Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter

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Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter

Todd Van Mechelen and Zubin Jacob

Phys. Rev. B 102, 155425 – Published 26 October 2020

Abstract

Chern-Simons theories have been very successful in explaining integer and fractional quantum Hall phases of matter, topological insulators, and Weyl semimetals. However, it remains an open question as to whether Chern-Simons theories can be adapted to topological photonics. We develop a viscous Maxwell-Chern-Simons theory to capture the fundamental physics of a topological electromagnetic phase of matter. We show the existence of a unique spin-1 skyrmion in the viscous Hall fluid arising from a photonic Zeeman interaction in momentum space. Our work bridges the gap between electromagnetic and condensed matter topological physics while also demonstrating the central role of photon spin-1 quantization in identifying new phases of matter.

Received 16 April 2020
Revised 4 October 2020
Accepted 6 October 2020

DOI:https://doi.org/10.1103/PhysRevB.102.155425

Physics Subject Headings (PhySH)

Exotic phases of matter Hall effect Polaritons Topological materials

Viscosity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Todd Van Mechelen and Zubin Jacob^*

Purdue University, School of Electrical and Computer Engineering, Birck Nanotechnology Center, West Lafayette, Indiana 47907, USA

^*zjacob@purdue.edu

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Issue

Vol. 102, Iss. 15 — 15 October 2020

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Images

Figure 1
(a) Summary of the four quantum Hall regimes. Hall quantization and plateauing behavior has been demonstrated in both static and dynamical regimes. However, topological electromagnetic phases $C_{e m} \neq 0$ are only realized in the dynamical $+$ viscous (nonlocal) regimes. (b) Overview of viscous Maxwell-Chern-Simons theory. The bulk topology is governed by a spin-1 photonic skyrmion in momentum space which arises from viscous Hall conductivity $σ_{x y} (k) = λ (κ - ξ k^{2})$ . The arrows represent the direction of the effective spin $\hat{d}$ of the photon. The boundary of the nontrivial phase $κ ξ > 0$ hosts topologically protected chiral photons which are linearly dispersing (massless).
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Figure 2
Bulk and edge dispersion of (a) continuum and (b) lattice models of viscous Maxwell-Chern-Simons theory. Cyan and magnetic lines are positive- and negative-energy topological bands while the black line is the chiral edge state. (a) Parameters are $κ = ξ = 1$ in the continuum theory $a \to 0$ . (b) Parameters are $κ a = ξ / a = 1$ in the lattice theory $a \neq 0$ .
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Figure 3
Topological phase diagrams for (a) continuum and (b) lattice models of viscous Maxwell-Chern-Simons theory. $C_{e m} = \pm 2, 0$ is the photonic Chern number of the positive energy band $ω > 0$ for different parameters. $κ$ and $ξ$ are the Chern-Simons and viscous Chern-Simons coupling, respectively. $a$ is the lattice constant of a square grid. $κ a^{2} = 0, 4 ξ, 8 ξ$ denote the phase transition lines in the lattice model. These correspond to points of accidental degeneracy, where the band gap closes at $k = Γ, X / Y, M$ , respectively. Importantly, conventional MCS theory $ξ = 0$ always corresponds to a topologically trivial phase $C_{e m} = 0$ in the lattice regularization.
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Figure 4
(a) Unit cell of a square lattice with the primitive Wigner-Seitz cell shown in yellow. (b), (c), (d) The Brillouin zone of the three phases $C_{e m} = \pm 2, 0$ in the lattice regularized theory. $κ > 0$ and $ξ > 0$ are chosen positive such that (b) and (c) label type-I and type-II photonic skyrmions, respectively. (d) The photonic ferromagnet. The eigenvalue at high-symmetry points denotes the sign of the Maxwell-Chern-Simons mass $j_{m} = sgn (Λ) = \pm 1$ , which determines the spin-1 representation—if the field is right ( $+ 1$ ) or left ( $- 1$ ) circularly polarized. The two nontrivial phases possess skyrmion numbers of $N = \pm 1$ corresponding to a spin-1 Chern number of $C_{e m} = 2 N = \pm 2$ .
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Figure 5
The two boundary conditions for the viscous Hall fluid that minimize the surface variation $δ S_{s} = 0$ at $x = 0$ . (a) Schematic of the truncated atomic lattice at $x = 0$ . (b), (c) Plots of the normalized energy density $u = | \vec{F} |^{2} = {| E |}^{2} + {| B_{z} |}^{2}$ of the chiral photonic edge state. The parameters are $κ a = 0.1$ , $ξ / a = 0.2$ , and $k_{y} a = 0.1$ as a demonstration. (b) The Dirichlet (open) boundary condition $\vec{F} (0) = 0$ has zero measure at $x = 0$ . (c) The natural boundary condition $v_{x} \vec{F} (0) = 0$ is more localized at the surface and resembles an evanescent wave.
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