Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Cornean, Horia D.; Monaco, Domenico; Teufel, Stefan

doi:10.1142/s0129055x17300011

“…where T 2 + denotes the set of points in T 2 with non-negative k 1 coordinate [11,7]. Remember that the Berry connection depends on the choice of a Bloch frame: for the above formula to be well-posed one must require that the Bloch frame be timereversal symmetric, in a sense to be specified in the next Subsection.…”

Section: Bloch Bundle Berry Connection and Berry Curvaturementioning

confidence: 99%

“…An alternative construction makes use of the parallel transport associated to the family of projectors P(k), see e. g. [7]. The modification of Ψ into Φ is performed by successive extensions, first at certain highsymmetry points, then along the edges that connect them, and finally on the whole R 2 .…”

Section: Obstruction Theorymentioning

confidence: 99%

“…(see e. g. [7,Lemma 2.12]). On E 1 and E 3 , the maps k → ξ (k) are actually periodic, since the second condition in (25) implies that at the time-reversal invariant momenta k λ the matrix X(k λ ) must be symplectic and thus of unit determinant.…”

Section: Extension To the Face: A Topological Obstructionmentioning

confidence: 99%

“…Giving a full account of the geometric nature of this invariant has been a primary objective for mathematical physicists in the last decade, and a plethora of mathematical tools has been used in this endeavour, ranging from K-theory to homotopy theory, from functional analysis to noncommutative geometry, from equivariant cohomology to operator theory. We refer to [10,25,7] for recent accounts on the ever-growing literature on the subject.…”

Section: Introductionmentioning

confidence: 99%

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Michelangeli¹,

Dell’Antonio²

2017

Springer INdAM Series

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The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in Z 2 . We illustrate how both the Chern number c ∈ Z and the Fu-Kane-Mele invariant δ ∈ Z 2 of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

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“…Giving a full account of the geometric nature of this invariant has been a primary objective for mathematical physicists in the last decade, and a plethora of mathematical tools has been used in this endeavour, ranging from K-theory to homotopy theory, from functional analysis to noncommutative geometry, from equivariant cohomology to operator theory. We refer to [10,25,7] for recent accounts on the ever-growing literature on the subject.The purpose of this contribution is to express both the Chern number and the Fu-Kane-Mele Z 2 index of 2-dimensional topological insulators in a common frame-…”

mentioning

confidence: 99%

Chern and Fu–Kane–Mele Invariants as Topological Obstructions

Monaco

¹

2017

Advances in Quantum Mechanics

Self Cite

5

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The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in Z 2 . We illustrate how both the Chern number c ∈ Z and the Fu-Kane-Mele invariant δ ∈ Z 2 of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame. Key words: Topological insulators, quantum Hall effect, quantum spin Hall effect, Chern numbers, Fu-Kane-Mele invariants, obstruction theory.Later theoretical investigations showed that a topological phenomenon underlies this quantization: the integer n in the above formula was shown to be the first Chern number of a vector bundle, naturally associated to the quantum system [27,1,2].The only role played by the magnetic field in quantum Hall systems is that of breaking time-reversal symmetry: if the system were time-reversal symmetric, then the Hall conductivity would vanish, and the system would remain in an insulating state. This fact was clarified by Haldane [14], who showed that non-trivial topological phases can be displayed also in absence of a magnetic field, thus initiating the field of Chern insulators [3,5]. Picking up on the work by Haldane, Fu, Kane and Mele [18,11,12] later introduced a model which still displays a topological phase even if time-reversal symmetry is preveserved, and is by now recognized as a milestone in the history of topological insulators. The phenomenon that the model proposed to illustrate is that of the quantum spin Hall effect, which differs from the quantum Hall effect in that the external magnetic field is replaced by spin-orbit interactions (exactly to preserve time-reversal symmetry), and spin rather than charge currents flow on the boundary of the sample. From the point of view of topological phases, the peculiarity of this phenomenon is that, contrary to what happens for Chern and quantum Hall insulators, one can only distinguish between the trivial (insulating) and non-trivial (quantum spin Hall) phase: the label is then assigned by a Z 2 -valued topological index. Giving a full account of the geometric nature of this invariant has been a primary objective for mathematical physicists in the last decade, and a plethora of mathematical tools has been used in this endeavour, ranging from K-theory to homotopy theory, from functional analysis to noncommutative geometry, from equivar...

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Rainbow Trapping with Engineered Topological Corner States and Cavities in Photonic Crystals

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This work presents a pioneering photonic crystal (PC) heterostructure design exploiting tailored topological corner states and cavities to unleash a fascinating topological rainbow effect. This effect arises from the strategic integration of a nontrivial topological PC with sharp corners within a trivial PC matrix, resulting in a heterostructure rich in corner states and cavities. The critical innovation lies in manipulating the sector angle of circular columns, granting dynamic control over the rainbow effect and light localization. This manipulation induces distinct group velocities for different light frequencies, leading to their separation and localization at specific corner states. This remarkable “rainbow trapping” phenomenon manifests as highly confined light exhibiting exceptional resilience against disorder. These findings illuminate a pathway toward crafting next‐generation photonic devices boasting unparalleled functionalities. The reconfigurable rainbow trapping holds immense potential for applications in wavelength division multiplexing, optical sensing, and even venturing into quantum information processing.

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Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Cited by 28 publications

References 42 publications

Advances in Quantum Mechanics

Advances in Quantum Mechanics

Chern and Fu–Kane–Mele Invariants as Topological Obstructions

Rainbow Trapping with Engineered Topological Corner States and Cavities in Photonic Crystals

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