arXiv:1909.05868 · Physical review. B./Physical review. B

Multipolar topological field theories: Bridging higher order topological insulators and fractons

Yizhi You, F. J. Burnell, Taylor L. Hughes

Published Jun 16, 2021• 75 citations

Abstract

The concepts of multipolar insulators and multipole symmetries have hinted at tantalizing connections between the disparate fields of higher-order topological insulators and fracton phases. Here, the authors show that these connections are not superficial, but instead arise from a unified field theory description of the topological response of these systems to external fields, similar to the quantized Hall conductance of Chern insulators, and the quantized axion magnetoelectric polarizability in 3D topological insulators. These response theories are built from the recently introduced higher-rank tensor gauge potentials, and can be interpreted, in some scenarios, as the topological response to ordinary electric and magnetic field gradients.

DOI: 10.1103/physrevb.103.245128

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arXiv:1909.05868 · Physical review. B./Physical review. B

Multipolar topological field theories: Bridging higher order topological insulators and fractons

Yizhi You, F. J. Burnell, Taylor L. Hughes

Published Jun 16, 2021• 75 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.103.245128

Related papers

Field theory of higher-order topological crystalline response, generalized global symmetries and elasticity tetrads

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Andrew Stewart Caldin

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A_{ij}

(symmetric tensor) with gauge transformation

\delta A_{ij} = \partial_i \partial_j \alpha

; fractonic Chern-Simons action:

S = \frac{k}{4\pi} \int \epsilon^{\mu\nu\rho} A_{\mu} \partial_\nu A_{\rho}

generalized to higher-rank; fracton charge conservation:

\partial_i \partial_j J_{ij} = 0

; dipole moment conservation:

\int x^i \rho

constant. | CONNECTION: Crystallographic group symmetries (e.g., cubic, hexagonal) constrain allowed higher-rank tensor gauge fields; lattice structures (e.g., square, triangular) yield subdimensional particle mobility; no direct golden ratio or base-60 link found. | DEPTH: 8 — Fractonic Chern-Simons theory unifies higher-rank gauge theory with topological order, revealing new quantum phase Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

A time-reversal invariant topological insulator is defined by its topological magnetoelectric response that is robust against disorder. The response formula, defined on a Brillouin torus, defines a

mathbb{Z}_2

invariant and classifies the topological phase. However, in the presence of disorder or the magnetic field, the notion of Brillouin torus is destroyed and the response formula is no longer well-defined. This has been a challenging open problem, and it is essental in defining a topological insulator. This thesis proposes a topological response theory that is free from this fundamental deficiency. We derived the magnetoelectric response formula in position space for a generic three dimensional model under disorder and finite magnetic field. For time-reversal invariant systems, we connected the result to the 2nd Chern number in Noncommutative Geometry. We developed the noncommutative theory of Chern numbers and showed that the quantization of the magnetoelectric response is robust against disorder. Numerical studies on serveral disodered topological models in 1D and 3D are presented.

F[\phi] = r\phi^2 + u\phi^4 + \cdots

, broken symmetry group

G \to H

; topological order: ground-state degeneracy

\text{GSD} \sim e^{cL}

(fractons) or

\text{GSD} \sim L^g

(toric code,

g

= genus); Chern–Simons action

S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A \wedge dA + \frac{2}{3}A^3)

; fracton gauge theory:

S = \int (B_{ij}\partial_t A_{ij} - \text{...})

with symmetric tensor gauge fields

A_{ij}

. | CONNECTION: Fracton phases exhibit sub-dimensional particle motion — point excitations confined to lines or planes — mirroring the geometry of root systems and lattice symmetries (e.g., cubic lattices with

C_4

rotational symmetry); the

k

level in Chern–Simons theory is quantized (integer), linking to mo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com