Bitan Roy, S. Das Sarma
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 $d=3$.
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A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potent…
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 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…
Disorder-induced instability of a Weyl nodal loop semimetal towards a diffusive topological metal with protected multifractal surface states
本文通过数值精确方法证明外尔节线环半金属在微小无序下会失稳为拓扑扩散金属，其表面态呈现多重分形结构且数量先增后减。
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…
Diffusion, a ubiquitous phenomenon in nature, is a consequence of particle number conservation and locality, in systems with sufficient damping. In this paper we consider diffusive processes in the bulk of Weyl semimetals, which are exotic quantum materials, recently of considerable interest. In order to do this, we first explicitly implement the analytical scheme by which disorder with anisotropic scattering amplitude is incorporated into the diagrammatic response-function formalism for calculating the `diffuson'. The result thus obtained is consistent with transport coefficients evaluated fr…
The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of disordered $\mathbb{Z}_2$ topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel s…
Quantum phase transitions of three-dimensional (3D) Weyl semimetals (WSMs) subject to uncorrelated on-site disorder are investigated through quantum conductance calculations and finite-size scaling of localization length. Contrary to previous claims that a direct transition from a WSM to a diffusive metal (DM) occurs, an intermediate phase of Chern insulator (CI) between the two distinct metallic phases should exist due to internode scattering that is comparable to intranode scattering. The critical exponent of localization length is $ν\simeq 1.3$ for both the WSM-CI and CI-DM transitions, in …
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 (…
We study the electronic transport in the lowest Landau level of disordered two-dimensional semimetals placed in a homogeneous perpendicular magnetic field. The material system is modeled by the Bernevig-Hughes-Zhang Hamiltonian, which has zero energy Landau modes due to the material's intrinsic Berry curvature. These turn out to be crucially important for the density of states and the static conductivity of the disordered system. We develop an analytical approach to the diffusion and conductivity based on a self-consistent equation of motion for the mean squared displacement. The obtained valu…
Semi-Dirac semimetals have received enthusiastic research both theoretically and experimentally in the recent years. Due to the anisotropic dispersion, its physical properties are highly direction-dependent. In this work we employ the Feynman diagrammatic perturbation theory to study the transport properties in quantum diffusive regime. The magneto-conductivity with quantum interference corrections is derived, which demonstrate the weak localization effect in the semi-Dirac semimetal. Furthermore, the origin of anomalous Hall conductivity is also clarified, where both the intrinsic and side-ju…
Two-dimensional Disordered Projected Branes: Stability and Quantum Criticality via Dimensional Reduction
本研究通过Schur分解从三维立方晶格构建二维无序投影膜，证实其可重现三维体系的量子相图与相近的临界指数，为传统二维晶格无法实现的稳定金属、半金属相提供了新的量子全息体系，并指出光子晶格可用于实验验证。
We study the quantum phase diagram of a three dimensional non-interacting Dirac semimetal in the presence of either quenched axial or scalar potential disorder, by calculating the average and the typical density of states as well as the inverse participation ratio using numerically exact methods. We show that as a function of the disorder strength a half-filled (i.e. undoped) Dirac semimetal displays three distinct ground states, namely an incompressible semimetal, a compressible diffusive metal, and a localized Anderson insulator, in stark contrast to a conventional dirty metal that only supp…
Disordered Parity Anomalous Semimetal
本文研究拓扑绝缘体半磁结构与窄带隙弱拓扑绝缘体薄膜中无序驱动的拓扑相变，证明强无序可引发宇称反常半金属到扩散金属的转变并在平庸绝缘相中诱导拓扑相，推进了对该类相变的理解。
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 low-energy behaviors of gapless double- and triple-Weyl fermions caused by the interplay of long-range Coulomb interaction and quenched disorder are studied by performing a renormalization group analysis. It is found that an arbitrarily weak disorder drives the double-Weyl semimetal to undergo a quantum phase transition into a compressible diffusive metal, independent of the disorder type and the Coulomb interaction strength. In contrast, the nature of the ground state of triple-Weyl fermion system relies sensitively on the specific disorder type in the noninteracting limit: The system is …
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 investigate the mutual influence of tilt, disorder, and Coulomb interaction in a type-I Dirac semimetal (DSM) with $x$-direction tilt by performing a renormalization group analysis. The interplay between disorder and ordinary tilt generates an effective tilt along the $x$-direction, which is the physically observable one. There exist two types of disorder which increase the effective tilt and drive a phase transition from the DSM phase to the diffusive metal phase. The diffusive phase transition stops the increase of the effective tilt and the surface of the original Dirac cone in the diffu…
Surface quantum critical phenomena in disordered Dirac semimetals
该研究利用重整化群与 ε 展开分析半无限 Dirac 半金属的表面量子相变，给出表面相变、超常相变及特殊相变点的临界性质。
A clean noninteracting parabolic semimetal is characterized by quadratic band touching between the conduction and the valence bands at isolated diabolic points in the Brillouin zone and describes a fermionic quantum critical system with dynamic exponent z=2. We consider the stability of such a semimetal against electronic interaction and quenched disorder using a perturbative renormalization group analysis for two and three spatial dimensions. For the noninteracting problem infinitesimally weak disorder leads to an Anderson insulator and a diffusive metal respectively in two and three dimensio…
Connecting the avoided quantum critical point to the magic-angle transition in three-dimensional Weyl semimetals
该研究理论上探讨了三维 Weyl 半金属中随机与准周期势的相互作用，揭示了无序引起的避免量子临界点在两者并存时变为交叉线，并终止于准周期极限下的魔角临界点，同时指出该现象可在冷原子实验中观测。
Weyl semimetals are gapless quasi-topological materials with a set of isolated nodal points forming their Fermi surface. They manifest their quasi-topological character in a series of topological electromagnetic responses including the anomalous Hall effect. Here we study the effect of disorder on Weyl semimetals while monitoring both their nodal/semi-metallic and topological properties through computations of the localization length and the Hall conductivity. We examine three different lattice tight-binding models which realize the Weyl semimetal in part of their phase diagram and look for un…
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…
Four-component massive and massless Dirac fermions in the presence of long range Coulomb interaction and chemical potential disorder exhibit striking fermionic quantum criticality. For an odd number of flavors of Dirac fermions, the sign of the Dirac mass distinguishes the topological and the trivial band insulator phases, and the gapless semi-metallic phase corresponds to the quantum critical point that separates the two. Up to a critical strength of disorder, the semi-metallic phase remains stable, and the universality class of the direct phase transition between two insulating phases is unc…
From transfer-matrix calculation of localization lengths and their finite-size scaling analyses, we evaluate critical exponents of the Anderson metal-insulator transition in three dimensional (3D) orthogonal class with particle-hole symmetry, class CI, as $ν=1.16\pm 0.02$. We further study disorder-driven quantum phase transitions in the 3D nodal line Dirac semimetal model, which belongs to class BDI, and estimate critical exponent as $ν=0.80\pm 0.02$. From a comparison of the critical exponents, we conclude that a disorder-driven re-entrant insulator-metal transition from the topological insu…
In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the $U(N)$ Gross-Neveu model in the limit of $N \to 0$. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is cont…
In electronic band structure of solid state material, two band touching points with linear dispersion appear in pair in the momentum space. When they annihilate with each other, the system undergoes a quantum phase transition from three-dimensional Weyl semimetal (WSM) phase to a band insulator phase such as Chern band insulator (CI) phase. The phase transition is described by a new critical theory with a `magnetic dipole' like object in the momentum space. In this paper, we reveal that the critical theory hosts a novel disorder-driven quantum multicritical point, which is encompassed by three…
We theoretically study the stability of three dimensional Dirac semimetals against short-range electron-electron interaction and quenched time-reversal symmetric disorder (but excluding mass disorder). First we focus on the clean interacting and the noninteracting dirty Dirac semimetal separately, and show that they support two distinct quantum critical points. Using renormalization group techniques, we find that while interaction driven quantum critical points are \emph{Gaussian} (mean-field) in nature, describing quantum phase transitions into various broken symmetry phases, the ones control…
Entanglement signatures of topological phase transitions in a dirty Weyl semimetal
本研究探讨无序外尔半金属经历外尔节点湮灭、无序驱动金属化两种拓扑相变时，纠缠谱与纠缠熵的演化，明确无序对拓扑特征及雷尼熵标度的影响。
Anomalous Quantum Criticality at a Continuous Metal-Insulator Transition
本文提出了一种解析理论，揭示了Falicov-Kimball模型中异常量子临界性在连续金属-绝缘体转变中的表现及影响。
We studied effects of disorder in a three dimensional layered Chern insulator. By calculating the localization length and density of states numerically, we found two distict types of metallic phases between Anderson insulator and Chern insulator; one is diffusive metallic (DM) phase and the other is renormalized Weyl semimetal (WSM) phase. We show that longitudinal conductivity at the zero energy state remains finite in the renormalizd WSM phase as well as in the DM phase, while goes to zero at a semimetal-metal quantum phase transition point between these two. Based on the Einstein relation c…
The Weyl semimetals are topologically protected from a gap opening against weak disorder in three dimensions. However, a strong disorder drives this relativistic semimetal through a quantum transition towards a diffusive metallic phase characterized by a finite density of states at the band crossing. This transition is usually described by a perturbative renormalization group in $d=2+\varepsilon$ of a $U(N)$ Gross-Neveu model in the limit $N \to 0$. Unfortunately, this model is not multiplicatively renormalizable in $2+\varepsilon$ dimensions: An infinite number of relevant operators are requi…
Three-dimensional random electron systems undergo quantum phase transitions and show rich phase diagrams. Examples of the phases are the band gap insulator, Anderson insulator, strong and weak topological insulators, Weyl semimetal, and diffusive metal. As in the previous paper on two-dimensional quantum phase transitions [J. Phys. Soc. Jpn. vol. 85, 123706 (2016)], we use an image recognition algorithm based on a multilayered convolutional neural network to identify which phase the eigenfunction belongs to. The Anderson model for localization-delocalization transition, the Wilson--Dirac model…
We construct and solve a two-dimensional, chirally symmetric model of Dirac cones subjected to a quasiperiodic modulation. In real space, this is realized with a quasiperiodic hopping term. This hopping model, as we show, at the Dirac node energy has a rich phase diagram with a semimetal-to-metal phase transition at intermediate amplitude of the quasiperiodic modulation, and a transition to a phase with a diverging density of states and sub-diffusive transport when the quasiperiodic hopping is strongest. We further demonstrate that the semimetal-to-metal phase transition can be characterized b…
We study three-dimensional Dirac fermions with weak finite-range scalar potential disorder. In the clean system, the density of states vanishes quadratically at the Dirac point. Disorder is known to be perturbatively irrelevant, and previous theoretical work has assumed that the Dirac semimetal phase, characterized by a vanishing density of states, survives at weak disorder, with a finite disorder phase transition to a diffusive metal with a non-vanishing density of states. In this paper we show that nonperturbative effects from rare regions, which are missed by conventional disorder-averaged …
Two-dimensional Dirac semimetal with tilted Dirac cone has recently attracted increasing interest. Tilt of Dirac cone can be realized in a number of materials, including deformed graphene, surface state of topological crystalline insulator, and certain organic compound. We study how Dirac cone tilting affects the low-energy properties by presenting a renormalization group analysis of the Coulomb interaction and quenched disorder. Random scalar potential or random vector potential along the tilting direction cannot exist on its own as it always dynamically generates a new type of disorder, whic…
Three-dimensional Dirac and Weyl semimetals exhibit a disorder-induced quantum phase transition between a semimetallic phase at weak disorder and a diffusive-metallic phase at strong disorder. Despite considerable effort, both numerically and analytically, the critical exponents $ν$ and $z$ of this phase transition are not known precisely. Here we report a numerical calculation of the critical exponent $ν=1.47\pm0.03$ using a minimal single-Weyl node model and a finite-size scaling analysis of conductance. Our high-precision numerical value for $ν$ is incompatible with previous numerical studi…
Direct topological insulator transitions in three dimensions are destabilized by non-perturbative effects of disorder
本文研究了三维拓扑绝缘体在无序下的相图，发现非微扰稀有态效应会破坏Dirac半金属临界点，使其转变为有限范围的扩散金属相，并讨论了其对实验的意义。
Random electron systems show rich phases such as Anderson insulator, diffusive metal, quantum and anomalous quantum Hall insulator, Weyl semimetal, as well as strong/weak topological insulators. Eigenfunctions of each matter phase have specific features, but due to the random nature of systems, judging the matter phase from eigenfunctions is difficult. Here we propose the deep learning algorithm to capture the features of eigenfunctions. Localization-delocalization transition as well as disordered Chern insulator-Anderson insulator transition is discussed.
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, …
Upper critical dimension for dirty Weyl semimetal-to-metal quantum phase transitions
本研究通过数值和场论方法确定脏外尔半金属-金属量子相变的下临界维度为2、上临界维度为4，并揭示其关联长度指数与动力学标度指数的普适行为。
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