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 Berry phase for higher-order symmetry-protected topological phases

Araki, Hiromu; Mizoguchi, Tsuyoshi; Hatsugai, Yasuhiro

doi:10.1103/physrevresearch.2.012009

Cited by 71 publications

(52 citation statements)

References 66 publications

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“…Moreover, the phases are related by 4 i=1 γ i = 0 (mod 2π). The higher-order Zak (Berry) phases we introduce are closely related to the geometric phases proposed by Araki et al [28]. However, in contrast to the construction in [28], our bulk Hamiltonian ĤOBC remains independent of θ and our gauge choice creates a twist of the Hamiltonian Ĥi (θ) without introducing flux in the bulk.…”

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confidence: 83%

“…The underlying Zak (Berry) phases are quantized in the symmetric HOSPT phases and serve as topological invariants of the latter. The invariants we define are similar to those introduced by Araki et al [28], but without the necessity to introduce magnetic flux in the bulk -hence they yield a definite value for any gapped phase in the thermodynamic limit. Our approach allows to directly relate the quantized corner charge in an HOSPT [10,12,33] to the Zak (Berry) phase, giving new physical meaning to the latter.…”

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confidence: 90%

“…Higher-order topological invariants have been proposed for both band insulators in a single-particle picture (HOTIs) and interacting quantum many-body systems (HOSPTs) protected by crystalline symmetries [8,9,13,14,16,[26][27][28][29][30][31][32]]. Yet, there is an ongoing debate on which of the proposed quantities constitute true bulk invariants and how exactly the multipole polarization can be calculated in extended systems with periodic boundary conditions [11,13].…”

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confidence: 99%

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Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

Wienand,

Horn,

Aidelsburger

et al. 2021

Preprint

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The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems (SPTs), quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here we show that quantized Thouless pumps connecting C4-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically non-trivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multi-pole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.

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confidence: 83%

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confidence: 90%

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confidence: 99%

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Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

Wienand,

Horn,

Aidelsburger

et al. 2021

Preprint

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show abstract

“…HOTIs are now actively studied and the classification of HOTIs has also been proposed [26,41,53]. Generalizing the bulk-boundary correspondence, relations between some gapped topology and corner states are much discussed [5,64,67].…”

Section: Introductionmentioning

confidence: 99%

Classification of topological invariants related to corner states

Hayashi

2021

Lett Math Phys

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We discuss some bulk-surface gapped Hamiltonians on a lattice with corners and propose a periodic table for topological invariants related to corner states aimed at studies of higher-order topological insulators. Our table is based on four things: (1) the definition of topological invariants, (2) a proof of their relation with corner states, (3) computations of K-groups and (4) a construction of explicit examples.

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“…The presence of these unique topological matter is usually guaranteed by the coexistence of crystal and non-spatial symmetries, and their classifications go beyond the tenfold way of first-order topological insulators and superconductors [ 9 , 10 , 11 , 12 ]. Besides great theoretical efforts in the study of higher-order topological insulators [ 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ], superconductors [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 ] and semimetals [ 49 , 50 , 51 , 52 , 53 , 54 ], HOTPs have also been observed in solid state materials [ 55 , 56 ,…”

Section: Introductionmentioning

confidence: 99%

Floquet Second-Order Topological Phases in Momentum Space

Zhou

2021

Nanomaterials

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Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers are shown to be measurable from the chiral dynamics of wave packets. Under open boundary conditions, multiple quartets Floquet corner modes with zero and π quasienergies emerge in the system and coexist with delocalized bulk states at the same quasienergies, forming second-order Floquet topological bound states in the continuum. The number of these corner modes is further counted by the bulk topological invariants according to the relation of bulk-corner correspondence. Our findings thus extend the study of HOTPs to momentum-space lattices and further uncover the richness of HOTPs and corner-localized bound states in continuum in Floquet systems.

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ZQ Berry phase for higher-order symmetry-protected topological phases

Cited by 71 publications

References 66 publications

Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

Classification of topological invariants related to corner states

Floquet Second-Order Topological Phases in Momentum Space

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