An index theorem for split-step quantum walks
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An index theorem for split-step quantum walks Yasumichi Matsuzawa1 Received: 1 July 2019 / Accepted: 6 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Split-step quantum walks are models of supersymmetric quantum walk, and thus, their Witten indices can be defined. We prove that the Witten index of a split-step quantum walk coincides with the difference between the winding numbers of functions corresponding to the right limit of coins and the left limit of coins. As a corollary, we give an alternative derivation of the index formula for split-step quantum walks, which is recently obtained by Suzuki and Tanaka. Keywords Quantum walks · Supersymmetry · Chiral symmetry · Index theorem · Witten index · Fredholm index · Winding number
1 Introduction and main result Suzuki [13] introduced the notion of supersymmetric quantum walk in an abstract way. Such a quantum walk is defined by a pair of two unitary self-adjoint operators and C on a Hilbert space H . The time evolution of the system is described by the unitary operator U := C. Then, U has a chiral symmetry , that is U = U ∗ . Thus, the self-adjoint operator Q := Im(U ) = (U − U ∗ )/2i, which is called a supercharge, satisfies Q + Q = 0. Since is unitary self-adjoint, we can decompose H and Q as 0 Q ∗+ H = ker ( − 1) ⊕ ker ( + 1), Q = , Q+ 0 where Q + is a bounded operator from ker ( − 1) to ker ( + 1). Suzuki defined the Witten index of the pair (, C) by ind(, C) := dim ker Q + − dim ker Q ∗+ .
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Yasumichi Matsuzawa [email protected] Department of Mathematics, Faculty of Education, Shinshu University, 6-Ro, Nishi-nagano, Nagano 380-8544, Japan 0123456789().: V,-vol
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Y. Matsuzawa
If Q + is a Fredholm operator, then the pair (, C) is called a Fredholm pair. In this case, the Witten index is nothing but the Fredholm index of Q + . We denote by index A the Fredholm index of a Fredholm operator A. Then, we have ind(, C) = index Q + , whenever (, C) is a Fredholm pair. Very recently, Suzuki and Tanaka [14] have computed the Witten index of a splitstep quantum walk [7]. A split-step quantum walk is a pair (, C) of two unitary self-adjoint operators and C on the Hilbert space 2 (Z) ⊕ 2 (Z) defined as follows: Let L be the left-shift operator on 2 (Z), that is, (Lψ)(x) := ψ(x + 1),
ψ ∈ 2 (Z), x ∈ Z.
Let a : Z → R, b : Z → C be functions such that a(x)2 + |b(x)|2 = 1 for each x ∈ Z. We identify these functions with the corresponding multiplication operators on 2 (Z). Then, the shift operator and the coin operator C of the system are defined by p qL , q¯ L ∗ − p
:=
C :=
a b∗ , b −a
where ( p, q) ∈ R × C are scalars satisfying p 2 + |q|2 = 1. This definition is a generalization of Kitagawa’s split-step quantum walk [7]. For details, see [5, Example 2.2] or [6, Section 2]. Since and C are unitary self-adjoint, the pair (, C) defines a supersymmetric quantum walk. In this paper, we suppose that the limits a(±∞) := lim a(x), x→±∞
b(±∞) := lim b(x) x→±∞
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