Scattering theory of topological phases in discrete-time quantum walks

Tarasinski, Brian; Asbóth, János K.; Dahlhaus, J. P.

doi:10.1103/physreva.89.042327

Cited by 66 publications

(82 citation statements)

References 41 publications

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“…We have introduced an effective numerical tool (the cloning trick) which allows us to calculate the scattering amplitudes for the split-step walk and efficiently calculate the topological invariants proposed in Ref. [32]. We then showed theoretically and investigated numerically various localization-delocalization transitions that occur whenever this system is tuned to a critical point at a topological phase transition.…”

Section: Discussionmentioning

confidence: 99%

“…Since disorder breaks translation invariance, the bulk topological invariants can no longer be obtained as k-space winding numbers. There are two alternative approaches to the topological invariants for the disordered case: one based on the scattering matrix [31,32] and one based on a reformulation of the winding number in real space, recently used for the disordered SSH model [39]. In the following we detail the first approach.…”

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

“…To define a scattering matrix, we have to use open boundary conditions, with two translationally invariant leads attached to the scattering region [32]. To obtain leads that host the right number of propagating modes, we omit the rotations in semi-infinite parts of the system, so the time-step operator there simply reads U = S ↓ S ↑ , as shown in Fig.…”

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

“…Chiral symmetry ensures that the reflection amplitudes at these energies are real and the topological invariants for a bulk are given by [32] …”

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

“…These invariants can be expressed as winding numbers of the bulk time-step operator over a part of the tim step [28][29][30] or using a generalization of the scattering theory of topological phases [31,32]. In this respect quantum walks are representative of the extreme limit of periodically driven systems [33][34][35][36], as is the (closely related) quantum kicked rotator [37].…”

Section: Introductionmentioning

confidence: 99%

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Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Rakovszky

Asbóth

2015

Phys. Rev. A

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Quantum walks are promising for information processing tasks because in regular graphs they spread quadratically more rapidly than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, with their wave function staying within a finite region even in the infinite time limit, with a probability exponentially close to 1. It is thus important to understand when a quantum walk will be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a one-dimensional quantum walk-the split-step walk-to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization transition occurs, and the walk spreads subdiffusively to infinity. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk and find that the disorder-induced Anderson localization and delocalization transitions are governed by the topological phases of the quantum walk.

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Section: Discussionmentioning

confidence: 99%

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

“…Chiral symmetry ensures that the reflection amplitudes at these energies are real and the topological invariants for a bulk are given by [32] …”

Section: Lyapunov Exponents and Topological Invariants By Cloningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Rakovszky

Asbóth

2015

Phys. Rev. A

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The Topological Classification of One-Dimensional Symmetric Quantum Walks

Cedzich

Geib

Grünbaum

et al. 2017

Ann. Henri Poincaré

120

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We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at 1 or −1 (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.

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Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

Cedzich

Geib

Grünbaum

et al. 2021

Commun. Math. Phys.

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This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $$\pm 1$$ ± 1 . The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

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Scattering theory of topological phases in discrete-time quantum walks

Cited by 66 publications

References 41 publications

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

The Topological Classification of One-Dimensional Symmetric Quantum Walks

Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

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