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Nontrivial topological invariant of bulk electronic wavefunctions in two-dimensional quantum crystals leaves its footprints on the edge, dislocation, and corner modes. Here we investigate non-unitary time dynamics of these topological modes in square lattice-based open quantum systems in which the time-dependent Hamiltonian smoothly interpolates between topologically distinct insulators across band gap closing quantum critical points. The temporal dynamics of these modes is described by a Lindblad equation in which the instantaneous Hamiltonian plays the role of the Lindblad operator, thereby allowing the environment to couple with the system through the energy channels (weak measurement protocol). We show that in the presence of such a real time ramp, the survival probability of these modes decreases (increases) in short (long) time scale where the dephasing (quantum Zeno) effect dominates with the increasing amplitude of the system-to-environment coupling, for both slow and fast ramps from a topological to a normal insulating state. For a reverse course of the time evolution, the revival or condensation probability of nucleating such topological modes, otherwise absent in the initial system, increases for stronger system-to-environment coupling. This phenomenon can be attributed to the strong decoherence of the initial mixed state among all the energy eigenstates of the final Hamiltonian which also includes the topological modes, causing their enhanced condensation probability. Our findings can be germane to real open topological materials with time-tunable band gap, and should be applicable to open topological crystals of arbitrary dimension and belonging to any symmetry class., Comment: 18 Pages, 17 Figures

Depending on the lattice geometry, the nearest-neighbor (NN) tight-binding model for free fermions gives rise to particle-hole symmetric emergent Dirac liquid, Fermi liquid, and flat bands near the half-filling or zero-energy on bipartite Euclidean and hyperbolic lattices, respectively embedded on the flat and negatively curved spaces. Such noninteracting electronic fluids are characterized by a vanishing, a finite, and a diverging density of states near half-filling, respectively. A non-Hermitian generalization of this scenario resulting from an imbalance of the hopping amplitudes in the opposite directions between any pair of NN sites continues to accommodate a real eigenvalue spectrum over an extended non-Hermitian parameter regime. Most importantly, it reduces the band width without altering the characteristic scaling of the density of states close to the zero-energy. Here, we show that on two-dimensional bipartite Euclidean and hyperbolic lattices such a non-Hermiticity catalyzes the formation of both charge-density-wave and spin-density-wave orders at weaker (in comparison to the counterparts in conventional or Hermitian systems) NN Coulomb and on-site Hubbard repulsions, respectively. These two ordered states correspond to staggered patterns of average electronic density and spin between the NN sites, respectively, and both cause insulation in half-filled systems. We arrive at these conclusions by combining biorthogonal quantum mechanics and lattice-based self-consistent numerical mean-field analysis in the Hartree channel. We discuss the scaling of the associated mass gaps near the zero-energy with the non-Hermitian parameter, and also address the finite size scaling of the order parameters specifically on hyperbolic lattices with open boundary conditions. A robust general mathematical criterion for the proposed non-Hermitian catalysis mechanism for ordered phases is showcased., Comment: 25 Pages, 17 Figures, and 4 Tables (For full Abstract see manuscript)

We formulate a field theoretic description for $d$-dimensional interacting nodal semimetals, featuring dispersion that scales with the linear ($n$th) power of momentum along $d_L$ ($d_M$) mutually orthogonal directions around a few isolated points in the reciprocal space with $d_L+d_M=d$, and residing at the brink of isotropic insulation, described by $N_b$-component bosonic order parameter fields. The resulting renormalization group (RG) procedure, tailored to capture the associated quantum critical phenomena, is controlled by a `small' parameter $\epsilon=2-d_M$ and $1/N_f$, where $N_f$ is the number of identical fermion copies (flavor number). When applied to three-dimensional interacting general Weyl semimetals ($d_L=1$ and $d_M=2$), characterized by the Abelian monopole charge $n>1$, living at the shore of the axionic insulation ($N_b=2$), a leading order RG analysis suggests Gaussian nature of the underlying quantum phase transition, around which the critical exponents assume mean-field values. A traditional field theoretic RG analysis yields same outcomes for simple Weyl semimetals ($n=1$, $d_L=3$, and $d_M=0$). Consequently, emergent marginal Fermi liquids showcase only logarithmic corrections to physical observables at intermediate scales of measurements., Comment: 6 Pages, 1 Figure (Supplemental Material as Ancillary file)

From lattice-regularized models, devoid of any non-Hermitian (NH) skin effects, here we compute the electrical ($\sigma_{xy}$), thermal ($\kappa_{xy}$), and spin ($\sigma^{sp}_{xy}$) Hall, and the electrical ($G_{xx}$) and thermal ($G^{th}_{xx}$) longitudinal conductivities for appropriate NH planar topological insulators and superconductors related to all five non-trivial Altland-Zirbauer symmetry classes in their Hermitian limits. These models feature real eigenvalues over an extended NH parameter regime, only where the associated topological invariants remain quantized. In this regime, the NH quantum anomalous and spin Hall insulators show quantized $\sigma_{xy}$ and $G_{xx}$, respectively, the NH $p+ip$ ($p \pm ip$) pairing shows half-quantized $\kappa_{xy}$ ($G^{th}_{xx}$), while the NH $d+id$ pairing shows quantized $\kappa_{xy}$ and $\sigma^{sp}_{xy}$ in the clean and weak disorder (due to random pointlike charge impurities) regimes. We compute these quantities in experimentally realizable suitable six-terminal setups using the Kwant software package. But, in the strong disorder regime, all these topological responses vanish and with the increasing non-Hermiticity in the system this generic phenomenon occurs at weaker disorder., Comment: Published version in PRB (Letter): 7 pages, 5 figures (Supplemental material as ancillary file)

The energy spectra of linearly dispersing gapless spin-3/2 Dirac fermions display birefringence, featuring two effective Fermi velocities, thus breaking the space-time Lorentz symmetry. Here, we consider a non-Hermitian (NH) generalization of this scenario by introducing a masslike anti-Hermitian birefringent Dirac operator to its Hermitian counterpart. The resulting NH operator shows real eigenvalue spectra over an extended NH parameter regime, and a combination of non-spatial and discrete rotational symmetries protects the gapless nature of such quasiparticles. However, at the brink of dynamic mass generation, triggered by Hubbardlike local interactions, the birefringent parameter always vanishes under coarse grain due to Yukawa-type interactions with scalar bosonic order-parameter fluctuations. The resulting quantum critical state is, therefore, described by two decoupled copies of spin-1/2 Dirac fermions with a unique terminal Fermi velocity, which is equal to the bosonic order-parameter velocity, thereby fostering an emergent space-time Lorentz symmetry. Furthermore, depending on the internal algebra between the anti-Hermitian birefringent Dirac operator and the candidate mass order, the system achieves the emergent Yukawa-Lorentz symmetry either by maintaining its non-Hermiticity or by recovering a full Hermiticity. We discuss the resulting quantum critical phenomena and possible microscopic realizations of the proposed scenarios., Comment: 12 Pages, 3 Figures and 1 Table

Fractal lattices, featuring the self-similarity symmetry, are often geometric descents of parent crystals, possessing all their discrete symmetries (such as rotations and reflections) except the translational ones. Here, we formulate three different general approaches to construct real space Hamiltonian on a fractal lattice starting from the Bloch Hamiltonian on the parent crystal, fostering for example strong and crystalline topological insulators resulting from the interplay between the nontrivial geometry of the underlying electronic wave functions and the crystal symmetries. As a demonstrative example, we consider a generalized square lattice Chern insulator model and within the framework of all three methods we successfully showcase incarnations of strong and crystalline Chern insulators on the Sierpi\'nski carpet fractal lattices. The proposed theoretical framework thus lays a generic foundation to build a tower of topological phases on the landscape of fractal lattices., Comment: Published version in PRB as a Letter: 6 Pages, 3 Figures and 1 Table (Change in authorship and Supplemental Material as ancillary file)

Three-dimensional doped Dirac insulators foster simply connected (in both topological and trivial regimes) and annular (deep inside the topological regime) Fermi surfaces (FSs) in the normal state, and allow on-site repulsions among fermions with opposite spin ($U_1$) and parity ($U_2$) eigenvalues. From an unbiased leading-order (one-loop) renormalization group analysis, controlled by a suitable $\epsilon$ expansion, we show that this system develops a strong propensity toward the nucleation of scalar $s$-wave and odd-parity pseudoscalar $p$-wave pairings, favored by repulsive $U_1$ and $U_2$ interactions, respectively, irrespective of the underlying FS topology. Our results can be pertinent for the observed superconductivity in various doped narrow gap semiconductors, and the theoretical foundation can readily be applied to investigate similar phenomenon in various doped topological materials., Comment: Published version in PRB Letter: 6 Pages, 1 Figure, 2 Tables (Supplemental Material as ancillary file)

Altermagnets are crystallographic rotational symmetry breaking spin-ordered states, possessing a net zero magnetization despite manifesting Kramers non-degenerate bands. Here, we show that momentum-independent local spin nematic orders in monolayer, Bernal bilayer and rhombohedral trilayer graphene give rise to $p$-wave, $d$-wave and $f$-wave altermagnets, respectively, thereby inheriting topology of linear, quadratic and cubic free fermion band dispersions that are also described in terms of angular momentum $\ell=1,\; 2$ and $3$ harmonics in the reciprocal space. The same conclusions also hold inside a spin-triplet nematic superconductor, featuring Majorana altermagnets. Altogether, these findings highlight the importance of electronic band structure in identifying such exotic magnetic orders in quantum materials. We depict the effects of in-plane magnetic fields on altermagnets, and propose novel spin-disordered alter-valleymagnets in these systems., Comment: 6 Pages and 1 Figure: Modified Title, Streamlined Presentation (Supplemental Material as ancillary file)

We propose a general principle of constructing non-Hermitian (NH) operators for insulating and gapless topological phases in any dimension ($d$) that over an extended NH parameter regime feature real eigenvalues and zero-energy topological boundary modes, when in particular their Hermitian cousins are also topological. However, the topological zero modes disappear when the NH operators accommodate complex eigenvalues. These systems are always devoid of NH skin effects, thereby extending the realm of the bulk-boundary correspondence to NH systems in terms of solely the left or right zero-energy boundary localized eigenmodes. We showcase these general and robust outcomes for NH topological insulators in $d=1,2$ and $3$, encompassing their higher-order incarnations, as well as for NH topological Dirac, Weyl and nodal-loop semimetals. Possible realizations of proposed NH topological phases in designer materials, optical lattices and classical metamaterials are highlighted., Comment: 8 Pages, 5 Figures

We develop an effective quantum electrodynamics for non-Hermitian (NH) Dirac materials interacting with photons. These systems are described by nonspatial symmetry protected Lorentz invariant NH Dirac operators, featuring two velocity parameters $v_{_{\rm H}}$ and $v_{_{\rm NH}}$ associated with the standard Hermitian and a masslike anti-Hermitian Dirac operators, respectively. They display linear energy-momentum relation, however, in terms of an effective Fermi velocity $v_{_{\rm F}}=\sqrt{v^2_{_{\rm H}}-v^2_{_{\rm NH}}}$ of NH Dirac fermions. Interaction with the fluctuating electromagnetic radiation then gives birth to an emergent Lorentz symmetry in this family of NH Dirac materials in the deep infrared regime, where the system possesses a unique terminal velocity $v_{_{\rm F}}=c$, with $c$ being the speed of light. While in two dimensions such a terminal velocity is set by the speed of light in the free space, dynamic screening in three spatial dimensions permits its nonuniversal values. Manifestations of such an emergent spacetime symmetry on the scale dependence of various physical observables in correlated NH Dirac materials are discussed., Comment: 19 Pages, 3 Figures: Published version in Journal of High Energy Physics

Nontrivial bulk topological invariants of quantum materials can leave their signatures on charge, thermal and spin transports. In two dimensions, their imprints can be experimentally measured from well-developed multiterminal Hall bar arrangements. Here, we numerically compute the low temperature ($T$) thermal ($\kappa_{xy}$) and zero temperature spin ($\sigma^{sp}_{xy}$) Hall conductivities, and longitudinal thermal conductance ($G^{th}_{xx}$) of various prominent two-dimensional fully gapped topological superconductors, belonging to distinct Altland-Zirnbauer symmetry classes, namely $p+ip$ (class D), $d+id$ (class C) and $p \pm ip$ (class DIII) paired states, in mesoscopic six-terminal Hall bar setups from the scattering matrix formalism using Kwant. In both clean and weak disorder limits, the time-reversal symmetry breaking $p+ip$ and $d+id$ pairings show half-quantized and quantized $\kappa_{xy}$ [in units of $\kappa_0=\pi^2 k^2_B T/(3h)$], respectively, while the latter one in addition accommodates a quantized $\sigma^{sp}_{xy}$ [in units of $\sigma^{sp}_0=\hbar/(8 \pi)$]. By contrast, the time-reversal invariant $p \pm ip$ pairing only displays a quantized $G^{th}_{xx}$ at low $T$ up to a moderate strength of disorder. In the strong disorder regime, all these topological responses ($\kappa_{xy}$, $\sigma^{sp}_{xy}$, and $G^{th}_{xx}$) vanish. Possible material platforms hosting such paired states and manifesting these robust topological thermal and spin responses are discussed., Comment: Published version in PRB: 13 pages, 4 figures, 1 Table

Traditional topological materials belong to different Altland-Zirnbauer symmetry classes (AZSCs) depending on their non-spatial symmetries. Here we introduce the notion of hybrid symmetry class topological insulators (HSCTIs): A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute. The boundaries of HSCTIs can also harbor TIs, typically affiliated with an AZSC different from the parent ones. As such, a fusion between planar quantum spin Hall and vertical Su-Schrieffer-Heeger insulators gives birth to a three-dimensional HSCTI, accommodating quantum anomalous Hall insulators and quantized Hall conductivity on the top and bottom surfaces. We extend this construction to encompass crystalline HSCTI and topological superconductors, and beyond three dimensions. Possible (meta)material platforms to harness HSCTIs are discussed., Comment: 6 pages, 5 figures: Supplemental Material as Ancillary file

Due to the linearly vanishing density of states, emergent massless Dirac quasiparticles resulting from the free fermion motion in a family of two-dimensional half-filled bipartite hyperbolic lattices feature dynamic mass generation through quantum phase transitions only for sufficiently strong finite-range Coulomb repulsion. As such, strong nearest-neighbor Coulomb repulsion ($V$) favors the nucleation of a charge-density-wave (CDW) order with a staggered pattern of average fermionic density between two sublattices of bipartite hyperbolic lattices. Considering a collection of spinless fermions (for simplicity), here we show that application of strong external magnetic fields by virtue of producing a \emph{finite} density of states near the zero energy triggers the condensation of the CDW order even for \emph{infinitesimal} $V$. The proposed curved space magnetic catalysis mechanism is operative for uniform and inhomogeneous (bell-shaped) magnetic fields. We present scaling of the CDW order with the total flux enclosed by hyperbolic Dirac materials for a wide range of (especially subcritical) $V$., Comment: Published version in PRB: 10 Pages, 7 Figures (change in authorship)

Topological lattice defects, such as dislocations and grain boundaries (GBs), are ubiquitously present in the bulk of quantum materials and externally tunable in metamaterials. In terms of robust modes, localized near the defect cores, they are instrumental in identifying topological crystals, featuring the hallmark band inversion at a finite momentum (translationally active type). Here we show that GB superlattices in both two-dimensional and three-dimensional translationally active higher-order topological insulators harbor a myriad of dispersive modes that are typically placed at finite energies, but always well-separated from the bulk states. However, when the Burgers vector of the constituting edge dislocations points toward the gapless corners or hinges, both second-order and third-order topological insulators accommodate self-organized emergent topological metals near the zero energy (half-filling) in the GB mini Brillouin zone. We discuss possible material platforms where our proposed scenarios can be realized through the band-structure and defect engineering., Comment: Published version: 7 Pages, 4 Figures (Supplementary Information as Ancillary file)

Free electrons hopping on hyperbolic lattices embedded on a curved space of negative curvature can foster (a) Dirac liquids, (b) Fermi liquids, and (c) flat bands, respectively characterized by a vanishing, constant, and divergent density of states near the half-filling. From numerical self-consistent Hartree-Fock analyses, here we show that the nearest-neighbor and on-site Coulomb repulsions respectively give rise to charge-density-wave and antiferromagnetic orders featuring staggered patterns of average electronic density and magnetization in all these systems, when the hyperbolic tessellation is accomplished by periodic arrangements of even $p$-gons. Both quantum orders dynamically open mass gaps near the charge neutrality point. Only on hyperbolic Dirac materials these orderings take place via quantum phase transitions (QPTs) beyond critical interactions, which however decreases with increasing curvature, showcasing curvature induced weak coupling QPTs. We present scaling of these masses with the corresponding interaction strengths and display their spatial variations., Comment: 6 Pages, 3 Figures & 1 Table (Supplementary Material as Ancillary file)

Driven quantum materials often feature emergent topology, otherwise absent in static crystals. Dynamic bulk-boundary correspondence, encoded by nondissipative gapless modes residing near the Floquet zone center and/or boundaries, is its most prominent example. Here we show that topologically robust gapless dispersive modes appear along the grain boundaries, embedded in the interior of Floquet topological crystals, when the Floquet-Bloch band inversion occurring at a finite momentum (${\bf K}^{\rm Flq}_{\rm inv}$) and the Burgers vector (${\bf b}$) of the constituting array of dislocations satisfy ${\bf K}^{\rm Flq}_{\rm inv} \cdot {\bf b}=\pi$ (modulo $2 \pi$). Such nondissipative gapless states can be found near the center and/or edges of the Floquet Brillouin zone, irrespective of the drive protocol. We showcase these general outcomes for two-dimensional driven time-reversal symmetry breaking insulators. Promising experimental platforms hosting such dynamic topological dispersive bands in real materials are discussed., Comment: Published version: 10 Pages and 6 Figures (Supplementary Materials as Ancillary File)

Immersed in external magnetic fields ($B$), buckled graphene constitutes an ideal tabletop setup, manifesting a confluence of time-reversal symmetry (${\mathcal T}$) breaking Abelian ($B$) and ${\mathcal T}$-preserving strain-induced internal axial ($b$) magnetic fields. In such a system, here we numerically compute two-terminal conductance ($G$), and four- as well as six-terminal Hall conductivity ($\sigma_{xy}$) for spinless fermions. On a flat graphene ($b=0$), the $B$ field produces quantized plateaus at $G=\pm |\sigma_{xy}|=(2n+1) e^2/h$, where $n=0,1,2, \cdots$. The strain induced $b$ field lifts the two-fold valley degeneracy of higher Landau levels and leads to the formation of additional even-integer plateaus at $G=\pm |\sigma_{xy}|= (2,4,\cdots)e^2/h$, when $B>b$. While the same sequence of plateaus is observed for $G$ when $b>B$, the numerical computation of $\sigma_{xy}$ in Hall bar geometries in this regime becomes unstable. A plateau at $G=\sigma_{xy}=0$ always appears with the onset of a charge-density-wave order, causing a staggered pattern of fermionic density between two sublattices of the honeycomb lattice., Comment: Published version in PRB: 7 pages, 4 figures: Supplemental Material as Ancillary file

Topological classification of quantum solids often (if not always) groups all trivial atomic or normal insulators (NIs) into the same featureless family. As we argue here, this is not necessarily the case always. In particular, when the global phase diagram of electronic crystals harbors topological insulators with the band inversion at various time-reversal invariant momenta ${\bf K}^{\rm TI}_{\rm inv}$ in the Brillouin zone, their proximal NIs display noninverted band-gap minima at ${\bf K}^{\rm NI}_{\rm min}={\bf K}^{\rm TI}_{\rm inv}$. In such systems, once topological superconductors nucleate from NIs, the inversion of the Bogoliubov de Gennes bands takes place at ${\bf K}^{\rm BdG}_{\rm inv}={\bf K}^{\rm NI}_{\rm min}$, inheriting from the parent state. We showcase this (possibly general) proposal for two-dimensional time-reversal symmetry-breaking insulators. Then distinct quantized thermal Hall conductivity and responses to dislocation lattice defects inside the paired states (tied with ${\bf K}^{\rm BdG}_{\rm inv}$ or ${\bf K}^{\rm NI}_{\rm min}$), in turn unambiguously identify different parent atomic NIs., Comment: Published version in PRB as a Letter: 7 pages, 4 figures (Supplemental Material as Ancillary file)

Linearly dispersing Rarita-Schwinger-Weyl (RSW) fermions featuring two Fermi velocities are the key constituents of itinerant spin-3/2 quantum materials. When doped, RSW metals sustain two Fermi surfaces (FSs), around which one fully gapped $s$-wave and five \emph{mixed-parity} local pairings can take place. The intraband components of four mixed-parity pairings support point nodes at the poles of two FSs, only around which long-lived quasiparticles live. For weak (strong) pairing amplitudes ($\Delta$), gapless north and south poles belonging to the same (different) FS(s) get connected by \emph{polar hairs}, one-dimensional line nodes occupying the region between two FSs. The remaining one, by contrast, supports four nodal rings in between two FSs, symmetrically placed about their equators, but only when $\Delta$ is small. For large $\Delta$, this paired state becomes fully gapped. The transition temperature and pairing amplitudes follow the BCS scaling. We explicitly showcase these outcomes for a rotationally symmetric RSW metal, and contrast our findings when the system possesses an enlarged Lorentz symmetry and with those in spin-3/2 Luttinger materials., Comment: Published version: 6 Pages, 5 Figures, 1 Table (Supplementary Materials as Ancillary File)

Bulk dislocation lattice defects are instrumental in identifying translationally active topological insulators (TATIs), featuring band inversion at a finite momentum (${\bf K}_{\rm inv}$). As such, TATIs host robust gapless modes around the dislocation core, when the associated Burgers vector ${\bf b}$ satisfies ${\bf K}_{\rm inv} \cdot {\bf b}=\pi$ (modulo $2 \pi$). From the time evolution of appropriate density matrices, we show that when a TATI via a real time ramp enters into a trivial or translationally inert topological insulating phase, devoid of gapless dislocation modes, the signatures of the preramp defect modes survive for a long time. More intriguingly, as the system ramps into a TATI phase from any translationally inert insulator, signature of the dislocation mode dynamically builds up near its core, which is prominent for slow ramps. We exemplify these generic outcomes for two-dimensional time-reversal symmetry breaking insulators. Proposed dynamic responses at the dislocation core can be experimentally observed in quantum crystals, optical lattices and metamaterials with time a tunable band gap., Comment: Published version in PRB: 6 Pages, 4 Figures: Supplementary as Ancillary file

Crystalline graphene heterostructures, namely Bernal bilayer and rhombohedral trilayer graphene, subject to electric displacement fields, display a rich confluence of competing orders, resulting in a valley-degenerate, spin-polarized half-metal at moderate doping, and a spin- and valley-polarized quarter-metal at low doping. Here we show that the annular Fermi surface of such a quarter-metal can be susceptible toward the nucleation of a unique spin and valley polarized superconducting state, accommodating interlayer Cooper pairs that break the translational symmetry, giving rise to a Kekul\'e or columnar pair-density-wave. The superconducting ground state produces isolated Fermi pockets of neutral Majorana fermions, featuring a three-fold rotational symmetry, resulting in power-law scaling physical observables with temperature ($T$), such as specific heat $C_v \sim T$., Comment: 6 Pages, 2 Figures, 1 Table: Supplementary Materials as Ancillary File

Fractional Dirac materials (FDMs) feature a fractional energy-momentum relation $E(\vec{k}) \sim |\vec{k}|^{\alpha}$, where $\alpha \; (<1)$ is a real noninteger number, in contrast to that in conventional Dirac materials with $\alpha=1$. Here we analyze the effects of short- and long-range Coulomb repulsions in two- and three-dimensional FDMs. Only a strong short-range interaction causes nucleation of a correlated insulator that takes place through a quantum critical point. The universality class of the associated quantum phase transition is determined by the correlation length exponent $\nu^{-1}=d-\alpha$ and dynamic scaling exponent $z=\alpha$, set by the band curvature. On the other hand, the fractional dispersion is protected against long-range interaction due to its nonanalytic structure. Rather, a linear Dirac dispersion gets generated under coarse graining, and the associated Fermi velocity increases logarithmically in the infrared regime, thereby yielding a two-fluid system. Altogether, correlated FDMs unfold a rich landscape accommodating unconventional emergent many-body phenomena., Comment: Published Version in Physical Review Research as a Letter (6 Pages, 3 Figures; Supplemental Material as Ancillary file)

Topological insulators, featuring bulk-boundary correspondence, have been realized on a large number of noncrystalline materials, among which amorphous network, quasicrystals and fractal lattices are the most prominent ones. By contrast, topological superconductors beyond the realm of quantum crystals are yet to be harnessed, as their nucleation takes place around a well-defined Fermi surface with a Fermi momentum, the existence of which rests on the underlying translational symmetry. Here we identify a family of noncrystalline Dirac materials, devoid of time-reversal (${\mathcal T}$) and translational symmetries, on which a suitable local or on-site pairing yields topological superconductors. We showcase this outcome on all the above mentioned noncrystalline platforms embedded in a two-dimensional flat space. The resulting noncrystalline topological superconductors possess quantized topological invariants (Bott index and local Chern marker) and harbor robust one-dimensional Majorana edge modes, analogs of ${\mathcal T}$-odd $p+ip$ pairing in noncrystalline materials., Comment: Published version in PRB: 8 Pages, 5 Figures

Non-Hermitian (NH) crystals, quasicrystals and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the \emph{inner skin effect} (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showcase this observation by implementing prominent models for NH insulators and superconductors on representative planar Sierpinski carpet fractal lattices. They accommodate both first-order and second-order ISEs at inner edges and corners, respectively, for charged as well as neutral Majorana fermions. Furthermore, over extended parameter regimes ISEs are tied with nontrivial bulk topological invariants, yielding intrinsic ISEs. With the recent success in engineering NH topological phases on highly tunable metamaterial platforms, such as photonic and phononic lattices, as well as topolectric circuits, the proposed ISEs can be observed experimentally at least on fractal metamaterials with periodic boundary condition., Comment: Published Version: 9 Pages, 5 Figures (Supplementary Materials as Ancillary File)

Nature harbors crystals of dimensionality ($d$) only up to three. Here we introduce the notion of \emph{projected topological branes} (PTBs): Lower-dimensional branes embedded in higher-dimensional parent topological crystals, constructed via a geometric cut-and-project procedure on the Hilbert space of the parent lattice Hamiltonian. When such a brane is inclined at a rational or an irrational slope, either a new lattice periodicity or a quasicrystal emerges. The latter gives birth to topoquasicrystals within the landscape of PTBs. As such PTBs are shown to inherit the hallmarks, such as the bulk-boundary, bulk-dislocation correspondences and topological invariant, of the parent topological crystals. We exemplify these outcomes by focusing on two-dimensional parent Chern insulators, leaving its signatures on projected one-dimensional (1D) topological branes in terms of localized endpoint, dislocation modes and the local Chern number. Finally, by stacking 1D projected Chern insulators, we showcase the imprints of three-dimensional Weyl semimetals in $d=2$, namely the Fermi arc surface states and bulk chiral zeroth Landau level, responsible for the chiral anomaly. Altogether, the proposed PTBs open a realistic avenue to harness higher-dimensional ($d>3$) topological phases in laboratory., Comment: Published version: 10 Pages, 7 Figures (Supplementary Information as Ancillary file)

Motivated by recent experiments [H. Zhou, \emph{et al.}, Science {\bf 375}, 774 (2022) and S. C. de la Barrera, \emph{et al.}, arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by Hubbard repulsion driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion driven quantum anomalous Hall (QAH) orders, lifts the four-fold degeneracy of electronic bands, yielding a quarter metal for small doping, as also observed in ABC trilayer graphene. With the disappearance of the QAH order, electronic bands recover two-fold valley degeneracy, thereby forming a conventional or compensated (with majority and minority carriers) half-metal at moderate doping, depending on the relative strength of LP and LAF. At even higher doping and for weak $D$-field only LAF survives and the Fermi surface recovers four-fold degeneracy. We also show that a pure repulsive electronic interaction mediated triplet $f$-wave pairing emerges from a parent correlated nematic liquid or compensated half-metal when an in-plane magnetic field is applied to the system., Comment: Published version: 6 Pages, 3 Figures (Supplementary: as ancillary file)

Combining mean-field and renormalization group analyses, here we unveil the nature of recently observed superconductivity and parent metallic states in chemically doped rhombohedral trilayer graphene, subject to external electric displacement fields ($D$) [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 434 (2021)]. We argue that close to the charge neutrality, on site Hubbard repulsion favors layer antiferromagnet, which when combined with the $D$-field induced layer polarization, produces a spin-polarized, but valley-unpolarized half-metal, conducive to the nucleation of spin-triplet $f$-wave pairing (SC2). At larger doping valence bond order emerges as a prominent candidate for isospin coherent paramagent, boosting condensation of spin-singlet Cooper pairs in the $s$-wave channel (SC1), manifesting a "selection rule" among competing orders. Responses of these paired states to displacement and in-plane magnetic fields show qualitative similarities with experimental observation. With the onset of the quantum anomalous Hall order, the valley degeneracy of half-metal gets lifted, forming a quarter-metal at lower doping [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 429 (2021)]., Comment: 5 Pages, 4 Figures (Supplementary Materials as Ancillary file): Published version

Electronic materials harbor a plethora of exotic quantum phases, ranging from unconventional superconductors to non-Fermi liquids, and, more recently, topological phases of matter. While these quantum phases in integer dimensions are well characterized by now, their presence in fractional dimensions remains vastly unexplored. Here, we theoretically show that a special class of crystalline phases, namely, higher-order topological phases that via an extended bulk-boundary correspondence feature robust gapless modes on lower dimensional boundaries, such as corners and hinges, can be found on a representative family of fractional materials: \emph{quantum fractals}. To anchor this general proposal, we demonstrate realizations of second-order topological insulators and superconductors, supporting charged and neutral Majorana corner modes, on planar Sierpi\'{n}ski carpet and triangle fractals, respectively. These predictions can be experimentally tested on designer electronic fractal materials, as well as on various highly tunable metamaterial platforms, such as photonic and acoustic lattices., Comment: Published version: 6+epsilon Pages, 3 Figures (Supplementary Material as ancillary file)

We identify time-reversal symmetry breaking mixed-parity superconducting states that feature eight Majorana corner modes in properly cleaved three-dimensional cubic crystals. Namely, when an odd-parity isotropic $p$-wave pairing coexists with cubic symmetry preserving even-parity octupolar $d_{x^2-y^2}+i d_{3z^2-r^2}$ pairing, the gapless surface Majorana modes of the former get localized at the eight corners, thus yielding an \emph{intrinsic} third-order topological superconductor (TOTSC). A cousin $d_{xy}+id_{3z^2-r^2}$ pairing also accommodating eight corner Majorana modes, by virtue of breaking the cubic symmetry, in contrast, yields an \emph{extrinsic} TOTSC. We identify a doped octupolar (topological or trivial) Dirac insulator as a suitable platform to sustain such unconventional superconductors, realized from an intraunit cell pairing. Finally, we argue that the proposed TOTSC can be experimentally realizable in NaCl and other structurally similar compounds under high pressure., Comment: Published Version: 6 Pages, 3 Figures (Supplementary Materials: as ancillary file)

The traditional bulk-boundary correspondence assuring robust gapless modes at the edges and surfaces of insulating and nodal topological materials gets masked in non-Hermitian (NH) systems by the skin effect, manifesting an accumulation of a macroscopic number of states near such interfaces. Here we show that dislocation lattice defects are immune to such skin effect or at most display a \emph{weak} skin effect (depending on its relative orientation with the Burgers vector), and as such they support robust topological modes in the bulk of a NH system, specifically when the parent Hermitian phase features band inversion at a finite momentum. However, the dislocation modes gradually lose their support at their core when the system approaches an exceptional point, and finally melt into the boundary of the system across the NH band gap closing. We explicitly demonstrate these findings for a two-dimensional NH Chern insulator, thereby establishing that dislocation lattice defects can be instrumental to experimentally probe pristine NH topology., Comment: Published version: 6 Pages, 4 Figures (Supplemental Material: as Ancillary file)

Performing a leading-order renormalization group analysis, here we compute the effects of generic local or short-range electronic interactions in monolayer and Bernal bilayer graphene. Respectively in these two systems chiral quasiparticles display linear and biquadratic band touching, leading to linearly vanishing and constant DOS. Consequently, the former system remains stable for weak enough local interactions, and supports a variety of ordered phases only beyond a critical strength of interactions. By contrast, ordered phases can nucleate for sufficiently weak interactions in bilayer graphene. By tuning the strength of all symmetry allowed local interactions, we construct various cuts of the phase diagram at zero and finite temperature and chemical doping. Typically, at zero doping insulating phases (such as charge-density-wave, antiferromagnet, quantum anomalous and spin Hall insulators) prevail at the lowest temperature, while gapless nematic or smectic liquids stabilize at higher temperatures. On the other hand, at finite doping the lowest temperature ordered phase is occupied by a superconductor. Besides anchoring such an organizing principle among the candidate ordered phases, we also establish a selection rule between them and the interaction channel responsible for the breakdown of linear or biquadrtic chiral nodal Fermi liquid. In addition, we also demonstrate the role of the normal state band structure in selecting the pattern of symmetry breaking from a soup of preselected incipient competing orders. As a direct consequence of the selection rule, while an antiferromagnetic phase develops in undoped monolayer and bilayer graphene, the linear (biquadratic) band dispersion favors condensation of a spin-singlet nematic (translational symmetry breaking Kekul\'e) superconductor in doped monolayer (bilayer) graphene, when the on site Hubbard repulsion dominates in these systems., Comment: 40 Pages, 13 Figures, 5 Tables (Published Version): extended abstract, see manuscript

Owing to comparably strong spin-orbit coupling and Hubbard interaction, iridium based 227 pyrochlore oxides harbor a rich confluence of competing magnetic orders and emergent multipolar Weyl quasiparticles. Here we show that this family of materials is predominantly susceptible toward the nucleation of electronic noncoplanar all-in all-out (AIAO) and three-in one-out (3I1O) orders, respectively transforming under the singlet $A_{2u}$ and triplet $T_{1u}$ representations, supporting octupolar and dipolar Weyl fermions, and favored by strong on-site Hubbard and nearest-neighbor ferromagnetic interaction. Furthermore, a coplanar magnetic order generically appears as an intermediate phase between them. This order transforms under the triplet $T_{2u}$ representation and also hosts octupolar Weyl fermions. With the AIAO and 3I1O phases possibly being realized in (Nd$_{1-x}$Pr$_{x}$)$_2$Ir$_2$O$_7$ when $x=0$ and 1, respectively, the intervening $T_{2u}$ order can in principle be found at an intermediate doping ($0

Electronic bands featuring nontrivial bulk topological invariant manifest through robust gapless modes at the boundaries, e.g., edges and surfaces. As such this bulk-boundary correspondence is also operative in driven quantum materials. For example, a suitable periodic drive can convert a trivial insulator into a Floquet topological insulator (FTI) that accommodates nondissipative dynamic gapless modes at the interfaces with vacuum. Here we theoretically demonstrate that dislocations, ubiquitous lattice defects in crystals, can probe FTIs as well as unconventional $\pi$-trivial insulator in the bulk of driven quantum systems by supporting normal and anomalous modes, localized near the defect core. Respectively, normal and anomalous dislocation modes reside at the Floquet zone center and boundaries. We exemplify these outcomes specifically for two-dimensional (2D) Floquet Chern insulator and $p_x+ip_y$ superconductor, where the dislocation modes are respectively constituted by charged and neutral Majorana fermions. Our findings should be therefore instrumental in probing Floquet topological phases in the state-of-the-art experiments in driven quantum crystals, cold atomic setups, and photonic and phononic metamaterials through bulk topological lattice defects., Comment: 8 Pages and 5 Figures (Supplementary Information as Ancillary File)

Following a general protocol of periodically driving static first-order topological phases (supporting surface states) with suitable discrete symmetry breaking Wilson-Dirac masses, here we construct a hierarchy of higher-order Floquet topological phases in three dimensions. In particular, we demonstrate realizations of both second-order and third-order Floquet topological states, respectively supporting dynamic hinge and corner modes at zero quasienergy, by periodically driving their static first-order parent states with one and two discrete symmetry breaking Wilson-Dirac mass(es). While the static surface states are characterized by codimension $d_c=1$, the resulting dynamic hinge (corner) modes, protected by \emph{antiunitary} spectral or particle-hole symmetries, live on the boundaries with $d_c=2$ $(3)$. We exemplify these outcomes for three-dimensional topological insulators and Dirac semimetals, with the latter ones following an arbitrary spin-$j$ representation., Comment: Published version: 6 Pages, 4 Figures

We compute the effects of strong Hubbardlike local electronic interactions on three-dimensional four-component massless Dirac fermions, which in a noninteracting system possess a microscopic global U(1)$\otimes$SU(2) chiral symmetry. A concrete lattice realization of such chiral Dirac excitations is presented, and the role of electron-electron interactions is studied by performing a field theoretic renormalization group (RG) analysis, controlled by a \emph{small} parameter $\epsilon$ with $\epsilon=d-1$, about the lower-critical one spatial dimension. Besides the noninteracting Gaussian fixed point, the system supports four quantum critical and four bicritical points at nonvanishing interaction couplings $\sim \epsilon$. Even though the chiral symmetry is absent in the interacting model, it gets restored (either partially or fully) at various RG fixed points as emergent phenomena. A representative cut of the global phase diagram displays a confluence of scalar and pseudoscalar excitonic and superconducting (such as the $s$-wave and $p$-wave) mass ordered phases, manifesting restoration of (a) chiral U(1) symmetry between two excitonic masses for repulsive interactions and (b) pseudospin SU(2) symmetry between scalar or pseudoscalar excitonic and superconducting masses for attractive interactions. Finally, we perturbatively study the effects of \emph{weak} rotational symmetry breaking on the stability of various RG fixed points., Comment: Published Version: 28 Pages, 3 Figures, 4 Tables

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of \emph{unity}, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the \emph{shift capacitor coupling} between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, \emph{topolectric circuits}, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables us to implement a wide class of topological models through rather simple topolectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, and second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the \emph{stacked topolectric circuit} construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac and Weyl semimetals, respectively displaying surface and hinge localized impedance., Comment: Published version: 24 Pages, 19 Figures

Topological materials occupy the central stage in the modern condensed matter physics because of their robust metallic edge or surface states protected by the topological invariant, characterizing the electronic band structure in the bulk. Higher-order topological (HOT) states extend this usual bulk-boundary correspondence, so they host the modes localized at lower-dimensional boundaries, such as corners and hinges. Here we theoretically demonstrate that dislocations, ubiquitous defects in crystalline materials, can probe higher-order topology, recently realized in various platforms. We uncover that HOT insulators respond to dislocations through symmetry protected finite-energy in-gap electronic modes, localized at the defect core, which originate from an interplay between the orientation of the HOT mass domain wall and the Burgers vector of the dislocation. As such, these modes become gapless only when the Burgers vector points toward lower-dimensional gapless boundaries. Our findings are consequential for the systematic probing of the extended bulk-boundary correspondence in a broad range of HOT crystals, and photonic and phononic or mechanical metamaterials through the bulk topological lattice defects., Comment: Published version: 17 Pages, 6 Figures & 3 Tables

The integer quantum Hall states at fillings $\nu = 0$ and $|\nu| = 1$ in monolayer graphene have drawn much attention as they are generated by electron-electron interactions. Here we explore aspects of the $\nu = 0$ and $|\nu| = 1$ quantum Hall states relevant for experimental samples. In particular, we study the effects of finite extent and finite temperature on the $\nu = 0$ state and finite temperature for the $\nu = 1$ state. For the $\nu = 0$ state we consider the situation in which the bulk is a canted antiferromagnet and use parameters consistent with measurements of the bulk gap to study the edge states in tilted magnetic fields in order to compare with experiment [A. F. Young et al., Nature 505, 528 (2014)]. When spatial modulation of the order parameters is taken into account, we find that for graphene placed on boron nitride, the gap at the edge closes for magnetic fields comparable to those in experiment, giving rise to edge conduction with $G \sim 2e^2/h$ while the bulk gap remains almost unchanged. We also study the transition into the ordered state at finite temperature and field. We determine the scaling of critical temperatures as a function of magnetic field, $B$, and distance to the zero field critical point and find sublinear scaling with magnetic field for weak and intermediate strength interactions, and $\sqrt{B}$ scaling at the coupling associated with the zero field quantum critical point. We also predict that critical temperatures for $\nu = 0$ states should be an order of magnitude higher than those for $|\nu| = 1$ states, consistent with the fact that the low temperature gap for $\nu = 0$ is roughly an order of magnitude larger than that for $|\nu| = 1$., Comment: 14 pages, 12 figures

We study two-dimensional fermionic quadratic band touching (QBT) systems in the presence of vortex and skyrmion of insulating and superconducting masses. A prototypical example of such systems is the Bernal bilayer graphene that supports eight zero-energy modes in the presence of a mass vortex with the requisite U(1) symmetry. Inside the vortex core, additional ten masses that close an SO(5) algebra can develop local expectation values by splitting the zero modes in five and ten different ways by lifting its SO(4) and SU(2) chiral symmetries, respectively. In particular, each SU(2) chiral symmetry can be broken by three distinct copies of chiral-triplet mass orders, giving rise to the notion of the color or flavor degeneracy among the competing orders. By contrast, a skyrmion of three anticommuting masses supports additional six masses in its core, and possesses an SU(2) isospin quantum number, besides the usual generalized U(1) charge. Consequently, charge $4e$ Kekule pair-density-waves can develop in the skyrmion core of N\'eel layer antiferromagnet, while a skyrmion of quantum spin Hall insulator in addition supports a mundane $s$-wave pairing. We also analyze the internal algebra of competing orders in the core of these defects on checkerboard or Kagome lattice that supports only a single copy of QBT., Comment: 14 Pages, 4 Figures, 2 Tables

The presence or absence of certain symmetries in the normal state (NS) also determines the symmetry of the Cooper pairs. Here we show that parity (${\mathcal P}$) and time-reversal (${\mathcal T}$) odd Dirac insulators (trivial or topological) or metals, sustain a local or intra-unit cell pairing that supports corner (in $d=2$) or hinge (in $d=3$) modes of Majorana fermions and stands as a higher-order topological superconductor (HOTSC), when the NS additionally breaks discrete four-fold ($C_4$) symmetry. Although these outcomes does not rely on the existence of a Fermi surface, around it (when the system is doped) the HOTSC takes the form of a mixed parity, ${\mathcal T}$-odd (due to the lack of ${\mathcal P}$ and ${\mathcal T}$ in the NS, respectively) $p+id$ pairing, where the $p$($d$)-wave component stems from the Dirac nature of quasiparticles (lack of $C_4$ symmetry) in the NS. Thus, when strained, magnetically ordered Dirac materials, such as doped magnetic topological insulators (MnBi$_2$Te$_4$), can harbor HOTSCs, while the absence of an external strain should be conducive for the axionic $p+is$ pairing., Comment: 5 Pages, 3 Figures, 2 Tables: Published version

We analyze the stability of time-reversal (${\mathcal T}$) and lattice four-fold ($C_4$) symmetry breaking three-dimensional higher-order topological (HOT) Dirac semimetals (DSMs) and the associated one-dimensional hinge modes in the presence of random pointlike charge impurities. Complementary real space numerical and momentum space renormalization group (RG) analyses suggest that a HOTDSM, while being a stable phase of matter for sufficiently weak disorder, undergoes a continuous quantum phase transition into a trivial metal at finite disorder. However, the corresponding critical exponents (numerically obtained from the scaling of the density of states) are extremely close to the ones found in a dirty, but first-order DSM that on the other hand preserves ${\mathcal T}$ and $C_4$ symmetries, and support two Fermi arc surface states. This observation suggests an emergent \emph{superuniversality} (insensitive to symmetries) in the entire family of dirty DSMs, as also predicted by a leading-order RG analysis. As a direct consequence of the bulk-boundary correspondence, the hinge modes in a system with open boundaries gradually fade away with increasing randomness, and completely dissolve in the trivial metallic phase at strong disorder., Comment: Published version: 12 Pages, 5 Figures, 1 Table

We address the emergent quantum critical phenomena for (pseudo)spin-3/2 birefringent fermions, featuring two effective Fermi velocities, when they reside close to itinerant Mott transitions realized through spontaneous symmetry breaking and triggered by strong local or Hubbardlike repulsive interactions. Irrespective of the nature of the mass orderings that produce fully gapped quasiparticle spectra in the ordered phase, which otherwise can be grouped into three classes, the system always possesses a \emph{unique} terminal velocity near the corresponding quantum critical point. The associated critical regime accommodates a relativistic non-Fermi liquid of strongly coupled collective bosonic and spin-1/2 Dirac excitations with vanishing weight of the quasiparticle pole. These conclusions are also operative near superconducting critical points. Therefore, relativistic non-Fermi liquid possibly constitutes a robust superuniversal description for the entire family of strongly correlated arbitrary half-integer spin Dirac materials., Comment: 6 Pages, 3 Figures and 1 Table: Published Version

Electronic materials can sustain a variety of unusual, but symmetry protected touchings of valence and conduction bands, each of which is identified by a distinct topological invariant. Well-known examples include linearly dispersing pseudo-relativistic fermions in monolayer graphene, Weyl and nodal-loop semimetals, biquadratic (bicubic) band touching in bilayer (trilayer) graphene, as well as mixed dispersions in multi-Weyl systems. Here we show that depending on the underlying band curvature, the shear viscosity in the collisionless regime displays a unique power-law scaling with frequency at low temperatures, bearing the signatures of the band topology, which are distinct from the ones when the system resides at the brink of a topological phase transition into a band insulator. Therefore, besides the density of states (governing specific heat, compressibility) and dynamic conductivity, shear viscosity can be instrumental to pin nodal topology in electronic materials., Comment: Published Version: 6 Pages, 2 Figures (Supplementary as Ancillary file)

Recently, the notion of topological phases of matter has been extended to higher-order incarnations, supporting gapless modes on even lower dimensional boundaries, such as corners and hinges. We here identify a collection of cubic spin-3/2 fermions with biquadratic touching of Kramers degenerate valence and conduction bands as a platform to strain-engineer higher-order topological (HOT) phases: external uniaxial strain gives birth to a HOT Dirac semimetal or an insulator, depending on its sign, featuring topological \emph{hinge} modes in the strain direction. The insulator in fact exhibits \emph{mixed} topology, and in addition supports edge modes on orthogonal planes. These outcomes are germane when the external strain is applied along one of the $C_{4v}$ or coordinate axes, as well as $C_{3v}$ or body-diagonal, directions. Our findings place HgTe, gray-Sn, 227 pyrochlore iridates and half-Heusler compounds at the frontier of strain-engineered electronic HOT phases., Comment: 6 Pages, 1 Figure (Supplemental Material as ancillary file): Published version

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an \emph{antiunitary} operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a \emph{weak} anomalous Hall insulator and a spin-orbital density wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes., Comment: 5.5 Pages, 4 Figures: Published version

We construct the effective field theory for time-reversal symmetry breaking multi-Weyl semimetals (mWSMs), composed of a single pair of Weyl nodes of (anti-)monopole charge $n$, with $n=1,2,3$ in crystalline environment. From both the continuum and lattice models, we show that a mWSM with $n>1$ can be constructed by placing $n$ flavors of linearly dispersing simple Weyl fermions (with $n=1$) in a bath of an $SU(2)$ non-Abelian static background gauge field. Such an $SU(2)$ field preserves certain crystalline symmetry (four-fold rotational or $C_4$ in our construction), but breaks the Lorentz symmetry, resulting in nonlinear band spectra (namely, $E \sim (p^2_x + p^2_y)^{n/2}$, but $E \sim |p_z|$, for example, where momenta ${\bf p}$ is measured from the Weyl nodes). Consequently, the effective field theory displays $U(1) \times SU(2)$ non-Abelian anomaly, yielding anomalous Hall effect, its non-Abelian generalization, and various chiral conductivities. The anomalous violation of conservation laws is determined by the monopole charge $n$ and a specific algebraic property of the $SU(2)$ Lie group, which we further substantiate by numerically computing the regular and "isospin" densities from the lattice models of mWSMs. These predictions are also supported from a strongly coupled (holographic) description of mWSMs. Altogether our findings unify the field theoretic descriptions of mWSMs of arbitrary monopole charge $n$ (featuring $n$ copies of the Fermi arc surface states), predict signatures of non-Abelian anomaly in table-top experiments, and pave the route to explore anomaly structures for multi-fold fermions, transforming under arbitrary half-integer or integer spin representations., Comment: 21 pages, 10 figures: Accepted version

Higher-order topological~(HOT) states,~hosting topologically protected modes on lower-dimensional boundaries,~such as hinges and corners, have recently extended the realm of the static topological phases.~Here we demonstrate the possibility of realizing a two-dimensional \emph{Floquet} second-order topological insulator, featuring corner-localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q^{\rm Flq}_{xy}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete fourfold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking Dirac mass perturbation.~Furthermore, we show that $Q^{\rm Flq}_{xy}$ becomes independent of the choice of origin as the system approaches the thermodynamic limit.~We also analyze the dynamics of a corner mode after a sudden quench, when the $C_4$ and ${\mathcal T}$ symmetry breaking perturbation is switched off, and find that the corresponding survival probability displays periodic appearances of complete, partial and no revival for long time, encoding the signature of corner modes in a QSHI.~Our protocol is sufficiently general to explore the territory of dynamical HOT phases in insulators and gapless systems., Comment: 5.5 Pages, 5 Figures: Published version

We analyze the possibility of emergent quantum multicritical points (MCPs) with enlarged chiral symmetry, when strongly interacting gapless Dirac fermions acquire comparable propensity toward the nucleation of Kekul\'{e} valence-bond solid (KVBS) and charge-density-wave ($N_b=1$) or $s$-wave pairing ($N_b=2$) or anti-ferromagnet ($N_b=3$) in honeycomb lattice, where $N_b$ counts the number of bosonic order parameter components. Besides the cubic terms present in the order parameter description of KVBS due to the breaking of a discrete $Z_3$ symmetry, quantum fluctuations generate new cubic vertices near the high symmetry MCPs. All cubic terms are strongly relevant at the bare level near three spatial dimensions, about which we perform a leading order renormalization group analysis of coupled Gross-Neveu-Yukawa field theory. We show that due to non-trivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an $O(2+N_b)$ symmetric MCP, near two spatial dimensions, where $N_b=1,2,3$. Therefore, MCPs with an enlarged $O(2+N_b)$ symmetry near KVBS ordering are stable., Comment: 6 Pages, 2 Figures: Published version in Phys. Rev. B (Supplementary: As ancillary file)