Recent years have seen a growing interest in topological phases beyond the standard paradigm of gapped isolated systems. One recent direction is to explore topological features in non-Hermitian systems that are commonly used as effective descriptions of open systems. Another direction explores the fate of topology at critical points, where the bulk gap collapses. One interesting observation is that both systems, though very different, share certain topological features. For instance, both systems can host half-integer quantized winding numbers and have very similar entanglement spectra. Here we make this similarity explicit by showing the equivalence of topological invariants in critical systems with non-Hermitian point-gap phases, in the presence of sublattice symmetry. Also, the corresponding entanglement spectra show the same topological features. This correspondence may carry over to other features and even be helpful to deepen our understanding of non-Hermitian systems using our knowledge of critical systems and vice versa.

Motivated by several claims of spin-orbit-driven spin-liquid physics in hexagonal Ba3Ti3-xIrxO9 hosting Ir2O9 dimers, we report on resonant inelastic x-ray scattering (RIXS) at the Ir L3 edge for different x. We demonstrate that magnetism in Ba3Ti3-xIrxO9 is governed by an unconventional realization of strong disorder, where cation disorder affects the character of the local moments. RIXS interferometry, studying the RIXS intensity over a broad range of transferred momentum q, is ideally suited to assign different excitations to different Ir sites. We find pronounced Ir-Ti site mixing. Both ions are distributed over two crystallographically inequivalent sites, giving rise to a coexistence of quasimolecular singlet states on Ir2O9 dimers and spin-orbit-entangled j=1/2 moments of 5d5Ir4+ ions. RIXS reveals different kinds of strong magnetic couplings for different bonding geometries, highlighting the role of cation disorder for the suppression of long-range magnetic order in this family of compounds.

The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.

Various types of topological phenomena at criticality are currently under active research. In this paper we suggest to generalize the known topological quantities to finite temperatures, allowing us to consider gapped and critical (gapless) systems on the same footing. It is then discussed that the quantization of the topological indices, also at critically, is retrieved by taking the low-temperature limit. This idea is explicitly illustrated on a simple case study of chiral critical chains where the quantization is shown analytically and verified numerically. The formalism is also applied for studying robustness of the topological indices to various types of disordering perturbations.

The mixed-valent iridate Ba3InIr2O9 has been discussed as a promising candidate for quantum spin-liquid behavior. The compound exhibits Ir4.5+ ions in face-sharing IrO6 octahedra forming Ir2O9 dimers with three t(2g) holes per dimer. Our results establish Ba3InIr2O9 as a cluster Mott insulator. Strong intradimer hopping delocalizes the three t(2g) holes in quasimolecular dimer states while interdimer charge fluctuations are suppressed by Coulomb repulsion. The magnetism of Ba3InIr2O9 emerges from spin-orbit entangled quasimolecular moments with yet unexplored interactions, opening up a new route to unconventional magnetic properties of 5d compounds. Using single-crystal x-ray diffraction we find the monoclinic space group C2/c already at room temperature. Dielectric spectroscopy shows insulating behavior. Resonant inelastic x-ray scattering reveals a rich excitation spectrum below 1.5 eV with a sinusoidal dynamical structure factor that unambiguously demonstrates the quasimolecular character of the electronic states. Below 0.3 eV, we observe a series of excitations. According to exact diagonalization calculations, such low-energy excitations reflect the proximity of Ba3InIr2O9 to a hopping-induced phase transition based on the condensation of a quasimolecular spin-orbit exciton. The dimer ground state roughly hosts two holes in a bonding j = 1/2 orbital and the third hole in a bonding j = 3/2 orbital.

In Ca₂RuO₄, the competition of spin-orbit coupling ζ and tetragonal crystal-field splitting Δ_CF has been discussed controversially for many years. The orbital occupation depends on Δ_CF/ζ, which allows us to address this ratio via the optical spectral weights of the lowest intersite Mott-Hubbard excitations. We study the optical conductivity of Ca₂Ru_0.99Ti_0.01O₄ in the range of 0.75–5 eV by ellipsometry, using the large single crystals that can be grown for small Ti concentrations. Based on a local multiplet calculation, our analysis results in 2.4≤Δ_CF/ζ≲4 at 15 K. The dominant crystal field yields a ground state close to xy orbital order but spin-orbit coupling is essential for a quantitative description of the properties. Furthermore, we observe a pronounced decrease of Δ_CF with increasing temperature, as expected based on the reduction of octahedral distortions.

The bulk polarization is a Z(2) topological invariant characterizing noninteracting systems in one dimension with chiral or particle-hole symmetries. We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it. In the symmetric case, the known relation between the bulk polarization and the number of virtual topological edge modes is recovered. We use the bulk polarization to compute Chern numbers in one and two dimensions, which illuminates their known relation to the entanglement spectrum. Furthermore, we discuss an alternative bulk polarization that can carry more information about the surface spectrum than the conventional one and can simplify the calculation of Chern numbers.

Chiral spin liquids (CSLs) are time-reversal-symmetry-breaking ground states of frustrated quantum magnets that show no long-range magnetic ordering but instead exhibit topological order and fractional excitations. Their realization in simple and tractable microscopic models has, however, remained an open challenge for almost two decades until it was realized that Kitaev models on lattices with odd-length loops are natural hosts for such states, even in the absence of a time-reversal-symmetry-breaking magnetic field. Here we report on the formation of CSLs in a three-dimensional Kitaev model on a hypernonagon lattice composed of nine-site loops, which differ from their widely studied two-dimensional counterparts; namely, they exhibit a crystalline ordering of the Z(2) gauge fluxes and thereby break some of the underlying lattice symmetries. We study the formation of these unconventional CSLs via extensive quantum Monte Carlo simulations and demonstrate that they are separated from the featureless paramagnet at high temperatures by a single first-order phase transition at which both time-reversal and lattice symmetries are simultaneously broken. Using variational approaches for the ground state, we explore the effect of varying the Kitaev couplings and find at least five distinct CSL phases, all of which possess crystalline ordering of the Z(2) gauge fluxes. For some of these phases, the complementary itinerant Majorana fermions exhibit gapless band structures with topological features such as Weyl nodes or nodal lines in the bulk and Fermi arc or drumhead surface states.

We propose a Ginzburg-Landau theory for a large and important part of the abelian quantum Hall hierarchy, including the prominently observed Jain sequences. By a generalized "flux attachment" construction we extend the Ginzburg-Landau-Chern-Simons composite boson theory to states obtained by both quasielectron and quasihole condensation, and express the corresponding wave functions as correlators in conformal field theories. This yields a precise identification of the relativistic scalar fields entering these correlators in terms of the original electron field.

The honeycomb compound alpha-RuCl3 is widely discussed as a proximate Kitaev spin-liquid material. This scenario builds on spin-orbit entangled j = 1/2 moments arising for a t(2g)(5) electron configuration with strong spin-orbit coupling lambda and a large cubic crystal field. The actual low-energy electronic structure of alpha-RuCl3, however, is still puzzling. In particular, infrared absorption features at 0.30, 0.53, and 0.75 eV seem to be at odds with a j = 1/2 scenario. Also the energy of the spin-orbit exciton, the excitation from j = 1/2 to 3/2, and thus the value of lambda, are controversial. Combining infrared and Raman data, we show that the infrared features can be attributed to single, double, and triple spin-orbit excitons. We find lambda = 0.16 eV and Delta = 42(4) meV for the observed noncubic crystal-field splitting, supporting the validity of the j = 1/2 picture for alpha-RuCl3. The unusual strength of the double excitation is related to the underlying hopping interactions, which form the basis for dominant Kitaev exchange.