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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 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: 5 Pages, 1 Figure, 2 Tables (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

Lorentz space–time symmetry represents a unifying feature of the fundamental forces, typically manifest at sufficiently high energies, while in quantum materials it emerges in the deep low-energy regime. However, its fate in quantum materials coupled to an environment thus far remained unexplored. We here introduce a general framework of constructing symmetry-protected Lorentz-invariant non-Hermitian (NH) Dirac semimetals (DSMs), realized by invoking masslike anti-Hermitian Dirac operators to its Hermitian counterpart. Such NH DSMs feature purely real or imaginary isotropic linear band dispersion, yielding a vanishing density of states. Dynamic mass orderings in NH DSMs thus take place for strong Hubbard-like local interactions through a quantum phase transition, hosting a non-Fermi liquid, beyond which the system becomes an insulator. We show that depending on the internal Clifford algebra between the NH Dirac operator and candidate mass order-parameter, the resulting quantum-critical fluid either remains coupled with the environment or recovers full Hermiticity by decoupling from the bath, while always enjoying an emergent Yukawa-Lorentz symmetry in terms of a unique terminal velocity. We showcase the competition between such mass orderings, their hallmarks on quasi-particle spectra in the ordered phases, and the relevance of our findings for correlated designer NH Dirac materials., QC 20240613

Lorentz space–time symmetry represents a unifying feature of the fundamental forces, typically manifest at sufficiently high energies, while in quantum materials it emerges in the deep low-energy regime. However, its fate in quantum materials coupled to an environment thus far remained unexplored. We here introduce a general framework of constructing symmetry-protected Lorentz-invariant non-Hermitian (NH) Dirac semimetals (DSMs), realized by invoking masslike anti-Hermitian Dirac operators to its Hermitian counterpart. Such NH DSMs feature purely real or imaginary isotropic linear band dispersion, yielding a vanishing density of states. Dynamic mass orderings in NH DSMs thus take place for strong Hubbard-like local interactions through a quantum phase transition, hosting a non-Fermi liquid, beyond which the system becomes an insulator. We show that depending on the internal Clifford algebra between the NH Dirac operator and candidate mass order-parameter, the resulting quantum-critical fluid either remains coupled with the environment or recovers full Hermiticity by decoupling from the bath, while always enjoying an emergent Yukawa-Lorentz symmetry in terms of a unique terminal velocity. We showcase the competition between such mass orderings, their hallmarks on quasi-particle spectra in the ordered phases, and the relevance of our findings for correlated designer NH Dirac materials.

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 the 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., QC 20231006

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 the 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.

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

Lorentz spacetime symmetry represents a unifying feature of the fundamental forces, typically manifest at sufficiently high energies, while in quantum materials it emerges in the deep low-energy regime. However, its fate in quantum materials coupled to an environment thus far remained unexplored. We here introduce a general framework of constructing symmetry-protected Lorentz invariant non-Hermitian (NH) Dirac semimetals (DSMs), realized by invoking masslike anti-Hermitian Dirac operators to its Hermitian counterpart. Such NH DSMs feature purely real or imaginary isotropic linear band dispersion, yielding a vanishing density of states. Dynamic mass orderings in NH DSMs thus take place for strong Hubbardlike local interactions through a quantum phase transition, hosting a non-Fermi liquid, beyond which the system becomes an insulator. We show that depending on the internal Clifford algebra between the NH Dirac operator and candidate mass order-parameter, the resulting quantum-critical fluid either remains coupled with the environment or recovers full Hermiticity by decoupling from the bath, while always enjoying an emergent Yukawa-Lorentz symmetry in terms of a unique terminal velocity. We showcase the competition between such mass orderings, their hallmarks on quasiparticle spectra in the ordered phases, and the relevance of our findings for correlated designer NH Dirac materials., Comment: Published Version: 10 Pages, 3 Figures & 1 Table (Supplemental Material 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)

Fractional Dirac materials (FDMs) feature a fractional energy-momentum relation E(k) <^> |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.

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)

Nature harbors crystals of dimensionality (d) only up to three. Here we introduce the notion of 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 and 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 modes, 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.

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)

Nature harbors crystals of dimensionality (d) only up to three. Here we introduce the notion of 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 and 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 modes, 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. Authors introduce a new class of topological materials, namely projected topological branes that are holographic images of higher-dimensional topological crystals, and feature either emergent crystalline or aperiodic quasicrystalline order. They manifest bulk-boundary and bulk-lattice defect correspondences of parent crystals and open a realistic route to harness four and higher-dimensional topological crystals in three-dimensional world., QC 20220929

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)

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

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)

In this thesis we study a variety of two- and three-dimensional (2D and 3D, respectively) nodal semimetals, subjected to local electronic interactions or disorder. Such systems constitute a minimal model for various real materials and capture a plethora of interesting physical phenomena therein. Our methodology includes an unbiased renormalization group analysis controlled by epsilon expansions about the appropriate lower critical dimension, mean-field analysis, as well as complementary numerical analyses. First, we focus on emergent symmetries at various infrared unstable quantum critical points, appearing in a renormalization group flow of interaction couplings. We investigate a 3D chiral Dirac semimetal, which in a noninteracting system enjoys a microscopic U(1)⊗SU(2) global symmetry. Though the chiral symmetry is absent in the interacting model, it gets restored (partially or fully) at various fixed points as emergent phenomena. Subsequently, we study a collection of 3D interacting effective spin-3/2 biquadratic Luttinger fermions, and demonstrate the emergence of full rotational symmetry between the distinct nematic sectors (namely Eg and T2g ) of the corresponding octahedral group. We then investigate the effects of electronic interactions at zero and finite temperature and chemical doping in a collection of (i) 2D Dirac and Luttinger fermions, constituting the linearly and quadratically dispersing low-energy excitations in monolayer and bilayer graphene, respectively, and (ii) 3D Luttinger fermions, describing a biquadratic touching of Kramers degenerate conduction and valence bands, relevant in the normal state of 227 pyrochlore iridates, and half-Heusler compounds, for example. These systems exhibit a plethora of competing broken symmetry phases (both magnetic and superconducting) when tuning the strength of interactions, temperature, and chemical doping. In this context we propose the selection rules, identifying the broken symmetry phases promoted by a gi

In this thesis we study a variety of two- and three-dimensional (2D and 3D, respectively) nodal semimetals, subjected to local electronic interactions or disorder. Such systems constitute a minimal model for various real materials and capture a plethora of interesting physical phenomena therein. Our methodology includes an unbiased renormalization group analysis controlled by epsilon expansions about the appropriate lower critical dimension, mean-field analysis, as well as complementary numerical analyses. First, we focus on emergent symmetries at various infrared unstable quantum critical points, appearing in a renormalization group flow of interaction couplings. We investigate a 3D chiral Dirac semimetal, which in a noninteracting system enjoys a microscopic U(1)⊗SU(2) global symmetry. Though the chiral symmetry is absent in the interacting model, it gets restored (partially or fully) at various fixed points as emergent phenomena. Subsequently, we study a collection of 3D interacting effective spin-3/2 biquadratic Luttinger fermions, and demonstrate the emergence of full rotational symmetry between the distinct nematic sectors (namely Eg and T2g ) of the corresponding octahedral group. We then investigate the effects of electronic interactions at zero and finite temperature and chemical doping in a collection of (i) 2D Dirac and Luttinger fermions, constituting the linearly and quadratically dispersing low-energy excitations in monolayer and bilayer graphene, respectively, and (ii) 3D Luttinger fermions, describing a biquadratic touching of Kramers degenerate conduction and valence bands, relevant in the normal state of 227 pyrochlore iridates, and half-Heusler compounds, for example. These systems exhibit a plethora of competing broken symmetry phases (both magnetic and superconducting) when tuning the strength of interactions, temperature, and chemical doping. In this context we propose the selection rules, identifying the broken symmetry phases promoted by a gi

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(x2-y2) + id(3z2-r2) pairing, the gapless surface Majorana modes of the former get localized at the eight corners, thus yielding an intrinsic third-order topological superconductor (TOTSC). A cousin d(xy) + id(3z2-r2) pairing also accommodating eight corner Majorana modes, by virtue of breaking the cubic symmetry, in contrast, yields an 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., QC 20211126

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)

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)

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(x2-y2) + id(3z2-r2) pairing, the gapless surface Majorana modes of the former get localized at the eight corners, thus yielding an intrinsic third-order topological superconductor (TOTSC). A cousin d(xy) + id(3z2-r2) pairing also accommodating eight corner Majorana modes, by virtue of breaking the cubic symmetry, in contrast, yields an 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.

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., QC 20210813

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 unity, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the shift capacitor coupling between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, 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 dc=m. Finally, we subscribe to the stacked topolectric circuit construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac, and Weyl semimetals, respectively, displaying surface- and hinge-localized impedance., QC 20210526

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 dc=1, the resulting dynamic hinge (corner) modes, protected by antiunitary spectral or particle-hole symmetries, live on the boundaries with dc=2(3). We exemplify these outcomes for three-dimensional topological insulators and Dirac semimetals, with the latter ones following an arbitrary spin-j representation., QC 20210601

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.

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 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.

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 unity, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the shift capacitor coupling between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, 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 stacked topolectric circuit construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac, and Weyl semimetals, respectively, displaying surface- and hinge-localized impedance.

In this thesis we study a variety of two- and three-dimensional (2D and 3D, respectively) nodal semimetals, subjected to local electronic interactions or disorder. Such systems constitute a minimal model for various real materials and capture a plethora of interesting physical phenomena therein. Our methodology includes an unbiased renormalization group analysis controlled by epsilon expansions about the appropriate lower critical dimension, mean-field analysis, as well as complementary numerical analyses. First, we focus on emergent symmetries at various infrared unstable quantum critical points, appearing in a renormalization group flow of interaction couplings. We investigate a 3D chiral Dirac semimetal, which in a noninteracting system enjoys a microscopic U(1)⊗SU(2) global symmetry. Though the chiral symmetry is absent in the interacting model, it gets restored (partially or fully) at various fixed points as emergent phenomena. Subsequently, we study a collection of 3D interacting effective spin-3/2 biquadratic Luttinger fermions, and demonstrate the emergence of full rotational symmetry between the distinct nematic sectors (namely Eg and T2g ) of the corresponding octahedral group. We then investigate the effects of electronic interactions at zero and finite temperature and chemical doping in a collection of (i) 2D Dirac and Luttinger fermions, constituting the linearly and quadratically dispersing low-energy excitations in monolayer and bilayer graphene, respectively, and (ii) 3D Luttinger fermions, describing a biquadratic touching of Kramers degenerate conduction and valence bands, relevant in the normal state of 227 pyrochlore iridates, and half-Heusler compounds, for example. These systems exhibit a plethora of competing broken symmetry phases (both magnetic and superconducting) when tuning the strength of interactions, temperature, and chemical doping. In this context we propose the selection rules, identifying the broken symmetry phases promoted by a gi

In this thesis we study a variety of two- and three-dimensional (2D and 3D, respectively) nodal semimetals, subjected to local electronic interactions or disorder. Such systems constitute a minimal model for various real materials and capture a plethora of interesting physical phenomena therein. Our methodology includes an unbiased renormalization group analysis controlled by epsilon expansions about the appropriate lower critical dimension, mean-field analysis, as well as complementary numerical analyses. First, we focus on emergent symmetries at various infrared unstable quantum critical points, appearing in a renormalization group flow of interaction couplings. We investigate a 3D chiral Dirac semimetal, which in a noninteracting system enjoys a microscopic U(1)⊗SU(2) global symmetry. Though the chiral symmetry is absent in the interacting model, it gets restored (partially or fully) at various fixed points as emergent phenomena. Subsequently, we study a collection of 3D interacting effective spin-3/2 biquadratic Luttinger fermions, and demonstrate the emergence of full rotational symmetry between the distinct nematic sectors (namely Eg and T2g ) of the corresponding octahedral group. We then investigate the effects of electronic interactions at zero and finite temperature and chemical doping in a collection of (i) 2D Dirac and Luttinger fermions, constituting the linearly and quadratically dispersing low-energy excitations in monolayer and bilayer graphene, respectively, and (ii) 3D Luttinger fermions, describing a biquadratic touching of Kramers degenerate conduction and valence bands, relevant in the normal state of 227 pyrochlore iridates, and half-Heusler compounds, for example. These systems exhibit a plethora of competing broken symmetry phases (both magnetic and superconducting) when tuning the strength of interactions, temperature, and chemical doping. In this context we propose the selection rules, identifying the broken symmetry phases promoted by a gi

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 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 identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two- and three-dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by an outer crystalline boundary, the system continues to host corner states, yielding amorphous HOT insulators. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system becomes a trivial insulator when amorphousness plagues the entire system. These outcomes are further substantiated by computing the quadrupolar (octupolar) moment in two (three) dimensions. Therefore, HOT phases can be realized in amorphous solids, when wrapped by a thin (lithographically grown, for example) crystalline layer. Our findings suggest that crystalline topological phases can be realized even in the absence of local crystalline symmetry., QC 20210125

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 Hubbard-like 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 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., QC 20210120

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 pseudorelativistic 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., QC 20200611

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 pseudorelativistic 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.

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 identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two- and three-dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by an outer crystalline boundary, the system continues to host corner states, yielding amorphous HOT insulators. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system becomes a trivial insulator when amorphousness plagues the entire system. These outcomes are further substantiated by computing the quadrupolar (octupolar) moment in two (three) dimensions. Therefore, HOT phases can be realized in amorphous solids, when wrapped by a thin (lithographically grown, for example) crystalline layer. Our findings suggest that crystalline topological phases can be realized even in the absence of local crystalline symmetry.

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)

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

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

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)

Flat electronic bands can accommodate a plethora of interaction-driven quantum phases, since kinetic energy is quenched therein and electronic interactions therefore prevail. Twisted bilayer graphene, near the so-called "magic angles", features slow Dirac fermions close to the charge-neutrality point that persist up to high energies. Starting from a continuum model of slow but strongly interacting Dirac fermions, we show that with increasing chemical doping away from the charge-neutrality point, a time-reversal symmetry breaking, valley pseudospin-triplet, topological p + ip superconductor gradually sets in, when the system resides at the brink of an antiferromagnetic ordering (due to Hubbard repulsion), in qualitative agreement with recent experimental findings. The p ip paired state exhibits quantized spin and thermal Hall conductivities, and polar Kerr and Faraday rotations. Our conclusions should also be applicable for other correlated two-dimensional Dirac materials., QC 20190409

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a D-dimensional first-order or regular topological phase involves m Hermitian matrices that anticommute with additional p - 1 mutually anticommuting matrices, it is conceivable to realize an nth-order HOT phase, where n = 1, ..., p, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An nth-order HOT phase accommodates zero modes on a surface with codimension n. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo)spin-1/2 or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class., QC 20190125

Flat electronic bands can accommodate a plethora of interaction-driven quantum phases, since kinetic energy is quenched therein and electronic interactions therefore prevail. Twisted bilayer graphene, near the so-called magic angles, features slow Dirac fermions close to the charge-neutrality point that persist up to high energies. Starting from a continuum model of slow but strongly interacting Dirac fermions, we show that with increasing chemical doping away from the charge-neutrality point, a time-reversal symmetry breaking, valley pseudospin-triplet, topological p + ip superconductor gradually sets in, when the system resides at the brink of an antiferromagnetic ordering (due to Hubbard repulsion), in qualitative agreement with recent experimental findings. The p ip paired state exhibits quantized spin and thermal Hall conductivities, and polar Kerr and Faraday rotations. Our conclusions should also be applicable for other correlated two-dimensional Dirac materials.

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a D-dimensional first-order or regular topological phase involves m Hermitian matrices that anticommute with additional p - 1 mutually anticommuting matrices, it is conceivable to realize an nth-order HOT phase, where n = 1, ..., p, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An nth-order HOT phase accommodates zero modes on a surface with codimension n. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo)spin-1/2 or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

We analyze the possibility of emergent quantum multicritical points (MCPs) with enlarged chiral symmetry, when strongly interacting gapless Dirac fermions acquire a comparable propensity toward the nucleation of Kekule valence-bond solid (KVBS) and charge-density-wave (N-b = 1) or s-wave pairing (N-b = 2) or antiferromagnet (N-b = 3) in a 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 nontrivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an O(2 + N-b) symmetric MCP, at leas 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.

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