The fluctuations-driven continuous quantum criticality has sparked tremendous interest in condensed matter physics. It has been verified that the gapless fermions fluctuations can change the nature of phase transition at criticality. In this paper, we study the fermionic quantum criticality with enlarged Ising$\times$Ising fluctuations in honeycomb lattice materials. The Gross-Neveu-Yukawa theory for the multicriticality between the semimetallic phase and two ordered phases that break Ising symmetry is investigated by employing perturbative renormalization group approach. We first determine the critical range in which the quantum fluctuations may render the phase transition continuous. We find that the Ising criticality is continuous only when the flavor numbers of four-component Dirac fermions $N_f\geq1/4$. Using the $ε$ expansion in four space-time dimensions, we then study the Ising$\times$Ising multicriticality stemming from the symmetry-breaking electronic instabilities. We analyze the underlying fixed-point structure and compute the critical exponents for the Ising$\times$Ising Gross-Neveu-Yukawa universality class. Further, the correlation scaling behavior for the fermion bilinear on the honeycomb lattice at the multicritical point are also briefly discussed.
The observation of quantum criticality in diverse classes of strongly correlated electron systems has been instrumental in establishing ordering principles, discovering new phases, and identifying the relevant degrees of freedom and interactions. At focus so far have been insulators and metals. Semimetals, which are of great current interest as candidate phases with nontrivial topology, are much less explored in experiments. Here we study the Kondo semimetal CeRu$_4$Sn$_6$ by magnetic susceptibility, specific heat, and inelastic neutron scattering experiments. The power-law divergence of the m…
We analyze the scaling behavior at and near a quantum critical point separating a semimetallic from a superfluid phase. To this end we compute the renormalization group flow for a model of attractively interacting electrons with a linear dispersion around a single Dirac point. We study both ground state and finite temperature properties. In two dimensions, the electrons and the order parameter fluctuations exhibit power-law scaling with anomalous scaling dimensions. The quasi-particle weight and the Fermi velocity vanish at the quantum critical point. The order parameter correlation length tur…
We consider the fermionic quantum criticality of anisotropic nodal point semimetals in $d = d_L + d_Q$ spatial dimensions that disperse linearly in $d_L$ dimensions, and quadratically in the remaining $d_Q$ dimensions. When subject to strong interactions, these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking. Such quantum critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields. We analyze the universal scaling in the physically relevant spatial dimensions, genera…
Pezzini et al. reported an unconventional mass enhancement in topological nodal line semimetal ZrSiS (Nat. Phys. 14, 178 (2018), whose origin remains puzzling. In this material, strong short-range interactions might induce excitonic particle-hole pairs. Here we study the renormalization of fermion velocities and find that the mass enhancement in ZrSiS can be well understood if we suppose that ZrSiS is close to the quantum critical point between semimetal and excitonic insulator. Near this quantum critical point, the fermion velocities are considerably reduced by excitonic quantum fluctuation, …
When a metal undergoes continuous quantum phase transition, the correlation length diverges at the critical point and the quantum fluctuation of order parameter behaves as a gapless bosonic mode. Generically, the coupling of this boson to fermions induces a variety of unusual quantum critical phenomena, such as non-Fermi liquid behavior and various emergent symmetries. Here, we perform a renormalization group analysis of the semimetal-superconductor quantum criticality in a three-dimensional anisotropic Weyl semimetal. Surprisingly, distinct from previously studied quantum critical systems, th…
We study superconductivity in a three-dimensional zero-density Dirac semimetal in proximity to a ferroelectric quantum critical point. We find that the interplay of criticality, inversion-symmetry breaking, and Dirac dispersion gives rise to a robust superconducting state at the charge-neutrality point, where no Fermi surface is present. Using Eliashberg theory, we show that the ferroelectric quantum critical point is unstable against the formation of a ferroelectric density wave (FDW), whose fluctuations, in turn, lead to a first-order superconducting transition. Surprisingly, long-range supe…
We study the stability of three-dimensional incompressible Weyl semimetals in the presence of random quenched charge impurities. Combining numerical analysis and scaling theory we show that in the presence of sufficiently weak randomness (i) Weyl semimetal remains stable, while (ii) double-Weyl semimetal gives rise to compressible diffusive metal where the mean density of states at zero energy is finite. At stronger disorder, Weyl semimetal undergoes a quantum phase transition and enter into a metallic phase. Mean density of states at zero energy serves as the order parameter and displays sing…
Disorder in Weyl semimetals and superconductors is surprisingly subtle, attracting attention and competing theories in recent years. In this brief review, we discuss the current theoretical understanding of the effects of short-ranged, quenched disorder on the low energy-properties of three-dimensional, topological Weyl semimetals and superconductors. We focus on the role of non-perturbative rare region effects on destabilizing the semimetal phase and rounding the expected semimetal-to-diffusive metal transition into a cross over. Furthermore, the consequences of disorder on the resulting natu…
We study a simple model of three-dimensional fermions close to a quadratic band touching point, built from the celebrated Luttinger single-particle Hamiltonian and an attractive contact interaction between the particles. Such a system displays a novel quantum critical point between the semimetallic and an s-wave superconducting phase at which the low-energy "Luttinger fermions" are inextricably coupled to the order parameter fluctuations. The quantum critical point is perturbatively accessible near four spatial dimensions, where it features nontrivial scaling with dynamical exponent 1<z<2 and …
We numerically study the effect of short ranged potential disorder on massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus on the question of the proposed quantum critical point separating the semimetal and diffusive metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian ($H$) and exactly calculate the density of states (DOS) near zero energy, using a combination of Lanczos on $H^2$ and the kernel polynomial method on $H$. We establish the existence of two distinct types of low energy eigenstates contributing to the disordere…
Exotic physics often emerges around quantum criticality in metallic systems. Here we explore the nature of topological phase transitions between 3D double-Weyl semimetals and insulators (through annihilating double-Weyl nodes with opposite chiralities) in the presence of Coulomb interactions. From renormalization-group (RG) analysis, we find a non-Fermi-liquid quantum critical point (QCP) between the double-Weyl semimetals and insulators when artificially neglecting short-range interactions. However, it is shown that this non-Fermi-liquid QCP is actually unstable against nematic ordering when …
In this paper we discuss the N$\acute{e}$el and Kekul$\acute{e}$ valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction $g(\barψ_iΓ_{ij}ψ_j)^2$ that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling $g$ is greater than a critical value $g_c$. The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling,…
Quantum criticality, a manifestation of emergent scale invariance in electron wavefunctions arises from intricate many-body quantum entanglement. One of the natural venues for the criticality is clean undoped Dirac semimetals, known as a marginally-renormalized critical phase. The ground state is only slightly modified from the Slater-type product wavefunction because the scatterings from weak disorder and Coulomb interactions are suppressed. Here, using the renormalization group (RG) analysis, we show that a novel class of quantum criticality appears in Dirac semimetals when the disorder stre…
We show that the asymptotic infinite dimensional enlarged gauge symmetries constructed for QED are anomalous in Dirac and Weyl semimetals. This symmetry is particularly important in particle physics for its analogy with the Bondi-Metzner-Sachs (BMS) symmetries in gravity, as well as for its connection with QED soft IR theorems. This leads to observable effects, because of the induction of a new current in the material, which carries a memory of the BMS symmetry precursor.
Pyrochlore systems ($A_{2}B_{2}O_{7}$) with $A$-site rare-earth local moments and $B$-site $5d$ conduction electrons offer excellent material platforms for the discovery of exotic quantum many-body ground states. Notable examples include U(1) quantum spin liquid of the local moments and semimetallic non-Fermi liquid of the conduction electrons. Here we investigate emergent quantum phases and their transitions driven by the Kondo lattice coupling between such highly entangled quantum ground states. Using the renormalization group method, it is shown that weak Kondo lattice coupling is irrelevan…
Unconstrained compact lattice QED$_{2+1}$ coupled to phonons: Gauss sectors, orthogonal semimetal, and deconfined criticality
Institut für Theoretische Physik und Astrophysik, Universität Würzburg（维尔茨堡大学理论物理与天体物理研究所）；Würzburg-Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群ctd.qmat）
We investigate the unconventional quantum phase transitions in Weyl semimetals. The emergent boson fields, coupling with the Weyl fermion bilinears, contain a Wess-Zumino-Witten term or topological $Θ$ term inherited from the momentum space monopoles carried by Weyl points. Three types of unconventional quantum critical points will be studied in order: (1) The transition between two distinct symmetry breaking phases whose criticality is beyond Landau's paradigm. (2) The transition between a symmetry breaking state to a topological ordered state. (3) The transition between $3d$ topological orde…
We establish a scenario where fluctuations of new degrees of freedom at a quantum phase transition change the nature of a transition beyond the standard Landau-Ginzburg paradigm. To this end we study the quantum phase transition of gapless Dirac fermions coupled to a $\mathbb{Z}_3$ symmetric order parameter within a Gross-Neveu-Yukawa model in 2+1 dimensions, appropriate for the Kekulé transition in honeycomb lattice materials. For this model the standard Landau-Ginzburg approach suggests a first order transition due to the symmetry-allowed cubic terms in the action. At zero temperature, howev…
Separating between ordinary (OHE) and anomalous (AHE) Hall effect in the paramagnetic phase of Mn$_{1-x}$Fe$_{x}$Si reveals OHE sign inversion associated with the hidden quantum critical (QC) point $x^*\sim0.11$. The semimetallic behavior at intermediate Fe content leads to verifiable predictions in the field of fermiology, magnetic interactions and QC in Mn$_{1-x}$Fe$_{x}$Si. The change of electron and hole concentrations is considered as a driving force for tuning the QC regime in Mn$_{1-x}$Fe$_{x}$Si via modifying of RKKY exchange interaction within the Heisenberg model of magnetism.
Fermion-induced quantum critical points (FIQCPs) were recently discovered at the putatively first-order transitions between two-dimensional (2D) Dirac semimetals and the Kekule valence bond solids on the honeycomb lattice by sign-free quantum Monte Carlo simulations [Nature Communications 8, 314, (2017)]. Here, we investigate possible FIQCP in 3D topological Weyl semimetals at a $Z_3$ symmetry-breaking transition that is putatively first-order according to the Landau criterion. We construct a lattice model featuring 3D double-Weyl fermions (monopole charges $\pm$2) and we show that $Z_3$ nodal…
We numerically investigate the critical behavior of the Hubbard model on the honeycomb and the $π$-flux lattice, which exhibits a direct transition from a Dirac semimetal to an antiferromagnetically ordered Mott insulator. We use projective auxiliary-field quantum Monte Carlo simulations and a careful finite-size scaling analysis that exploits approximately improved renormalization-group-invariant observables. This approach, which is successfully verified for the three-dimensional XY transition of the Kane-Mele-Hubbard model, allows us to extract estimates for the critical couplings and the cr…
Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)$\times$U(1) symmetry. Using large-scale auxiliary-field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly-interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The…
Progress in the understanding of quantum critical properties of itinerant electrons has been hindered by the lack of effective models which are amenable to controlled analytical and numerically exact calculations. Here we establish that the disorder driven semimetal to metal quantum phase transition of three dimensional massless Dirac fermions could serve as a paradigmatic toy model for studying itinerant quantum criticality, which is solved in this work by exact numerical and approximate field theoretic calculations. As a result, we establish the robust existence of a non-Gaussian universalit…
We compare two crystallographic phases of the low-dimensional WP$_2$ to better understand features of electron-electron and electron-phonon interactions in topological systems. The topological $β$-phase, a Weyl semimetal with a giant magneto-resistance, shows a larger intensity of electronic Raman scattering compared to the topologically trivial $α$-phase. This intensity sharply drops for $T < T^* = 20$ K which evidences a crossover in the topological phase from marginal quasiparticles to a coherent low temperature regime. In contrast, the non-topological $α$-phase shows more pronounced signat…
We formulate the effective Gross-Neveu-Yukawa theory of the semimetal-insulator transitions on the honeycomb lattice and compute its quantum critical behavior near three (spatial) dimensions. We find that at the critical point Dirac fermions do not survive as coherent excitations and that the $\sim 1/r$ tail of the weak Coulomb interaction is an irrelevant coupling. The emergent Lorentz invariance near criticality implies a universal ratio of the low-temperature specific heats of the metallic and the rotational-symmetry-broken insulating phase.
Strain tuning is increasingly being recognized as a clean tuning parameter to induce novel behavior in quantum matter. Motivated by the possibility of straining graphene up to $20$ percent, we investigate novel quantum criticality due to interplay between strain-induced anisotropic band structure and critical antiferromagnetic spin fluctuations (AFSF) in this setting. We detail how this interplay drives $(i)$ a quantum phase transition (QPT) between the Dirac-semimetal-incoherent pseudogapped metal-correlated insulator as a function of strain ($ε$), and $(ii)$ critical AFSF-driven divergent ne…
Magnetic fluctuations and electrons couple in intriguing ways in the vicinity of zero temperature phase transitions - quantum critical points - in conducting materials. Quantum criticality is implicated in non-Fermi liquid behavior of diverse materials, and in the formation of unconventional superconductors. Here we uncover an entirely new type of quantum critical point describing the onset of antiferromagnetism in a nodal semimetal engendered by the combination of strong spin-orbit coupling and electron correlations, and which is predicted to occur in the iridium oxide pyrochlores. We formula…
The field theory of the semimetal-superconductor quantum phase transition for graphene and surface states of topological insulators is presented. The Lagrangian possesses the global U(1) symmetry, with the self-interacting complex bosonic order-parameter and the massless Dirac fermions coupled through a Yukawa term. The same theory also governs the quantum critical behavior of graphene near the transition towards the bond-density-wave (Kekule) insulator. The local U(1) gauged version of the theory which describes the quantum semimetal-superconductor transition in the ultimate critical regime i…
Existing theoretical works differ on whether three-dimensional Dirac and Weyl semimetals are stable to a short-range-correlated random potential. Numerical evidence suggests the semimetal to be unstable, while some field-theoretic instanton calculations have found it to be stable. The differences go beyond method: the continuum field-theoretic works use a single, perfectly linear Weyl cone, while numerical works use tight-binding lattice models which inherently have band curvature and multiple Weyl cones. In this work, we bridge this gap by performing exact numerics on the same model used in a…
Recent discoveries of a new type of quantum criticality arising from Yb-valence fluctuations in Yb-based metal in periodic crystal and quasicrystal have opened a new class of quantum critical phenomena in correlated electron systems. To clarify whether this new concept can be generalized to other rare-earth-based semimetal and insulator, we study SmS which exhibits golden-black phase transition under pressure. By constructing the model for SmS, we show that Coulomb repulsion between 4f and 5d orbitals at Sm drives first-order valence transition (FOVT) and semimetal-to-insulator transition (MIT…
We study the Hubbard model on the honeycomb lattice in the vicinity of the quantum critical point by means of a multiband formulation of the Dual Fermion approach. Beyond the strong local correlations of the dynamical mean field, critical fluctuations on all length scales are included by means of a ladder diagram summation. Analysis of the susceptibility yields an estimate of the critical interaction strength of the quantum phase transition from a paramagnetic semimetal to an antiferromagnetic insulator, in good agreement to other numerical methods. We further estimate the crossover temperatur…
We analyze the quantum phase transition between a semimetal and a superfluid in a model of attractively interacting fermions with a linear dispersion. The quantum critical properties of this model cannot be treated by the Hertz-Millis approach since integrating out the fermions leads to a singular Landau-Ginzburg order parameter functional. We therefore derive and solve coupled renormalization group equations for the fermionic degrees of freedom and the bosonic order parameter fluctuations. In two spatial dimensions, fermions and bosons acquire anomalous scaling dimensions at the quantum criti…
We analyze emergent quantum multi-criticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders, which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $O(S_1)$ and $O(S_2)$ symmetries in the ordered phase. Performing a renormalization group analysis by using the $ε=(3-d)$ exp…
Quantum chaos in many-body systems may be characterized by the Lyapunov exponent defined as the exponential growth rate of out-of-time-order correlators (OTOC). So far Lyaponov exponents around various quantum critical points (QCP) remain largely unexplored. Here, we investigate the Lyapunov exponent around QCPs of the Gross-Neveu (GN) model with $N$ flavors of Dirac fermions in (2+1) dimensions. Around the GN quantum phase transition between a Dirac semimetal and a gapped insulator breaking $Z_2$ symmetry (e.g., inversion symmetry of the honeycomb lattice), we find that the Lyaponov exponent …
Topological quantum phase transitions intrinsically intertwine self-similarity and topology of many-electron wave-functions, and divining them is one of the most significant ways to advance understanding in condensed matter physics. Our focus is to investigate an unconventional class of the transitions between insulators and Dirac semimetals whose description is beyond conventional pseudo relativistic Dirac Hamiltonian. At the transition without the long-range Coulomb interaction, the electronic energy dispersion along one direction behaves like a relativistic particle, linear in momentum, but…
Supersymmetry (SUSY) interchanges bosons and fermions but no direct evidences of it have been revealed in nature yet. In this letter, we observe that fluctuating pair density waves (PDW) consist of two complex order parameters which can be superpartners of the unavoidably-doubled Weyl fermions in three-dimensional lattice models. We construct explicit fermionic lattice models featuring 3D Weyl fermions and show that PDW is the leading instability via a continuous phase transition as short-range interactions exceed a critical value. Using renormalization group, we theoretically show that N=2 sp…
Quantum spin liquids (QSL) are exotic phases of matter that host fractionalized excitations. It is difficult for local probes to characterize QSL, whereas quantum entanglement can serve as a powerful diagnostic tool due to its non-locality. The kagome antiferromagnetic Heisenberg model is one of the most studied and experimentally relevant models for QSL, but its solution remains under debate. Here, we perform a numerical Aharonov-Bohm experiment on this model and uncover universal features of the entanglement entropy. By means of the density-matrix renormalization group, we reveal the entangl…
We perform large-scale quantum Monte Carlo simulations of SLAC fermions on a two-dimensional square lattice at half filling with a single Dirac cone with $N=2$ spinor components and repulsive on-site interactions. Despite the presence of a sign problem, we accurately identify the critical interaction strength $U_c = 7.28 \pm 0.02$ in units of the hopping amplitude, for a continuous quantum phase transition between a paramagnetic Dirac semimetal and a ferromagnetic insulator. Using finite-size scaling, we extract the critical exponents for the corresponding $N=2$ chiral Ising Gross-Neveu univer…
