In planar tilted Dirac cone systems, the tilt parameter can be made space-dependent by either a perpendicular displacement field, or by chemical substitution in certain systems. We show that the symmetric partial derivative of the tilt parameter generates non-Abelian synthetic gauge fields in these systems. The small velocity limit of these gauge forces corresponds to Rashba and Dresselhaus spin-orbit couplings. At the classical level, the same symmetric spatial derivatives of tilt contribute to conservative, Lorentz-type and friction-like forces. The velocity dependent forces are odd with respect to tilt and therefore have opposite signs in the two valleys when the system is inversion symmetric. Furthermore, toggling the chemical potential between the valence and conduction bands reverses the sign of the all these classical forces, which indicates these forces couple to the electric charge of the carriers. As such, these forces are natural extensions of the electric and magnetic forces in the particular geometry of the tilted Dirac cone systems.
The notion of chiral symmetry for the conventional Dirac cone is generalized to include the tilted Dirac cones, where the generalized chiral operator turns out to be non-hermitian. It is shown that the generalized chiral symmetry generically protects the zero modes (n=0 Landau level) of the Dirac cone even when tilted. The present generalized symmetry is equivalent to the condition that the Dirac Hamiltonian is elliptic as a differential operator, which provides an explicit relevance to the index theorem.
The polarization function of electrons with the tilted Dirac cone found in organic conductors is studied using the tilted Weyl equation. The dynamical property is explored based on the analytical treatment of the particle-hole excitation. It is shown that the polarization function as the function of both the frequency and the momentum exhibits cusps and nonmonotonic structures. The polarization function depends not only on the magnitude but also the direction of the external momentum. These properties are characteristic of the tilted Dirac cone, and are contrast to the isotropic case of graphe…
Spin lattice relaxation rate is investigated for 3D tilted cone Weyl semimetals (TCWSMs). The nuclear spin relaxation rate is presented as a function of temperature and tilt parameter. We find that the relaxation rate behaves as $(1-ζ^2)^{-α}$ with $α\approx 9$ where $0\le ζ< 1$ is the tilt parameter. We demonstrate that such a strong enhancement for $ζ\lesssim 1$ that gives rise to very fast relaxation rates, is contributed by the combined effect of a new hyperfine interactions arising from the tilt itself, and the anisotropy of the ellipsoidal Fermi surface. Extracting an effective density o…
Recently, the existence of Dirac/Weyl cones in three dimensional systems has been demonstrated experimentally. While in high energy physics the isotropy of the Dirac/Weyl cones is guaranteed by relativistic invariance, in condensed matter systems corrections to this can occur, one possible type being a tilt. In this paper we study the effect of of tilted Weyl cones in collective effects. We study both the opticql conductivity as well as the polarization function. We also investigate the perturbative effect of long-range Coulomb interactions using a renormalization group calculation. We find th…
The interband optical absorption of linearly polarised light by two-dimensional (2D) semimetals hosting tilted and anisotropic Dirac cones in the bandstructure is analysed theoretically. Super-critically tilted (type-II) Dirac cones are characterised by an absorption that is highly dependent on the incident photon polarisation and frequency, and is tunable by changing the Fermi level with a back-gate voltage. Type-II Dirac cones exhibit open Fermi surfaces and large regions of the Brillouin zone where the valence and conduction bands sit either above or below the Fermi level. As a consequence,…
The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an…
We theoretically investigate the indirect signatures of the tilted anisotropic Dirac cones on Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction in a two dimensional polymorph of boron atoms. Unlike the case of isotropic non-tilted Dirac material-graphene, here we observe that the tilting of the Dirac cones exhibits a significant impact on the RKKY exchange interaction in terms of the suppression of oscillation frequency. The reason can be attributed to the behavior of the Fermi level and the corresponding density of states with respect to the tilting parameter. The direct measurement o…
We study the effects of pseudo-magnetic fields on Weyl semimetals with over-tilted Weyl cones, or type II cones. We compare the phenomenology of the resulting pseudo-Landau levels in the type II Weyl semimetal to the known case of type I cones. We predict that due to the nature of the chiral Landau level resulting from a magnetic field, a pseudo-magnetic field, or their combination, the optical conductivity can be utilized to detect a type II phase and deduce the direction of the tilt. Finally, we discuss ways to engineer homogeneous and inhomogeneous type II semimetals via generalizations of …
Using first principles pseudopotential density functional theory calculations, we find that terminating zigzag graphene nanoribbons (ZGNR) with monovalent alkali atoms at a reduced concentration has a dramatic impact on their properties. In particular, using sodium atoms for the saturation of ZGNR edges at half the concentration of edge-carbon atoms make it a one dimensional, perfect semimetal, where the valance and conduction bands meet at only a single, Dirac-like point. Unlike pristine graphene, the Dirac-"cones" of Na-ZGNR is not symmetric with respect to wave vector, but rather it is tilt…
We calculate conductance and noise for quantum transport at the nodal point for arbitrarily tilted and anisotropic Dirac or Weyl cones. Tilted and anisotropic dispersions are generic in absence of certain discrete symmetries, such as particle-hole and lattice point group symmetries. Whereas anisotropy affects the conductance g, but leaves the Fano factor F (the ratio of shot noise power and current) unchanged, a tilt affects both g and F. Since F is a universal number in many other situations, this finding is remarkable. We apply our general considerations to specific lattice models of straine…
Transverse electric (TE) modes can not propagate through the conducting solids. This is because the continuum of particle-hole excitations of conductors contaminates with the TE mode and dampes it out. But in solids hosting tilted Dirac cone (TDC) that admit a description in terms of a modified Minkowski spacetime, the new spacetime structure remedies this issue and therefore a tilted Dirac cone material (TDM) supports the propagation of an undamped TE mode which is sustained by density fluctuations. The resulting TE mode propagates at fermionic velocities which strongly confines the mode to t…
While it has been well-known that the chirality is an important symmetry for Dirac-fermion systems that gives rise to the zero-mode Landau level in graphene, here we explore whether this notion can be extended to tilted Dirac cones as encountered in organic metals. We have found that there exists a "generalized chiral symmetry" that encompasses the tilted Dirac cones, where a generalized chiral operator $γ$, satisfying $γ^{\dagger} H + Hγ=0$ for the Hamiltonian $H$, protects the zero mode. We can use this to show that the $n=0$ Landau level is delta-function-like (with no broadening) by extend…
We study the collective modes of a low-energy continuum model of the quasi-two-dimensional electron liquid in a layer of the organic compound $α-$(BEDT-TTF)$_2$I$_3$ in a perpendicular magnetic field. As testified by zero magnetic field transport experiments and \textit{ab initio} theory, this material hosts both massless and massive low-energy carriers, the former being described by tilted and anisotropic Dirac cones. The polarizability of these cones is anisotropic, and two sets of magnetoplasmon modes occur between any two cyclotron resonances. We show that the tilt of the cones causes a un…
Polymorph of 8-Pmmn borophene exhibits anisotropic tilted Dirac cones. In this work, we explore the consequences of the tilted Dirac cones in magnetotransport properties of a periodically modulated borophene. We evaluate modulation induced diffusive conductivity by using linear response theory in low temperature regime. The application of weak modulation (electric/magnetic or both) gives rise to the magnetic field dependent non-zero oscillatory drift velocity which causes Weiss oscillation in the longitudinal conductivity at low magnetic field. The Weiss oscillation is studied in presence of a…
We determine the antisymmetric current-current response for a pair of (type I) tilted Weyl cones with opposite chirality. We find that the dynamical chiral magnetic effect depends on the magnitude of the tilt and on the angle between the tilting direction and the wave vector of the magnetic field. Additionally, the chiral magnetic effect is shown to be closely related to the presence of an intrinsic anomalous Hall effect with a current perpendicular to the tilting direction and the electric field. We investigate the nonanalytic long-wavelength limit of the corresponding transport coefficients.
The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure. When the dispersion around the Dirac points is tilted, the emergence of intricate transport phenomena has been predicted, such as modified Klein tunnelling, intrinsic anomalous Hall effects and ferrimagnetism. However, Dirac materials are rare, particularly with tilted Dirac cones. Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of ex…
Massless Dirac fermions occur as low-energy modes in several quasi-two-dimensional condensed matter systems such as graphene, the surface of bulk topological insulators, and in layered organic semiconductors. When the rotational symmetry in such systems is reduced either by an in-plane electric field or an intrinsic tilt of the Dirac cones, the allowed dipolar optical transitions evolve from a few selected transitions into a wide fan of interband transitions. We show that the Lorentz covariance of the low-energy carriers allows for a concise analysis of the emerging magneto-optical properties.…
We analyze the chiral symmetry breaking in a planar four-fermion model with non-null chemical potential, temperature and including the effect of the tilt of the Dirac cone. The system is modeled with a $(2 + 1)$-dimensional Gross-Neveu-like interaction model in the context of the generalized Weyl Hamiltonian and its phase structure is studied in the mean-field and large-$N$ approximations. Possible applications of the results obtained, e.g., in connection to graphene, are discussed. We also discuss the effect of an external magnetic field applied to the system, which can give rise to the appea…
Nodal line in single-component molecular conductor [Pd(dddt)_2] has been examined to understand the tilted Dirac cone on the non-coplanar loop. In the previous work [J. Phys. Soc. Jpn. 87, 113701 (2018)], the velocity of the cone was calculated at respective Dirac points on the nodal loop based on our first-principles band structure calculations, which was a new method to derive an effective Hamiltonian with a 2 x 2 matrix. However, the Dirac cones on the nodal line are fully reproduced only at symmetric points. In the present paper, we show that our improved method well reproduces reasonable …
We study the conductivity of two-dimensional graphene-type materials with tilted cones as well as their three-dimensional Weyl counterparts and show that a covariant quantum Boltzmann equation is capable of providing an accurate description of these materials' transport properties. The validity of the covariant Boltzmann approach is corroborated by calculations within the Kubo formula. We find a strong anisotropy in the conductivities parallel and perpendicular to the tilt direction upon increase of the tilt parameter $η$, which can be interpreted as the boost parameter of a Lorentz transforma…
To simulate the dynamics of massless Dirac fermions in curved spacetimes with one, two, and three spatial dimensions we construct tight-binding Hamiltonians with spatially varying hoppings. These models represent tilted Weyl semimetals where the tilting varies with position, in a manner similar to the light cones near the horizon of a black hole. We illustrate the gravitational analogies in these models by numerically evaluating the propagation of wave packets on the lattice and then comparing them to the geodesics of the corresponding curved spacetime. We also show that the motion of electron…
We investigate a generalized two-dimensional Weyl Hamiltonian, which may describe the low-energy properties of mechanically deformed graphene and of the organic compound alpha-(BEDT-TTF)_2I_3 under pressure. The associated dispersion has generically the form of tilted anisotropic Dirac cones. The tilt arises due to next-nearest-neighbor hopping when the Dirac points, where the valence band touches the conduction band, do not coincide with crystallographic high-symmetry points within the first Brillouin zone. Within a semiclassical treatment, we describe the formation of Landau levels in a stro…
We investigate a longitudinal conductivity of a two-dimensional relativistic electron gas with a tilted Dirac cone in magnetic field. It is demonstrated that the conductivity behaves differently in the directions parallel and perpendicular to the tilting of the cone. At high magnetic fields the conductivity at non-zero Landau levels in the direction perpendicular to the tilting modifies non-trivially, in contrast to the parallel case. At zero temperature the crossover of the conductivity at the Dirac point from high to low magnetic field is studied numerically. It is found that that the tiltin…
Weyl semimetals (WSM) exhibit chiral anomaly in their magnetotransport due to broken conservation laws. Here, we analyze the magnetotransport of WSM in the presence of the time-reversal symmetry-breaking tilt parameter. The analytical expression for the magnetoconductivity is derived in the small tilt limit using the semiclassical Boltzmann equation. We predict a planar Hall current which flows transverse to the electric field and in the plane containing magnetic and electric fields and scales linearly with the tilt parameter. A tilt-induced transverse conductivity is also present in the case …
The possible quantum Hall ferromagnet at a filling factor $ν=0$ is investigated for the zero-energy (N=0) Landau level of the two dimensional massless Dirac fermions in $α$-(BEDT-TTF)$_2$I$_3$ under pressure with tilted cones and a twofold valley degeneracy resulting from time-reversal symmetry. In the case of the Dirac cones without tilting, the long-range Coulomb interaction in the N=0 Landau level exhibits the SU(2) valley-pseudo-spin symmetry even to the order $O(a/l_{\rm H})$, in contrast to $N \ne 0$ Landau levels, where $a$ and $l_{\rm H}$ represent the lattice constant and the magnetic…
We investigate high-order harmonic generations (HHGs) under the comparison of Weyl cones in two types. Due to the hyperboloidal electron pocket structure, strong noncentrosymmetrical generations in high orders are observed around a single type-II Weyl point, especially at frequency zero. Such remarkable DC signal is proved to have attributions from the intraband transition after spectral decomposition. Under weak pulse electric field , the linear optical response of a non-tilted Weyl cone is consistent with the Kubo theory. With more numerical simulations, we conclude the non-zero chemical pot…
The enchanting Dirac fermions in graphene stimulated us to seek for other two-dimensional (2D) Dirac materials, and boron monolayers may be a good candidate. So far, a number of monolayer boron sheets have been theoretically predicted, and three have been experimentally prepared. However, none of them possesses Dirac electrons. Herein, by means of density functional theory (DFT) computations, we identified a new boron monolayer, namely hr-sB, with two types of Dirac fermions coexisting in the sheet: one type is related to Dirac nodal lines traversing Brillouin zone (BZ) with velocities approac…
Isolated Weyl cones in a disordered environment do not show the phenomenon of Anderson localization due to the abscence of backscattering processes. However, besides the conventional three dimensional diffusive metal, an additional semimetallic phase can form. In this paper we study the effect of tilt on the physics of disorder in isolated Weyl cones. Our main finding is that while the semimetallic phase remains a stable phase, tilt decreases the size of the semimetallic region. Conversely, disorder increases the effective tilt of the quasiparticle excitations.
The effect of the Dirac cone tilt of anisotropic two-dimensional materials on the time evolution of coherent electron states in the presence of electric and magnetic fields is studied. We propose a canonical transformation that maps the anisotropic Dirac-Weyl Hamiltonian with tilted Dirac cones to an effective and isotropic Dirac Hamiltonian under these fields. In this way, the well-known Landau-level spectra and wave functions allow calculating the Wigner matrix representation of Landau and coherent states. We found a valley dependency in the behavior of the Wigner function for both Landau an…
A new type of angular oscillations of the high-frequency conductivity for conductors with a band-contact line has been predicted. The effect is caused by groups of charge carriers near the self-intersection points of the Fermi surface, where the electron energy spectrum is near-linear and can be described by anisotropic Dirac cone model. The amplitude of the resonance peaks satisfies the simple sum rule. The ease in changing the degree of anisotropy of the Dirac cone due to the angle of inclination of the magnetic field makes the considered type of oscillations attractive for experimental obse…
We present a detail theoretical study of the Drude weight and optical conductivity of 8-$Pmmn$ borophene having tilted anisotropic Dirac cones. We provide exact analytical expressions of $xx$ and $yy$ components of the Drude weight as well as maximum optical conductivity. We also obtain exact analytical expressions of the minimum energy ($ε_1$) required to trigger the optical transitions and energy ($ε_2$) needed to attain maximum optical conductivity. We find that the Drude weight and optical conductivity are highly anisotropic as a consequence of the anisotropic Dirac cone. The optical condu…
Gapless Weyl semimetals (WSMs) are a novel class of topological materials that host massless Weyl fermions as their low-energy excitations. When the Weyl cone is tilted, the Lorentz invariance is broken and the Lifshitz transition drives the system from type-I WSMs into type-II WSMs. Here, based on the lattice model, we perform a systematic numerical study of the effect of on-site disorder on the diagonal magnetoconductivity in tilted WSMs.We use the self-consistent Born approximation to calculate the disorder-induced self-energy and then apply the Kubo-Streda formula to calculate the diagonal…
We introduce the notion of a band-inverted, topological semimetal in two-dimensional nonsymmorphic crystals. This notion is materialized in the monolayers of MTe$_2$ (M $=$ W, Mo) if spin-orbit coupling is neglected. We characterize the Dirac band touching topologically by the Wilson loop of the non-Abelian Berry gauge field. An additional feature of the Dirac cone in monolayer MTe$_2$ is that it tilts over in a Lifshitz transition to produce electron and hole pockets, a type-II Dirac cone. These pockets, together with the pseudospin structure of the Dirac electrons, suggest a unified, topolog…
We investigate how tilting affects the off-diagonal, dissipationless response of a pair of chirally imbalanced Weyl cones to various external perturbations. The pair of chirally imbalanced Weyl cones can be described as a chiral electron fluid, that can flow with a velocity field that contains vorticity. Upon applying an external magnetic field, we obtain the so-called magnetovortical linear-response matrix that relates electric and heat currents to the magnetic field (chiral magnetic effect) and the vorticity (chiral vortical effect). We show how this reponse matrix becomes anisotropic upon t…
Anomalous acoustic plasmons in two-dimensional over-tilted Dirac bands
Sichuan Normal University（四川师范大学）；Beijing Forestry University（北京林业大学）；High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院高磁场科学中心）；School of Physics, Beijing Institute of Technology（北京理工大学物理学院）
Tilting the Weyl cone breaks the Lorentz invariance and enriches the Weyl physics. Here, we report the observation of a magnetic-field-antisymmetric Seebeck effect in a tilted Weyl semimetal, Co$_3$Sn$_2$S$_2$. Moreover, it is found that the Seebeck effect and the Nernst effect are antisymmetric in both the in-plane magnetic field and the magnetization. We attribute these exotic effects to the one-dimensional chiral anomaly and phase space correction due to the Berry curvature. The observation is further reproduced by a theoretical calculation, taking into account the orbital magnetization.
The doubling of massless Dirac fermions on two-dimensional lattices is theoretically studied. It has been shown that the doubling of massless Dirac fermions on a lattice with broken chiral symmetry is topologically protected even when the Dirac cone is tilted. This is due to the generalized chiral symmetry defined for lattice systems, where such models can be generated by a deformation of the chiral-symmetric lattice models. The present paper shows for two-band lattice models that this is a general way to produce systems with the generalized chiral symmetry in that such systems can always be t…
We study the anomalous Nernst and thermal Hall effects in a linearized low-energy model of a tilted Weyl semimetal, with two Weyl nodes separated in momentum space. For inversion symmetric tilt, we give analytic expressions in two opposite limits: for a small tilt, corresponding to a type-I Weyl semimetal, the Nernst conductivity is finite and independent of the Fermi level, while for a large tilt, corresponding to a type-II Weyl semimetal, it acquires a contribution depending logarithmically on the Fermi energy. This result is in a sharp contrast to the nontilted case, where the Nernst respon…
Using the path integral formulation in Euclidean space, we extended the calculation of the abelian chiral anomalies in the case of Lorentz violating theories by considering a new fermionic correction term provided by the standard model extension, which arises in the continuous Hamiltonian of a weakly tilted Weyl semimetal, and whose cones have opposite tilting. We found that this anomaly is insensitive to the tilting parameter, retaining its well-known covariant form. This independence on the Lorentz violating parameters is consistent with other findings reported in the literature. The initial…
The polymorph of ${\rm 8-Pmmn}$ borophene is an anisotropic Dirac material with tilted Dirac cones at two valleys. The tilting of the Dirac cones at two valleys are in opposite direction, which manifests itself via valley dependent Landau levels in presence of an in-plane electric field (Hall field). The sensitivity of the Landau levels on valley index causes valley polarized magnetotransport properties in presence of a Hall field, which is in contrast to the monolayer graphene with isotropic non-tilted Dirac cones. The longitudinal conductivity and Hall conductivity are evaluated by using lin…
We calculate the A.C. optical response to circularly polarized light of a Weyl semimetal (WSM) with varying amounts of tilt of the Dirac cones. Both type-I and II (overtilted) WSM are considered in a continuum model with broken time reversal (TR) symmetry. The Weyl nodes appear in pairs of equal energies but of opposite momentum and chirality. For type-I the response of a particular node to right (RHP) and left (LHP) hand polarized light are distinct only in a limited range of photon energy $Ω$, $\frac{2}{1+C_{2}/v}<\fracΩμ<\frac{2}{1-C_{2}/v}$ with $μ$ the chemical potential and $C_{2}$ the t…
