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1
KLAUDER, J. R., and E. ONOFRI. "LANDAU LEVELS AND GEOMETRIC QUANTIZATION." International Journal of Modern Physics A 04, no. 15 (1989): 3939–49. http://dx.doi.org/10.1142/s0217751x89001606.
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The geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specific polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy sector of a generalized theory in which all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.
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Galasso, Andrea, and Mauro Spera. "Remarks on the geometric quantization of Landau levels." International Journal of Geometric Methods in Modern Physics 13, no. 10 (2016): 1650122. http://dx.doi.org/10.1142/s021988781650122x.
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In this note, we resume the geometric quantization approach to the motion of a charged particle on a plane, subject to a constant magnetic field perpendicular to the latter, by showing directly that it gives rise to a completely integrable system to which we may apply holomorphic geometric quantization. In addition, we present a variant employing a suitable vertical polarization and we also make contact with Bott’s quantization, enforcing the property “quantization commutes with reduction”, which is known to hold under quite general conditions. We also provide an interpretation of translational symmetry breaking in terms of coherent states and index theory. Finally, we give a representation theoretic description of the lowest Landau level via the use of an [Formula: see text]-equivariant Dirac operator.
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de Melo, J. Lemos, I. A. Pedrosa, and C. Furtado. "Coherent states of Landau–Aharonov–Casher levels." International Journal of Modern Physics B 30, no. 06 (2016): 1650022. http://dx.doi.org/10.1142/s0217979216500223.
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In this paper, we study the coherent states of Landau–Aharonov–Casher (LAC) levels. These LAC levels are an analogue of the Landau quantization for neutral particles. Afterwards, we investigate some properties of the coherent states, evaluate the uncertainty product and derive the wave functions for our problem. We also discuss the coherent states for the Landau–He–McKellar–Wilkens (LHMW) levels by using the Maxwell duality.
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Gao, Yang, and Qian Niu. "Zero-field magnetic response functions in Landau levels." Proceedings of the National Academy of Sciences 114, no. 28 (2017): 7295–300. http://dx.doi.org/10.1073/pnas.1702595114.
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We present a fresh perspective on the Landau level quantization rule; that is, by successively including zero-field magnetic response functions at zero temperature, such as zero-field magnetization and susceptibility, the Onsager’s rule can be corrected order by order. Such a perspective is further reinterpreted as a quantization of the semiclassical electron density in solids. Our theory not only reproduces Onsager’s rule at zeroth order and the Berry phase and magnetic moment correction at first order but also explains the nature of higher-order corrections in a universal way. In applications, those higher-order corrections are expected to curve the linear relation between the level index and the inverse of the magnetic field, as already observed in experiments. Our theory then provides a way to extract the correct value of Berry phase as well as the magnetic susceptibility at zero temperature from Landau level fan diagrams in experiments. Moreover, it can be used theoretically to calculate Landau levels up to second-order accuracy for realistic models.
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RIBEIRO, L. R., K. BAKKE, and C. FURTADO. "LANDAU QUANTIZATION FOR A NEUTRAL PARTICLE IN THE PRESENCE OF TOPOLOGICAL DEFECTS." International Journal of Modern Physics: Conference Series 18 (January 2012): 101–4. http://dx.doi.org/10.1142/s2010194512008288.
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In this short communication, we study the Landau levels in the non-relativistic quantum dynamics of a neutral particle which possesses a permanent magnetic dipole moment interacting with an external electric field in curved spacetime background with the presence or absence of a torsion field. We show that the presence of the topological defect breaks the infinite degeneracy of the Landau levels.
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Onorato, P. "Landau levels for relativistic particles: Einstein–Brillouin–Keller quantization approach." Physics Letters A 376, no. 46 (2012): 3525–29. http://dx.doi.org/10.1016/j.physleta.2012.10.007.
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BAKKE, K., and C. FURTADO. "RELATIVISTIC LANDAU–AHARONOV–CASHER QUANTIZATION IN TOPOLOGICAL DEFECT SPACE–TIME." International Journal of Modern Physics D 19, no. 01 (2010): 85–96. http://dx.doi.org/10.1142/s0218271810016221.
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In this paper we study the Landau levels arising within the relativistic dynamics of a neutral particle which possesses a permanent magnetic dipole moment interacting with an external electric field in the curved space–time background with the presence of a torsion field. We use the Aharonov–Casher effect to couple this neutral particle with the electric field in this curved background. The eigenfunction and eigenvalues of the Hamiltonian are obtained. We show that the presence of the topological defect breaks the infinite degeneracy of the relativistic Landau levels arising in this system. We study the nonrelativistic limit of the eigenvalues and compare these results with cases studied earlier.
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Landry, Alexandre, and Fayçal Hammad. "Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case." Universe 7, no. 5 (2021): 144. http://dx.doi.org/10.3390/universe7050144.
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We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. Two independent approaches are used to solve the relevant Schrödinger equation within the Newtonian approximation. It is found that both approaches yield qualitatively similar results within their respective approximations. With the goal of clarifying some results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, but impractical, way to use such an approach; a way which does in principle yield a consistent quantization condition. We discuss how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level by deriving the corrections brought by Yukawa-like and power-law deviations from the inverse-square law. The full relativistic regime is also examined in detail.
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Heckotter, J., J. Thewes, D. Frohlich, M. Abmann, and M. Bayer. "Landau-level quantization of the yellow excitons in cuprous oxide." Физика твердого тела 60, no. 8 (2018): 1585. http://dx.doi.org/10.21883/ftt.2018.08.46252.20gr.
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AbstractLately, the yellow series of P -excitons in cuprous oxide could be resolved up to the principal quantum number n = 25. Adding a magnetic field, leads to additional confinement normal to the field. Thereby, the transition associated with the exciton n is transformed into the transition between the electron and hole Landau levels with quantum number n , once the associated magnetic length becomes smaller than the related exciton Bohr radius. The magnetic field of this transition scales roughly as n ^–3. As a consequence of the extended exciton series, we are able to observe Landau level transitions with unprecedented high quantum numbers of more than 75.
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Wendler, Florian, Andreas Knorr, and Ermin Malic. "Ultrafast carrier dynamics in Landau-quantized graphene." Nanophotonics 4, no. 3 (2015): 224–49. http://dx.doi.org/10.1515/nanoph-2015-0018.
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AbstractIn an external magnetic field, the energy of massless charge carriers in graphene is quantized into non-equidistant degenerate Landau levels including a zero-energy level. This extraordinary electronic dispersion gives rise to a fundamentally new dynamics of optically excited carriers. Here, we review the state of the art of the relaxation dynamics in Landau-quantized graphene focusing on microscopic insights into possible many-particle relaxation channels.We investigate optical excitation into a non equilibrium distribution followed by ultrafast carrier- carrier and carrier-phonon scattering processes. We reveal that surprisingly the Auger scattering dominates the relaxation dynamics in spite of the non-equidistant Landau quantization in graphene. Furthermore, we demonstrate how technologically relevant carrier multiplication can be achieved and discuss the possibility of optical gain in Landau-quantized graphene. The provided microscopic view on elementary many-particle processes can guide future experimental studies aiming at the design of novel graphene-based optoelectronic devices, such as highly efficient photodetectors, solar cells, and spectrally broad Landau level lasers.
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Kordyukov, Yuri A. "Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian." Journal of Spectral Theory 12, no. 1 (2022): 143–67. http://dx.doi.org/10.4171/jst/397.
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Nimyi, I. O., S. G. Sharapov, and V. P. Gusynin. "WKB Energy levels in gapped graphene under crossed electromagnetic fields." Low Temperature Physics 51, no. 5 (2025): 588–95. https://doi.org/10.1063/10.0036517.
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We consider a single layer of graphene subjected to a magnetic field H applied perpendicular to the layer and an in-plane constant radial electric field E. The Dirac equation for this configuration does not admit analytical solutions in terms of known special functions. Using the Wentzel–Kramers–Brillouin (WKB) approximation, we demonstrate that for gapped graphene the Bohr–Sommerfeld quantization condition for eigenenergies includes an additional valley-dependent geometrical phase. When this term is accounted for, the WKB approximation exhibits good agreement with results from the exact diagonalization method except for the lowest Landau level.
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López, Alexander, Bertrand Berche, John Schliemann, Francisco Mireles, and Benjamin Santos. "Photoinduced polarization enhancement on biased bilayer graphene in the Landau level regime." Journal of Physics: Condensed Matter 31, no. 49 (2019): 495703. https://doi.org/10.1088/1361-648x/ab3c3c.
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We investigate the charge carrier dynamics in bilayer graphene subject to monochromatic laser irradiation within the Landau level quantization regime. Even though the radiation field does not lift the energy degeneracy of the lowest Landau levels (LLs), it nevertheless has a strong effect on the photoinduced pseudospin polarization response for higher LLs (n 2). Our results show that the photoinduced bandgaps lead to a finite response of the averaged pseudospin polarization with nontrivial oscillating behavior. It is shown that the contribution from these higher LL transitions turns out to be crucial to achieve an enhanced photoinduced polarization in radiated bilayer graphene. The experimental feasibility of our findings is also discussed.
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DE OCA, A. CABO MONTES, R. GONZALEZ FELIPE, and AURORA PÉREZ MARTINEZ. "FUNCTIONAL QUANTIZATION AROUND GW STATES OF FQHE." Modern Physics Letters B 06, no. 26 (1992): 1649–64. http://dx.doi.org/10.1142/s0217984992001356.
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Generating functional equations related to the Greiter-Wilczek ground states in FQHE are written. The Ward identities of the magnetic translation and rotational invariances allow us to determine the spatial structure of the electron propagator as in usual states. The electromagnetic linear responses again show that the Strêda formula in terms of the derivative over the real magnetic field gives the Hall conductivity. The coefficient of the Chern-Simons effective action is also determined by this expression. Calculations of the energies in the Hartree-Fock approximation are presented up to three filled Landau Levels and all the related filling factors. Under natural assumptions, the Hall resistivity versus the magnetic field curve is determined. It resembles quantitatively the experimental results for the most relevant filling fractions.
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Monarkha, Yu P. "Microwave-induced intersubband scattering and magneto-oscillation phenomena in an inhomogeneous 2D electron gas on liquid helium." Low Temperature Physics 49, no. 8 (2023): 942–54. http://dx.doi.org/10.1063/10.0020161.
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The theoretical description of microwave-induced intersubband scattering in a 2D electron gas exposed to a magnetic field on liquid helium with the inhomogeneous distribution of subband excitation energies is presented. We demonstrate that the alignment of the staircases of Landau levels in the ground and the first excited subbands differently affects the microwave resonant absorption and in-plane magnetoconductivity. In contrast with previously used models, the microwave excitation rate is shown to have a strong dependence on the magnetic field with sharp minima at the Landau level alignment points even in a heavily inhomogeneous 2D electron system. The inhomogeneous distribution of transverse-quantization eigenfrequencies changes the lineshape of the intersubband resonance and substantially broadens conductivity magneto-oscillations. These results explain unexpected experimental observations reported previously.
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KUNDU, ARITRA, and BANIBRATA MUKHOPADHYAY. "MASS OF HIGHLY MAGNETIZED WHITE DWARFS EXCEEDING THE CHANDRASEKHAR LIMIT: AN ANALYTICAL VIEW." Modern Physics Letters A 27, no. 15 (2012): 1250084. http://dx.doi.org/10.1142/s0217732312500848.
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In recent years a number of white dwarfs have been observed with very high surface magnetic fields. We can expect that the magnetic field in the core of these stars would be much higher (~1014 G ). In this paper, we analytically study the effect of high magnetic field on relativistic cold electron, and hence its effect on the stability and the mass–radius relation of a magnetic white dwarf. In strong magnetic fields, the equation of state of the Fermi gas is modified and Landau quantization comes into play. For relatively very high magnetic fields (with respect to the average energy density of matter) the number of Landau levels is restricted to one or two. We analyze the equation of states for magnetized electron degenerate gas analytically and attempt to understand the conditions in which transitions from the zeroth Landau level to first Landau level occurs. We also find the effect of the strong magnetic field on the star collapsing to a white dwarf, and the mass–radius relation of the resulting star. We obtain an interesting theoretical result that it is possible to have white dwarfs with mass more than the mass set by Chandrasekhar limit.
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WÓJS, ARKADIUSZ, LESZEK BRYJA, and MAREK POTEMSKI. "EFFECTS OF IONIZED IMPURITIES ON BINDING AND RECOMBINATION OF POSITIVE AND NEGATIVE QUASI-TWO-DIMENSIONAL MAGNETO-TRIONS." International Journal of Modern Physics B 23, no. 12n13 (2009): 2964–68. http://dx.doi.org/10.1142/s0217979209062633.
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Binding energies of negative and positive trions in high magnetic fields are compared. Simultaneous inclusion of several Landau levels and quantum well subbands in exact numerical diagonalization allowed quantitative description of the coupling between in-plane dynamics (governed by interplay of cyclotron quantization and Coulomb interactions) and single-particle excitations in the normal direction. Symmetric and asymmetric GaAs quantum wells of different widths were considered, as well as the effect of a possible binding of trions by sparse ionized impurities nearby the quantum well.
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Farias, B., and C. Furtado. "An analog of magnetic oscillations for neutral atoms with induced electric dipole." International Journal of Modern Physics B 32, no. 16 (2018): 1850206. http://dx.doi.org/10.1142/s0217979218502065.
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We show that the analog of de Haas–van Alphen effect (adHvA) can be generated in a two-dimensional (2D) atomic gas by the interaction between an induced electric dipole and an electromagnetic field. When the neutral atoms move in a properly designed field configuration, we have a quantization similar to the Landau levels. The effective magnetic field dependence of the oscillation amplitude of the the energy and the analog magnetization of the atomic cloud are studied. We also estimate the area of the Fermi circle of the atomic gas.
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Kuznetsova, I. A., D. N. Romanov, and O. V. Savenko. "Galvanomagnetic properties of a thin metal film considering dimensional quantization and diffuse surface scattering of electrons." Physica Scripta 98, no. 1 (2022): 015839. http://dx.doi.org/10.1088/1402-4896/acad38.
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Abstract The Expressions for the transverse magnetoresistance coefficient and the Hall coefficient of a film are analytically obtained and analyzed. Quantum dimensional effect is taken into account, but Landau levels are not considered. The film zone structure is spherically symmetric and obeys the parabolic law. Galvanomagnetic coefficients are calculated from the Liouville (von Neumann) equation taking into account diffuse boundary conditions for the distribution function.
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DAHAN, PINCHAS. "EXCITON BOUND TO MAGNETIC IMPURITIES IN TWO DIMENSION SYSTEMS UNDER STRONG MAGNETIC FIELD." International Journal of Modern Physics B 21, no. 08n09 (2007): 1460–65. http://dx.doi.org/10.1142/s0217979207043014.
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The possibility of excitons bound to magnetic impurities in a two-dimensional system quantized by a strong magnetic field is considered. We show that the Landau quantization leads to dramatic changes in the structure of these bound excitons. Such changes are due primarily to a lower symmetry, which lifts the selection rules in the hybridization integrals, and in turn leads to a threshold van Hove singularity of the density of states in a two-dimensional system. Hence, a bound exciton with zero angular momentum, m = 0, is captured by a spin selective potential of the deep impurity levels.
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Vitória, R. L. L., and H. Belich. "Effects of a Landau-Type Quantization Induced by the Lorentz Symmetry Violation on a Dirac Field." Advances in High Energy Physics 2020 (January 22, 2020): 1–7. http://dx.doi.org/10.1155/2020/4208161.
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Inspired by the extension of the Standard Model, we analyzed the effects of the spacetime anisotropies on a massive Dirac field through a nonminimal CPT-odd coupling in the Dirac equation, where we proposed a possible scenario that characterizes the breaking of the Lorentz symmetry which is governed by a background vector field and induces a Landau-type quantization. Then, in order to generalize our system, we introduce a hard-wall potential and, for a particular case, we determine the energy levels in this background. In addition, at the nonrelativistic limit of the system, we investigate the effects of the Lorentz symmetry violation on thermodynamic aspects of the system.
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Rau, Peter B., and Ira Wasserman. "Magnetohydrodynamic stability of magnetars in the ultrastrong field regime I: the core." Monthly Notices of the Royal Astronomical Society 506, no. 3 (2021): 4632–53. http://dx.doi.org/10.1093/mnras/stab1538.
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ABSTRACT We study magnetohydrodynamic stability of neutron star core matter composed of neutrons, protons, and leptons threaded by a magnetar-strength magnetic field 1014–1017 G, where quantum electrodynamical effects and Landau quantization of fermions are important. Stability is determined using the Friedman–Schutz formalism for the canonical energy of fluid perturbations, which we calculate for a magnetizable fluid with H ≠ B. Using this and the Euler–Heisenberg–Fermi–Dirac Lagrangian for a strongly magnetized fluid of Landau-quantized charged fermions, we calculate the local stability criteria for a neutron star core with a spherical axisymmetric geometry threaded by a toroidal field, accounting for magnetic and composition gradient buoyancy. We find that, for sufficiently strong fields B ≳ 1015 G, the magnetized fluid is unstable to a magnetosonic-type instability with growth times of the order of 10−3 s. The instability is triggered by sharp changes in the second-order field derivative of the Euler–Heisenberg–Fermi–Dirac Lagrangian that occur where additional Landau levels start being populated. These sharp changes are divergent at zero temperature, but are finite for non-zero temperature, so realistic neutron star core temperatures 5 × 107 K < T < 5 × 108 K are used. We conjecture that this mechanism could promote the formation of magnetic domains as predicted by Blandford and Hernquist and Suh and Mathews.
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Shrivastava, Keshav N. "Electron Spin in Quantum Hall Effect in AlxGa1-xas: D. C. Tsui's Data." Applied Mechanics and Materials 110-116 (October 2011): 3097–102. http://dx.doi.org/10.4028/www.scientific.net/amm.110-116.3097.
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The Hall resistivity in the layers of AlxGa1-xAs/Al0.32Ga0.68As is found to show plateaus at certain fractions which depend on the effective charge. The Hall resistivity formula ρxy=h/e2has been modified to ρxy=h/[(1/2) ge2] so that the effective charge of the electron becomes, e*=(1/2) ge. The plateaus occur at the effective charge determined by g = (2j+1)/(2l+1). Some of the plateaus are explained to arise from the g values while some others require the use of Landau levels. The flux quantization is modified to include the effect of spin. When the samples are doped with aluminium, the clusters of Al atoms occur in the GaAs resulting into electron clusters in which the spin is NS with S=1/2 and N=101. The electron clusters form a temperature dependent plateau in the Hall resistivity.
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GAO, ZHI FU, NA WANG, QIU HE PENG, XIANG DONG LI, and YUAN JIE DU. "PRESSURE OF DEGENERATE AND RELATIVISTIC ELECTRONS IN A SUPERHIGH MAGNETIC FIELD." Modern Physics Letters A 28, no. 36 (2013): 1350138. http://dx.doi.org/10.1142/s0217732313501381.
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Based on our previous work, we deduce a general formula for pressure of degenerate and relativistic electrons, Pe, which is suitable for superhigh magnetic fields, discuss the quantization of Landau levels of electrons, and consider the quantum electrodynamic (QED) effects on the equations of states (EOSs) for different matter systems. The main conclusions are as follows: Pe is related to the magnetic field B, matter density ρ, and electron fraction Ye; the stronger the magnetic field, the higher the electron pressure becomes; the high electron pressure could be caused by high Fermi energy of electrons in a superhigh magnetic field; compared with a common radio pulsar, a magnetar could be a more compact oblate spheroid-like deformed neutron star (NS) due to the anisotropic total pressure; and an increase in the maximum mass of a magnetar is expected because of the positive contribution of the magnetic field energy to the EOS of the star.
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VARMA, RAM K. "Macro-quantization of the guiding centre motion of charged particles in a magnetic field." Journal of Plasma Physics 79, no. 2 (2012): 197–213. http://dx.doi.org/10.1017/s0022377812000840.
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AbstractThis review describes the results of investigations on charged particle dynamics in a magnetic field carried out over a number of years. The studies have unravelled the existence of some very surprising and unusual phenomena. Though existing on the macro-scale, they are found to be of quantum origin, and are thereby not covered by the Lorentz equation, which has been regarded conventionally as the descriptor of electrodynamic phenomena on the macro-scale. These novel phenomena have been found to be attributed to the ‘quantum modulation’ of the de Broglie wave along the magnetic field. This is brought about through the scattering-induced transition across Landau levels, leading to the modulation of the plane wave state along the field as a result of the entanglement between the parallel and perpendicular degrees of freedom. These findings were motivated by the predictions of a formalism developed by the author and include such unusual phenomena as (i) macro-scale matter wave interference effects and (ii) the detection of curl-free vector potential also on the macro-scale, both attributed to quantum modulation which is a matter wave on the macro-scale. The formalism is thus described as ‘macro-quantization of guiding centre motion’.
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Rau, Peter B., and Ira Wasserman. "Numerical Simulation of Electron Magnetohydrodynamics with Landau-quantized Electrons in Magnetar Crusts." Astrophysical Journal 979, no. 2 (2025): 154. https://doi.org/10.3847/1538-4357/ad9dea.
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Abstract In magnetar crusts, magnetic fields are sufficiently strong to confine electrons into a small to moderate number of quantized Landau levels. This can have a dramatic effect on the crust's thermodynamic properties, generating field-dependent de Haas–van Alphen oscillations. We previously argued that the large-amplitude oscillations of the magnetic susceptibility could enhance the ohmic dissipation of the magnetic field by continuously generating small-scale, rapidly dissipating field features. This could be important to magnetar field evolution and contribute to their observed higher temperatures. To study this, we performed quasi-3D numerical simulations of electron MHD in a representative volume of neutron star crust matter, for the first time including the magnetization and magnetic susceptibility resulting from Landau quantization. We find that the potential enhancement in the ohmic dissipation rate due to this effect can be a factor ∼3 for temperatures of the order of 108 K, and ∼4.5 for temperatures of the order of 5 × 107 K, depending on the magnetic field configuration. The nonlinear Hall term is crucial to this amplification: without it, the magnetic field decay is only enhanced by a factor ≲2 even at 5 × 107 K. These effects generate a high wavenumber plateau in the magnetic energy spectrum associated with the small-scale de Haas–van Alphen oscillations. Our results suggest that this mechanism could help explain the magnetar heating problem, though due to the effect's temperature-dependence, full magneto-thermal evolution simulations in a realistic stellar model are needed to judge whether it is viable explanation.
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Moskalenko, Sveatoslav A., Igor V. Podlesny, and Ion A. Zubac. "Bound states of the two-dimensional magnetoexcitons taking into account the Rashba spin-orbit coupling." Moldavian Journal of Physical Sciences 1-2, no. 19 (2020): 11–44. https://doi.org/10.5281/zenodo.4118641.
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Molecular-type bound states of two-dimensional (2D) magnetoexcitons supplementarily subjected to the action of an external electric field perpendicular to the layer and parallel to a strong magnetic field are studied. The electron and hole wave functions have different numbers of the Landau quantization levels for different spin projections. The Rashba spin–orbit coupling (RSOC) is characterized by first-order chirality terms for electrons and third-order chirality terms for heavy holes in GaAs-type quantum wells. In this case, the external electric field gives rise to the nonparabolic dispersion law of heavy holes and a nonmonotonous dependence of their energy levels on the magnetic field strength. The lowest spinor-type wave functions of electrons and holes are used to construct the one- and two-magnetoexciton wave functions, determine their normalization conditions, and deduce the Hamiltonian of the Coulomb electron–electron interaction in the presence of the RSOC. It contains electron and hole imprint factors, the difference of which determines the affinities of the electron–hole pairs to interact between themselves. The average value of the Coulomb interaction Hamiltonian makes it possible to calculate the mean energy value per two pairs forming the molecule versus parameter a of the wave function, which determines the relative motion of two magnetoexcitons in the frame of the bound state. The next numerical calculations will show whether the stable bound state of the molecular type or the metastable bound state does exist.
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KOMA, TOHRU. "REVISITING THE CHARGE TRANSPORT IN QUANTUM HALL SYSTEMS." Reviews in Mathematical Physics 16, no. 09 (2004): 1115–89. http://dx.doi.org/10.1142/s0129055x04002230.
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We re-examine the charge transport induced by a weak electric field in two-dimensional quantum Hall systems in a finite, periodic box at very low temperatures. Our model covers random vector and electrostatic potentials and electron–electron interactions. The resulting linear response coefficients consist of the time-independent term σxy corresponding to the Hall conductance and the linearly time-dependent term γsy · t in the transverse and longitudinal directions s=x,y in a slow switching limit for adiabatically applying the initial electric field. The latter terms γsy · t are due to the acceleration of the electrons by the uniform electric field in the finite and isolated system, and so the time-independent term σyy corresponding to the diagonal conductance which generates dissipation of heat always vanishes. The well-known topological argument yields the integral and fractional quantization of the averaged Hall conductance [Formula: see text] over gauge parameters under the assumption that there exists a spectral gap above the ground state. In addition to this fact, we show that the averaged acceleration coefficients [Formula: see text] vanish under the same assumption. In the non-interacting case, the spectral gap between the neighboring Landau levels persists if the vector and the electrostatic potentials together satisfy a certain condition, and then the Hall conductance σxy without averaging exhibits the exact integral quantization with the vanishing acceleration coefficients in the infinite volume limit. We also estimate their finite size corrections. In the interacting case, the averaged Hall conductance [Formula: see text] for a non-integer filling of the electrons is quantized to a fraction not equal to an integer under the assumption that the potentials satisfy certain conditions in addition to the gap assumption. We also discuss the relation between the fractional quantum Hall effect and the Atiyah–Singer index theorem for non-Abelian gauge fields.
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Chang, Yao-Wen, and Yia-Chung Chang. "Theory of magnetic-field effect on trions in two-dimensional materials." Journal of Chemical Physics 157, no. 4 (2022): 044104. http://dx.doi.org/10.1063/5.0096834.
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In this work, we present a theoretical method to study the effect of magnetic field on trions in two-dimensional materials. The trion is modeled by a three-particle Schrödinger equation and the magnetic-field interaction is included by means of a vector potential in symmetric gauge. By using a coordinate transformation and a unitary transformation, the trion Hamiltonian can be converted into the sum of a translational term describing the Landau quantization for the trion center-of-mass motion, an internal term describing the trion binding, and a translational–internal coupling term depending linearly on the magnetic-field strength. The trion eigenenergy and wavefunction can then be calculated efficiently by using a variational method, and the quantum numbers of trions in magnetic fields can be assigned. The eigenenergies, binding energies, and correlation energies of three trion branches, which correspond to the ground-state trion and two excited-state trions solved from the trion Hamiltonian in zero magnetic field, are studied numerically in finite magnetic fields. The present method is applied to study the magnetic-field dependence of trion energy levels in hole-doped WSe2 monolayers. The binding energies and correlation energies of positive trions in WSe2 are investigated over a range of magnetic fields up to 25 T.
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KOROLEV, ANDREI V., and MICHAEL A. LIBERMAN. "EXCITONS IN A HIGH MAGNETIC FIELD." International Journal of Modern Physics B 10, no. 07 (1996): 729–75. http://dx.doi.org/10.1142/s0217979296000313.
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A high magnetic field, such that the distance between the Landau levels exceeds the binding energy of an exciton, gives an opportunity to create various new states of matter, i.e. exciton crystal, molecular compexes, Bose–Einstein condensation an exciton gas in a semiconductor, depending on the dimensionality of the system. We consider the problem of excitonic interaction in a semiconductor in its multi-electron formulation, starting from the second-quantization representation of the Hamiltonian of interacting electrons and holes in a high magnetic field. A system of excitons in its ground state in a high magnetic field is similar to a weakly non-ideal Bose gas; whereas the excited states may be strongly bounded. It is shown that different types of exciton complexes in a quasi-one-dimensional semiconductor quantum wire, from crystals to molecular chains, can be obtained both by varying the direction and intensity of the magnetic field and by changing the exciton density. The existence and the stability of the Bose condensate in a bulk semiconductor due to an essential decrease of the interaction between excitons and an increase of their binding energy in a high magnetic field are established at a high density of excitons. Existence of the built-in condensate of excitons in a broad density range significantly changes the excitation spectrum of coupled excitons and photons in a high magnetic field and results in a number of interesting optical phenomena.
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Ozerin, A. Yu, and L. A. Falkovsky. "Berry phase, semiclassical quantization, and Landau levels." Physical Review B 85, no. 20 (2012). http://dx.doi.org/10.1103/physrevb.85.205143.
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Aggarwal, Srishty, Banibrata Mukhopadhyay, and Gianluca Gregori. "Relativistic Landau quantization in non-uniform magnetic field and its applications to white dwarfs and quantum information." SciPost Physics 11, no. 5 (2021). http://dx.doi.org/10.21468/scipostphys.11.5.093.
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We investigate the two-dimensional motion of relativistic cold electrons in the presence of `strictly’ spatially varying magnetic fields satisfying, however, no magnetic monopole condition. We find that the degeneracy of Landau levels, which arises in the case of the constant magnetic field, lifts out when the field is variable and the energy levels of spin-up and spin-down electrons align in an interesting way depending on the nature of change of field. Also, the varying magnetic field splits Landau levels of electrons with zero angular momentum from positive angular momentum, unlike the constant field which only can split the levels between positive and negative angular momenta. Exploring Landau quantization in non-uniform magnetic fields is a unique venture on its own and has interdisciplinary implications in the fields ranging from condensed matter to astrophysics to quantum information. As examples, we show magnetized white dwarfs, with varying magnetic fields, involved simultaneously with Lorentz force and Landau quantization affecting the underlying degenerate electron gas, exhibiting a significant violation of the Chandrasekhar mass-limit; and an increase in quantum speed of electrons in the presence of a spatially growing magnetic field.
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Bakke, Knut, Lincoln Ribeiro, and Claudio Furtado. "Landau quantization for an induced electric dipole in the presence of topological defects." Open Physics 8, no. 6 (2010). http://dx.doi.org/10.2478/s11534-010-0006-z.
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AbstractIn this contribution we investigate the non-relativistic quantum dynamics of induced electric dipoles in the presence of a topological defect. We propose an analog of Landau quantization for neutral atoms, where a electric dipole is induced by the electromagnetic field configuration. We investigate this system in the presence of a topological defect and show that it breaks the infinite degeneracy of Landau levels.
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Hwang, Yoonseok, Jun-Won Rhim, and Bohm-Jung Yang. "Geometric characterization of anomalous Landau levels of isolated flat bands." Nature Communications 12, no. 1 (2021). http://dx.doi.org/10.1038/s41467-021-26765-z.
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AbstractAccording to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.
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Fuchs, Jean-Noël, Frédéric Piéchon, and Gilles Montambaux. "Landau levels, response functions and magnetic oscillations from a generalized Onsager relation." SciPost Physics 4, no. 5 (2018). http://dx.doi.org/10.21468/scipostphys.4.5.024.
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A generalized semiclassical quantization condition for cyclotron orbits was recently proposed by Gao and Niu , that goes beyond the Onsager relation . In addition to the integrated density of states, it formally involves magnetic response functions of all orders in the magnetic field. In particular, up to second order, it requires the knowledge of the spontaneous magnetization and the magnetic susceptibility, as was early anticipated by Roth . We study three applications of this relation focusing on two-dimensional electrons. First, we obtain magnetic response functions from Landau levels. Second we obtain Landau levels from response functions. Third we study magnetic oscillations in metals and propose a proper way to analyze Landau plots (i.e. the oscillation index nn as a function of the inverse magnetic field 1/B1/B) in order to extract quantities such as a zero-field phase-shift. Whereas the frequency of 1/B1/B-oscillations depends on the zero-field energy spectrum, the zero-field phase-shift depends on the geometry of the cell-periodic Bloch states via two contributions: the Berry phase and the average orbital magnetic moment on the Fermi surface. We also quantify deviations from linearity in Landau plots (i.e. aperiodic magnetic oscillations), as recently measured in surface states of three-dimensional topological insulators and emphasized by Wright and McKenzie .
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Chong, Su Kong, Chao Lei, Seng Huat Lee, et al. "Anomalous Landau quantization in intrinsic magnetic topological insulators." Nature Communications 14, no. 1 (2023). http://dx.doi.org/10.1038/s41467-023-40383-x.
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AbstractThe intrinsic magnetic topological insulator, Mn(Bi1−xSbx)2Te4, has been identified as a Weyl semimetal with a single pair of Weyl nodes in its spin-aligned strong-field configuration. A direct consequence of the Weyl state is the layer dependent Chern number, $$C$$ C . Previous reports in MnBi2Te4 thin films have shown higher $$C$$ C states either by increasing the film thickness or controlling the chemical potential. A clear picture of the higher Chern states is still lacking as data interpretation is further complicated by the emergence of surface-band Landau levels under magnetic fields. Here, we report a tunable layer-dependent $$C$$ C = 1 state with Sb substitution by performing a detailed analysis of the quantization states in Mn(Bi1−xSbx)2Te4 dual-gated devices—consistent with calculations of the bulk Weyl point separation in the doped thin films. The observed Hall quantization plateaus for our thicker Mn(Bi1−xSbx)2Te4 films under strong magnetic fields can be interpreted by a theory of surface and bulk spin-polarised Landau level spectra in thin film magnetic topological insulators.
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Yuting Yang, Xinyue Qian, and Liwei Shi. "Manipulation of electromagnetic waves induced by pseudomagnetic fields in two dimensional photonic crystals." Acta Physica Sinica, 2023, 0. http://dx.doi.org/10.7498/aps.72.20222242.
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Under strong magnetic field, many interesting phenomena can occur in the electronic system, for example quantization of Landau levels and quantum Hall effect. However, photons do not carry charge, and can not have many properties induced by external magnetic fields. Recently, the pseudomagnetic field being an artificial synthetic gauge field, has attracted intense research interest in classical wave systems, in which the propagation of wave can be manipulated like in real magnetic field. The photonic crystal is an optical structure composed of periodic material distributions and provides a good platform to study the control of electromagnetic waves. In this work, we construct a uniform pseudomagnetic field by introducing uniaxial linear gradient deformation of metallic rods in a two-dimensional photonic crystal. The strong pseudomagnetic field leads to the quantization of photonic Landau levels in photonic crystals. The sublattice polarization of n=0 Landau level is also demonstrated in our simulations. Unlike the real magnetic field, the pseudomagnetic fields of photonic crystal is opposite in two inequivalent energy valleys, and the time-reversal symmetry of the system is not broken. Our designed gradient photonic crystals support the transport of edge state in the gap between <i>n</i>=0 and <i>n</i>=±1 Landau levels. The edge state can propagate unidirectionally when is excited by a chiral source. When a gaussian beam impinges on the photonic crystal, the propagating path of two splitting beams can be controlled, which gives rise to the bend of two beams. Two photonic crystals with opposite pseudomagnetic fields are assembled together, and the interesting phenomenon of "snake-state" can be obtained. Our proposal opens a new way for the design of information processing devices by manipulating electromagnetic waves.
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Chaudhary, Rozina, Asma Afzal, Stefaan Poedts, Shahid Ali та Hassan A. Shah. "Signatures of quantized magnetic field on the nonlinear Landau damping of transverse electromagnetic waves". Physica Scripta, 14 липня 2023. http://dx.doi.org/10.1088/1402-4896/ace79e.
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Abstract A quantum kinetic approach along with the Landau theory of quantization (LQ) is utilized to study the impact of the magnetic field on the nonlinear Landau damping (NLD) of transverse electromagnetic (EM) waves in a degenerate electron-ion plasma. The gyratory motion of fermions around the magnetic field (H) lines gets quantized into the Landau levels and consequently the associated Fermi-Dirac distribution function becomes modified with the fermion cyclotron frequency under the limit lℏω_{ce}-ε_{Fe}≫k_{B}T_{e}, where l is the orbital quantum number with all other standard notations. In this context, the density oscillations due to electrons are calculated in the presence of the LQ parameter η(=ℏω_{ce}/ε_{Fe}<1) and ion density perturbations are computed using the framework of Maxwell distribution. A new type of kinetic nonlinear Shrödinger equation is derived in the presence of η, which involves nonlocal nonlinear term responsible for the NLD of EM waves. Furthermore, longitudinal wave modes are investigated to account for quantization parameter η. The LQ is also shown to absorb oscillation spectra of the linear ion-acoustic mode. The present findings might be helpful to understand the impact of the H field on the nonlinear interaction of EM waves with astrophysical plasmas, e.g., in the atmosphere of neutron star the presence of quantized magnetic field is more common.&#xD;&#xD;
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Jiang, Linfeng, Honghui Wang, Zizhen Zhou, et al. "Landau Quantization and Lifshitz Transition‐Modulated High Magneto‐Thermoelectric Performance." Advanced Energy Materials, September 13, 2024. http://dx.doi.org/10.1002/aenm.202402613.
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AbstractOne of the necessary conditions for achieving high thermoelectric (TE) performance is the large longitudinal (Sxx) and transverse thermopower (Syx), which are closely related to the Fermi level position and the energy‐dependent density of states (DOS) at the Fermi level. Herein, it is demonstrated that HfTe5 has both high‐mobility induced significant quantum oscillations with an ultra‐low quantum limit of 2.7 T, and a significant Lifshitz transition where the Fermi level shows strong temperature dependence. The Landau levels yielded by the magnetic field distort the DOS with many equally‐spaced delta functions and significantly enhance the Sxx and Syx. The drastically enhanced magneto‐longitudinal and transverse TE performance in HfTe5 is achieved under an ultra‐low magnetic field of 1.4 T, which can be easily realized by a permanent magnet. Moreover, the Lifshitz transition further modulates its TE and magneto‐TE performance. Ultimately, the peak values of magneto‐longitudinal and transverse power factors (PF) reach 64 µW cm−1 K−2 at 150 K and 0.9 T and 19 µW cm−1 K−2 at 80 K and 1.4 T, respectively, which are comparable to the best TE materials such as Bi2Te3. This work provides novel methods and significant guidance for the development of TE and magneto‐TE materials.
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Yaning, Wang, Gao Xiang, Yang Kaining, et al. "Data For : Quantum Hall phase in graphene engineered by interfacial charge coupling." October 31, 2022. https://doi.org/10.5281/zenodo.7046671.
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The quantum Hall effect can be substantially affected by interfacial coupling between the host two-dimensional electron gases and the substrate, and has been predicted to give rise to exotic topological states. Yet the understanding of the underlying physics and the controllable engineering of this interaction remains challenging. Here we demonstrate the observation of an unusual quantum Hall effect, which differs markedly from that of the known picture, in graphene samples in contact with an antiferromagnetic insulator CrOCl equipped with dual gates. Two distinct quantum Hall phases are developed, with the Landau levels in monolayer graphene remaining intact at the conventional phase, but largely distorted for the interfacial-coupling phase. The latter quantum Hall phase is even present close to the absence of a magnetic field, with the consequential Landau quantization following a parabolic relation between the displacement field and the magnetic field. This characteristic prevails up to 100 K in a wide effective doping range from 0 to 10<sup>13</sup> cm<sup>−2</sup>.
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Espoukeh, Pakhshan, and Hossein Fakhri. "Dynamical and structural symmetries for the highest Landau levels on the AdS 2." Open Physics 10, no. 2 (2012). http://dx.doi.org/10.2478/s11534-011-0113-5.
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AbstractThe aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.
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Kumari, Anamika, Harsha Silotia, Sunit Das, et al. "Observation of the Magnetic Field Induced Fermi Surface Expansion in Aperiodic Quantum Oscillations." Advanced Functional Materials, June 16, 2025. https://doi.org/10.1002/adfm.202422986.
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AbstractMagnetic field‐induced quantum oscillations in resistivity have been extensively used to explore Fermi surfaces in quantum materials. This is enabled by the robustness of the Fermi surface to a magnetic field and the validity of the semiclassical Onsager's quantization relation for Landau levels (LLs). Challenging this conventional understanding, evidence of magnetic field‐induced expansion of the Fermi surface is presented. This is captured by our observations of magnetic field induced aperiodic quantum oscillations in resistivity at the conducting interfaces of LaVO3‐KTaO3 (LVO‐KTO) and EuO‐KTaO3 (EO‐KTO). It is showed that these aperiodic oscillations occur in systems with a small Fermi surface, where the magnetic susceptibility‐induced corrections to the Fermi surface become substantial. Physically, these arise from the magnetic susceptibility‐induced modifications in the Free energy, resulting in the generalization of the semiclassical quantization rule for LLs. The magnetic field‐induced Fermi surface expansion is corroborated via the measurements of nonlinear Landau fan diagrams, deviations in the Lifshitz–Kosevich (LK) fit of longitudinal magnetoresistivity, and the magnetic field dependence of the effective cyclotron mass extracted from transport measurements. These findings provide a new perspective and fresh insights into the intriguing physics of magnetic field‐based probes of the Fermi surface.
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Emerson B.S. Corrêa, M. G. Lima, and S. C. Q. Arruda. "The Eigenfunction Method to Calculate the Klein-Gordon Propagator in an Inhomogeneous Magnetic Field." Contemporary Mathematics, August 25, 2022, 343–52. http://dx.doi.org/10.37256/cm.3320221556.
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In this paper, we use the Eigenfunction method (Ritus'method) to calculate the Klein-Gordon propagator under an external magnetic field with an exponential damping factor b, in a four-dimensional Euclidean space. We write the propagator of scalar field in terms of plane waves and Laguerre polynomials. Also the eigenvalues associated to differential operator show the quantization in the xy-plane through the Landau levels. We apply the calculated propagator to the mass parameter of an interacting boson system under an exponentially decaying magnetic field.
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Shih, Po-Hsin, Thi-Nga Do, Godfrey Gumbs, Danhong Huang, Hai Duong Pham, and Ming-Fa Lin. "Rich Magnetic Quantization Phenomena in AA Bilayer Silicene." Scientific Reports 9, no. 1 (2019). http://dx.doi.org/10.1038/s41598-019-50704-0.
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Abstract The rich magneto-electronic properties of AA-bottom-top (bt) bilayer silicene are investigated using a generalized tight-binding model. The electronic structure exhibits two pairs of oscillatory energy bands for which the lowest conduction and highest valence states of the low-lying pair are shifted away from the K point. The quantized Landau levels (LLs) are classified into various separated groups by the localization behaviors of their spatial distributions. The LLs in the vicinity of the Fermi energy do not present simple wave function modes. This behavior is quite different from other two-dimensional systems. The geometry symmetry, intralayer and interlayer atomic interactions, and the effect of a perpendicular magnetic field are responsible for the peculiar LL energy spectra in AA-bt bilayer silicene. This work provides a better understanding of the diverse magnetic quantization phenomena in 2D condensed-matter materials.
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Ghatak, Kamakhya P. "On the Photoemission from Ternary and Quaternary Alloys Systems Under Strong Magnetic Quantization." MRS Proceedings 216 (1990). http://dx.doi.org/10.1557/proc-216-469.
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ABSTRACTWe study the photomission from ternary and quanternary alloys systems by formulating the respective expressions considering the spin & broadening of Landau levels. It is found taking n-Hg1−x CdxTe and In1−xGaxAsyP1−y lattice matched to InP as examples of ternary and quaternary alloys respectively that the photoemission exhibits oscillatory magnetic field dependence and increases with increasing electron concentration in spiky manner for both the cases. The numerical magnitudes of photoemission in ternary alloys are greater than that of the quaternary systems and the theoretical formulations are in agreement with the experimental observations as reported elsewhere.
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Hattori, Koichi, Hidetoshi Taya, and Shinsuke Yoshida. "Di-lepton production from a single photon in strong magnetic fields: vacuum dichroism." Journal of High Energy Physics 2021, no. 1 (2021). http://dx.doi.org/10.1007/jhep01(2021)093.
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Abstract We study di-lepton production from a single photon in the presence of a strong constant magnetic field. By the use of the Ritus-basis formalism, we analytically evaluate the photon-to-di-lepton conversion vertex with fully taking into account the non-perturbative interactions between the produced fermions and the strong magnetic field. We show that the di-lepton spectrum becomes anisotropic with respect to the magnetic-field direction and depends on the photon polarization as a manifestation of the vacuum dichroism in a strong magnetic field. According to the energy conservation in the presence of the Landau quantization, not only the transverse momentum of the produced fermions but also the longitudinal momentum is discretized, and the di-lepton spectrum exhibits spike structures as functions of the incident photon energy and the magnetic field strength. We also show that the di-lepton production is strictly prohibited for massless fermions in the lowest Landau levels as an analogue of the so-called helicity suppression.
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Tawfik, Abdel Nasser, and Abdel Magied Diab. "Chiral magnetic properties of QCD phase-diagram." European Physical Journal A 57, no. 6 (2021). http://dx.doi.org/10.1140/epja/s10050-021-00501-z.
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AbstractThe QCD phase-diagram is studied, at finite magnetic field. Our calculations are based on the QCD effective model, the SU(3) Polyakov linear-sigma model (PLSM), in which the chiral symmetry is integrated in the hadron phase and in the parton phase, the up-, down- and strange-quark degrees of freedom are incorporated besides the inclusion of Polyakov loop potentials in the pure gauge limit, which are motivated by various underlying QCD symmetries. The Landau quantization and the magnetic catalysis are implemented. The response of the QCD matter to an external magnetic field such as magnetization, magnetic susceptibility and permeability has been estimated. We conclude that the parton phase has higher values of magnetization, magnetic susceptibility, and permeability relative to the hadron phase. Depending on the contributions to the Landau levels, we conclude that the chiral magnetic field enhances the chiral quark condensates and hence the chiral QCD phase-diagram, i.e. the hadron-parton phase-transition likely takes place, at lower critical temperatures and chemical potentials.
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Wang, Yaning, Xiang Gao, Kaining Yang, et al. "Quantum Hall phase in graphene engineered by interfacial charge coupling." Nature Nanotechnology, November 21, 2022. http://dx.doi.org/10.1038/s41565-022-01248-4.
Full textAbstract:
AbstractThe quantum Hall effect can be substantially affected by interfacial coupling between the host two-dimensional electron gases and the substrate, and has been predicted to give rise to exotic topological states. Yet the understanding of the underlying physics and the controllable engineering of this interaction remains challenging. Here we demonstrate the observation of an unusual quantum Hall effect, which differs markedly from that of the known picture, in graphene samples in contact with an antiferromagnetic insulator CrOCl equipped with dual gates. Two distinct quantum Hall phases are developed, with the Landau levels in monolayer graphene remaining intact at the conventional phase, but largely distorted for the interfacial-coupling phase. The latter quantum Hall phase is even present close to the absence of a magnetic field, with the consequential Landau quantization following a parabolic relation between the displacement field and the magnetic field. This characteristic prevails up to 100 K in a wide effective doping range from 0 to 1013 cm−2.
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N. Shrivastava, Keshav, Ithnin Abdul Jalil, Norhasliza Yusof, and Hasan Abu Kassim. "The quantum Hall effect: spin-charge locking." Malaysian Journal of Fundamental and Applied Sciences 2, no. 1-2 (2014). http://dx.doi.org/10.11113/mjfas.v2n1-2.15.
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In two-dimensional electron gas when a large magnetic field is applied in one direction and an electric field perpendicular to it, there is a current in a direction perpendicular to both. This current is called the Hall effect. It remained without quantization until 1980 when it was found that the quantization leads to correct measurement of h/e2. Therefore the quantized Hall effect was further studied at high magnetic fields where fractional quantization was found. The fractional charge can arise from the “incompressibility” in the flux quantization. Laughlin wrote a wave function, the excitations of which are fractionally charged quasiparticles. This wave function comes in competition with charge density waves but for a few fractions it does give the ground state. If “incompressibility” is not considered and it is allowed to be compressible, the fractional charge can arise from the angular momentum which appears in the Bohr magneton in the form of g values. Usually the positive spin is considered but we consider both the positive as well as the negative values so that there is a spin-charge coupling. The values thus calculated for the fractional charge agree with the experimental data on the quantum Hall effect. We have followed this subject for a long time and hence have reviewed the subject. There are several interesting concepts which we learn from this subject. The concept of the Hall effect is quite clear particularly when combined with the flux quantization. We learn about the Landau levels and hence the boson character of electrons in two dimensions. We learn that charge becomes a vector quantity and there is spincharge coupling.
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Zhang, He-Xia, Jin-Wen Kang, and Ben-Wei Zhang. "Thermoelectric properties of the (an-)isotropic QGP in magnetic fields." European Physical Journal C 81, no. 7 (2021). http://dx.doi.org/10.1140/epjc/s10052-021-09409-w.
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AbstractThe Seebeck effect and the Nernst effect, which reflect the appearance of electric fields along x-axis and along y-axis ($$E_{x}$$ E x and $$E_{y}$$ E y ), respectively, induced by the thermal gradient along x-axis, are studied in the QGP at an external magnetic field along z-axis. We calculate the associated Seebeck coefficient ($$S_{xx}$$ S xx ) and Nernst signal (N) using the relativistic Boltzmann equation under the relaxation time approximation. In an isotropic QGP, the influences of magnetic field (B) and quark chemical potential ($$\mu _{q}$$ μ q ) on these thermoelectric transport coefficients are investigated. In the presence (absence) of weak magnetic field, we find $$S_{xx}$$ S xx for a fixed $$\mu _{q}$$ μ q is negative (positive) in sign, indicating that the dominant carriers for converting heat gradient to electric field are negatively (positively) charged quarks. The absolute value of $$S_{xx}$$ S xx decreases with increasing temperature. Unlike $$S_{xx}$$ S xx , the sign of N is independent of charge carrier type, and its thermal behavior displays a peak structure. In the presence of strong magnetic field, due to the Landau quantization of transverse motion of (anti-)quarks perpendicular to magnetic field, only the longitudinal Seebeck coefficient ($$S_{zz}$$ S zz ) exists. Our results show that the value of $$S_{zz}$$ S zz at a fixed $$\mu _{q}$$ μ q in the lowest Landau level (LLL) approximation always remains positive. Within the effect of high Landau levels, $$S_{zz}$$ S zz exhibits a thermal structure similar to that in the LLL approximation. As the Landau level increases further, $$S_{zz}$$ S zz decreases and even its sign changes from positive to negative. The computations of these thermoelectric transport coefficients are also extended to a medium with momentum-anisotropy induced by initial spatial expansion as well as strong magnetic field.