The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially in Phys. Rev. Lett. 52, 18 (1984). In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. By calculating the effective action, we reveal that the Newtonian mass term acts like a parity-breaking element of a high-energy regularization scheme. As such, it is directly related to the parity anomaly. In addition, the calculation of the effective action allows us to determine the finite frequency correction to the DC Hall conductivity of a QAH insulator. We derive that the leading order AC correction contains a term proportional to the torsional Hall viscosity. This paves the way to measure this non-dissipative transport coefficient via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.
In an analogy to the odd-dimensional case we define the parity anomaly as the part of the one-loop effective action for fermions associated with spectral asymmetry of the Dirac operator. This quantity is computed directly on four-dimensional manifolds with boundary and related to the Chern-Simons current on the boundary. Despite a quite unusual Chern-Simons level obtained, the action is gauge invariant and passes all consistency checks.
Nodal-line semimetals are topological semimetals characterized by one-dimensional band-touching loops protected by the combined symmetry of inversion $\mathcal{P}$ and time-reversal $\mathcal{T}$ in absence of spin-orbit coupling. These nodal loops can be understood as a one-parameter family of Dirac points exhibiting the parity anomaly associated to $\mathcal{P}*\mathcal{T}$ symmetry. We find that the parity anomaly also appears in the non-linear optical response of these systems in an analogous way to the linear response transport. We analyze the presence of a tilting term in the Hamiltonian…
The low-energy physics of two-dimensional Quantum Anomalous Hall insulators like (Hg,Mn)Te quantum wells or magnetically doped (Bi,Sb)Te thin films can be effectively described by two Chern insulators, including a Dirac, as well as a momentum-dependent mass term. Each of those Chern insulators is directly related to the parity anomaly of planar quantum electrodynamics. In this work, we analyze the finite temperature Hall conductivity of a single Chern insulator in 2+1 space-time dimensions under the influence of a chemical potential and an out-of-plane magnetic field. At zero magnetic field, t…
Parity anomaly is a quantum mechanical effect that the parity symmetry in a two-dimensional classical action is failed to be restored in any regularization of the full quantum theory and is characterized by a half-quantized Hall conductivity. Here we propose a scheme to explore the experimental signature from parity anomaly in the measurement of optical Hall conductivity, in which the optical Hall conductivity is nearly half quantized for a proper range of frequency. The behaviors of optical Hall conductivity are studied for several models, which reveal the appearance of half-quantized Hall co…
Recent experimental progress in condensed matter physics enables the observation of signatures of the parity anomaly in two-dimensional Dirac-like materials. Using effective field theories and analyzing band structures in external out-of-plane magnetic fields (orbital fields), we show that topological properties of quantum anomalous Hall (QAH) insulators are related to the parity anomaly. We demonstrate that the QAH phase survives in orbital fields, violates the Onsager relation, and can be therefore distinguished from a quantum Hall (QH) phase. As a fingerprint of the QAH phase in increasing …
The parity-preserving massive QED3 exhibits vanishing gauge coupling beta-function and is parity and infrared anomaly free at all orders in perturbation theory. Parity is not an anomalous symmetry, even for the parity-preserving massive QED3, in spite of some claims about the possibility of a perturbative parity breakdown, called parity anomaly. The proof is done by using the algebraic renormalization method, which is independent of any regularization scheme, based on general theorems of perturbative quantum field theory.
The physics of a parity anomaly, potentially observable in a narrow-gap semiconductor, is revisited. Fradkin, Dagotto, and Boyanovsky have suggested that a Hall current of anomalous parity can be induced by a Peierls distortion on a domain wall. I argue that a perturbation inducing the parity anomaly must break the time reversal symmetry, which rules out the Peierls distortion as a potential cause. I list all possible perturbations that can generate the anomaly.
Graphene is a monolayer of carbon atoms packed into a hexagon lattice to host two pairs of massless two-dimensional Dirac fermions in the absence of or with negligible spin-orbit coupling. It is known that the existence of non-zero electric polarization in reduced momentum space which is associated with a hidden chiral symmetry will lead to the zero-energy flat band of zigzag nanoribbon. The Adler-Bell-Jackiw chiral anomaly or non-conservation of chiral charges at different valleys can be realized in a confined ribbon of finite width. In the laterally diffusive regime, the finite-size correcti…
We study the Josephson effect in a quantum spin Hall system coupled to a localized magnetic impurity. As a consequence of the fermion parity anomaly, the spin of the combined system of impurity and spin-Hall edge alternates between half-integer and integer values when the superconducting phase difference across the junction advances by $2π$. This leads to characteristic differences in the splittings of the spin multiplets by exchange coupling and single-ion anisotropy at phase differences, for which time-reserval symmetry is preserved. We discuss the resulting $8π$-periodic (or $\mathbb{Z}_4$)…
A topological superconductor ring is uniquely characterized by a switch in the ground state fermion number parity upon insertion of one superconducting flux quantum - a direct consequence of the topological `parity anomaly.' Despite the many other tantalizing signatures and applications of topological superconductors, this fundamental, defining property remains to be observed experimentally. Here we propose definitive detection of the fermion parity switch from the charging energy, temperature, and tunnel barrier dependence of the flux periodicity of two-terminal conductance of a floating supe…
Conductivity of a disorder-free intrinsic graphene is studied to the first order in the long-range Coulomb interaction and is found to be σ=σ_0(1+0.01 g), where 'g' is the dimensionless ("fine structure") coupling constant. The calculations are performed using three different methods: i) electron polarization function, ii) Kubo formula for the conductivity, iii) quantum transport equation. Surprisingly, these methods yield different results unless a proper ultraviolet cut-off procedure is implemented, which requires that the interaction potential in the effective Dirac Hamiltonian is cut-off a…
A granular system slightly below the percolation threshold is a collection of finite metallic clusters, characterized by wide spectrum of sizes, resistances, and charging energies. Electrons hop from cluster to clusters via short insulating "links" of high resistance. At low temperatures all clusters are Coulomb blockaded and the dc-conductivity is exponentially suppressed. At lowest T the leading transport mechanism is variable range cotunneling via largest (critical) clusters, leading to the modified Efros-Shklovsky law. At intermediate temperatures the principal suppression of the conductiv…
We describe the formation of highly degenerate, Landau-level-like amplified states in a strained photonic honeycomb lattice in which amplification breaks the sublattice symmetry. As a consequence of the parity anomaly, the zeroth Landau level is localized on a single sublattice and possesses an enhanced or reduced amplification rate. The spectral properties of the higher Landau levels are constrained by a generalized time-reversal symmetry. In the setting of two-dimensional photonic crystal lasers, the anomaly directly affects the mode selection and lasing threshold while in three-dimensional …
We present a theory of magnetotransport phenomena related to the chiral anomaly in Weyl semimetals. We show that conductivity, thermal conductivity, thermoelectric and the sound absorption coefficients exhibit strong and anisotropic magnetic field dependences. We also discuss properties of magneto-plasmons and magneto-polaritons, whose existence is entirely determined by the chiral anomaly. Finally, we discuss the conditions of applicability of the quasi-classical description of electron transport phenomena related to the chiral anomaly.
Lattice deformations act on the low-energy excitations of Dirac materials as effective axial vector fields. This allows to directly detect quantum anomalies of Dirac materials via the response to axial gauge fields. We investigate the parity anomaly in Dirac nodal line semimetals induced by lattice vibrations, and establish a topological piezoelectric effect; i.e., periodic lattice deformations generate topological Hall currents that are transverse to the deformation field. The currents induced by this piezoelectric effect are dissipationless and their magnitude is completely determined by the…
We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semi-metals. Finally, using dimens…
It is commonly believed that the current response of an electron fluid to a mechanical force (such as an electric field) or to a ``statistical force" (e.g., a gradient of chemical potential) are governed by a single linear transport coefficient - the electric conductivity. We argue that this is not the case in anomalous Hall materials. In particular, we find that transverse (Hall) currents manifest two distinct Hall responses governed by an unconventional {\it transverse} Einstein relation that captures an anomalous relation between the Hall conductivity and the Hall diffusion constant. We giv…
The low energy quasiparticle dispersion of various narrow gap and gapless semiconductors are respectively described by three dimensional massive and massless Dirac fermions. The three dimensional Dirac spinor structure admits a time-reversal invariant, odd parity and Lorentz pseudoscalar topological superconducting state. Here we derive the effective field theory of this topological paired state for massless Dirac fermions in the presence of a fluctuating Zeeman term, which appears as a chiral gauge field. The effective theory consists of a mixed electromagnetic and chiral anomaly term in the …
Recent developments in solid state physics give a prospect to observe the parity anomaly in (2+1)D massive Dirac systems. Here we show, that the quantum anomalous Hall (QAH) state in orbital magnetic fields originates from the Dirac mass term and induces an anomalous four-current related to the parity anomaly. This differentiates the QAH from the quantum Hall (QH) state for the experimentally relevant case of an effective constant density (seen by the gate). A direct signature of QAH phase in magnetic fields is a long $σ_{xy}= e^2/h$ ($σ_{xy}= -e^2/h$) plateau in Cr$_x$(Bi$_{1-y}$Sb$_y$)$_{2-x…
We demonstrate that, apart from the chiral anomaly, Dirac semimetals possess another quantum anomaly, which we call the mirror anomaly, and which manifests in a singular response of the Dirac semimetal to an applied magnetic field. Namely, the anomalous Hall conductivity exhibits step-function singularities when the field is rotated. We show that this phenomenon is closely analogous to the parity anomaly of (2+1)-dimensional Dirac fermions, but with mirror symmetry, which we demonstrate emerges near any Dirac point at a time reversal invariant momentum, replacing the parity symmetry.
We show that the presence of parity-odd terms in the conductivity (or, in other words, in the polarization tensor of Dirac quasiparticles in graphene) leads to rotation of polarization of the electromagnetic waves passing through suspended samples of graphene. Parity-odd Chern-Simons type contributions appear in external magnetic field, giving rise to a quantum Faraday effect (though other sources of parity-odd effects may also be discussed). The estimated order of the effect is well above the sensitivity limits of modern optical instruments.
The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length xi). We calculate the maximum (critical) current I_c that can flow without dissipation along a single edge, going beyond the short-junction restriction L<<xi of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than xi. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction I_c is the same with or without pa…
A three-dimensional topological insulator features a two-dimensional surface state consisting of a single linearly-dispersive Dirac cone. Under broken time-reversal symmetry, the single Dirac cone is predicted to cause half-integer quantization of Hall conductance, which is a manifestation of the parity anomaly in quantum field theory. However, despite various observations of quantization phenomena, the half-integer quantization has been elusive because a pair of equivalent Dirac cones on two opposing surfaces are simultaneously measured in ordinary experiments. Here we demonstrate the half-in…
We study entanglement entropy for parity-violating (time-reversal breaking) quantum field theories on $\mathbb{R}^{1,2}$ in the presence of a domain wall between two distinct parity-odd phases. The domain wall hosts a 1+1-dimensional conformal field theory (CFT) with non-trivial chiral central charge. Such a CFT possesses gravitational anomalies. It has been shown recently that, as a consequence, its intrinsic entanglement entropy is sensitive to Lorentz boosts around the entangling surface. Here, we show using various methods that the entanglement entropy of the three-dimensional bulk theory …
From the Kubo formula for transport in a tilted Weyl semimetal we calculate the absorptive part of the dynamic conductivity for both right and left handed circular polarized light. These depend on the real part of the longitudinal conductivity and the imaginary part of the transverse (Hall) conductivity. We include the effect of the chiral anomaly which pumps charge from negative to positive chirality node when the usual ${\bs E}\cdot{\bs B}$ term is included in the electrodynamics and obtain analytic expressions. To calculate the Hall angle we further provide expressions for the imaginary par…
Parity Anomalous Semimetal with Minimal Conductivity Induced by an In-Plane Magnetic Field
Huawei Technologies Co., Ltd.（华为技术有限公司）；School of Sciences, Great Bay University（大湾区大学理学院）；Department of Physics and State Key Laboratory of Optical Quantum Materials, The University of Hong Kong（香港大学物理系及光学量子材料国家重点实验室）
We theoretically consider the effect of plasmon collective modes on the frequency-dependent conductivity of graphene in the presence of the random static potential of charged impurities. We develop an equation of motion approach suitable for the relativistic Dirac electrons in graphene that allows analytical high-frequency asymptotic solution in the presence of both disorder and interaction. We show that the presence of the acoustic plasmon pole (i.e. the plasmon frequency vanishing at long wavelengths as the square-root of wavevector) in the inverse dynamical dielectric function of graphene g…
Quantum anomalies in Weyl semimetal (for either ${\bf E}\cdot{\bf B} \neq 0$ or ${\bf \nabla}T\cdot{\bf B} \neq 0$) leads to chiral charge and energy pumping between the opposite chirality nodes. This results in chiral charge and energy imbalance between the Weyl nodes which manifests in several intriguing magneto-transport phenomena. Here, we investigate the role of electrical-, thermal-, and gravitational chiral anomaly on magneto-transport in Weyl semimetals. We predict the planar Ettinghausen and Righi-Leduc effect to be a distinct signature of these quantum anomalies. We also demonstrate …
We modify a kinetic theory of massless fermions to incorporate the effects of the conformal anomaly. Working in a collisionless regime, we emulate the conformal anomaly via a momentum-dependent electric coupling. In this prescription, the conformal anomaly leads to a hedgehog-like structure in the momentum space similar to the Berry phase associated with the axial anomaly. The interplay between the axial and conformal anomalies generates the axial current, proportional to the helicity flow of the electromagnetic background. The corresponding conductivity is determined by the running of the ele…
The quantum spin Hall effect arises due to band inversion in topological insulators, and has the defining characteristic that it hosts helical edge channels at zero magnetic field, leading to a finite spin Hall conductivity. The spin Hall conductivity is understood as the difference of the contributions of two spin states. In the effective four-band BHZ model, these two spin states appear as two uncoupled blocks in the Hamiltonian matrix. However, this idea breaks down if additional degrees of freedom are considered. The two blocks cannot be identified by proper spin $S_z$ or total angular mom…
A lattice-based model exhibits an unusual conductivity when it is subjected to both a static magnetic field and electromagnetic radiation. This conductivity anomaly may explain some aspects of the recently observed "zero-resistance states". PACS: 72.40+w, 73.40-c, 73.63 Keywords: Zero-resistance states, negative conductivity, lattice model
By making use of the Kubo formula, we calculate the conductivity of Dirac and Weyl semimetals in a magnetic field. We find that the longitudinal (along the direction of the magnetic field) magnetoresistivity is negative at sufficiently large magnetic fields for {\it both} Dirac and Weyl semimetals. The physical reason of this phenomenon is intimately connected with the dimensional spatial reduction $3 \to 1$ in the dynamics of the lowest Landau level. The off-diagonal component of the transverse (with respect to the direction of the magnetic field) conductivity in Weyl semimetals contains an a…
Torsion can cause various anomalies in various dimensions, including the $\left(3+1\right)$-dimensional $[(3+1)D]$ Nieh-Yan anomaly, the $\left(2+1\right)$D Hughes-Leigh-Fradkin (HLF) parity anomaly, and the $\left(3+1\right)$D, $\left(1+1\right)$D chiral energy-momentum anomaly. We study these anomalies from the Hamiltonian approach. We derive the $\left(1+1\right)$D chiral energy-momentum anomaly from the single-body Hamiltonian. We then show how other torsional anomalies can be related to the $\left(1+1\right)$D chiral energy-momentum anomaly in a straightforward way. Finally, the Nieh-Yan …
The anomalous thermal conductivity in spin chains observed in experiments is studied for the low temperature regime. In the effective dynamics with most realistic perturbations, the so-called Umklapp terms is irrelevant to reduce mean free path in the energy transport at even finite temperatures. This is consistent with large conductivities found in recent experiments. The Drude weight which is the prefactor in the divergent conductivity is calculated, and the temperature dependence is discussed.
We introduce a new generic model of a deformed Composite Fermion-Fermi Surface (CF-FS) for the Fractional Quantum Hall Effect near $/nu=1/2$ in the presence of a periodic density modulation. Our model permits us to explain recent Surface Acoustic Wave observations of anisotropic anomalies [1,2] in sound velocity and attenuation- appearance of peaks and anisotropy - which originate from contributions to the conductivity tensor due to regions of the CF-FS which are flattened by the applied modulation. The calculated magnetic field and wave vector dependence of the CF conductivity,velocity shift …
In this paper we critically reconsider the Casimir repulsion between surfaces that carry the Chern-Simons interaction (corresponding to the Hall type conductivity). We present a derivation of the Lifshitz formula valid for arbitrary planar geometries and discuss its properties. This analysis allows us to resolve some contradictions in the previous literature. We compute the Casimir energy for two surfaces that have constant longitudinal and Hall conductivities. The repulsion is possible only if both surfaces have Hall conductivities of the same sign. However, there is a critical value of the l…
We study constraints imposed by the Galilean invariance on linear electromagnetic and elastic responses of two-dimensional gapped systems in background magnetic field. Exact relations between response functions following from the Ward identities are derived. In addition to viscosity-conductivity relations known in literature we find new relations between the density-curvature response and the chiral central charge.
We introduce a new generic model of a deformed Composite Fermion-Fermi Surface (CF-FS) for the Fractional Quantum Hall Effect near $ν=1/2$ in the presence of a periodic density modulation. Our model permits us to explain the recent surface acoustic wave observations of anisotropic anomalies[R.L Willett et al,PRL vol78,4478 (1997);J.H.Smet et al, PRL vol 80,4538 (1998)] in sound velocity and attenuation--appearance of peaks and anisotropy-- which originate from contributions to the conductivity tensor due to regions of the CF-FS which are flattened by the applied modulation. The calculated magn…
We consider the properties of the type II Weyl semimetals at low temperatures basing on the particular tight - binding model. In the presence of electric field directed along the line connecting the Weyl points of opposite chirality the occupied states flow along this axis giving rise to the creation of electron - hole pairs. The electrons belong to a vicinity of one of the two type II Weyl points while the holes belong to the vicinity of the other. This process may be considered as the manifestation of the chiral anomaly that exists without any external magnetic field. It may be observed expe…
The current-voltage characteristics through a metallic quantum dot which is well coupled to a metallic lead are measured. It is shown that the I-V curves are composed of two contributions. One is a suppression of the tunneling conductivity at the Fermi level and the second is an oscillating feature which shifts with gate voltage. The results indicate that Zero-Bias-Anomaly and Coulomb Blockade phenomena coexist in an asymmetric strongly coupled quantum dot.
A crossover theory connecting Altshuler-Aronov electron-electron interaction corrections and Luttinger liquid behavior in quasi-1D disordered conductors has been formulated. Based on an interacting non-linear sigma model, we compute the tunneling density of states and the interaction correction to the conductivity, covering the full crossover.
Static and Ensemble-Dependent Thermodynamics of the Strain-Induced Parity Anomaly in Gapped Graphene
Yerevan Physics Institute（埃里温物理研究所）；Physics Department, New York City College of Technology, The City University of New York（纽约市立大学纽约城市学院物理系）；Institut für Physik, Universität Augsburg（奥格斯堡大学物理研究所）
We introduce a theoretical framework for computing transport coefficients for complex materials. As a first example, we resolve long-standing inconsistencies between experiment and theory pertaining to the conductivity and Hall mobility for amorphous silicon and show that the Hall sign anomaly is a consequence of localized states. Next, we compute the AC conductivity of amorphous polyanaline. The formalism is applicable to complex materials involving defects and band-tail states originating from static topological disorder and extended states. The method may be readily integrated with current …
