Topological Invariants in Fermi Systems with Time-Reversal Invariance

Avron, J. E.; Sadun, Lorenzo; Segert, Jan; Simon, Barry

doi:10.1103/physrevlett.61.1329

Cited by 110 publications

(183 citation statements)

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confidence: 99%

“…Chern numbers are defined for energy levels which have a fixed degree of degeneracy for all Hamiltonians in the family. The Chern numbers for quadrupoles with j s f are defined and have been previously computed [4].In this paper, we will calculate the second Chern numbers for all quadrupole systems with half-odd-integer spin. In fact, every topological invariant of twodimensional complex vector bundles over S4 is a function of the second Chern number, i.e.…”

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confidence: 99%

“…these bundles are classified by the second Chern number up to topological equivalence. It will be shown in [5] that the second Chern numbers are indeed well defined for all half-odd-integer j .An energy level can be specified by the eigenvalue of a particular Hamiltonian. It is convenient to take the quadrupole Hamiltonian Qo = J : -$J2 which commutes with J 3 .…”

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confidence: 99%

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confidence: 99%

“…The degenerate, or non-Abelian, case has received much attention [2]. In the class of time-reversal-invariant fermionic Hamiltonians, which have Kramers degeneracy [3], the quadrupole systems Zap QapJaJp for half-odd-integer spin are in many ways the analogues of Berry's examples [4]. The relevant topological invariants are the first Chern number over a 2-sphere for Berry's examples, and the second Chern number over a 4-sphere for the quadrupoles.…”

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Chern numbers for fermionic quadrupole systems

Sadun

Segert

1989

J. Phys. A: Math. Gen.

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Abstract. We analyse families of quantum quadrupole Hamiltonians H = Z,, QnpJuJp for half-odd-integer spin, and calculate the second Chern numbers of the energy levels. Each non-zero integer occurs only a finite number of times. The adiabatic time evolution, the non-Abelian generalisation of Berry's phase, is different for each system, in contrast to Berry's example. The j = $ and j = f cases previously analysed are the only ones with self-dual curvatures and SO(5) symmetry.Geometrical and topological techniques applied to the study of time-dependent quantum Hamiltonians have recently generated much interest [l]. Berry's examples of a family of Hamiltonians of the form B . J display the diversity of phenomena. The degenerate, or non-Abelian, case has received much attention [2]. In the class of time-reversal-invariant fermionic Hamiltonians, which have Kramers degeneracy [3], the quadrupole systems Zap QapJaJp for half-odd-integer spin are in many ways the analogues of Berry's examples [4]. The relevant topological invariants are the first Chern number over a 2-sphere for Berry's examples, and the second Chern number over a 4-sphere for the quadrupoles. Chern numbers are defined for energy levels which have a fixed degree of degeneracy for all Hamiltonians in the family. The Chern numbers for quadrupoles with j s f are defined and have been previously computed [4].In this paper, we will calculate the second Chern numbers for all quadrupole systems with half-odd-integer spin. In fact, every topological invariant of twodimensional complex vector bundles over S4 is a function of the second Chern number, i.e. these bundles are classified by the second Chern number up to topological equivalence. It will be shown in [5] that the second Chern numbers are indeed well defined for all half-odd-integer j .An energy level can be specified by the eigenvalue of a particular Hamiltonian. It is convenient to take the quadrupole Hamiltonian Qo = J : -$J2 which commutes with J 3 . The energy level can then be labelled by ( j , mT), where j is the total angular momentum and m: is the eigenvalue of J:. We shall refer to this Hamiltonian as the north pole, and to minus this Hamilitonian as the south pole. The level can alternately be labelled by ( j , mB), where m', is the eigenvalue of J: at the south pole, with m B = j ++ -mT.Second Chern numbers over the 4-sphere will be calculated as the integral of the 4-form o = -Tr(fl A fl)/87r2, where fl is the curvature of the connection on the eigenstate bundle given by adiabatic time evolution [ 6 ] . As a first step we will reduce the integral of a general rotationally invariant 4-form over the 4-sphere of unit quadrupoles

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Chern numbers for fermionic quadrupole systems

Sadun

Segert

1989

J. Phys. A: Math. Gen.

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Ultrafast and Local Optoelectronic Transport in Topological Insulators

Kiemle

Seifert

Holleitner

et al. 2020

Physica Status Solidi (b)

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Recently, topological insulators (TIs) were discovered as a new class of materials representing a subset of topological quantum matter. While a TI possesses a bulk band gap similar to an ordinary insulator, it exhibits gapless states at the surface featuring a spin‐helical Dirac dispersion. Due to this unique surface band structure, TIs may find use in (opto)spintronic applications. Herein, optoelectronic methods are discussed to characterize, control, and read‐out surface state charge and spin transport of 3D TIs. In particular, time‐ and spatially‐resolved photocurrent microscopy at near‐infrared excitation can give fundamental insights into charge carrier dynamics, local electronic properties, and the interplay between bulk and surface currents. Furthermore, possibilities of applying such ultrafast optoelectronic methods to study Berry curvature‐related transport phenomena in topological semimetals are discussed.

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From the adiabatic theorem of quantum mechanics to topological states of matter

Budich

Trauzettel²

2013

Physica Rapid Research Ltrs

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Abstractmagnified imageOwing to the enormous interest the rapidly growing field of topological states of matter (TSM) has attracted in recent years, the main focus of this review is on the theoretical foundations of TSM. Starting from the adiabatic theorem of quantum mechanics which we present from a geometrical perspective, the concept of TSM is introduced to distinguish gapped many body ground states that have representatives within the class of non‐interacting systems and mean field superconductors, respectively, regarding their global geometrical features. These classifying features are topological invariants defined in terms of the adiabatic curvature of these bulk insulating systems. We review the general classification of TSM in all symmetry classes in the framework of K‐Theory. Furthermore, we outline how interactions and disorder can be included into the theoretical framework of TSM by reformulating the relevant topological invariants in terms of the single particle Green's function and by introducing twisted boundary conditions, respectively. We finally integrate the field of TSM into a broader context by distinguishing TSM from the concept of topological order which has been introduced to study fractional quantum Hall systems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Topological Invariants in Fermi Systems with Time-Reversal Invariance

Abstract: %'e discuss topological invariants

Cited by 110 publications

References 13 publications

Chern numbers for fermionic quadrupole systems

Chern numbers for fermionic quadrupole systems

Ultrafast and Local Optoelectronic Transport in Topological Insulators

From the adiabatic theorem of quantum mechanics to topological states of matter

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