By means of slave-boson theory for the Hubbard model on the checkerboard lattice, we calculate dynamical altermagnetic spin susceptibilities from the metallic to the Mott-insulating regime. We track magnon dispersion and lifetime renormalization, allowing us to uncover a crossover from a chirality-selective dissipation of magnon modes to coherent yet strongly deformed chiral magnon branches across the metal insulator transition. Our formalism lends itself to a quantitative description of collective spin dynamics in correlated altermagnets.

Topolectrical circuits are electrical networks that encode topological band theory. Their physical behaviour is set by how the components are wired together, not by where they are placed. This wiring-centred design delivers defect tolerance, directional signal flow without bulky magnetic parts, and amplified response at boundaries. Conventional electronics obtain these only by adding isolators, feedback or calibration after the circuit is laid out. In this Review, we identify which capabilities are engineering-ready and which remain confined to the laboratory. We cover the underlying physics, the resulting sensors and engineering applications, and cross-disciplinary uses in quantum simulation, artificial intelligence-assisted design and curved-space geometries. We then examine how co-integration with memristive devices makes the platform adaptive and history-dependent. We close with the hardware routes towards deployable electronics, from monolithic chips to flexible substrates and body-worn systems. Topolectrical circuits route signals one way without magnets and keep working when components fail. These capabilities come from how the components are wired together, not from added parts. This Review covers their physics, applications across sensing, wireless power and communications, and the path towards integrated, flexible and adaptive electronics.

We investigate the emergence of altermagnetic (AM) phases on the square-kagome lattice. Our analysis reveals that matrix element effects due to an orthogonal sublattice weight decomposition of Fermi level eigenstates known as sublattice interference enable decoupled magnetic ordering tendencies on distinct sublattices. Depending on which sublattice undergoes a magnetic instability, we identify a d_xy-type AM phase and a d_x^2-y^2-type AM phase originating from different sublattice polarization patterns. Using the Kotliar-Ruckenstein slave boson formalism we explore the stability of these AM phases as a function of interaction strength. Our findings demonstrate that sublattice-selective magnetic instabilities provide a versatile route to engineer the nature of AM order.

RuO_{2} has emerged as a prototypical candidate for altermagnetism. In the face of daunting evidence for magnetic order in the bulk, the focus naturally shifted to surfaces and ultrathin films, where Coulomb interactions are dimensionally quenched and electron correlations strongly enhanced. Here, we examine atomically ordered, ultrathin RuO_{2}(110) grown on Ru(0001) using a combination of scanning tunneling microscopy (STM), density functional theory, and density matrix renormalization group methods. We observe a nonmagnetic charge order that is imprinted by the incommensurate moiré stacking with the substrate and enhanced by the electronic Fermi-surface scattering within the flat-band surface state. We further identify a nonmagnetic, metastable c(2×2) surface reconstruction that arises from surface phonon softening and can be toggled reversibly via STM tip manipulation. Spin-polarized STM measurements, however, reveal no evidence of magnetic order on the RuO_{2}(110) surface. Our findings of a nonmagnetic charge modulation position ultrathin RuO_{2}(110) as an intriguing platform for exploring moiré-assisted electronic orders.

The kagome superconductor family AV3Sb5 (A=Cs, Rb, K) provides a rich platform for exploring diverse electronic symmetry breaking phenomena, including superconductivity and various forms of density wave orders. Although these compounds share the identical lattice structure in the normal state, they exhibit distinct forms of symmetry breaking upon entering the charge density wave (CDW) phase, and the microscopic origin of which remain elusive. Here, we investigate the lattice and charge degrees of freedom in AV3Sb5 using angle-resolved polarized Raman spectroscopy. Our comprehensive polarization-resolved measurements reveal that the lifting of the twofold-degeneracy of the E2g phonon mode in the CDW phase-previously reported only in CsV3Sb5 with a 3 GHz splitting-also appears ubiquitously in the other two compounds. In contrast, the collective CDW excitations exhibit markedly different polarization dependences depending on the alkali-metal species. These distinct behaviors in the lattice and charge channels provide crucial insight into the enigmatic material-dependent symmetry breaking phenomena that appear in the CDW phase. Furthermore, our experiments, together with first-principles calculations and an effective Hamiltonian model, shed light on the nature of the charge order structure in AV3Sb5 kagome superconductors.

Triangular-lattice quantum spin-liquid insulators are observed to undergo transitions to superconductivity under pressure or doping, and exhibit enhanced terahertz conductivity when driven by mid-infrared light. We present a general theoretical framework for the emergence of superconducting and charge-ordered phases from a U(1) Dirac spin liquid with fermionic spinons, as well as from its gapped ℤ_2 and chiral descendants. Numerical studies have provided substantial evidence for these spin-liquid states. The spin-liquid phase hosts fractionalized Dirac spinons coupled to an emergent gauge field, whereas the superconducting and charge-ordered phases are conventional, with neither fractionalized excitations nor emergent gauge dynamics. The transition between these phases is driven by the Higgs condensation of spinless charge-e bosonic chargons (“doublons” and “holons”). We show that the projective symmetry group of the Dirac spinons uniquely determines the symmetry and dispersion of the chargons, allowing us to construct an effective low-energy theory near the chargon band minima. Gauge-invariant composites of the chargon Higgs fields provide the order parameters characterizing the phases. The resulting phase diagram contains a rich variety of ordered states, including d+id superconductivity, charge-density waves, bond-density waves, and pair-density waves.

Recent discoveries in kagome materials have unveiled their capacity to harbor exotic quantum states, including intriguing charge density wave (CDW) and superconductivity. Notably, accumulating experimental evidence suggests time-reversal symmetry breaking within the CDW, hinting at the long-pursued loop current order (LCO). Despite extensive research efforts, achieving its model realization and understanding the mechanism through unbiased many-body simulations have remained both elusive and challenging. In this Letter, we develop a microscopic model for LCO on the spinless kagome lattice with nonlocal interactions, utilizing unbiased functional renormalization group calculations to explore ordering tendencies across all two-particle scattering channels. At the Van Hove filling, we identify sublattice interference to suppress onsite CDW order, leaving LCO, charge bond order, and nematic CDW state as the main competitors. Remarkably, a 2×2 LCO emerges as the many-body ground state over a significant parameter space with strong second nearest-neighbor repulsion, stemming from the unique interplay between sublattice characters and lattice geometry. The resulting electronic model with LCO bears similarities to the Haldane model and culminates in a quantum anomalous Hall state. We also discuss potential experimental implications for kagome metals.

We investigate the emergence of electronic order and unconventional superconductivity in M-valley moiré materials. Starting from a first-principles Wannier model of AB-stacked twisted SnSe2, we tackle the (gate-screened) long-ranged Coulomb interaction with functional renormalization group simulations resolving the momentum structure and energy scales of the leading Fermi surface instabilities. Upon doping an antiferromagnetic stripe state at half-filling (ν=3 electrons per moiré unit cell) of the moiré flat bands, magnetic order gives way to unconventional superconductivity mediated by valley-selective spin fluctuations: Large hole doping (ν≈1) leads to weak-coupling superconductors with various pairing symmetries, while slight electron- and hole-doping (ν≈2,4) stabilizes a spin-singlet, extended s-wave state that benefits from scattering between virtual particle and hole states that are detuned from the Fermi level. These findings establish M-point moiré materials as a quantum simulation platform with phenomenological parallels to the class of iron pnictide superconductors.

Three important driving forces for creating qualitatively new phases in quantum materials are the topology of the materials' electronic band structures, frustration in the electrons' motion or magnetic interactions, and strong correlations between their charge, spin, and orbital degrees of freedom. In very few material systems do all of these aspects come together to contribute on an equal footing to stabilize new electronic states with unprecedented properties; however the search for such systems can be guided by models of configurational motifs or key sublattices that can host such physics. One of the most fascinating structural motifs for realizing this rich interplay of frustration, electronic topology, and electron correlation effects is the kagome lattice. In this review, we provide an overview of the theoretical underpinnings driving the physics of kagome lattices, and we then discuss experimental progress in realizing novel states enabled by kagome networks in crystalline materials. Different material classes are discussed with an emphasis on the phenomenologies of their electronic states and how they map to interactions arising from their kagome lattices.

An effective continuum theory is constructed for the topological phase transition of excitons in quasi-twodimensional systems. These topological excitons crucially determine the optoelectronic properties because of their larger binding energies in two dimensions, as well as their topologically enhanced exciton transport. The core idea of this paper is that the essential physics determining the topological invariants across the phase transition is localized near N-fold band-crossing points (BCPs) in the interaction-induced exciton band structure. The construction of the continuum theory around these BCPs needs only the information of exciton states that build up these BCPs at both Q = 0 and finite-Q points, and not the numerically challenging solution of the Bethe-Salpeter equation over the full exciton Brillouin zone. This theory applies to systems with and without spin conservation. Our theory is illustrated in two specific examples: the transition-metal dichalcogenide twisted bilayer systems and the Bernevig-Hughes-Zhang model. These results offer a promising route toward studying complex systems, such as the room-temperature quantum spin Hall system bismuthene (Bi/SiC) and other twisted bilayer systems.

Geometrical frustration in quantum magnets provides a fertile setting for unconventional phases of matter, including quantum spin liquids (QSLs). The trellis lattice, with its complex site arrangements and edge-sharing triangular motifs, presents a promising platform for such physics. In this work, we undertake a comprehensive classification of all fully symmetric QSLs on the trellis lattice using the projective symmetry group approach within the Abrikosov-fermion representation. We find 7 U(1) and 25 Z2 short-ranged Ans & auml;tze and analyze the phase diagram in the mean-field parameter space, uncovering both gapped and Dirac QSLs as well as a semi-Dirac spin liquid that emerges at the level of projective symmetry group classification and mean-field band structure, in which the spinon dispersion is linear along one momentum direction but quadratic along the orthogonal one. We demonstrate that such dispersions can occur only at high-symmetry points in the Brillouin zone with C2v little groups and analyze their characteristic correlation signatures. Moreover, by optimizing over all symmetry-allowed mean-field states, we map out a phase diagram-featuring six distinct phases-of the nearest-neighbor Heisenberg Hamiltonian on the trellis lattice. Among these, we find four quasi-one-dimensional QSL phases, one dimer phase, and one Dirac QSL phase. Going beyond mean field, we also assess equal-time and dynamical spin structure factors of these phases using density-matrix renormalization group and Keldysh pseudofermion functional renormalization group calculations and compare qualitative momentum-space features of these spectra with those obtained at the mean-field level. Finally, we identify four cuprate and vanadate compounds as promising experimental realizations and provide spectroscopic predictions, based on first-principles Hamiltonians, as a guide for future neutron-scattering studies on these materials.

The U(1) Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED_3 with N_f=12 two-component Dirac fermions. The charge-one monopoles span a 924-dimensional complex representation. In the spin-singlet, zero-momentum, zero-C_6-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, C_6, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-N_f and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED_3 in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the N_f=12 Dirac spin liquid.

We propose metal-organic frameworks (MOFs) as a versatile platform for realizing the nonlinear Hall effect. We develop an analytical down-folding scheme that maps a broad class of MOFs onto a universal effective low-energy model. As representative examples, we analyze two C_3-symmetric frameworks: Cu-dicyanoanthracene and triphenyl-metal monolayer, demonstrating how their low-energy bands can be efficiently captured by a star-lattice geometry. First-principles calculations corroborate this mapping and show that both Fermi levels lie close to symmetry-protected Dirac points. Spin-orbit coupling or inversion-symmetry breaking gaps these Dirac cones, generating Berry-curvature hotspots near the Fermi level. Supported with symmetry-based indicators, these MOFs thus suggest themselves for strain and substrate engineering as well as doping to achieve a finite nonlinear Hall response. We formulate a synthesis-oriented strategy that implements the Dirac gap directly within the framework architecture without externally applied strain. Our results establish MOFs as a broadly designable platform for engineering Berry-curvature physics and nonlinear Hall transport.

We determine the free energy of strongly correlated electron systems in the example of the Hubbard model by calculating the contribution of spin and charge fluctuations to the Gutzwiller approximation mean field result. We employ the slave boson formulation of Kotliar and Ruckenstein in its spin-rotation invariant form in the usual continuous time approximation of the functional integral representation, corrected by "high frequency contributions" (SRIKR+). Previous method-related shortcomings are shown to be overcome when the correct operator ordering for the renormalized kinetic energy is used. The results for the ground state energy in the paramagnetic phase are in very good agreement with state-of-the-art results obtained by methods such as density matrix embedded theory (DMET), quantum Monte Carlo (QMC) and others. The leading low temperature behavior of the free energy allows to extract the quasiparticle effective mass, in particular its enhancement near a continuous phase transition into an ordered state. Our work demonstrates that the SRIKR+ method is competitive with the best available alternative methods and equips the slave-boson approach with an improved synoptic power to explore strongly correlated electron systems.

We develop a class of microscopic lattice models in which pair density wave order emerges in the asymptotically exact weak coupling limit. The underlying mechanism is due to non-symmorphic momentum symmetry implied by the models' projective space group representation of electrons. Our mechanism suggests moiré systems and extended s-wave altermagnets as potential hosts for pair density wave order.

Pair density wave (PDW) exhibits periodic amplitude and sign modulations of the superconducting order parameter. Such a pairing state has long been proposed to be highly sensitive to nonmagnetic scattering, but its experimental realization remains elusive. Here, we find a nonmagnetic PDW-breaking effect in a kagome superconductor, using designer atomic nonmagnetic impurities and high-precision scanning tunneling microscopy (STM) at a base temperature of 30 mK. We detect 2 × 2 pair density modulations by Josephson STM with a superconducting tip and 2 × 2 pairing gap modulations by normal STM. We find that the pairing modulations in both cases are substantially suppressed upon doping the kagome lattice with dilute isovalent nonmagnetic impurities, whereas the charge order and uniform superconductivity remain robust. We further identify the correlation between atomic dopants and the local suppression of PDW. We attribute these findings to a nonmagnetic pair-breaking effect, arising from the phase modulation of PDW in the kagome d-orbital. Taken together with its signatures in other state-of-the-art spectroscopy and transport measurements linked by theory, our findings support the ground state of the kagome superconductor as a correlated topological phase with superconducting loop currents.

We explore the effect of self-orthogonality at exceptional points (EPs) in non-Hermitian Parity-Time-symmetric systems. Using a driven three-band lattice model, we show that the Rabi frequency diverges as the system approaches an EP due to the coalescence of eigenstates. We demonstrate that this divergence manifests in experimentally accessible power oscillations, establishing a direct observable for self-orthogonality. Our results provide a pathway for probing EP physics in various metamaterial platforms.

Artificial gauge fields allow uncharged particles such as photons to mimic the behaviour of charged particles subjected to magnetic fields, providing a powerful platform to alter the effective dynamics and exploring topological physics. Topological exciton-polariton lasers have attracted considerable interest, but often require strong magnetic fields to realise propagating topological edge states. Here we experimentally realise the topological Hall effect in a micron-scale micropillar chain by using an artificial gauge field, exploiting the circular polarisation of polaritons as an artificial dimension. Careful rotational alignment of elliptical micropillars induces strictly polarisation-dependent edge-state propagation, demonstrating non-reciprocal transport of the polariton pseudospins. Our results demonstrate that the dimensionality limitation of topological interface states as well as requirements for strong external magnetic fields in coupled topological laser arrays can be overcome. Our results open new ways towards the implementation of topological polariton lattices and related optically active devices with additional artificial dimension.

We study an interacting kagome-lattice realization of a quadratic band-touching semimetal at 2/3 filling with onsite and nearest-neighbor repulsive interactions. Combining functional renormalization group and slave-boson approaches, we map its phase diagram from intermediate to strong coupling and uncover a hierarchy of unconventional electronic orders. The leading instabilities comprise loop-current order, spontaneous altermagnetism arising from a spin-Pomeranchuk instability, and spin-loop-current order with distinctive and largely unexplored properties. We demonstrate how the interplay of band kinematics, electronic interactions, and quantum geometry governs the selection of these phases. Our findings establish quadratic band-touching semimetals as a promising platform for unconventional symmetry breaking and suggest analogous phenomena in other parabolic semimetals.

We investigate the resilience of spinon quasiparticles in the J_1-J_2 zig-zag spin chain (J_2>0) from the viewpoint of momentum-space entanglement. For small J_2, we show that deconfined spinons survive well past the liquid-dimer transition before eventually collapsing towards the Majumdar-Ghosh point. In the highly frustrated zig-zag regime (J_2 ≫ |J_1|), we model the system as two coupled Heisenberg chains and by Fourier transforming each subchain individually, a framework we dub the double-spinon description. While continuum field theories predict that this decoupled phase is strictly unstable to any finite inter-chain coupling, our analysis reveals that the double-spinon description remains robust over an extensive parameter regime. Notably, we find a stark asymmetry in spinon stability reflecting the underlying renormalization group flow: ferromagnetic coupling (J_1 < 0) is marginally irrelevant and sustains fractionalization deep into the spiral phase, whereas antiferromagnetic coupling (J_1 > 0) is marginally relevant and drives confinement much earlier. The ultimate breakdown of this fractionalized description is driven by a continuum of inter-chain excitations which manifests itself as a sharp ground-state momentum shift distinct from macroscopic thermodynamic phase boundaries. Our results establish momentum cut entanglement analysis as a tool to trace the quasiparticle resilience of spinons, as we show that treating the zig-zag Heisenberg chain as two coupled SU(2)_1 Wess-Zumino-Witten models provides a theoretical framework for strongly frustrated quantum magnets applicable beyond the decoupled limit.