It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $arphi$ per unit area, then any spectral island $sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $mathcal{I}_b=M arphi$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau operator $H_b$ which also has a bounded $Z^2$-periodic electric potential. Assume that $H_b$ has a spectral island $sigma_b$ which remains isolated from the rest of the spectrum as long as $arphi$ lies in a compact interval $[arphi_1,arphi_2]$. Then $mathcal{I}_b=c_0+c_1arphi$ on such intervals, where the constant $c_0in mathbb{Q}$ while $c_1in Z$. The integer $c_1$ is the Chern character of the spectral projection onto the spectral island $sigma_b$. This result also implies that the Fermi projection on $sigma_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1 e0$ or $c_1=0$ and $arphi$ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.
Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians / Monaco, Domenico; Cornean, Horia; Moscolari, Massimo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 23:11(2021), pp. 3679-3705. [10.4171/JEMS/1079]
Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians
Monaco, Domenico
;Cornean, Horia;Moscolari, Massimo
2021
Abstract
It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $arphi$ per unit area, then any spectral island $sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $mathcal{I}_b=M arphi$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau operator $H_b$ which also has a bounded $Z^2$-periodic electric potential. Assume that $H_b$ has a spectral island $sigma_b$ which remains isolated from the rest of the spectrum as long as $arphi$ lies in a compact interval $[arphi_1,arphi_2]$. Then $mathcal{I}_b=c_0+c_1arphi$ on such intervals, where the constant $c_0in mathbb{Q}$ while $c_1in Z$. The integer $c_1$ is the Chern character of the spectral projection onto the spectral island $sigma_b$. This result also implies that the Fermi projection on $sigma_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1 e0$ or $c_1=0$ and $arphi$ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.File | Dimensione | Formato | |
---|---|---|---|
Cornean_Beyond-Diophantine_2021.pdf
accesso aperto
Tipologia:
Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
371.6 kB
Formato
Adobe PDF
|
371.6 kB | Adobe PDF |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.