The observed fractional quantum Hall (FQH) plateaus follow a recurring hierarchical structure that allows an understanding of complex states based on simpler ones. Condensing the elementary quasiparticles of an Abelian FQH state results in a new Abelian phase at a different filling factor, and this process can be iterated ad infinitum. We show that condensing clusters of the same quasiparticles into an Abelian state can instead realize non-Abelian FQH states. In particular, condensing quasiparticle pairs in the ν=2/3 Laughlin state yields the anti-Pfaffian phase at half filling. We moreover show that the successive condensation of Laughlin quasiparticles produces quantum Hall states whose fillings coincide with the most prominent plateaus in the first excited Landau level of GaAs. More generally, such condensation can realize any non-Abelian FQH state that admits a parton representation. This surprising result is supported by an exact analysis of explicit wave functions, field theory arguments, conformal-field theory constructions of trial states, and numerical simulations.

The insulating mixed-valent Ir+3.66 compound Ba4NbIr3O12 hosts two holes per Ir3O12 trimer unit. We address the electronic structure via resonant inelastic x-ray scattering (RIXS) at the Ir L3 edge and exact diagonalization. The holes occupy quasimolecular orbitals that are delocalized over a trimer. This gives rise to a rich intra-t2g excitation spectrum that extends from 0.5 eV to energies larger than 2 eV. Furthermore, it yields a strong modulation of the RIXS intensity as a function of the transferred momentum q. A clear fingerprint of the quasimolecular trimer character is the observation of two modulation periods, 2π/d and 2π/2d, where d and 2d denote the intratrimer Ir-Ir distances. We discuss how the specific modulation reflects the character of the wave function of an excited state. Our quantitative analysis shows that spin-orbit coupling λ of about 0.4 eV is decisive for the character of the electronic states, despite a large hopping ta1g of about 0.8 eV. The ground state of a single trimer is described very well by both holes occupying the bonding j=12 orbital, forming a vanishing quasimolecular moment with J=0.

We study the effect of disorder on 𝑍₂ quantum spin liquids with a Majorana Fermi line (respectively a surface in three dimensions), and we show that depending on the symmetries that are preserved on average, qualitatively different scenarios will occur. In particular, we identify the relevant non-Hermitian symmetries for which disorder will effectively split the Fermi line into two exceptional lines, with Re⁡(𝐸)=0 states filling the area in between. We demonstrate the different scenarios using both toy models as well as large-scale numerical simulations.

Recent quantum Hall experiments have observed "daughter states"next to several plateaus at half-integer filling factors in various platforms. These states were first proposed based on model wave functions for the Moore-Read state by Levin and Halperin. We show that these daughters and their parents belong to an extensive family tree that encompasses all pairing channels and permits a unified description in terms of weakly interacting composite fermions. Each daughter represents a bosonic integer quantum Hall state formed by composite-fermion pairs. The pairing of the parent dictates an additional number of filled composite-fermion Landau levels. We support our field-theoretic composite-fermion treatment by using the K-matrix formalism, analysis of trial wave functions, and a coupled-wire construction. Our analysis yields the topological orders, quantum numbers, and experimental signatures of all daughters of paired states at half-filling and "next-generation"even denominators. Crucially, no two daughters share the same two parents. The unique parentage implies that Hall conductance measurements alone could pinpoint the topological order of even-denominator plateaus. Additionally, we propose a numerically suitable trial wave function for one daughter of the SU(2)2 topological order, which arises at filling factor ν=611. Finally, our insights explain experimentally observed features of transitions in wide-quantum wells, such as suppression of the Jain states with the simultaneous development of half-filled and daughter states.

Quasimolecular orbitals in cluster Mott insulators provide a route to tailor exchange interactions, which may yield novel quantum phases of matter. We demonstrate the cluster Mott character of the lacunar spinel GaTa₄Se₈ using resonant inelastic x-ray scattering (RIXS) at the Ta L₃ edge. Electrons are fully delocalized over Ta4 tetrahedra, forming quasimolecular J_tet=3/2 moments. The modulation of the RIXS intensity as function of the transferred momentum q allows us to determine the cluster wave function, which depends on competing intracluster hopping terms that mix states with different character. This mixed wave function is decisive for the macroscopic properties since it affects intercluster hopping and exchange interactions and furthermore renormalizes the effective spin-orbit coupling constant. The versatile wave function, tunable via intracluster hopping, opens a new perspective on the large family of lacunar spinels and cluster Mott insulators in general.

The half-filled 𝑡_2⁢𝑔 shell of the configuration usually, in 𝐿⁢𝑆 coupling, hosts a 𝑆=3/2 ground state with quenched orbital moment. This state is not Jahn-Teller active. Sufficiently large spin-orbit coupling 𝜁 has been predicted to change this picture by mixing in orbital moment, giving rise to a sizable Jahn-Teller distortion. In 5⁢𝑑³K_2⁢ReCl₆ we study the electronic excitations using resonant inelastic x-ray scattering and optical spectroscopy. We observe on-site intra-𝑡_2⁢𝑔 excitations below 2 eV and corresponding overtones with two intra-𝑡_2⁢𝑔 excitations on adjacent sites, the Mott gap at 2.7 eV, 𝑡_2⁢𝑔-to-𝑒_𝑔 excitations above 3 eV, and charge-transfer excitations at still higher energy. The intra-𝑡_2⁢𝑔 excitation energies are a sensitive measure of 𝜁 and Hund's coupling 𝐽_H. The sizable value of 𝜁≈0.29eV places K₂⁢ReCl₆ into the intermediate coupling regime, but 𝜁/𝐽_H≈0.6 is not sufficiently large to drive a pronounced Jahn-Teller effect. We discuss the ground state wave function in a Kanamori picture and find that the 𝑆=3/2 multiplet still carries about 97% of the weight. However, the finite admixture of orbital moment allows for subtle effects. We discuss small temperature-induced changes of the optical data and find evidence for a lowering of the ground state by about 3 meV below the structural phase transitions.

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 5d5 Ir4+ 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.