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 nature of excitations, transport, and topology are reviewed. New results on a bipartite random hopping model are presented that confirm previous results in a $p+ip$ Weyl superconductor, demonstrating that particle-hole symmetry is insufficient to help stabilize the Weyl semimetal phase in the presence of disorder. The nature of the avoided transition in a model for a single Weyl cone in the continuum is discussed. We close with a discussion of open questions and future directions.
We correct erroneous conclusions from Ref.[1] regarding the values of various critical exponents, calculated to the two-loop order, and argue that $ε$-expansion near two spatial dimension, with $ε=d-2$ may not be reliable to address the critical properties of disorder driven Dirac semimetal-metal quantum phase transition in $d=3$.
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…
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…
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…
We study the effects of disorder on semi-infinite Weyl and Dirac semimetals where the presence of a boundary leads to the formation of either Fermi arcs/rays or Dirac surface states. Using a local version of the self-consistent Born approximation, we calculate the profile of the local density of states and the surface group velocity. This allows us to explore the full phase diagram as a function of boundary conditions and disorder strength. While in all cases we recover the sharp criticality in the bulk, we unveil a critical behavior at the surface of Dirac semimetals, which is smoothed out by…
Three dimensional Dirac semimetals are stable against weak potential disorder, but not against strong disorder. In the language of renormalization group, such stability stems from the irrelevance of weak disorder in the vicinity of the noninteracting Gaussian fixed point. However, beyond a threshold, potential disorder can take Dirac semimetals into a compressible diffusive metallic phase through a quantum phase transition (QPT), where density of states at zero energy, quasiparticle lifetime and metallic conductivity at $T=0$, are finite. Universal behavior of such unconventional QPT is descri…
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 (…
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…
The average density of states in a disordered three-dimensional Weyl system is discussed in the case of a continuous distribution of random scattering. Our result clearly indicate that the average density of states does not vanish, reflecting the absence of a critical point for a metal-insulator transition. This calculation supports recent suggestions of an avoided quantum critical point in the disordered three-dimensional Weyl semimetal. However, the effective density of states can be very small such that the saddle-approximation with a vanishing density of states might be valid for practical…
We theoretically study the single particle Green function of a three dimensional disordered Weyl semimetal using a combination of techniques. These include analytic $T$-matrix and renormalization group methods with complementary regimes of validity, and an exact numerical approach based on the kernel polynomial technique. We show that at any nonzero disorder, Weyl excitations are not ballistic: they instead have a nonzero linewidth that for weak short-range disorder arises from non-perturbative resonant impurity scattering. Perturbative approaches find a quantum critical point between a semime…
Breakthroughs in two-dimensional van der Waals heterostructures have revealed that twisting creates a moiré pattern that quenches the kinetic energy of electrons, allowing for exotic many-body states. We show that cold-atomic, trapped ion, and metamaterial systems can emulate the effects of a twist in many models from one to three dimensions. Further, we demonstrate at larger angles (and argue at smaller angles) that by considering incommensurate effects, the magic-angle effect becomes a single-particle quantum phase transition (including in a model for twisted bilayer graphene in the chiral l…
Texas Center for Superconductivity and Department of Physics, University of Houston（休斯顿大学物理系超导电性中心）；Department of Physics and Astronomy, Rice University（莱斯大学物理与天文学系）；Rice Center for Quantum Materials, Rice University（莱斯大学量子材料中心）
We study the Majorana surface states of higher-spin topological superconductors (TSCs) that could be realized in ultracold atomic systems or doped semimetals with spin-orbit coupling. As a paradigmatic example, we consider a model with p-wave pairing of spin-3/2 fermions that generalizes ${}^3\text{He-}B$. This model has coexisting linear and cubic dispersing Majorana surface bands. We show that these are unstable to interactions, which can generate a spontaneous surface thermal quantum Hall effect (TQHE). By contrast, nonmagnetic quenched disorder induces a surface conformal field theory (CFT…
We study the quantum criticality of the phase transition between the Dirac semimetal and the excitonic insulator in two dimensions. Even though the system has a semimetallic ground state, there are observable effects of excitonic pairing at finite temperatures and/or finite energies, provided that the system is in proximity to the excitonic insulating transition. To determine the quantum critical behavior, we consider three potentially important interactions, including the Yukawa coupling between Dirac fermions and the excitonic order parameter fluctuation, the long-range Coulomb interaction, …
We clarify novel forms of scaling functions of conductance, critical conductance distribution and localization length in a disorder-driven quantum phase transition between band insulator and Weyl semimetal phases. Quantum criticality of the phase transition is controlled by a clean-limit fixed point with spatially anisotropic scale invariance. We argue that the anisotropic scale invariance is reflected on unconventional scaling function forms in the quantum phase transition. We verify the proposed scaling function forms in terms of transfer-matrix calculations of conductance and localization l…
We study equilibration and ordering in the classical dipolar kagome Ising antiferromagnet, which we show behaves as a disorder-free fragile spin glass. By identifying an appropriate order parameter, we demonstrate a transition to the ordered state proposed by Chioar et al. [Phys. Rev. B 93, 214410 (2016)] with a 12-site unit cell that breaks time-reversal and sublattice symmetries, and further provide evidence that the nature of the transition is first order. Upon approaching the transition, the spin dynamics slow dramatically. The system readily falls out of equilibrium, overshooting the tran…
The gapless Bogoliubov-de Gennes (BdG) quasiparticles of a clean three dimensional spinless $p_x+ip_y$ superconductor provide an intriguing example of a thermal Hall semimetal (ThSM) phase of Majorana-Weyl fermions in class D of the Altland-Zirnbauer symmetry classification; such a phase can support a large anomalous thermal Hall conductivity and protected surface Majorana-Fermi arcs at zero energy. We study the effect of quenched disorder on such a topological phase with both numerical and analytical methods. Using the kernel polynomial method, we compute the average and typical density of st…
Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the a…
Eigenstate Transitions, Duality, and Anomalous Diffusion in a Quasiperiodic Qi-Wu-Zhang Chern Insulator
Center for Advanced Quantum Studies, School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院先进量子研究中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Key Laboratory of Multiscale Spin Physics, Ministry of Education（教育部多尺度自旋物理重点实验室）；Interdisciplinary Center for Theoretical Physics and Information Sciences, Fudan University（复旦大学理论物理与信息科学交叉中心）
