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Anisotropic scaling in 3D topological models: fromWeyl semimetal to Hopf insulator

S. Rufo, M. A. R. Griffith, Nei Lopes, M. A. Contínentino

Published May 11, 2021• 0 citations
Publisher

Abstract

Abstract A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent ν obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent θ = νx/νz. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for ν and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation να¯ zα¯ = 2q and θ = νx/νz = zz/zx for ν and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction α¯ . An anisotropic quantum hyperscaling relation is also obtained.

DOI
10.21203/rs.3.rs-515110/v1

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Abstract A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent ν obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent θ = νx/νz. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for ν and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation να¯ zα¯ = 2q and θ = νx/νz = zz/zx for ν and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction α¯ . An anisotropic quantum hyperscaling relation is also obtained.
A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent [Formula: see text] obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent [Formula: see text]. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for [Formula: see text] and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation [Formula: see text] and [Formula: see text] for [Formula: see text] and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction [Formula: see text]. An anisotropic quantum hyperscaling relation is also obtained.
The interplay among topology and interactions has been a promising path towards identifying novel phases of condensed matter beyond these predicted by the established classification paradigms. In the present work, we propose such a novel phase of matter by studying the fate of a three-dimensional Berry-dipole semimetal, lying at the topological quantum critical point separating a Hopf insulator from a trivial insulator, in the presence of long-range Coulomb interactions. Utilizing large-NfN_fNf​ analysis and an εεε-expansion within the renormalization-group scheme, we predict the emergence of a spatially anisotropic non-Fermi liquid with enhanced Berry-dipole moment. The corresponding scaling relations of certain physical observables are derived as functions of the probed energy and temperature scale, and we conclude by providing an observational test for probing the enhanced topological features of the anisotropic non-Fermi liquid.
Weyl semimetals are three-dimensional, topologically protected, gapless phases which show exotic phenomena such as Fermi arc surface states or negative magnetoresistance. It is an open question whether interparticle interactions can turn the topological semimetal into a topologically nontrivial Mott-insulating phase. We investigate an experimentally motivated model for Weyl physics of cold atoms in optical lattices, with the main focus on interaction effects and topological properties, by means of dynamical mean-field theory. We characterize topological phases by numerically evaluating the Chern number via the Ishikawa-Matsuyama formula for interacting phases. Within our studies, we find that the Chern numbers become trivial when interactions lead to insulating behavior. For a deeper understanding of the Weyl-semimetal-to-Mott-insulator topological phase transition, we evaluate the topological properties of quasiparticle bands as well as so-called blind bands. Finally, we consider a system with an open boundary along one spatial direction in order to study correlation effects of surface states. In a narrow regime close to the topological phase transition, we find a correlation-induced state in which the surface becomes metallic while the bulk is semimetallic.
The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of a disordered Z2 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 single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent ν and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.
We discuss the effects of disorder in time-reversal-invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in the symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field-theory analysis (the large-Nf{N}_{f}Nf​ expansion, where Nf{N}_{f}Nf​ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a nontrivial critical point separating a metallic phase and a Dirac semimetal. A similar-field theory approach is developed for disorder-driven transitions in symmetry classes AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the nonlinear sigma model supplemented with a topological term, which is a three-dimensional analog of the Pruisken term in the integer quantum Hall effect.
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, generalizing to a large number N f of fermion flavors for analytic control.Landau damping by gapless fermionic excitations gives rise to nonanalytic self-energy corrections to the bosonic order-parameter propagator that dominate the long-wavelength behavior.We show that perturbative momentum shell RG leads to nonuniversal, cutof-dependent results, as it does not correctly account for this nonanalytic structure.In turn, using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator can be deduced by enforcing that results are independent of the cutoff scheme.Using the soft cutoff RG with the dressed dynamical RPA boson propagator, we compute the exact critical exponents for anisotropic semi-Dirac fermions (d L = 1, d Q = 1) to leading order in 1/N f and to all loop orders.Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods, such as conformal bootstrap.Unlike in the relativistic case, where the UV-IR connection is reestablished at the upper critical dimension, nonanalytic IR contributions persist near the upper critical line 2d L + d Q = 4 of anisotropic nodal fermions.We present expansions in both the number of linear and quadratic dimensions.The corrections to critical exponents are nonanalytic in , with a functional form that depends on the starting point on the upper critical line.