Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

AbstractThis 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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Many-body topological invariants from randomized measurements in synthetic quantum matter

Science Advances ◽

10.1126/sciadv.aaz3666 ◽

2020 ◽

Vol 6 (15) ◽

pp. eaaz3666 ◽

Cited By ~ 5

Author(s):

Andreas Elben ◽

Jinlong Yu ◽

Guanyu Zhu ◽

Mohammad Hafezi ◽

Frank Pollmann ◽

...

Keyword(s):

Body Wave ◽

Complete Classification ◽

Theoretical Description ◽

Spin Model ◽

Topological Invariants ◽

Topological Phases ◽

Many Body ◽

Topological Quantum ◽

The Many ◽

Symmetry Protected

Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by site-resolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays.

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Photonic quantum walks with symmetry protected topological phases

10.1063/1.4998022 ◽

2017 ◽

Cited By ~ 1

Author(s):

A. Blanco-Redondo ◽

B. Bell ◽

M. Segev ◽

B. J. Eggleton

Keyword(s):

Quantum Walks ◽

Topological Phases ◽

Symmetry Protected

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Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

Quantum ◽

10.22331/q-2018-09-24-95 ◽

2018 ◽

Vol 2 ◽

pp. 95 ◽

Cited By ~ 5

Author(s):

C. Cedzich ◽

T. Geib ◽

C. Stahl ◽

L. Velázquez ◽

A. H. Werner ◽

...

Keyword(s):

Practical Method ◽

Quantum Walks ◽

Translation Invariance ◽

Translation Invariant ◽

Symmetric Quantum ◽

Space Projection ◽

Exponential Behaviour ◽

A Chain ◽

The One ◽

Locality Conditions

We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrödinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.

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Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

Physical Review X ◽

10.1103/physrevx.8.011055 ◽

2018 ◽

Vol 8 (1) ◽

Cited By ~ 26

Author(s):

Qing-Rui Wang ◽

Zheng-Cheng Gu

Keyword(s):

Complete Classification ◽

Three Dimensions ◽

Topological Phases ◽

General Group ◽

Symmetry Protected

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Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

SciPost Physics ◽

10.21468/scipostphys.11.2.024 ◽

2021 ◽

Vol 11 (2) ◽

Author(s):

Shenghan Jiang ◽

Meng Cheng ◽

Yang Qi ◽

Yuan-Ming Lu

Keyword(s):

Ground State ◽

Real Space ◽

Unit Cell ◽

Topological Phases ◽

Invariant System ◽

Translation Invariant ◽

Topological Excitations ◽

Spatial Dimensions ◽

Space Construction ◽

Symmetry Protected

We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

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