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 in-sulator, which remains critical on the surface for odd values of the invariant.

Disorder can localize the eigenstates of one-dimensional non-Hermitian systems, leading to an Anderson transition with a critical exponent of 1. We show that, due to the lack of energy conservation, the dynamics of individual, real-space wave packets follows a different behavior. Both transitions between localization and unidirectional amplification, as well as transitions between distinct propagating phases become possible. The critical exponent of the transition is close to 1/2 in propagating-propagating and (de)localization transitions.

The non-Hermitian skin effect is a phenomenon in which an extensive number of states accumulates at the boundaries of a system. It has been associated to nontrivial topology, with nonzero bulk invariants predicting its appearance and its position in real space. Here, we demonstrate that the non-Hermitian skin effect has weaker bulk-edge correspondence than topological insulators: when translation symmetry is broken by a single non-Hermitian impurity, skin modes are depleted at the boundary and accumulate at the impurity site, without changing any bulk invariant. Similarly, a single non-Hermitian impurity may deplete the states from a region of Hermitian bulk.

The study of crystalline solids and condensed matter physics at large concerns itself with the new behaviors and phases of matter exhibited by elementary particles, atoms, and molecules by virtue of being assembled into a structure. These phases arise from complex microscopic behaviors, which makes it is difficult to establish rigorous quantitative models. The analysis of certain phases is greatly simplified in the presence of symmetries. These symmetries can be non-spatial (time-reversal, particle-hole) or spatial (rotation, inversion, reflection). For example, topological phases ofmatter are easily characterized and classified by the symmetries of the system. Symmetries constrain the band structure of a system, and as a result produce certain quantized responses, such as surface modes on an otherwise insulating bulk. Since these surface modes are related to the symmetry of the bulk, this phenomenon is known as bulk-edge correspondence. So long as the symmetries protecting the topological phase are respected and the energy gap of the insulating bulk remains open, bulk-edge correspondence persists in the presence of disorder. This disorder can be non-structural (appliedmagnetic field), involve part of the structure (impurities) or the entire structure, such as in amorphous systems.....

Protection of topological surface states by reflection symmetry breaks down when the boundary of the sample is misaligned with one of the high symmetry planes of the crystal. We demonstrate that this limitation is removed in amorphous topological materials, where the Hamiltonian is invariant on average under reflection over any axis due to continuous rotation symmetry. We show that the edge remains protected from localization in the topological phase, and the local disorder caused by the amorphous structure results in critical scaling of the transport in the system. In order to classify such phases we perform a systematic search over all the possible symmetry classes in two dimensions and construct the example models realizing each of the proposed topological phases. Finally, we compute the topological invariant of these phases as an integral along a meridian of the spherical Brillouin zone of an amorphous Hamiltonian.