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arXiv:1802.01460

Diamagnetism of 2D-Fermions in the Strong Nonhomogeneous Static Magnetic Field $( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))$ : gas magnetization, ... and gas compressibility

Matej Hudák, O. Hudák

Published Oct 11, 2017• 0 citations
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Abstract

We study diamagnetism of a gas of fermions moving in a nonhomogeneous magnetic field $( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))$ The gas magnetization, the static magnetic susceptibility, the chemical potential and the gas compressibility are discussed and compared with the uniform field case. General need to study dynamics of electrons in different types of magnetic fields follows from a large number of experimental situations in which its understanding enables physicists to obtain new information.

DOI
10.48550/arxiv.1802.01460

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Diamagnetism of 2D-Fermions in the Strong Nonhomogeneous Static Magnetic Field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) : gas magnetization, ... and gas compressibility
We study diamagnetism of a gas of fermions moving in a nonhomogeneous magnetic field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) The gas magnetization, the static magnetic susceptibility, the chemical potential and the gas compressibility are discussed and compared with the uniform field case. General need to study dynamics of electrons in different types of magnetic fields follows from a large number of experimental situations in which its understanding enables physicists to obtain new information.
Abstract At low temperature and under weak magnetic field, non-interacting Fermi gases reveal both Pauli paramagnetism and Landau diamagnetism, and the magnitude of the diamagnetic susceptibility is 1/3 of that of the paramagnetic one. When the temperature is finite and the magnetic field is also finite, we demonstrate that the paramagnetism and diamagnetism start to deviate from the ratio 1/3. For understanding the magnetic properties of an ideal Fermi gas at quite low temperature and under quite weak magnetic field, we work out explicitly the third-order magnetic susceptibility in three cases, from intrinsic spin, orbital motion and in total. An interesting property is in third-order magnetic susceptibilities: when viewing individually, they are both diamagnetic, but in total it is paramagnetic.
Using results of our exact description of the spinless fermion motion in a nonhomogeneous magnetic field B=B(0,0,1/cosh2(x−x0δ)) {\bf B} = B( 0, 0, 1/cosh^{2}( \frac{x-x_{0}}{ δ})) B=B(0,0,1/cosh2(δx−x0​​)) we study a gas of these particles moving in this field. For lower densities ν<νc(B,δ) ν< ν_{c}(B, δ) ν<νc​(B,δ) the corresponding total energy is lower than that of the uniform field state. Thus when the density of anyons decreases a transition from the uniform statistical field state to the nonhomogeneous field state is predicted.
A charged Bose gas under confinements ensures a phase transition, in contrast with the homogeneous system in an external magnetic field. The present investigation concerns charged Bose gases trapped in a two-dimensional potential as well as neutral Bose gases rotating around the z axis. It is found that the systems exhibit quite different magnetic properties. A diamagnetization is determined in the charged system while the rotating system shows up an uncommon magnetic response. However, both models possess finite magnetic susceptibilities at vanishing field or at stopping rotation.
It is shown that a differential characterization of Sampson-Seitz methods for calculating the static responses of a normal fermion system at zero temperature yields expressions for these quantities which are identical with those deduced by Landau from a semiphenomenological basis and by the author, in several instances, from spherical, time-independent, many-body perturbation theory. This connection is explicitly demonstrated for the Galilean invariance, magnetic susceptibility, and compressibility of a normal fermion system with translation-invariant interactions. The calculation of the magnetic susceptibility and compressibility of a dense electron gas is examined from both points of view. In the case of the magnetic susceptibility, it is shown that the contribution to ${\ensuremath{\alpha}}_{c}$ from graphs with two virtual excitations vanishes to O(rs)O({r}_{s})O(rs​) in agreement with the Sampson-Seitz calculation of Brueckner and Sawada. Although this demonstration is trivial by Sampson-Seitz methods, it is only a consequence of detailed calculation in Landau's formulation. It is found that $(\frac{1}{K})(\frac{2}{\ensuremath{\rho}R})={\ensuremath{\alpha}}_{c}\ensuremath{-}0.0676\mathrm{ln}{r}_{s}+0.255$, where KKK is the compressibility, $\ensuremath{\rho}$ is the density, and RRR is the rydberg.
In this paper we have calculated the exact expression for the magnetic moment of a Fermi gas in a magnetic field (magnetized Fermi gas). The susceptibility obtained from our result becomes the classical Curie-Langevin law in the weak-field limit for a nondegenerate and nonrelativistic gas. The induced magnetic moment M\mathfrak{M}M was found to be proportional to the square of the electron density, but at higher densities M\mathfrak{M}M passes through a maximum and eventually becomes negative; hence a magnetized gas is first paramagnetic but later becomes diamagnetic. We have also shown that spontaneous magnetization cannot take place in a noninteracting electron gas.
We analyze potential nonanalytic terms in the Landau diamagnetic susceptibility, ${\ensuremath{\chi}}_{\text{dia}}$, at a finite temperature TTT and/or in-plane magnetic field HHH in a two-dimensional (2D) Fermi liquid. To do this, we express the diamagnetic susceptibility as ${\ensuremath{\chi}}_{\text{dia}}={(e/c)}^{2}{lim}_{Q\ensuremath{\rightarrow}0}{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$, where ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}$ is the transverse component of the static current-current correlator, and evaluate ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)$ for a system of fermions with Hubbard interaction to second order in Hubbard UUU by combining self-energy, Maki-Thompson, and Aslamazov-Larkin diagrams. We find that at T=H=0T=H=0T=H=0, the expansion of ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$ in UUU is regular, but at a finite TTT and/or HHH, it contains U2T{U}^{2}TU2T and/or U2∣H∣{U}^{2}|H|U2∣H∣ terms. Similar terms have been previously found for the paramagnetic Pauli susceptibility. We obtain the full expression for the nonanalytic $\ensuremath{\delta}{\ensuremath{\chi}}_{\text{dia}}(H,T)$ when both TTT and HHH are finite, and we show that the H/TH/TH/T dependence is similar to that for the Pauli susceptibility.