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Register a new userDisorder driven itinerant quantum criticality of three dimensional massless Dirac fermions
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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 universality class, and also construct the relevant low energy effective field theory that could guide the understanding of quantum critical scaling for many strange metals. Using the kernel polynomial method (KPM), we provide numerical results for the calculated dynamical exponent ($z$) and correlation length exponent ($ u$) for the disorder-driven semimetal (SM) to diffusive metal (DM) quantum phase transition at the Dirac point for several types of disorder, establishing its universal nature and obtaining the numerical scaling functions in agreement with our field theoretical analysis.
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We numerically study the effect of short ranged potential disorder on massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus on the question of the proposed quantum critical point separating the semimetal and diffusive metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian ($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 disordered density of states in the weak disorder semimetal regime. These are (i) typical eigenstates that are well described by linearly dispersing perturbatively dressed Dirac states, and (ii) nonperturbative rare eigenstates that are weakly-dispersive and quasi-localized in the real space regions with the largest (and rarest) local random potential. Using twisted boundary conditions, we are able to systematically find and study these two types of eigenstates. We find that the Dirac states contribute low energy peaks in the finite-size DOS that arise from the clean eigenstates which shift and broaden in the presence of disorder. On the other hand, we establish that the rare quasi-localized eigenstates contribute a nonzero background DOS which is only weakly energy-dependent near zero energy and is exponentially small at weak disorder. We find that the expected semimetal to diffusive metal quantum critical point is converted to an {it avoided} quantum criticality that is rounded out by nonperturbative effects, with no signs of any singular behavior in the DOS near the Dirac energy. We discuss the implications of our results for disordered Dirac and Weyl semimetals, and reconcile the large body of existing numerical work showing quantum criticality with the existence of the rare region effects.
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 $epsilon=(3-d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an emph{enlarged} $O(S_1+S_2)$ chiral symmetry. Such a fixed point acts as an exotic quantum multi-critical point (MCP), governing the emph{continuous} semimetal-insulator as well as insulator-insulator (for example antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $O(S_1)$ or $O(S_2)$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher dimensional Dirac fermions is also outlined.
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 states for the BdG quasiparticles; based on this, we construct the disordered phase diagram. We show for infinitesimal disorder, the ThSM is converted into a diffusive thermal Hall metal (ThDM) due to rare statistical fluctuations. Consequently, the phase diagram of the disordered model only consists of ThDM and thermal insulating phases. Nonetheless, there is a cross-over at finite energies from a ThSM regime to a ThDM regime, and we establish the scaling properties of the avoided quantum critical point which marks this cross-over. Additionally, we show the existence of two types of thermal insulators: (i) a trivial thermal band insulator (ThBI) [or BEC phase] supporting only exponentially localized Lifshitz states (at low energy), and (ii) a thermal Anderson insulator (AI) at large disorder strengths. We determine the nature of the two distinct localization transitions between these two types of insulators and ThDM.We also discuss the experimental relevance of our results for three dimensional, time reversal symmetry breaking, triplet superconducting states.
We study the properties of the avoided or hidden quantum critical point (AQCP) in three dimensional Dirac and Weyl semi-metals in the presence of short range potential disorder. By computing the averaged density of states (along with its second and fourth derivative at zero energy) with the kernel polynomial method (KPM) we systematically tune the effective length scale that eventually rounds out the transition and leads to an AQCP. We show how to determine the strength of the avoidance, establishing that it is not controlled by the long wavelength component of the disorder. Instead, the amount of avoidance can be adjusted via the tails of the probability distribution of the local random potentials. A binary distribution with no tails produces much less avoidance than a Gaussian distribution. We introduce a double Gaussian distribution to interpolate between these two limits. As a result we are able to make the length scale of the avoidance sufficiently large so that we can accurately study the properties of the underlying transition (that is eventually rounded out), unambiguously identify its location, and provide accurate estimates of the critical exponents $ u=1.01pm0.06$ and $z=1.50pm0.04$. We also show that the KPM expansion order introduces an effective length scale that can also round out the transition in the scaling regime near the AQCP.
The recent theoretical prediction and experimental realization of topological insulators (TI) has generated intense interest in this new state of quantum matter. The surface states of a three-dimensional (3D) TI such as Bi_2Te_3, Bi_2Se_3 and Sb_2Te_3 consist of a single massless Dirac cones. Crossing of the two surface state branches with opposite spins in the materials is fully protected by the time reversal (TR) symmetry at the Dirac points, which cannot be destroyed by any TR invariant perturbation. Recent advances in thin-film growth have permitted this unique two-dimensional electron system (2DES) to be probed by scanning tunneling microscopy (STM) and spectroscopy (STS). The intriguing TR symmetry protected topological states were revealed in STM experiments where the backscattering induced by non-magnetic impurities was forbidden. Here we report the Landau quantization of the topological surface states in Bi_2Se_3 in magnetic field by using STM/STS. The direct observation of the discrete Landau levels (LLs) strongly supports the 2D nature of the topological states and gives direct proof of the nondegenerate structure of LLs in TI. We demonstrate the linear dispersion of the massless Dirac fermions by the square-root dependence of LLs on magnetic field. The formation of LLs implies the high mobility of the 2DES, which has been predicted to lead to topological magneto-electric effect of the TI.
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