Isotropic 3D topological phases with broken time reversal symmetry

Abstract

Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a three-dimensional medium with average isotropy and inversion symmetry where time-reversal symmetry is systematically broken. We devise a model of an amorphous material with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the bonds. The presence of only average spatial symmetries -- continuous rotation and inversion -- is sufficient to protect a topological phase, yielding a statistical topological insulator. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant for the effective continuum model, which guarantees gapless surface spectrum on any surface with an odd number of Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.

Figures (9)

• Figure 1: The (a) bulk and (b)-(e) surface spectral functions of the amorphous tight-binding models. (b)-(c) The surface spectral functions of the $4\times 4$ model \ref{['eqn:real_space_model_simp_4x4']} and the doubled $8\times 8$ model \ref{['eqn:real_space_model_8x8']}. (d)-(e) the same models as (b)-(c) but with broken spatial (mirror and rotation) symmetries. Plot details are in App. \ref{['appendix:plots']}.
• Figure 2: Time-reversal symmetry breaking in a microscopic system with inversion and rotation symmetry. (a) A bond between $s$ and $p$ orbitals hosting four mid-bond $s$ orbitals (on plane shown in green) that host magnetic moments. (b) A section of a rock salt crystal structure made from the bond shown in (a). Red lines indicate nearest-neighbor hopping between $s$ and $p$ orbitals, dashed lines indicate second neighbor hopping between $s$ (purple) and $p$ (blue) orbitals, green lines indicate third neighbor hopping between $s$ and $p$ orbitals. (c) The bulk dispersion relation obtained from the crystal structure shown in (b) along the high-symmetry points of the face-centered cubic Brillouin zone. Different colors indicate different bands. (d) Bulk and surface dispersion of a 3D slab of the crystal. Darker color indicates a larger participation ratio. Plot details are in App. \ref{['appendix:plots']}.
• Figure 3: Topological phase transitions of the doubled class A amorphous tight-binding model \ref{['eqn:real_space_model_8x8']} as a function of chemical potentials $\mu_1 = -\mu_2$, using parameters $t_3(d) = 1.2 \exp(-d) + i \exp(-0.3d)$, $t_1(d) = -2 \exp(-d)$, $t_2(d) = 2 \exp(-d)$. Top panel: Bulk density of states, showing gap closings and gapped phases as a function of the chemical potential. Brighter colors denote higher density of states in arbitrary units. Bottom panel: Topological invariants $C_M$ (defined in \ref{['eqn:bcurv_invariant']} and $\nu_I$\ref{['eqn:inversion_invariant']}. Plots are offset for clarity.
• Figure 4: Topological phase transitions of the doubled class A amorphous tight-binding model \ref{['eqn:real_space_model_8x8']} as a function of chemical potentials $\mu_1 = -\mu_2 = \mu_3 = -\mu_4$, see Appendix \ref{['appendix:plots']} for the other parameters used. Top panel: Bulk density of states, showing several gap closings and gapped phases as a function of the chemical potential. Brighter colors denote higher density of states in arbitrary units. Bottom panel: Topological invariants $C_M$ and $\nu_I$. Plots are offset for clarity.
• Figure 5: Conductivity of translationally invariant and amorphous networks. (a) Schematic of the Chalker-Coddington model. Dashed links loop in the vertical direction to indicate periodic boundary conditions. Circular nodes indicate external nodes where modes enter and exit the network. Internal nodes are located at all solid line crossings. (b) Schematic of the amorphous network. Circular nodes indicate external nodes where modes enter and exit the network. Nodes internal to the network are located at all line crossings. (c) Schematic of modes in the doubled model. (d) Average conductivity of the networks as a function of network length and width $L$ and fits (dashed lines). Results are shown for the Chalker-Coddington (CC) network and amorphous network, with 1 mode per link (crosses) and 2 modes per link (diamonds). Plot details are in App. \ref{['appendix:plots']}.