Inspired by the discovery of a variety of correlated insulators in the moiré universe, controlled by interactions projected to a set of isolated bands with a narrow bandwidth, we examine here a partial sum rule associated with the inverse frequency-weighted optical conductivity restricted to low energies. Unlike standard sum rules that extend out to infinite frequencies, which include contributions from all interband transitions, we focus here on transitions associated only with the projected degrees of freedom. We analyze the partial sum rule in a nonperturbative but solvable limit for a variety of correlation-induced insulators. This includes (i) magic-angle twisted bilayer graphene at integer filling with projected Coulomb interactions, starting from the chiral flat-band limit and including realistic perturbations, (ii) fractional fillings of Chern-bands which support generalized Laughlin-like states, starting from a Landau level and including a periodic potential and magnetic-field, respectively, drawing connections to twisted ${\mathrm{MoTe}}_2$, and (iii) integer filling in toy models of nontopological flat bands with a tunable quantum geometry in the presence of repulsive interactions. The partial sum rule in all these examples is implicitly constrained by the form of the band quantum geometry via the low-lying excitation spectrum but is not related to it explicitly. For interacting Slater-determinant insulators, the partial sum rule is related to a “many-body projected quantum geometry”, obtained from the interaction-renormalized electronic bands. We also point out an intriguing connection between the partial sum rule and the quantum Fisher information associated with the projected many-body position operator. Finding alternative experimental routes that enable a direct measurement of the quantum geometry in correlated insulators remains an exciting and uncharted territory.

A quantum critical point (QCP) represents a continuous phase transition at absolute zero. In unconventional superconductors, enhanced superconducting transition temperature and magnetic fluctuation strength are often observed together, indicating magnetism-mediated superconductivity. This raises the question of whether quantum fluctuations in other degrees of freedom, such as charge, could similarly boost superconductivity. However, because charge is frequently intertwined with magnetism, isolating and understanding its specific role in Cooper pair formation pose a significant challenge. Here, we report persistent charge density fluctuations (CDFs) down to 15 K in the nonmagnetic superconductor ${\mathrm{Sr}}_{0.77}{\mathrm{Ba}}_{0.23}{\mathrm{Ni}}_2{\mathrm{As}}_2$, which lie near a proposed nematic QCP associated with a sixfold enhancement of superconductivity. Our results show that the quasielastic CDFs do not condense into resolution-limited Bragg peaks but, rather, display nonsaturated strength. The phonons associated with CDFs completely soften at 25 K, with their critical behavior described by the same mathematical framework as the antiferromagnetic Fermi liquid model, yielding a fitted Curie-Weiss temperature of $\theta \approx 0\mathrm{K}$. Additionally, we find that the nematic fluctuations are weakly coupled to the lattice, as evidenced by the absence of softening in nematic-coupled in-plane transverse acoustic phonons. Our discovery positions ${\mathrm{Sr}}_x{\mathrm{Ba}}_{1-x}{\mathrm{Ni}}_2{\mathrm{As}}_2$ as a promising candidate for charge-fluctuation-driven superconductivity.

Recent experimental reports of correlated physics in twisted homobilayer ${\mathrm{WSe}}_2$ have spurred interest in the interplay of electronic interactions and topology in this system. Here, we explore its phase diagram using the Hartree-Fock approximation within a three-orbital Wannier model of the bilayer. Our analysis reveals a dominant intervalley-coherent antiferromagnetic instability, whose stability in the space of twist angle, interaction strength, out-of-plane displacement field, and hole density is primarily set by nesting and commensurability. At large angles or low interaction-to-bandwidth ratios, the instability arises at hole densities above half filling near a van Hove line where the strong Fermi surface nesting occurs due to the flatness of the band in a region enclosing the van Hove and $\kappa$ points. Increasing interaction strength or decreasing the twist angle gradually shifts the ordered phase toward half filling, where the strongest antiferromagnetic order gets pinned due to commensurability effects that enable a full gap opening. The antiferromagnetic order parameter strongly couples to the layer polarization, which makes its transition to the normal state sharp in the strong-coupling limit and carries implications for collective modes. Our Hartree-Fock phase diagram reproduces key aspects of recent experiments and the reconstructed Fermi surfaces and DOS in the antiferromagnetic phase account for subtle transport signatures observed in these studies.

Conformal field theory underlies critical ground states of quantum many-body systems. While conventional conformal field theory is associated with positive central charges, nonunitary conformal field theory with complex-valued central charges has recently been recognized as physically relevant. Here, we demonstrate that complex-valued entanglement entropy characterizes complex conformal field theory and critical phenomena of open quantum many-body systems. This is based on non-Hermitian reduced density matrices constructed from the combination of right and left ground states. Applying the density matrix renormalization group to non-Hermitian systems, we numerically calculate the complex entanglement entropy of the non-Hermitian five-state Potts model, thereby confirming the scaling behavior predicted by complex conformal field theory.

The metallic delafossites host ultrahigh-mobility carriers in the bulk, while at their polar surfaces, intrinsic electronic reconstructions stabilize markedly distinct electronic phases, from charge-disproportionated insulators to Rashba-split heavy-hole gases and ferromagnetic metals. The understanding of these phases has been strongly informed by surface spectroscopic measurements, but previous studies have been complicated by the presence of spatially varying terminations of the material surface. Here, we demonstrate the potential of microscopic-area angle-resolved photoemission to overcome these challenges. Our measurements of the model compound ${\mathrm{PdCoO}}_2$ yield extremely high quality spectra of the electronic structure, which allows us to place stringent experimental constraints on the weak electron-phonon coupling in the bulk of ${\mathrm{PdCoO}}_2$, while revealing much stronger interactions at its surfaces. While the ${\mathrm{CoO}}_2$-terminated surface exhibits a conventional weak-coupling behavior, our measurements reveal surprising spectroscopic signatures of polaron formation at the Pd-terminated surface, despite its pronounced metallicity. Together, our findings reveal how mode- and symmetry-selective couplings can markedly tune the electron-phonon interactions in a single host material, here opening routes to stabilize surprisingly persistent polaronic quasiparticles.

The interconversion between spin and charge current lies at the core of spintronics. Although significant efforts have been made in the study of spin-to-charge conversion, a close observation of the dynamical process of spin-to-charge remains elusive. Here we explored $\mathrm{Ru}{\mathrm{O}}_2$ film thickness-dependent terahertz (THz) emission from the ultrafast charge current converted from $z$ spins, which are generated by incident circularly polarized lasers via optical orientation. The shift trend of the THz peak position with respect to the film thickness reveals that the conversion process occurs simultaneously with the $z$ spin propagation within the film, rather than at the $\mathrm{Ru}{\mathrm{O}}_2$ surface. A qualitative model of spin-to-charge conversion in $\mathrm{Ru}{\mathrm{O}}_2$ was employed to explain the shift trend, fitting the experimental data with numerical calculation. Our work not only helps improve the fundamental knowledge of spin-to-charge conversion, but also advances the development of $z$ spin-based ultrafast spintronic devices.

Whether a quantum critical point (QCP) demarcates the end of the pseudogap (PG) regime in hole-doped cuprates at a singular doping level $p^{*}\approx 0.19$ is still an open question. A crucial component of the puzzle is the manner in which the carrier density predicted by electronic structure calculations is recovered for $p>p^{*}$. Here, we use magnetic fields up to 67 T to suppress superconductivity down to 50 K, allowing us to simultaneously measure the low-temperature Hall number $n_{\text{H}}$ and the in-plane resistivity anisotropy ${\rho }_a/{\rho }_b$ in overdoped ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ single crystals. We confirm a previous finding [S. Badoux et al., Nature (London) 531, 210 (2016)] that $n_{\text{H}}$(50 K) exhibits a sharp increase below $p^{*}$ and use the measured resistivity anisotropy to obtain the planar carrier density $n_{pl}=n_{\text{H}}{({\rho }_a/{\rho }_b)}^{-1}$. The doping dependence of $n_{pl}$(50 K) reveals two key findings: (i) At optimal doping, $n_{pl}\approx p$, and (ii) the sharp rise in $n_{\mathrm{H}}\left(p\right)$ is softened to such an extent that the full Fermi volume ($n_{pl}=1+p$) is only partially recovered at $p^{*}$. The latter result disfavors a conventional QCP scenario in which the PG end point leads to a reconstructed Fermi surface.

Strong electron-electron interactions are known to significantly modify the electron-phonon coupling relative to the predictions of density functional theory, but this effect is challenging to calculate with realistic theories of strongly correlated materials. Here we define and calculate a version of the electron-phonon coupling applicable beyond band theory by combining first-principles density functional theory plus dynamical mean-field theory with finite-difference phonon perturbations, presenting results for several representative phonon modes in two materials of interest. In the three-orbital correlated metal ${\mathrm{SrVO}}_3$, we find that intra-V-$t_{2g}$-band correlation significantly increases the coupling of these electrons to a Jahn-Teller phonon mode that splits the degenerate orbital energies, while slightly reducing the coupling associated with a breathing phonon that couples to the charge on each V atom. In the infinite layer cuprate ${\mathrm{CaCuO}}_2$, we find that local correlation within the $d_{x^2-y^2}$ orbital derived band has a modest effect on coupling of near-Fermi-surface electrons to optical breathing modes. In both cases, the interaction correction to the electron-phonon coupling predicted by dynamical mean-field theory has a significant dependence on the electronic frequency, arising from a lattice-distortion dependence of the correlated electron dynamics, showing the inadequacy of the simple picture in which correlations change static local susceptibilities. We also show that the electron-phonon scattering and phonon lifetimes associated with these phonon modes are modified by the electronic correlation. Our findings shed light on the material- and mode-specific role of dynamical electronic correlation in electron-phonon coupling and highlight the importance of developing efficient computational methods for treating electron-phonon coupling in correlated materials.

We present the synthesis and characterization of ${\mathrm{GdZn}}_3{\mathrm{As}}_3$, a previously unreported variant of the $R{M}_3{X}_3$ family ($R$ = lanthanides; $M$ = Zn, Cd; $X$ = P, As), prepared in both single-crystal and polycrystalline forms. Unlike other $R{M}_3{X}_3$ compounds that crystallize in undistorted hexagonal structures, ${\mathrm{GdZn}}_3{\mathrm{As}}_3$ adopts a distorted ${\mathrm{ScAl}}_3{\mathrm{C}}_3$-type orthorhombic structure with $Cmcm$ space group. Magnetic measurements demonstrate that ${\mathrm{GdZn}}_3{\mathrm{As}}_3$ undergoes a ferromagnetic transition at the Curie temperature ($T_{\mathrm{C}}$) of 6.3 K, which is unique among known $R{M}_3{X}_3$ materials. This magnetic transition is further confirmed by specific heat and electrical resistivity measurements. Single-crystal ${\mathrm{GdZn}}_3{\mathrm{As}}_3$ displays metallic behavior with a pronounced resistivity peak near $T_{\mathrm{C}}$, which is strongly suppressed by magnetic fields, leading to significant negative magnetoresistance. Hall effect measurements reveal a low carrier density and a clear nonlinear anomalous Hall effect in ${\mathrm{GdZn}}_3{\mathrm{As}}_3$. Furthermore, both specific heat and resistivity data suggest the presence of additional magnetic transition(s) below $T_{\mathrm{C}}$, requiring further investigation. These results demonstrate that ${\mathrm{GdZn}}_3{\mathrm{As}}_3$ possesses distinct structural, magnetic, and electronic transport properties within the $R{M}_3{X}_3$ family, establishing it as an exceptional platform for investigating competing magnetic interactions in low-carrier-density rare-earth triangular-lattice systems.

Dynamic nuclear polarization (DNP) of the ${}^7\mathrm{Li}$ NMR lithium metal signal via an Overhauser mechanism was predicted and demonstrated over 70 years ago, and applied more recently at high fields and room temperature to study electrodeposited microstructural lithium metal. Here we further explore the requirements and limitations of the Overhauser effect for alkali metals. Following a theoretical treatment of the Overhauser mechanism, we explain why only relatively modest enhancements are obtained ($\sim 50$ compared to a theoretical maximum of 1700 for ${}^7\mathrm{Li}$) at a magnetic field strength of 9.4 T. Through a systematic investigation of the ${}^7\mathrm{Li}$ enhancement as a function of experimental parameters (magnetic field strength, $B_0,$ temperature, microwave power, and MAS frequency), we show that the experimental enhancements are limited by the short spin-lattice relaxation times ($T_{1e}$ and $T_{2e}$) of the electrons in lithium metal, and thus the degree of saturation of the electron spin resonance (ESR) transition. The enhancements of ${}^6\mathrm{Li}$ and ${}^{23}\mathrm{Na}$ are similarly limited by the degree of electron spin saturation, the effect being more pronounced for ${}^{23}\mathrm{Na}$; enhancements are improved by using higher microwave power and for ${}^{23}\mathrm{Na}$, with its temperature-dependent $T_{1e}$, by performing experiments at lower temperatures. Sample considerations, such as sample volume, metal particle sizes, and dilution of metal particles, are also important for obtaining high enhancements. Studies of the hyperpolarization mechanism of the spins in diamagnetic environment at the interfaces between lithium metal and its passivating film suggest that both spin diffusion and direct chemical exchange play a role, although the role of direct cross-relaxation with the conduction electrons requires further investigation. The hyperpolarization is then transferred to spins farther from the metal surface via spin diffusion, allowing only the spins within less than $\sim 10$ nm from the metal surface to be hyperpolarized. Based on the insights gained through this work, lithium metal possesses a possibly unique combination of beneficial qualities, making it a highly suitable system for Overhauser DNP. The high enhancements achieved for ${}^6\mathrm{Li}$ ($\sim 150$) using a 1.3 mm rotor are particularly promising for studying lithium metal samples extracted from batteries, making use of the generally higher resolution of ${}^6\mathrm{Li}$ over ${}^7\mathrm{Li}$.

The superconductor ${\mathrm{BaNi}}_2{\mathrm{As}}_2$ exhibits a soft-phonon-driven, incommensurate charge density wave (I-CDW) which is accompanied by a small orthorhombic structural phase transition. Upon further cooling, ${\mathrm{BaNi}}_2{\mathrm{As}}_2$ undergoes a first-order structural transition to a triclinic phase in which a commensurate CDW (C-CDW) appears. The relationship and interplay between the I-CDW, C-CDW and structural phase transitions have remained elusive. To investigate this issue, we present a complementary study of thermal diffuse x-ray scattering and inelastic x-ray scattering for phosphorus substituted ${\mathrm{BaNi}}_2{\left({\mathrm{As}}_{1-\mathrm{x}}{\mathrm{P}}_{\mathrm{x}}\right)}_2(x\lesssim 0.12$) and down to $2.2\mathrm{K}$. We show that most of the diffuse scattering signal can be well described by first-principles lattice dynamics calculations. Furthermore, we find that although phosphorus substitution rapidly suppresses the structural transition temperatures, the temperature dependence of the correlation length of the I-CDW fluctuations and the formation of Bragg-like superstructure peaks associated with long-range ordering of this order depends only weakly on the substitution level. Finally, we present the absence of signatures of the I-CDW to C-CDW or triclinic transition in the lattice dynamics, indicating that these instabilities are not (soft) phonon driven.

The realization of topological surface states and superconductivity within a single material platform is a crucial step toward achieving topologically nontrivial superconductivity. This can be achieved at an interface between a superconductor and a topological insulator or within a single material that intrinsically hosts both superconductivity and topological surface states. Here we use scanning tunneling microscopy to study ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x{\mathrm{Bi}}_2{\mathrm{Te}}_4$ crystals. Spectroscopic evidence reveals the coexistence of topological surface states and superconductivity on the same surface of the crystals. The Te-terminated surface exhibits a single U-shaped superconducting gap with a size of up to $311\mu \mathrm{eV}$, alongside Dirac bands outside the gap. Analysis of the vortex structure and differential conductance suggests weak-coupling $s$-wave superconductivity. The absence of observed zero modes suggests that shifting the Fermi level closer to the Dirac point of the topological bands is necessary to realize a topological superconducting state.

Diluted magnetic semiconductor (DMS) systems have been extensively studied in recent decades. DMSs provides a platform where charge transport and magnetic ordering phenomena exhibit unique interplays, together with possible applications to spin-dependent electronics (spintronics) devices. Initial development of ferromagnetic (FM) DMS systems centered around III-V semiconductors doped with dilute transition metals, such as (Ga,Mn)As, obtained by co-doping of spin and charge. More recently, independent spin and charge doping was first achieved in Li(Zn,Mn)As, a DMS system based on I-II-V semiconductor, with charge doping via variable Li concentrations and spin doping via iso-valent (Zn,Mn) substitutions. Although more than 30 new DMS systems with independent spin and charge doping have been synthesized since then, the main research emphasis has been put on development and characterization of systems with higher FM Curie temperature ($T_{\mathrm{C}}$) and different crystal structures suitable for possible formation of heterostructure devices. This article focuses on a new DMS material Na(Zn,Mn)Sb, which exhibits a spin glass (SG) ordering, together with metal-insulator transition (MIT) and colossal negative magnetoresistance (CMR) as a function of independent spin and charge doping and application of external magnetic fields. MIT and CMR phenomena are elucidated by magneto transport, magnetization, angle resolved photoemission spectroscopy (ARPES), and scanning tunneling microscopy (STM) measurements, and by band calculations which demonstrate development and disappearance of energy gap. Magnetic order and dynamic spin fluctuations are probed with muon spin relaxation (µSR) and magnetization, and the results for Na(Zn,Mn)Sb are compared to those from FM DMS systems Li(Zn,Mn)As, Li(Zn,Mn)P, and Li(Zn,Mn,Cu)As. First-principles calculations are performed for Na(Zn,Mn)Sb, (Ga,Mn)As and Li(Zn,Mn)P to highlight the roles of charge and spin doping on exchange interactions mediated by nearest neighbor super-exchange coupling and oscillatory Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling via conduction electrons. These studies reveal (1) MIT and CMR of Na(Zn,Mn)Sb manifest as a response to spin configurations as spin-driven transport phenomena; (2) a dynamic critical behavior is observed in SG transition of Na(Zn,Mn)Sb, in contrast to more first-order-like magnetic evolutions in other FM DMS systems; (3) charge doping supports FM coupling additive to direct AFM exchange interaction between nearest-neighbor Mn pairs; and (4) a widely different coercive fields seen in different families of FM and SG DMS systems can be explained by geometrical frustration of AFM interaction in underlying lattice for Mn spin network.

We explore the Fano resonance in ${\mathrm{ZrTe}}_5$, using terahertz Raman scattering measurements. We identified two closely spaced ${\mathrm{B}}_{2\mathrm{g}}$ phonon modes, ${\mathrm{B}}_{2\mathrm{g}}$ I and ${\mathrm{B}}_{2\mathrm{g}}$ II, around 9 meV and 11 meV, respectively. Interestingly, only ${\mathrm{B}}_{2\mathrm{g}}$ I exhibited the Fano resonance, an outcome of quantum interference between discrete phonon modes and continuous electronic excitations. This is consistent with the much stronger electron-phonon coupling of ${\mathrm{B}}_{2\mathrm{g}}$ I mode demonstrated by first-principles calculations. Additionally, temperature-dependent measurements highlight an enhanced Fano asymmetry at elevated temperatures, contributed by the thermal renormalization of the band structure and electron-phonon coupling. This study offers insights into the complex interrelation of electron-phonon coupling, thermal effects, and Fano resonances in topological materials.

Dimer models are well known as prototypes for locally constrained physics. They describe systems in which every site on a lattice must be attached to one dimer. Loop models extend this idea, with the constraint that two dimers must touch at each site. Here, we present a further generalization where every site must have three dimers attached—a trivalent network model. As concrete physical realizations, we discuss $d^3$ transition metal dichalcogenides in the 1T structure—materials with the structural formula ${MX}_2$ $(M=\mathrm{Tc}, \mathrm{Re})$ or ${AM}^{\prime }{X}_2$ $(A=\mathrm{Li} \mathrm{or} \mathrm{Na};$ $M^{\prime }=\mathrm{Mo}, \mathrm{W})$, where X is a chalcogen atom. These materials have a triangular layer of transition metal atoms, each with three valence electrons in $t_{2g}$ orbitals. Each atom forms valence bonds with three of its nearest neighbors. The geometry of the 1T structure imbues each bond with sharp orbital character. We argue that this imposes a “bending constraint” so that two dimers attached to the same site cannot be parallel. This leads to a highly structured space of configurations, with alternating bonds along each line of the underlying triangular lattice. There is no dynamics, as constraints forbid local rearrangements of dimers. We construct a phase diagram, identifying configurations that minimize potential energy. We find a rhombus-stripe phase that explains a distortion pattern seen across several materials. Remarkably, the local constraints in this model lead to a simple example of holography. The bonding configuration in the bulk is completely determined by the configuration at the boundary. We recast the model in terms of three Ising chains that are defined on the boundaries of a triangular cluster. As a testable prediction, we propose that a single impurity will generate long-ranged domain walls.

The thorough treatment of electron-lattice interactions from first principles is one of the main goals in condensed matter physics. While the commonly applied adiabatic Born-Oppenheimer approximation is sufficient for describing many physical phenomena, it is limited in its ability to capture meaningful features originating from nonadiabatic coupling effects. The exact factorization method, starting from the full Hamiltonian of electrons and nuclei, provides a way to systematically account for nonadiabatic effects. This formalism was recently developed into an ab initio density functional theory framework. Within this framework we develop here a perturbative approach to the electronic states in solid state materials. We derive exact-factorization-based perturbations of the Kohn-Sham states up to second order in the nuclear displacements. These nonadiabatic features in the calculated energy and wave function corrections are expressed in terms of readily available density functional perturbation theory components.

The realization of various qubit systems based on high-quality hybrid superconducting quantum devices is often achieved using semiconductor nanowires. For such hybrid devices, a good coupling between the superconductor and the conducting states in the semiconductor wire is crucial. GaAs/InAs core/shell nanowires with an insulating core and a conductive InAs shell fulfill this requirement, since the electronic states are strongly confined near the surface. However, maintaining a good crystal quality in the conducting shell is a challenge for this type of nanowire. In this work, we present phase-pure zinc-blende GaAs/InAs core/shell nanowires and analyze their low-temperature magnetotransport properties. We observe pronounced magnetic flux quantum periodic oscillations, which can be attributed to a combination of Aharonov-Bohm and Altshuler-Aronov-Spivak oscillations. From the gate and temperature dependence of the conductance oscillations, as well as from supporting theoretical transport calculations, we conclude that the conducting states in the shell are in the quasiballistic transport regime, with few scattering centers, but nevertheless leading to an Altshuler-Aronov-Spivak correction that dominates at small magnetic field strengths. Our results demonstrate that phase-pure zinc-blende GaAs/InAs core/shell nanowires represent a very promising alternative semiconductor-nanowire-based platform for hybrid quantum devices.

We examine the spectral properties of collective excitations with finite angular momentum $l$ for a system of interacting fermions near a Pomeranchuk quantum critical point, both in the Fermi liquid and non-Fermi liquid regimes. Previous studies found that deep in the Fermi liquid regime, the spectral functions for even and odd $l$ behave differently—the latter is suppressed compared to the former because of kinematic constraints on scattering processes. The main focus of our Letter is to understand how the spectral functions for even and odd $l$ evolve as the system enters the non-Fermi liquid regime. We obtain the full scaling function for the electron polarization bubble at arbitrary $l$, which interpolates between the Fermi liquid and non-Fermi liquid regimes. We show that collective excitations for all $l$ remain stable and causal throughout the crossover and right at the quantum critical point.

Fano resonance combs are characteristic of Rydberg autoionizing states and have recently been observed in Mie scattering by water droplets. We demonstrate the generality of this effect by revealing overlapping combs of Fano resonance in the extinction spectra of dielectric resonators using rings and split rings with rectangular cross sections as examples. An unambiguous interpretation of the series of overlapping Fano resonance combs is obtained using quasinormal mode theory.

Two quantum theories that look different but are secretly describing the same low-energy physics are said to be dual to each other. When realized in the topological holography formalism, duality corresponds to changing the gapped boundary condition on the top boundary of a topological field theory, which determines the symmetry of the system while not affecting the bottom boundary where all the dynamics take place. In this paper, we show that duality in the topological holography formalism can be realized with a sequential quantum circuit applied to the top boundary. As a consequence, the Hamiltonians before and after the duality mapping have exactly the same spectrum in the corresponding symmetry sectors, and the entanglement in the corresponding low-energy eigenstates differs by at most an area law term. These results reformulate the findings from [Lootens et al., arXiv:2311.01439] for dualities in $1+1D$ and extend them, using the topological holography framework, to higher dimensions.