Landau level

A charged particle in a uniform magnetic field has equally spaced orbital energy levels with guiding-centre degeneracy.

At fixed sheet density $n_s$, each resolved Landau level gains degeneracy $eB/h$ per unit area as the magnetic field grows. A partly filled ideal level has chemical potential at its energy; when it empties into lower levels, the chemical potential drops. This produces an inverse-field-periodic sawtooth in the clean zero-temperature model. At fixed field and increasing density the corresponding chemical potential is an upward staircase with density increment $eB/h$. Fixed reservoir chemical potential is a different boundary condition, in which the density responds instead. spin counting, disorder and temperature qualify these ideal pictures.

Disorder turns ideal sharp Landau levels into bands of states. In the integer quantum Hall effect, localized tails and extended states near level centres play different transport roles; a mobility gap need not be an empty spectral gap.

A quantum oscillation is an oscillatory contribution to a metal's response as a magnetic field moves quantized orbital levels past the Fermi energy. For a closed extremal Fermi surface cross-section of wavevector-space area $A_{\mathrm{ext}}$, Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash A_{\mathrm{ext}})$. Oscillations of magnetization and electrical resistivity are respectively the de Haas-van Alphen effect and Shubnikov-de Haas effect. Low temperature and long quantum lifetime are needed so thermal smearing and disorder do not wash out the Landau levels.

At half filling the nearest-neighbor band $E-E_0=-2t(\cos x+\cos y)-t\cos z$ has an electron contour at $z=\pi$ and a hole contour at $z=0$. Translation $(x,y)\mapsto (x+\pi ,y+\pi )$ maps the enclosed pockets into each other, proving equal regular Onsager frequencies for a field along $z$. Comparing an occupied slice with the complement of a hole orbit is not a comparison of two different enclosed planar orbits. Additional band terms or changed filling can lift this equality.

The de Haas-van Alphen effect is an oscillation of a metal's magnetization as changing magnetic field moves Landau levels through the Fermi energy. The associated Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash S_F)$ for an extremal Fermi-surface cross-section $S_F$. Oscillatory resistance is instead the Shubnikov-de Haas effect.

For a semiclassical closed electron orbit in a plane perpendicular to a uniform magnetic field, its wavevector-space area obeys $S_k(E_n)=2\pi eB(n+\gamma )/\hslash$. An ordinary parabolic band without an extra geometric phase has $\gamma =1/2$. The semiclassical equations rotate and scale the real-space orbit by $\hslash /(eB)$, giving $S_r=[\hslash /(eB){]}^2{S}_k$. For a parabolic band this is equivalent to $E_n=\hslash {\omega }_c(n+1/2)$.

For an electron in a uniform perpendicular magnetic field of magnitude $B>0$, the magnetic length sets the orbital length scale of a Landau level; $e$ is the positive elementary charge.

For a two-dimensional electron gas in a perpendicular magnetic field of magnitude $B>0$, the filling factor is the electron number per unit area $n_s$ divided by the orbital degeneracy per unit area $eB/h$ of one spin branch of a Landau level. Here $e$ is the positive elementary charge and $h$ the Planck constant. The nominal filling factor varies continuously with $n_s/B$; it need not be an integer.

One resolved spin branch of a two-dimensional Landau level has degeneracy $eB/h$ per area. Thus the total carrier density is $n_s=\nu eB/h$, where the filling factor $\nu$ counts all occupied spin and other internal branches, including partial filling. If consecutive unsplit orbital peaks change filling by an unresolved degeneracy $g_d$,For only spin degeneracy, $g_d=2$. Once spin splitting is resolved, adjacent individual branches differ by one rather than two; counting split subpeaks as unsplit orbital peaks gives a factor-two error.

For $2N$ noninteracting electrons in a planar sample of area $A$, neglecting Zeeman splitting leaves two spin states for each orbital. Each Landau level therefore holds $D=eBA/(\pi \hslash )$ electrons. When $N\le D\le 2N$, fill the lowest level with $D$ electrons and the next with $2N-D$; summing their energies gives the displayed concave quadratic in $B$. The energies at the two integer-filling endpoints are equal.

The $n=0$ level of the transverse harmonic oscillator in a uniform magnetic field, with energy $\hslash |qB|/(2m)$ before longitudinal kinetic energy or spin contributions.

A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency ${\omega }_c=|qB|/m$.

In a uniform magnetic field, a nonrelativistic particle of mass $m$ and nonzero charge $q$ completes one circular transverse orbit in time $T_c=2\pi /{\omega }_c$, where ${\omega }_c=|q|B/m$. Parallel velocity changes the helix pitch but not this period. In relativistic motion the constant Lorentz factor multiplies the period: $T_c=2\pi \gamma m/(|q|B)$. The direction of rotation depends on the charge sign; the period is positive.

Cyclotron resonance is resonant absorption by charged particles in a magnetic field when an applied oscillating electric field drives their cyclotron motion. For a parabolic band, the resonance frequency is ${\omega }_c=|q|B/m^{\ast }$, so measuring it determines the cyclotron effective mass. In the quantum description, transverse electric dipole transitions connect adjacent Landau levels, whose separation is $\hslash {\omega }_c$. For an electron and a field along $z$, only the circular polarization rotating in the electron's orbital sense drives the positive-frequency resonance. Scattering broadens the absorption peak; resolving it requires an orbital period short compared with the relaxation time.

An rf field restricted to a metal's skin layer can transfer energy coherently to electrons whose orbits return to that layer. With magnetic field parallel to the surface, resonance can occur at $\omega =n{\omega }_c$, producing harmonics at $B_n=m_c\omega /(ne)$. Surface loss or cavity impedance replaces simple bulk transmission as the measured signal.

For $\mathbf{B}=B{\mathbf{e}}_z$, the vector potential $\mathbf{A}=(0,Bx,0)$ is a Landau gauge. Translation invariance in $y$ turns the transverse Hamiltonian into a family of harmonic oscillators in $x$ whose centres are labelled by the conserved $y$-momentum.

In a planar region of area $A$, each spinless Landau level has guiding-centre degeneracyMagnetic periodic boundary conditions require this flux count to be an integer.

For a spin-one-half particle with the Pauli magnetic term and gyromagnetic factor two, the orbital energies $\hslash {\omega }_c(n+1/2)$ split into $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$. The lowest level has one spin branch; every positive level has two.

A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.

In uniform perpendicular electric and magnetic fields, a charged particle's guiding centre drifts with velocity ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/B^2$, independently of its charge and mass. For $\mathbf{E}=E{\mathbf{e}}_x$ and $\mathbf{B}=B{\mathbf{e}}_z$, this gives $v_y=-E/B$.

In Landau gauge with periodic boundary conditions of length $L_y$, adjacent guiding centres differ by $\Delta k_y=2\pi /L_y$. A perpendicular electric field tilts the energy by $-(E/B)\hslash k_y$, so adjacent states formerly in one degenerate Landau level are separated by $2\pi \hslash |E/B|/L_y$.

For a constant magnetic field,is the symmetric gauge.

For charge $e$, defineIn the symmetric gauge,

For a charged particle in a uniform magnetic field, the symmetric gauge $\mathbf{A}=\mathbf{B}\times \mathbf{x}/2$ makes the displayed magnetic pseudomomentum a conserved quantity when no other position-dependent potential breaks translation symmetry. It differs from the kinetic momentum $\mathbf{\pi }=\mathbf{p}-e\mathbf{A}=m\dot{\mathbf{x}}$. With $\mathbf{B}=B{\mathbf{e}}_3$, Hamilton's equations for $H=|\mathbf{p}-e\mathbf{A}{|}^2/(2m)$ give ${\dot{p}}_1=(eB/2){\dot{x}}_2$ and ${\dot{p}}_2=-(eB/2){\dot{x}}_1$, proving conservation of $K_1=p_1-eB{x}_2/2$ and $K_2=p_2+eB{x}_1/2$. The charge factor is necessary unless it has already been absorbed into the field convention. In quantum mechanics the transverse components need not commute, even though each commutes with this Hamiltonian operator; magnetic translation algebra records that distinction.

For charges $+e,-e$ and relative coordinate $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$, total magnetic pseudomomentum is conserved in a uniform magnetic field. In an appropriate symmetric-gauge Lagrangian differing by a total derivative, it is the canonical centre-of-mass momentum. Mechanical centre-of-mass momentum instead equals $\mathbf{K}+e\mathbf{B}\times \mathbf{r}$ and need not be constant. The sector $\mathbf{K}=0$ is not generally a stationary-centre-of-mass frame.

In the zero-pseudomomentum sector, a planar relative circular orbit has two signed angular-frequency branches. Here $\mu =m_1{m}_2/(m_1+m_2)$, $e^{\ast }=e(m_1-m_2)/[2(m_1+m_2)]$, $R$ is separation, and the Coulomb constant is absorbed into $e^2$. The centre of mass also circles: ${\mathbf{R}}_{\mathrm{cm}}={\mathbf{R}}_0+eB\mathbf{r}/[(m_1+m_2)\omega ]$. This distinction is essential when reconstructing individual particle orbits.

The kinetic momentum $\mathbf{\Pi }=\mathbf{p}-q\mathbf{A}$ equals $m\mathbf{v}$ for a nonrelativistic charged particle and is invariant under an electromagnetic gauge transformation.

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

For constant $B$,

Magnetic translations along two lattice vectors commute when their cell flux obeys

For a nonvanishing superconducting order parameter $\psi =\sqrt{n_s}{e}^{i\theta }$, $j=(q{n}_s/m)(\hslash \nabla \theta -qA)$. Single-valuedness gives phase winding $2\pi N$ around a closed curve, hence the fluxoid and flux quantization in a superconductor identity above. Magnetic flux alone is $Nh/q$ when the current integral vanishes. If the entire spanning surface is in a region where $D\psi =0$, the field vanishes there and the winding is zero; nonzero quantized flux requires a hole or a vortex core where that condition does not hold.

For uniform superconducting carrier density, the displayed combination of enclosed magnetic flux and the circulating supercurrent is quantized on a closed contour where the order parameter does not vanish. It follows by integrating the gauge-invariant phase gradient. Magnetic flux alone is quantized only when the current contribution on that contour is negligible.

The magnitude of one magnetic flux unit for an electron-pair order parameter, whose electric charge has magnitude $2e$. The enclosing contour must satisfy the flux condition in fluxoid and flux quantization in a superconductor; it is not a statement that every finite area inside the material contains an integer number of these units.