Homotopy and Quantization in Condensed Matter Physics

Abstract

It is shown that the integers found by Thouless

Elementary formula for the Hall conductivity of interacting systems

“…[2][3][4][5][6] For example, the relation between the quantized number C and the non-degenerate many-body ground state wave functions |Ψ(φ, ϕ) obeying twisted boundary conditions parametrized by the pair of angles 0 ≤ φ, ϕ ≤ 2π for a gas of electrons confined in twodimensional (position) space and subjected to a uniform magnetic field is C = − i 2π …”

Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands

“…24 We will call this the "sum rule". The current distribution among subbands, which is also quantized in each subband, was obtained by Thouless et al 11 In their famous paper, they found that the subband Hall conductivities are the integer-valued solutions of the Diophantine equation.…”

Noncommutative geometry for three-dimensional topological insulators

“…Here, the symbol CS (3) (3) to the noncommutative algebra obeyed by the components of the projected position operator through the noninteracting groundstate expectation value of their 3-bracket. Because the position operator and its projection are unbounded operators and because Wannier states may not be exponentially localized if the Bloch states have a topological character, 60,61 a regularization procedure is needed to compute Eq.…”

“…The index in this case is the sum of the (first) Chern numbers obtained for each of the fully filled Landau bands, and the associated physical observable is the Hall conductance. [2][3][4] The fractional quantum Hall effect (FQHE) 5 results from the effects of electron-electron interactions when the Landau levels are partially filled with electrons, for certain rational filling fractions. 6 More examples of topological states of matter that are comprised of noninteracting fermions have been discovered recently, [7][8][9][10][11][12][13][14][15][16] and have been classified according to discrete symmetries they respect or not, and the dimensionality of space in which the particles propagate.…”

“…The operator over which the trace is taken on the right-hand side of Eq. (A60) can be rewritten in a way that brings it to a linear combination of 3-brackets, thereby justifying the upper index (3) for the functional F (3) gauge invariant [A] over the manifold of su( N ) gauge fields. Indeed, we are allowed to reorder the 3 × 6 = 18 operators over which the trace is taken on the first three lines of the right-hand side of Eq.…”