note that if $C(\sharp)$ is diagonal, then $|p|\neq|a(\sharp)|=1$ obviously holds true. The claim follows. ∎

5.2 Proof of the index formula A2

From here on, we shall assume that $(\varGamma,C)$ is a Fredholm SUSYQW and prove A2 by considering the four coin types separately:

5.2.1 Type I coin

Proof of Equality A2.

If $C$ is of Type I, then A2 becomes

[TABLE]

since $|a(\sharp)|=1$ for each $\sharp=\mathrm{L},\mathrm{R}$ by assumption. In fact, 79 immediately follows 33. The claim follows. ∎

5.2.2 Type II coin

Proof of Equality A2.

If $C$ is of Type II, then A2 becomes

[TABLE]

since $|a(\mathrm{L})|=1$ by assumption. It follows from Theorem 2 that $d_{+}\neq d_{-}$ if and only if one of the following two conditions holds true:

[TABLE]

where the last condition is equivalent to $|a(\mathrm{R})|<\pm p$, since $|p|\neq|a(\mathrm{R})|.$ Thus 80 holds true. ∎

5.2.3 Type II’ coin

Proof of Equality A2.

If $C$ is of Type II’, then A2 becomes

[TABLE]

since $|a(\mathrm{R})|=1$ by assumption. It follows from Theorem 2 that $d_{+}\neq d_{-}$ if and only if one of the following two conditions holds true:

[TABLE]

where the last condition is equivalent to $|a(\mathrm{R})|<\mp p$, since $|p|\neq|a(\mathrm{R})|.$ ∎

5.2.4 Type III coin

Proof of Equality A2.

Let us assume that $C$ is of Type III coin operator. Note first that if $|a(\mathrm{L})|=|a(\mathrm{R})|,$ then $\mathrm{ind}\,(\varGamma,C)=0.$ Thus, we shall assume $|a(\mathrm{L})|\neq|a(\mathrm{R})|$ from here on. By the invariance principle 85, we shall assume without loss of generality that $a(\mathrm{R})<a(\mathrm{L})$ throughout, and so

[TABLE]

We show first that A2 holds true, under the assumption $|a(\mathrm{R})|<|a(\mathrm{L})|$ first. That is, we need to check

[TABLE]

Since $|a(\mathrm{R})|<|a(\mathrm{L})|$ and $a(\mathrm{R})<a(\mathrm{L}),$ we must always have $a(\mathrm{L})=|a(\mathrm{L})|.$ If $a(\mathrm{R})\geq 0,$ then

[TABLE]

On the other hand, if $a(\mathrm{R})<0,$ then

[TABLE]

It remains to prove that A2 holds true, under the other assumption $|a(\mathrm{L})|<|a(\mathrm{R})|.$ That is, we need to check

[TABLE]

As before invariance principle 85 allows us to assume without loss of generality that $a(\mathrm{R})<a(\mathrm{L}).$ Since $|a(\mathrm{L})|<|a(\mathrm{R})|$ and $a(\mathrm{R})<a(\mathrm{L}),$ we must always have $a(\mathrm{R})=-|a(\mathrm{R})|.$ If $a(\mathrm{L})<0,$ then

[TABLE]

On the other hand, if $a(\mathrm{L})\geq 0,$ then

[TABLE]

∎

6 Concluding Remarks

A somewhat natural question arises. Can we still define the Witten index, if a given SUSYQW fails to be Fredholm? The answer to this question turns out to be yes, and we shall give a brief account of how research towards this direction can be undertaken. In fact, the standard theory of supersymmetry is already capable of dealing with the Witten index which cannot be interpreted as the Fredholm index by making use of a certain trace formula. See, for example, BGGSS87 or Gesztesy-Simon88 . These papers provide a theoretical foundation in what follows.

Let $(\varGamma,C)$ be a one-dimensional split-step SUSYQW, and let $\epsilon$ be any unitary operator which gives diagonalisation of the shift operator as in 24. We can then consider the unitarily equivalent SUSYQW $(\epsilon^{*}\varGamma\epsilon,\epsilon^{*}C\epsilon)$ together with the new associated supercharge $Q_{\epsilon}:=\epsilon^{*}\varGamma\epsilon$ and superhamiltonian $H_{\epsilon}:=\epsilon^{*}H\epsilon=Q_{\epsilon}^{2}$ admitting:

[TABLE]

where the first equality follows from 28. We say that the triple $(\varGamma,C,\epsilon)$ is trace-compatible, if $H_{\epsilon_{+}}-H_{\epsilon_{-}}$ is a trace-class operator on $\ell^{2}(\mathbb{Z}).$ We can then define the Witten index of the triple $(\varGamma,C,\epsilon)$ by

[TABLE]

whenever the limit exists. It is not known to the authors whether or not Formula 82 depends on $\epsilon.$ As in BGGSS87 , if the SUSYQW $(\varGamma,C)$ turns out to be Fredholm, then the above limit exists, and we get

[TABLE]

where the left hand side does not depend on $\epsilon$ in this Fredholm case. That is, in principle, we should be able to recover A2 by simply evaluating the trace-formula 82. Research towards this direction is work in progress, and this will be part of the PhD dissertation of the second author. The present paper concludes with the following simple example;

Example 1**.**

Let $(\varGamma,C)$ be a one-dimensional split-step SUSYQW whose coin operator $C$ has the property that $b(x)=0$ for each $x\in\mathbb{Z}.$ With the notation introduced in Theorem 1 in mind, we obtain

[TABLE]

Then the superhamiltonian becomes

[TABLE]

This implies $H_{+}=H_{-},$ so that $(\varGamma,C,\epsilon)$ is trace-compatible and $\mathrm{ind}\,(\varGamma,C,\epsilon)=0.$

Appendix I Supplementary Material

An (abstract) supersymmetric quantum walk (SUSYQW) is a pair $(\varGamma,C)$ of two unitary involutions on a Hilbert space $\mathcal{H}.$

The Witten index for SUSYQWs turns out to enjoy the following two invariance principles, each of which will a significant role in this paper.

Theorem 1** (Invariance of the Witten index).**

The following two assertions hold true:

Unitary Invariance. Let $(\varGamma,C)$ and $(\varGamma^{\prime},C^{\prime})$ be two SUSYQWs that are unitarily equivalent in the sense $(\varGamma^{\prime},C^{\prime})=(\epsilon^{*}\varGamma\epsilon,\epsilon^{*}C\epsilon)$ for some unitary operator $\epsilon$ on $\mathcal{H}.$

Then $(\varGamma,C)$ is a Fredholm SUSYQW if and only if so is $(\epsilon^{*}\varGamma\epsilon,\epsilon^{*}C\epsilon).$ In this case,

[TABLE] 2. 2.

Topological Invariance. Let $(\varGamma,C)$ and $(\varGamma,C^{\prime})$ be two SUSYQWs sharing the same shift operator $\varGamma,$ and let $C-C^{\prime}$ be a compact operator. Then $(\varGamma,C)$ is a Fredholm SUSYQW if and only if so is $(\varGamma,C^{\prime}).$ In this case,

[TABLE]

Remark 2**.**

The invariance principle 83 can be used to classify SUSYQWs in the following precise sense. If the Witten indices associated with two given SUSYQWs $(\varGamma,C)$ and $(\varGamma^{\prime},C^{\prime})$ do not agree to each other, then they cannot be unitarily equivalent. This is, of course, analogous to the manner in which we use the homotopy/homology groups to prove that certain topological spaces are not homotopy equivalent.

Here is yet another important principle of the Witten index:

Theorem 3** ((Suzuki18, , Corollary 3.7)).**

If one of $(\varGamma,C),(-\varGamma,C),(\varGamma,-C)$ is a Fredholm SUSYQW, then so are the rest. In this case we have the following formulas:

[TABLE]

Acknowledgements.

This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University. A. S. was supported by JSPS KAKENHI Grant Number JP18K03327.