Topological field theory of time-reversal invariant insulators

Abstract

We show that the fundamental time reversal invariant (TRI) insulator exists in 4 + 1 dimensions, where the effective field theory is described by the 4 + 1 dimensional Chern-Simons theory and the topological properties of the electronic structure is classified by the second Chern number. These topological properties are the natural generalizations of the time reversal breaking (TRB) quantum Hall insulator in 2 + 1 dimensions. The TRI quantum spin Hall insulator in 2 + 1 dimensions and the topological insulator in 3 + 1 dimension can be obtained as descendants from the fundamental TRI insulator in 4 + 1 dimensions through a dimensional reduction procedure. The effective topological field theory, and the Z2 topological classification for the TRI insulators in 2 + 1 and 3 + 1 dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of novel and measurable phenomena, the most striking of which is the topological magneto-electric effect, where an electric field generates a magnetic field in the same direction, with an universal constant of proportionality quantized in odd multiples of the fine structure constant α = e 2 / c. Finally, we present a general classification of all topological insulators in various dimensions, and describe them in terms of a unified topological Chern-Simons field theory in phase space. ContentsA y = −E y t, A x = 0.

Ripple-modulated electronic structure of a 3D topological insulator

“…An intriguing consequence of our observations is that in a magnetic field, such periodic chemical potential variations can play host to 1D chiral metallic modes. Analogous to edge states in an integer quantum Hall system, 1D modes are predicted at the spatial boundaries between two regions with successive half-integer fillings 2,6 . These modes are interesting as practical realizations of 1D quantum wire states and can potentially be used to achieve dissipationless transport in 1D (Fig.…”

“…T he helical Dirac fermions that exist on the surface of topological insulators (TIs) are condensed matter analogues of relativistic fermions extensively studied in high-energy physics [1][2][3][4][5][6] . Many theoretical scenarios in high-energy physics such as Majorana fermions and axion electrodynamics are also anticipated to manifest experimental signatures in TIs 2, 7-9 .…”

Topological states in a ladder-like optical lattice containing ultracold atoms in higher orbital bands

“…This Hamiltonian belongs to the symmetry group G þ (U, C) and allows a Z 2 classification of its topological properties 25 . Even though time-reversal symmetry is absent, particle-hole symmetry still ensures that Berry phase g is quantized, with g mod 2p ¼ 0 or p defining the trivial and topological insulator, respectively 29 phase quantization is robust against the time-reversal symmetry breaking term D y , and this topological phase is protected by particle-hole symmetry. For D y greater than D c y , Berry phase vanishes and the system becomes a trivial band insulator.…”

“…At the critical point the band gap closes. Apart from the Berry phase, the topological distinction between H 0 ðkÞ and HðkÞ can also be seen from a gapped interpolation 29 as discussed in Supplementary Note 3. Besides probing the half charges on the boundaries, another signature for the critical point of the topological phase transition is the local density fluctuation, dr…”

Topological protection of bound states against the hybridization

“…Bulk-edge correspondence guarantees the existence of these edge channels, that is, they are protected by the gap and the non-trivial topology of the bulk states 9,10 . A recent remarkable advance in the study of threedimensional (3D) topological insulators, which support the helical Dirac fermion on the surface of the sample, provides another example of bound states belonging to the same category [11][12][13][14] .…”