scholarly journals Bulk-boundary correspondence in (3+1)-dimensional topological phases

2016 ◽  
Vol 94 (4) ◽  
Author(s):  
Xiao Chen ◽  
Apoorv Tiwari ◽  
Shinsei Ryu
Keyword(s):  
Topological Phases ◽  
Boundary Correspondence

scholarly journals The bulk-corner correspondence of time-reversal symmetric insulators

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Sander Kooi ◽  
Guido van Miert ◽  
Carmine Ortix
Keyword(s):  
Time Reversal ◽  
Real Space ◽  
Topological Phases ◽  
Spin Orbit Coupling ◽  
Topological Properties ◽  
Many Body ◽  
Boundary Correspondence ◽  
Berry Phases ◽  
The Many ◽  
Do So

AbstractThe topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.

scholarly journals Dynamical signatures of topological order in the driven-dissipative Kitaev chain

2019 ◽  
Vol 6 (2) ◽  
Author(s):  
Moos van Caspel ◽  
Sergio Enrique Tapias Arze ◽  
Isaac Pérez Castillo
Keyword(s):  
Steady State ◽  
Closed System ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Topological Phases ◽  
Topological Order ◽  
Superconducting Nanowires ◽  
Boundary Correspondence ◽  
Steady State Current ◽  
Different Types

We investigate the effects of dissipation and driving on topological order in superconducting nanowires. Rather than studying the non-equilibrium steady state, we propose a method to classify and detect dynamical signatures of topological order in open quantum systems. Bulk winding numbers for the Lindblad generator \hat{\mathcal{L}}ℒ̂ of the dissipative Kitaev chain are found to be linked to the presence of Majorana edge master modes – localized eigenmodes of \hat{\mathcal{L}}ℒ̂. Despite decaying in time, these modes provide dynamical fingerprints of the topological phases of the closed system, which are now separated by intermediate regions where winding numbers are ill-defined and the bulk-boundary correspondence breaks down. Combining these techniques with the Floquet formalism reveals higher winding numbers and different types of edge modes under periodic driving. Finally, we link the presence of edge modes to a steady state current.

scholarly journals Multiple scattering theory of non-Hermitian sonic second-order topological insulators

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
María Rosendo López ◽  
Zhiwang Zhang ◽  
Daniel Torrent ◽  
Johan Christensen
Keyword(s):  
Finite Element ◽  
Multiple Scattering ◽  
Scattering Theory ◽  
Spectral Dependence ◽  
Topological Insulators ◽  
Acoustic Energy ◽  
Topological Phases ◽  
Multiple Scattering Theory ◽  
Boundary Correspondence ◽  
Numerical Tool

Abstract Topological phases of sound enable unconventional confinement of acoustic energy at the corners in higher-order topological insulators. These unique states which go beyond the conventional bulk-boundary correspondence have recently been extended to non-Hermitian wave physics comprising finite crystal structures including loss and gain units. We use a multiple scattering theory to calculate these topologically trapped complex states that agree very well to finite element predictions. Moreover, our semi-numerical tool allows us to compute the spectral dependence of corner states in the presence of defects, illustrating the limits of the topological resilience of these confined non-Hermitian acoustic states.

scholarly journals Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

2019 ◽  
Vol 377 (3) ◽  
pp. 1761-1821 ◽  
Author(s):  
Alexander Alldridge ◽  
Christopher Max ◽  
Martin R. Zirnbauer
Keyword(s):  
Classification Scheme ◽  
Topological Phases ◽  
Free Fermion ◽  
Space Theory ◽  
Symmetry Classes ◽  
Boundary Class ◽  
Boundary Correspondence ◽  
Interacting Systems ◽  
Bulk System ◽  
The Many

Abstract Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$$^*$$ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $$K$$ K -theory (or $$KR$$ KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $$KR$$ KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.

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