The unrivaled robustness of topologically ordered states of matter against perturbations has immediate applications in quantum computing and quantum metrology, yet their very existence poses a challenge to our understanding of phase transitions. In particular, topological phase transitions cannot be characterized in terms of local order parameters, as it is the case with conventional symmetry-breaking phase transitions. Currently, topological order is mostly discussed in the context of nonlocal topological invariants or indirect signatures like the topological entanglement entropy. However, a comprehensive understanding of what actually constitutes topological order enabling precise quantitative statements is still lacking. Here we show that one can interpret topological order as the ability of a system to perform topological error correction. We find that this operational approach corresponding to a measurable quantity both lays the conceptual foundations for previous classifications of topological order and also leads to a successful classification in the hitherto inaccessible case of topological order in open quantum systems. We demonstrate the existence of topological order in open systems and their phase transitions to topologically trivial states. Our results demonstrate the viability of topological order in nonequilibrium quantum systems and thus substantially broaden the scope of possible technological applications.

We present a systematic study on the structural and magnetic phase transitions in quasi-two-dimensional (2D) material ${\mathrm{VBr}}_3$ through various experimental techniques. With decreasing the temperature, ${\mathrm{VBr}}_3$ first undergoes a structural phase transition at around 90 K and then changes into the antiferromagnetic (AFM) state below 26.5 K. Interestingly, in addition to the strong AFM coupling background below 26.5 K, our ${\mathrm{VBr}}_3$ single crystals also exhibit robust hysteresis loops with coercive field about 1.5 T at 2 K for $H//c$, but none for $H//ab$. This is a typical feature of canted AFM and should not originate from a possible ferromagnetic (FM) impurities phase, because FM materials such as ${\mathrm{VI}}_3$ always have clear hysteresis loops in all magnetic field directions. Magnetization measurements of angle for rotation in both $ab$ and $ac$ planes also suggest ${\mathrm{VBr}}_3$ as a canted AFM with easy axis slightly off the $c$ axis. Furthermore, detailed Raman spectroscopy measurements also reveal the structural phase transition at 90 K. More importantly, splitting of a doubly degenerated $E_{\mathrm{g}}$ mode is observed, which demonstrates threefold rotational symmetry breaking in the low-temperature phase. Considering the similar structural phase transition of isostructural materials, we conclude that ${\mathrm{VBr}}_3$ changes from $R\overline{3}$ structure to $C2/m$ structure with stacking order change as temperature decreases.

The quantum confinement of Rydberg excitons (REs) in quantum structures opens the way towards considering nonlinear interactions in such systems. We present a theoretical calculation of optical functions in the case of a nonlinear coupling between REs in a quantum well with an electromagnetic wave. Using the real density matrix approach (RDMA), the analytical expressions for a linear and nonlinear absorption are derived and numerical calculations for ${\mathrm{Cu}}_2\mathrm{O}$ quantum wells are performed. The results indicate the conditions in which quantum well confinement states can be observed in linear and nonlinear optical spectra. The Kerr nonlinearity and self-phase modulation in such a system are studied. The effect of the Rydberg blockade and the associated optical bleaching are also discussed and confronted with available experimental data.

We study the $S=\frac{1}{2}$ pyrochlore Heisenberg antiferromagnet in a magnetic field. Using large-scale density-matrix renormalization group calculations for clusters with up to 128 spins, we find indications of a finite triplet gap, causing a threshold field to nonzero magnetization in the magnetization curve. We obtain a robust saturation field consistent with a magnon crystal, although the corresponding $5/6$ magnetization plateau is very slim and possibly unstable. Most remarkably, there is a pronounced and apparently robust $1/2$ magnetization plateau where the ground state breaks the rotational symmetry of the lattice, exhibiting oppositely polarized spins on alternating kagome and triangular planes. Reminiscent of the kagome ice plateau of the pyrochlore Ising antiferromagnet known as spin ice, it arises via a much more subtle “quantum order by disorder” mechanism.

Crystallographically complex compounds containing $4f$ and $5f$ electrons often have peculiar chemical and physical properties. In this work we present discovery and characterization of a new material ${\mathrm{UHg}}_{6.4}$, located at the border to complex intermetallics. This material crystallizes in the ${\mathrm{LaHg}}_{6.4}$ structure type, which can be represented by La/U-centered polyhedra with coordination numbers of 13 and 14. Much like the ${\mathrm{LaHg}}_{6.4}$ analog, ${\mathrm{UHg}}_{6.4}$ shows strong crystallographic disorder, in particular for Hg atoms in the channels along [001]. The ${\mathrm{UHg}}_{6.4}$ compound orders antiferromagnetically below $T_{\text{N1}}=35.5\pm 1\mathrm{K}$ and displays another transition at $T_{\text{N2}}=47.3\pm 1\mathrm{K}$, which is likely also antiferromagnetic. Both transitions are only slightly affected by magnetic field. Given high air sensitivity of ${\mathrm{UHg}}_{6.4}$, an exceptional experimental environment for sample synthesis and characterization was necessary in order to comprehensively describe the chemical and physical properties of this system.

We report on terahertz high-harmonic generation in a Dirac semimetal as a function of the driving-pulse ellipticity and on a theoretical study of the field-driven intraband kinetics of massless Dirac fermions. Very efficient control of the third-harmonic yield and polarization state is achieved in electron-doped ${\mathrm{Cd}}_3{\mathrm{As}}_2$ thin films at room temperature. The observed tunability is understood as resulting from terahertz-field-driven intraband kinetics of the Dirac fermions. Our study paves the way for exploiting the nonlinear optical properties of Dirac matter for applications in signal processing and optical communications.

We investigate dissipationless magneto-Seebeck effect in three-dimensional Dirac and Weyl semimetals. The Hall resistivity ${\rho }_{yx}$ and thermoelectric Hall coefficient ${\alpha }_{xy}$ exhibit plateaus at the quantum limit, where electrons occupy only the zeroth Landau level. In this condition, quantum oscillation in the Seebeck coefficient $S_{xx}\approx {\rho }_{yx}{\alpha }_{xy}$ is suppressed, and the massless fermions are transformed into a Fermi liquid system. We show that the Seebeck coefficient at the quantum limit is expressed by the harmonic sum of Fermi wavelength and thermal de Broglie wavelength scaled by magnetic length.

Excitons in two-dimensional semiconductors provide a novel platform for fundamental studies of many-body interactions. In particular, dipolar interactions between spatially indirect excitons may give rise to strongly correlated phases of matter that so far have been out of reach of experiments. Here we show that excitonic few-body systems in atomically thin transition-metal dichalcogenides confined to a one-dimensional geometry undergo a crossover from a Tonks-Girardeau to a charge density wave regime. To this end, we take into account realistic system parameters and predict the effective exciton-exciton interaction potential. We find that the pair-correlation function contains key signatures of the many-body crossover already at small exciton numbers and show that photoluminescence spectra provide readily accessible experimental fingerprints of these strongly correlated quantum many-body states.

To determine the impact of including edge states on the phase diagram of a spinless Falicov-Kimball model on the Haldane lattice, a study of a corresponding ribbon geometry with zigzag edges is conducted. By varying the ribbon widths, the distinction between the effects connected to the mere presence of the edges and those originating from interference between the edge states is established. The local doping caused by the former is shown to give rise to a topologically trivial bulk insulator with metallic edge states. Additionally, it gives rise to a charge density wave (CDW) phase with mixed character of the subbands in various parts of the phase diagram. The impact of local doping on the CDW instability is also addressed. Two additional gapless phases are found, caused by the edges but with stability regions depending on the width of the ribbon.

The phase diagram of the Kondo lattice model is explored around the ferromagnetic region, including an external magnetic field, and allowing the coexistence of ferromagnetism and Kondo effect ($\mathrm{FM}+\mathrm{K}$). Considering the combined effect of temperature and pressure, the evolution of the $\mathrm{FM}+\mathrm{K}$ phase yields a wing structure phase diagram with a tricritical point and a ferromagnetic quantum critical end point. The evolution of the spin-selective Kondo insulator phase, inside the coexistence region, introduces another critical end point. The results show a correspondence with the experimental phase diagram of heavy-fermion compounds, in particular, the ferromagnetic superconductor ${\mathrm{UGe}}_2$.

Spin ice is a paradigmatic frustrated system famous for the emergence of magnetic monopoles and a large magnetic entropy at low temperatures. It exhibits unusual behavior in the presence of an external magnetic field as a result of the competition between the spin ice entropy and the Zeeman energy. Studies of this have generally focused on fields applied along high-symmetry directions: [111], [001], and [110]. Here we consider a model of spin ice with an external field in an arbitrary direction. We find that the Kasteleyn transition known for [001] fields appears also for general field directions and we calculate the associated Kasteleyn temperature $T_K$ as a function of field direction. $T_K$ is found to vanish, with a logarithmic dependence on field angle, approaching certain lines of special field directions. We further investigate the thermodynamic properties of spin ice for $T>T_K$, using a Husimi cactus approximation. As the system is cooled toward $T_K$, a large magnetic torque appears, tending to align the [001] crystal axis with the external field. The model also exhibits a rotational magnetocaloric effect: significant temperature changes can be obtained by adabiatically rotating the crystal relative to a fixed field.

The earliest known examples of quantum superposition were found in organic chemistry—in molecules that “resonate” among multiple arrangements of $\pi$-bonds. Small molecules such as benzene resonate among a few bond arrangements. In contrast, a large system that stretches over a macroscopic lattice may resonate among an extensive number of configurations. This is analogous to a quantum spin liquid as described by Anderson's resonating valence bond picture. In this article, we study the intermediate case of somewhat large molecules, the ${\mathrm{C}}_{20}$ and ${\mathrm{C}}_{60}$ fullerenes. We build a minimal description in terms of quantum dimer models (QDMs), allowing for local resonance processes and repulsion between proximate $\pi$-bonds. This allows us to characterize ground states, e.g., with ${\mathrm{C}}_{60}$ forming a superposition of 5828 dimer covers. Despite the large number of contributing dimer covers, the ground state shows strong dimer-dimer correlations. Going beyond the ground state, the full spectrum of ${\mathrm{C}}_{60}$ shows many interesting features. When repulsive terms are neglected, the energy values are placed symmetrically about zero. This reflection symmetry originates from a chiral symmetry of the Hamiltonian, which in turn originates from the local bond geometry of ${\mathrm{C}}_{60}$. This property also leads to a large number of protected zero-energy states. In general, the spectrum contains many scarlike states, corresponding to localized dimer-rearrangement dynamics. Resonance dynamics in QDMs can manifest in the behavior of defects, potentially binding defects via an effective attractive interaction. To test this notion, we introduce pairs of vacancies at all possible separations. Resonance energy reaches its lowest value when the vacancies are closest to one another. This suggests confinement of monomers, albeit within a finite cluster. We discuss qualitative pictures for understanding bonding in fullerenes, and we draw connections with results from quantum chemistry.

Although topological band theory has been used to discover and classify a wide array of novel topological phases in insulating and semimetal systems, it is not well suited to identifying topological phenomena in metallic or gapless systems. Here, we develop a theory of topological metals based on the system's spectral localizer and associated Clifford pseudospectrum, which can both determine whether a system exhibits boundary-localized states despite the presence of degenerate bulk bands and provide a measure of these states' topological protection even in the absence of a bulk band gap. We demonstrate the generality of this method across symmetry classes in two lattice systems, a Chern metal and a higher-order topological metal, and prove the topology of these systems is robust to relatively strong perturbations. The ability to define invariants for metallic and gapless systems allows for the possibility of finding topological phenomena in a broad range of natural, photonic, and other artificial materials that could not be previously explored.

We present the results of muon spin relaxation ($\mu \mathrm{SR}$) and neutron scattering on the Ce-based quasikagome lattice ${\mathrm{CeRh}}_{1-x}{\mathrm{Pd}}_x\mathrm{Sn}$ ($x=0.1$ to 0.75). Our zero-field (ZF) $\mu \mathrm{SR}$ results reveal the absence of both static long-range magnetic order and spin freezing down to 0.05 K in the single-crystal sample of $x=0.1$. The weak temperature-dependent plateaus of the dynamic spin fluctuations below 0.2 K in ZF-$\mu \mathrm{SR}$ together with its longitudinal-field (LF) dependence between 0 and 3 kG indicate the presence of dynamic spin fluctuations persisting even at $T=0.05\mathrm{K}$ without static magnetic order. On the other hand, the magnetic specific heat divided by temperature $C_{\text{4f}}/T$ increases as $-\text{log}T$ on cooling below 0.9 K, passes through a broad maximum at 0.13 K, and slightly decreases on further cooling. The ac susceptibility also exhibits a frequency-independent broad peak at 0.16 K, which is prominent with an applied field $H$ along the $c$ direction. We, therefore, argue that such a behavior for $x=0.1$ [namely, a plateau in spin relaxation rate ($\lambda$) below 0.2 K and a linear $T$ dependence in $C_{\text{4f}}$ below 0.13 K] can be attributed to a metallic spin-liquid-like ground state near the quantum critical point in the frustrated Kondo lattice. The LF-$\mu \mathrm{SR}$ study suggests that the out of kagome plane spin fluctuations are responsible for the spin-liquid (SL) behavior. Low-energy inelastic neutron scattering (INS) of $x=0.1$ reveals gapless magnetic excitations, which are also supported by the behavior of $C_{\text{4f}}$ proportional to $T^{1.1}$ down to 0.06 K. Our high-energy INS study shows very weak and broad scattering in $x=0$ and 0.1, which transforms into well-localized crystal-field excitations with increasing $x$. The ZF-$\mu \mathrm{SR}$ results for the $x=0.2$ polycrystalline sample exhibit similar behavior to that of $x=0.1$. A saturation of $\lambda$ below 0.2 K suggests a spin-fluctuating SL ground state down to 0.05 K. The ZF-$\mu \mathrm{SR}$ results for the $x=0.5$ sample are interpreted as a long-range antiferromagnetic (AFM) ground state below $T_{\text{N}}=0.8\mathrm{K}$, in which the AFM interaction of the enlarged moments probably overcomes the frustration effect. The long-range AFM ordering is also supported by the evolution of magnetic Bragg peaks in $x=0.75$ sample observed below 5 K in the neutron diffraction data.

We study the two-particle spectral functions and collective modes of weakly disordered superconductors using a disordered attractive Hubbard model on a square lattice. We show that the disorder-induced scattering between collective modes leads to a finite subgap spectral weight in the long-wavelength limit. In general, the spectral weight is distributed between the phase and the Higgs channels, but as we move towards half filling the Higgs contribution dominates. The inclusion of the density fluctuations lowers the frequency at which this mode occurs and results in the phase channel gaining a larger contribution to this subgap mode. Near half filling, the proximity of the system to the charge density wave instability leads to strong fluctuations of the effective disorder at the commensurate wave vector ($[\pi ,\pi ]$). We develop an analytical mode coupling approach where the pure Goldstone mode in the long-wavelength limit couples to the collective mode at $[\pi ,\pi ]$. This provides insight into the location and distribution of the two-particle spectral weights between the Higgs and the phase channels.

The superconducting order parameter of strontium ruthenate is the center of a lasting puzzle calling for theoretical studies that include the seldom-considered effects of spin-orbit coupling and the frequency-dependence of the order parameters. Here we generalize the frequency-dependent theory of superconductivity mediated by spin and charge fluctuations to include spin-orbit coupling in multiorbital systems and we characterize the superconducting states using the spin-parity-orbital-time $SPOT$ quantum numbers, group theory, and phase distributions in the complex plane. We derive a pseudospin formulation that maps the inter-pseudospin sector of the normal state Eliashberg equation to a pseudospin-diagonal one. Possible superconducting order parameters for strontium ruthenate are obtained starting from a realistic density-functional-theory normal state. We find that spin-orbit coupling leads to ubiquitous entanglement of spin and orbital quantum numbers, along with notable mixing between even- and odd-frequency correlations. We propose a phase diagram obtained from the temperature dependence of the leading and subleading symmetries in the pseudospin-orbital basis. An accidental degeneracy between leading inter-pseudospin symmetries in strontium ruthenate, ${\mathrm{B}}_{1g}^{+}$ and ${\mathrm{A}}_{2g}^{-}$, could resolve apparent experimental contradictions.