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Emergent phases often appear when the electronic kinetic energy is comparable to the Coulomb interactions. One approach to seek material systems as hosts of such emergent phases is to realize localization of electronic wavefunctions due to the geometric frustration inherent in the crystal structure, resulting in flat electronic bands. Recently, such efforts have found a wide range of exotic phases in the two-dimensional kagome lattice, including magnetic order, time-reversal symmetry breaking charge order, nematicity, and superconductivity. However, the interlayer coupling of the kagome layers disrupts the destructive interference needed to completely quench the kinetic energy. Here we demonstrate that an interwoven kagome network—a pyrochlore lattice—can host a three dimensional (3D) localization of electron wavefunctions. Meanwhile, the nonsymmorphic symmetry of the pyrochlore lattice guarantees all band crossings at the Brillouin zone X point to be 3D gapless Dirac points, which was predicted theoretically but never yet observed experimentally. Through a combination of angle-resolved photoemission spectroscopy, fundamental lattice model and density functional theory calculations, we investigate the novel electronic structure of a Laves phase superconductor with a pyrochlore sublattice, CeRu2. We observe evidence of flat bands originating from the Ce 4f orbitals as well as flat bands from the 3D destructive interference of the Ru 4d orbitals. We further observe the nonsymmorphic symmetry-protected 3D gapless Dirac cone at the X point. Our work establishes the pyrochlore structure as a promising lattice platform to realize and tune novel emergent phases intertwining topology and many-body interactions.

Kagome materials often host exotic quantum phases, including spin liquids, Chern gap, charge density wave, and superconductivity. Existing scanning microscopy studies of the kagome charge order have been limited to nonkagome surface layers. Here, we tunnel into the kagome lattice of FeGe to uncover features of the charge order. Our spectroscopic imaging identifies a 2×2 charge order in the magnetic kagome lattice, resembling that discovered in kagome superconductors. Spin mapping across steps of unit cell height demonstrates the existence of spin-polarized electrons with an antiferromagnetic stacking order. We further uncover the correlation between antiferromagnetism and charge order anisotropy, highlighting the unusual magnetic coupling of the charge order. Finally, we detect a pronounced edge state within the charge order energy gap, which is robust against the irregular shape fluctuations of the kagome lattice edges. We discuss our results with the theoretically considered topological features of the kagome charge order including unconventional magnetism and bulk-boundary correspondence.

Despite concerted efforts, the problem of dilute local moments embedded in a correlated conduction electron host such as Nd$_{1-x}$Ce$_x$CuO$_2$ persists due to lack of analytically controllable models. Here, we address the question: how do local moments couple to correlated but integrable hosts? We describe the conduction electrons by the model of Hatsugai-Kohmoto (HK) which has undergone a recent resurgence arising from its exact solvability, existence of Luttinger surfaces, and connections to Sachdev-Ye-Kitaev (SYK) thermodynamics. We derive an exact low energy "Kondo-HK" Hamiltonian and show the existence of additional spin-exchange coupling that is relevant in the renormalization group (RG) sense. This term is ferromagnetic and does not vanish at low energies yielding an algebraic enhancement of the Kondo temperature. "Poor man's" scaling of couplings exhibits an exotic step-like RG flow between UV-IR fixed points attributed to severely restricted scattering phase space. This phenomenon is analogous to the flow of central charge in Zamolodchikov's diagonal resonance scattering in integrable quantum field theories., Comment: 8 pages, 2 figures

The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature ($T=0$). The equivalent question of pairing instability on a Luttinger surface (LS) -- a contour of zeros of the propagator -- instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid (NFL) and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the $q\rightarrow \infty$ SYK model with $q$-body interactions \textcolor{black}{close to the LS}, and (ii) the enforcement of gauge invariance results in a $\frac{1}{\sqrt{\tau}}$ ($\tau\sim T^{-1}$) behavior of the fluctuation propagator near the QCP, as is a feature of the fundamental SYK fermion., Comment: 7+ pages, 3 figures

Despite concerted efforts, the problem of dilute local moments embedded in a correlated conduction electron host such as Nd$_{1-x}$Ce$_x$CuO$_2$ persists due to lack of analytically controllable models. Here, we address the question: how do local moments couple to correlated but integrable hosts? We describe the conduction electrons by the model of Hatsugai-Kohmoto (HK) which has undergone a recent resurgence arising from its exact solvability, existence of Luttinger surfaces, and connections to Sachdev-Ye-Kitaev (SYK) thermodynamics. We derive an exact low energy "Kondo-HK" Hamiltonian and show the existence of additional spin-exchange coupling that is relevant in the renormalization group (RG) sense. This term is ferromagnetic and does not vanish at low energies yielding an algebraic enhancement of the Kondo temperature. "Poor man's" scaling of couplings exhibits an exotic step-like RG flow between UV-IR fixed points attributed to severely restricted scattering phase space. This phenomenon is analogous to the flow of central charge in Zamolodchikov's diagonal resonance scattering in integrable quantum field theories., Comment: 8 pages, 2 figures

Superconductivity results from an instability of the Fermi surface -- contour of \textit{poles} of the single particle propagator -- to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model \textit{Luttinger} surface, or contour of \textit{zeros} of the Green function. At zero temperature ($\beta \rightarrow \infty$) and a critical interaction strength ($u_{c\infty}$) characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\omega}}$ type, van-Hove singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit ($\beta u \gg 1$), the leading order fluctuation free energy terms in the normal state of this NFL-SC transition resemble the equivalent $\left(O(1)\right)$ terms of the Sachdev-Ye-Kitaev (SYK) model. This free energy contribution takes a simple form $-\beta F = \beta u_{c\infty} - \gamma~\text{ln}\left(\beta u_{c \infty}\right)$ where $\gamma$ is a constant equal to $\frac{1}{2}$. Weak impurity scattering ($\tau \gg \beta^{-1}$) leaves the low-energy spectral density unaffected, but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy-models exhibiting gravity duals., Comment: 16 pages, 6 figures including Appendices

Superconductivity results from an instability of the Fermi surface -- contour of \textit{poles} of the single particle propagator -- to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model \textit{Luttinger} surface, or contour of \textit{zeros} of the Green function. At zero temperature ($\beta \rightarrow \infty$) and a critical interaction strength ($u_{c\infty}$) characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\omega}}$ type, van-Hove singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit ($\beta u \gg 1$), the leading order fluctuation free energy terms in the normal state of this NFL-SC transition resemble the equivalent $\left(O(1)\right)$ terms of the Sachdev-Ye-Kitaev (SYK) model. This free energy contribution takes a simple form $-\beta F = \beta u_{c\infty} - \gamma~\text{ln}\left(\beta u_{c \infty}\right)$ where $\gamma$ is a constant equal to $\frac{1}{2}$. Weak impurity scattering ($\tau \gg \beta^{-1}$) leaves the low-energy spectral density unaffected, but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy-models exhibiting gravity duals., Comment: 16 pages, 6 figures including Appendices

We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not find any universal property of the Butterfly velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butterfly velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter., Comment: v1: 24 pages, 9 figures; v2: minor additions in main text and references; v3: minor changes in presentation (jhep version)

We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not find any universal property of the Butterfly velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butterfly velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter., Comment: v1: 24 pages, 9 figures; v2: minor additions in main text and references; v3: minor changes in presentation (jhep version)

We study a single-channel Kondo effect using a recently developed holographic large-$N$ technique. In order to obtain resistivity of this model, we introduce a probe field. The gravity dual of a localized fermionic impurity in 1+1-dimensional host matter is constructed by embedding a localized 2-dimensional Anti-de Sitter (\ads{2})-brane in the bulk of \ads{3}. This helps us construct an impurity charge density which acts as a source to the bulk equation of motion of the probe gauge field. The functional form of the charge density is obtained independently by solving the equations of motion for the fields confined to the \ads{2}-brane. The asymptotic solution of the probe field is dictated by the impurity charge density, which in turn, affects the current-current correlation functions, and hence the resistivity. Our choice of parameters tunes the near-boundary impurity current to be marginal, resulting in a $\log T$ behavior in the UV resistivity, as is expected for the Kondo problem. The resistivity at the IR fixed point turns out to be zero, signaling a complete screening of the impurity., Comment: v2: 15 pages, typos corrected, few more clarifications added

Motivated by recent bounds for charge diffusion in critical matter, we investigate the question: What sets the scale for charge diffusion in a scale-invariant system? To make our statements precise, we analyze the diffusion pole in an exactly solvable model for a Mott transition in the presence of a long-range interaction term. To achieve scale invariance, we limit our discussion to the flat-band regime. We find in this limit that the diffusion pole which would normally obtain at finite energy is pushed to zero energy resulting in a vanishing of the diffusion constant. This occurs even in the presence of interactions in certain limits, indicating the robustness of this result to the inclusion of a scale in the problem. Consequently, scale-invariance precludes any reasonable definition of the diffusion constant. Nonetheless, we do find that a scale can be defined, all be it, irrelevant to diffusion, which is the product of the squared band velocity and the density of states., Comment: 9 pages including appendices, 1 figure

We study a single-channel Kondo effect using a recently developed holographic large-$N$ technique. In order to obtain resistivity of this model, we introduce a probe field. The gravity dual of a localized fermionic impurity in 1+1-dimensional host matter is constructed by embedding a localized 2-dimensional Anti-de Sitter (\ads{2})-brane in the bulk of \ads{3}. This helps us construct an impurity charge density which acts as a source to the bulk equation of motion of the probe gauge field. The functional form of the charge density is obtained independently by solving the equations of motion for the fields confined to the \ads{2}-brane. The asymptotic solution of the probe field is dictated by the impurity charge density, which in turn, affects the current-current correlation functions, and hence the resistivity. Our choice of parameters tunes the near-boundary impurity current to be marginal, resulting in a $\log T$ behavior in the UV resistivity, as is expected for the Kondo problem. The resistivity at the IR fixed point turns out to be zero, signaling a complete screening of the impurity., Comment: v2: 15 pages, typos corrected, few more clarifications added

Motivated by recent bounds for charge diffusion in critical matter, we investigate the question: What sets the scale for charge diffusion in a scale-invariant system? To make our statements precise, we analyze the diffusion pole in an exactly solvable model for a Mott transition in the presence of a long-range interaction term. To achieve scale invariance, we limit our discussion to the flat-band regime. We find in this limit that the diffusion pole which would normally obtain at finite energy is pushed to zero energy resulting in a vanishing of the diffusion constant. This occurs even in the presence of interactions in certain limits, indicating the robustness of this result to the inclusion of a scale in the problem. Consequently, scale-invariance precludes any reasonable definition of the diffusion constant. Nonetheless, we do find that a scale can be defined, all be it, irrelevant to diffusion, which is the product of the squared band velocity and the density of states., Comment: 9 pages including appendices, 1 figure