Landau Levels Of Fermions Research Articles

Heavy-fermion representation for twisted bilayer graphene systems

We construct a heavy fermion representation for twisted bilayer graphene (TBG) systems. Two local orbitals (per spin/valley) are analytically found, which are exactly the maximally localized zero modes of the continuum Hamiltonian near the AA-stacking center. They have similar properties to the Wannier functions found in a recent study, but also have a clear interpretation as the zeroth pseudo Landau levels (ZLL) of Dirac fermions under the uniform strain field created by twisting. The electronic states of TBG can be viewed as the hybridization between these ZLL orbitals and other itinerant states, which can be obtained following the standard procedure of orthogonalized plane wave method. The ``heavy fermion'' model for TBG separates the strongly correlated components from the itinerant components and provides a solid base for the comprehensive understanding of the exotic physics in TBG.

Relativistic collapse of Landau levels of Kane fermions in crossed electric and magnetic fields

Using an elegant model involving only $\Gamma_{6c}$ and $\Gamma_{8v}$ bands, massless Kane fermions were defined as the particles associated with the peculiar band structure of gapless HgCdTe crystals. Although their dispersion relation resembles that of a pseudo-spin-1 Dirac semimetal, these particles were originally considered to be hybrids of pseudospin-1 and -1/2 fermions. Here we unequivocally find that by considering an additional $\Gamma_{7c}$ conduction band inherent in HgCdTe crystals, the Kane fermions are ultimately two nested Dirac particles. This observation allows the direct application of Lorentz transformations to describe the relativistic behavior of these particles in crossed electric and magnetic fields. By studying the relativistic collapse of their Landau levels at different orientations between the crossed fields and the main crystallographic axes, we demonstrate that the Kane fermions strikingly decay into two independent Dirac particles with increasing of electric field. Our results provide new insight into semi-relativistic effects in narrow-gap semiconductors in crossed electric and magnetic fields.

Experimental signatures of the chiral anomaly in Dirac–Weyl semimetals

We provide a review of recent experimental results on the chiral anomaly in Dirac/Weyl semimetals. After a brief introduction, we trace the steps leading to the prediction of materials that feature protected 3D bulk Dirac nodes. The chiral anomaly is presented in terms of charge pumping between the chiral Landau levels of Weyl fermions in parallel electric and magnetic fields. The related chiral magnetic effect and chiral zero sound are described. Current jetting effects, which present major complications in experiments on the longitudinal magnetoresistance, are carefully analyzed. We describe a recent test that is capable of distinguishing these semiclassical artifacts from intrinsic quantum effects. Turning to experiments, we review critically the longitudinal magnetoresistance experiments in the Dirac/Weyl semimetals Na3Bi, GdPtBi, ZrTe5 and TaAs. Alternate approaches to the chiral anomaly, including experiments on non-local transport, thermopower, thermal conductivity and optical pump-probe response are reviewed. In the Supplement, we provide a brief discussion of the chiral anomaly in the broader context of high energy physics and relativistic quantum field theory, as well the anomaly's starring role at the nexus of quantum physics and differential geometry.

Multiple crossings of Landau levels of two-dimensional fermions in double HgTe quantum wells

The double quantum well systems consisting of two HgTe layers separated by a tunnel-transparent barrier are expected to manifest a variety of phase states including two-dimensional gapless semimetal and two-dimensional topological insulator. The presence of several subbands in such systems leads to a rich filling factor diagram in the quantum Hall regime. We have performed magnetotransport measurements of the HgTe-based double quantum wells in both gapless and gapped state and observed numerous crossings between the Landau levels belonging to different subbands. We analyze the Landau level crossing patterns and compare them to the results of theoretical calculations.

Chiral magnetic effect for chiral fermion system * * R.-H. F. is Supported by the National Natural Science Foundation of China (11847220); D.-F. H. is in part Supported by the National Natural Science Foundation of China (11735007, 11890711)

The chiral magnetic effect is concisely derived by employing the Wigner function approach in the chiral fermion system. Subsequently, the chiral magnetic effect is derived by solving the Landau levels of chiral fermions in detail. The second quantization and ensemble average leads to the equation of the chiral magnetic effect for righthand and lefthand fermion systems. The chiral magnetic effect arises uniquely from the contribution of the lowest Landau level. We carefully analyze the lowest Landau level and find that all righthand (chirality is +1) fermions move along the direction of the magnetic field, whereas all lefthand (chirality is −1) fermions move in the opposite direction of the magnetic field. Hence, the chiral magnetic effect can be explained clearly using a microscopic approach.

Matter Chern Simons theories in a background magnetic field

We study large N 2+1 dimensional fermions in the fundamental representation of an SU(N)k Chern Simons gauge group in the presence of a uniform background magnetic field for the U (1) global symmetry of this theory. The magnetic field modifies the Schwinger Dyson equation for the propagator in an interesting way; the product between the self energy and the Greens function is replaced by a Moyal star product. Employing a basis of functions previously used in the study of non-commutative solitons, we are able to exactly solve the Schwinger Dyson equation and so determine the fermion propagator. The propagator has a series of poles (and no other singularities) whose locations yield a spectrum of single particle energies at arbitrary t’ Hooft coupling and chemical potential. The usual free fermion Landau levels spectrum is shifted and broadened out; we compute the shifts and widths of these levels at arbitrary t’Hooft coupling. As a check on our results we independently solve for the propagators of the conjecturally dual theory of Chern Simons gauged large N fundamental Wilson Fisher bosons also in a background magnetic field but this time only at zero chemical potential. The spectrum of single particle states of the bosonic theory precisely agrees with those of the fermionic theory under Bose-Fermi duality.

Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase

We propose that the electronic structure of twisted bilayer graphene (TBG) can be understood as Dirac fermions coupled with opposite pseudo magnetic fields generated by the moir\'e pattern. The two low-energy flat bands from each monolayer valley originate from the two zeroth pseudo Landau levels of Dirac fermions under such opposite effective magnetic fields, which have opposite sublattice polarizations and carry opposite Chern numbers $\pm1$, giving rise to helical edge states in the gaps below and above the low-energy bulk bands near the first magic angle. We argue that small Coulomb interactions would split the eight-fold degeneracy (including valley and physical spin) of these zeroth pseudo Landau levels, and may lead to insulating phases with non-vanishing Chern numbers at integer fillings. Besides, we show that all the high-energy bands below or above the flat bands are also topologically nontrivial in the sense that for each valley the sum of their Berry phases is quantized as $\pm\pi$. Such quantized Berry phases give rise to nearly flat edge states, which are dependent on truncations on the moir\'e length scale. Our work provides a complete and clear picture for the electronic structure and topological properties of TBG, and has significant implications on the natrue of the correlated insulating phase observed in experiments.

Magnetic dispersion of Dirac fermions in graphene under inhomogeneous field profiles

We address the exact relations for the energy spectrum of massive Dirac fermions of graphene for a class of position-dependent magnetic field profiles, which could be created under inhomogeneous strain fields, with possibility of inducing a Dirac gap. We then show that in the linear regime our results correspond well to the earlier approaches for the massless case. We also prove that our approach for obtaining the relativistic Landau levels of Dirac fermions created by a constant magnetic field together with a spatially varying normal pseudo-magnetic profile, which varies as $ 1/x^2$ , recovers well the earlier results in this regard. It turns out that, in this case, the associated Landau energy levels are obtained by considering two shape-invariant potentials which belong to the family of isotonic oscillators with a set of equally spaced energy levels.

Non-abelian 3D bosonization and quantum Hall states

Bosonization dualities relate two different Chern-Simons-matter theories, with bosonic matter on one side replaced by fermionic matter on the other. We first describe a more general class of non-Abelian bosonization dualities. We then explore the non-relativistic physics of these theories in the quantum Hall regime. The bosonic theory lies in a condensed phase and admits vortices which are known to form a non-Abelian quantum Hall state. We ask how this same physics arises in the fermionic theory. We find that a condensed boson corresponds to a fully filled Landau level of fermions, while bosonic vortices map to fermionic holes. We confirm that the ground state of the two theories is indeed described by the same quantum Hall wavefunction.

Energy loss of a nonaccelerating quark moving through a strongly coupledN=4super Yang-Mills vacuum or plasma in strong magnetic field

Using AdS/CFT correspondence, we find that a massless quark moving at the speed of light $v=1$, in arbitrary direction, through a strongly coupled $\mathcal{N}=4$ super Yang-Mills (SYM) vacuum at $T=0$, in the presence of strong magnetic field $\mathcal{B}$, loses its energy at a rate linearly dependent on $\mathcal{B}$, i.e., $\frac{dE}{dt}=-\frac{\sqrt{\lambda}}{6\pi}\mathcal{B}$. We also show that a heavy quark of mass $M\neq 0$ moving at near the speed of light $v^2=v_{*}^2=1-\frac{4\pi^2 T^2}{\mathcal{B}}\simeq1$, in arbitrary direction, through a strongly coupled $\mathcal{N}=4$ SYM plasma at finite temperature $T\neq 0$, in the presence of strong magnetic field $\mathcal{B}\gg T^2$, loses its energy at a rate linearly dependent on $\mathcal{B}$, i.e., $\frac{dE}{dt}=-\frac{\sqrt{\lambda}}{6\pi}\mathcal{B}v_{*}^2\simeq-\frac{\sqrt{\lambda}}{6\pi}\mathcal{B}$. Moreover, we argue that, in the strong magnetic field $\mathcal{B}\gg T^2$ (IR) regime, $\mathcal{N}=4$ SYM and adjoint QCD theories (when the adjoint QCD theory has four flavors of Weyl fermions and is at its conformal IR fixed point $\lambda=\lambda^*$) have the same microscopic degrees of freedom (i.e., gluons and lowest Landau levels of Weyl fermions) even though they have quite different microscopic degrees of freedom in the UV when we consider higher Landau levels. Therefore, in the strong magnetic field $\mathcal{B}\gg T^2$ (IR) regime, the thermodynamic and hydrodynamic properties of $\mathcal{N}=4$ SYM and adjoint QCD plasmas, as well as the rates of energy loss of a quark moving through the plasmas, should be the same.

Microscopic model for the magnetic-field-driven breakdown of the dissipationless state in the integer and fractional quantum Hall effect

Intra Landau level thermal activation, from localized states in the tail, to delocalized states above the mobility edge in the same Landau level, explains the $B_c(T)$ (half width of the dissipationless state) phase diagram for a number of different quantum Hall samples with widely ranging carrier density, mobility and disorder. Good agreement is achieved over $2-3$ orders of magnitude in temperature and magnetic field for a wide range of filling factors. The Landau level width is found to be independent of magnetic field. The mobility edge moves, in the case of changing Landau level overlap to maintain a sample dependent critical density of states at that energy. An analysis of filling factor $\nu=2/3$ shows that the composite Fermion Landau levels have exactly the same width as their electron counterparts. An important ingredient of the model is the Lorentzian broadening with long tails which provide localized states deep in the gap which are essential in order to reproduce the robust high temperature $B_c(T)$ phase observed in experiment.

Massive and massless Dirac fermions in Pb1-xSnxTe topological crystalline insulator probed by magneto-optical absorption.

Dirac fermions in condensed matter physics hold great promise for novel fundamental physics, quantum devices and data storage applications. IV-VI semiconductors, in the inverted regime, have been recently shown to exhibit massless topological surface Dirac fermions protected by crystalline symmetry, as well as massive bulk Dirac fermions. Under a strong magnetic field (B), both surface and bulk states are quantized into Landau levels that disperse as B1/2, and are thus difficult to distinguish. In this work, magneto-optical absorption is used to probe the Landau levels of high mobility Bi-doped Pb0.54Sn0.46Te topological crystalline insulator (111)-oriented films. The high mobility achieved in these thin film structures allows us to probe and distinguish the Landau levels of both surface and bulk Dirac fermions and extract valuable quantitative information about their physical properties. This work paves the way for future magnetooptical and electronic transport experiments aimed at manipulating the band topology of such materials.

Jack 3/5 State from Two-Body Interaction

We investigate the Read–Rezayi parafermion state of correlated electrons at the fractional Landau level filling ν = 3/5. It is a Jack polynomial generated by contact four-body repulsion. We show by exact diagonalization that it is also emerges from a suitable short-range two-body interaction. We find that it closely matches Coulomb ground state in the second Landau level of non-relativistic fermions, and thus possibly describes the ν = 13/5 (and, by conjugation, ν = 12/5) fractional quantum Hall effect in GaAs.

Mirror symmetry and the half-filled Landau level

We study the dynamics of the half-filled zeroth Landau level of Dirac fermions using mirror symmetry, a supersymmetric duality between certain pairs of $2+1$-dimensional theories. We show that the half-filled zeroth Landau level of a pair of Dirac fermions is dual to a pair of Fermi surfaces of electrically-neutral composite fermions, coupled to an emergent gauge field. Thus, we use supersymmetry to provide a derivation of flux attachment and the emergent Fermi liquid-like state for the lowest Landau level of Dirac fermions. We find that in the dual theory the Coulomb interaction induces a dynamical exponent $z=2$ for the emergent gauge field, making the interactions classically marginal. This enables us to map the problem of $2+1$-dimensional Dirac fermions in a finite transverse magnetic field, interacting via a strong Coulomb interaction, into a perturbatively controlled model. We analyze the resulting low-energy theory using the renormalization group and determine the nature of the BCS interaction in the emergent composite Fermi liquid.

Fractional quantum Hall effect in a dilute magnetic semiconductor

We report the observation of the fractional quantum Hall effect in the lowest Landau level of a two-dimensional electron system (2DES), residing in the diluted magnetic semiconductor Cd(1-x)Mn(x)Te. The presence of magnetic impurities results in a giant Zeeman splitting leading to an unusual ordering of composite fermion Landau levels. In experiment, this results in an unconventional opening and closing of fractional gaps around filling factor v = 3/2 as a function of an in-plane magnetic field, i.e. of the Zeeman energy. By including the s-d exchange energy into the composite Landau level spectrum the opening and closing of the gap at filling factor 5/3 can be modeled quantitatively. The widely tunable spin-splitting in a diluted magnetic 2DES provides a novel means to manipulate fractional states.

Landau-Level Spectroscopy of Relativistic Fermions with Low Fermi Velocity in theBi2Te3Three-Dimensional Topological Insulator

X-band microwave spectroscopy is applied to study the cyclotron resonance in Bi(2)Te(3) exposed to ambient conditions. With its help, intraband transitions between Landau levels of relativistic fermions are observed. The Fermi velocity equals to 3260 m/s, which is much lower than has been reported in the literature for samples cleaved in vacuum. Simultaneous observation of bulk Shubnikov-de Haas oscillations by contactless microwave spectroscopy allows determination of the Fermi-level position. Occupation of topological surface states depends not only on bulk Fermi level but also on the surface band bending.

Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels

Photons propagating in strong magnetic fields are subject to a phenomenon called the “vacuum birefringence” where refractive indices of two physical modes both deviate from unity and are different from each other. We compute the vacuum polarization tensor of a photon in a static and homogeneous magnetic field by utilizing Schwinger’s proper-time method, and obtain a series representation as a result of double integrals analytically performed with respect to proper-time variables. The outcome is expressed in terms of an infinite sum of known functions which is plausibly interpreted as summation over all the Landau levels of fermions. Each contribution from infinitely many Landau levels yields a kinematical condition above which the contribution has an imaginary part. This indicates decay of a sufficiently energetic photon into a fermion–antifermion pair with corresponding Landau level indices. Since we do not resort to any approximation, our result is applicable to the calculation of refractive indices in the whole kinematical region of a photon momentum and in any magnitude of the external magnetic field.

Spinful composite fermions in a negative effective field

In this paper we study fractional quantum Hall composite fermion wavefunctions at filling fractions \nu = 2/3, 3/5, and 4/7. At each of these filling fractions, there are several possible wavefunctions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wavefunctions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion states at filling factors \nu = 2+2/3, 2+3/5, and 2+4/7. The latter two cases we predict to be spin polarized even at zero Zeeman energy.

Isotropic Landau levels of Dirac fermions in high dimensions

We generalize the Landau levels of two-dimensional Dirac fermions to three dimensions and above with the full rotational symmetry. Similarly to the two-dimensional case, there exists a branch of zero energy Landau levels of fractional fermion modes for the massless Dirac fermions. The spectra of other Landau levels distribute symmetrically with respect to the zero energy scaling with the square root of the Landau-level indices. This mechanism is a nonminimal coupling of Dirac fermions to the background fields. This high dimensional relativistic Landau-level problem is a square-root problem of its previous studied nonrelativistic version investigated in Li and Wu [arXiv:1103.5422 (2011)].

Observation of Landau levels in potassium-intercalated graphite under a zero magnetic field

The charge carriers in graphene are massless Dirac fermions and exhibit a relativistic Landau-level quantization in a magnetic field. Recently, it has been reported that, without any external magnetic field, quantized energy levels have been also observed from strained graphene nanobubbles on a platinum surface, which were attributed to the Landau levels of massless Dirac fermions in graphene formed by a strain-induced pseudomagnetic field. Here we show the generation of the Landau levels of massless Dirac fermions on a partially potassium-intercalated graphite surface without applying external magnetic field. Landau levels of massless Dirac fermions indicate the graphene character in partially potassium-intercalated graphite. The generation of the Landau levels is ascribed to a vector potential induced by the perturbation of nearest-neighbour hopping, which may originate from a strain or a gradient of on-site potentials at the perimeters of potassium-free domains.