In the tight-binding approximation, an Iwatsuka magnetic field is modeled by a function on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with constant, but distinct values in the two parts of the lattice separated by a straight line of slope <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \in [-\infty ,\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mi>∞</mi> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, the K-theory of the magnetic <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras generated by an Iwatsuka magnetic field for any possible <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is computed. One interesting aspect concerns the analysis of the behavior of the system in the transition from rational to irrational <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. It turns out that when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is irrational, the magnetic hull associated with the flux operator forms a Cantor set. On the other hand, for rational <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> this set coincides with the two-point compactification of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>. This characterization, along with the use of the Pimsner-Voiculescu exact sequence, is the main ingredient for the computation of the K-theory. Once the K-theory is known, with the use of the index theory one can deduce the bulk-interface correspondence for tight-binding Hamiltonians subjected to an Iwatsuka magnetic field. Notably, it occurs that the topological quantization of the interface currents remains independent of the slope <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> (as physically expected).

On the K-theory of magnetic algebras: Iwatsuka case

期刊论文
发表日期：

2026年1月24日

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Giuseppe De Nittis,
Jaime Gomez,
Danilo Polo Ojito

摘要

In the tight-binding approximation, an Iwatsuka magnetic field is modeled by a function on $\mathbb {Z}^2$ $Z^{2}$ with constant, but distinct values in the two parts of the lattice separated by a straight line of slope $\alpha \in [-\infty ,\infty ]$ $α \in [- \infty, \infty]$ . In this paper, the K-theory of the magnetic $C^*$ $C^{*}$ -algebras generated by an Iwatsuka magnetic field for any possible $\alpha $ $α$ is computed. One interesting aspect concerns the analysis of the behavior of the system in the transition from rational to irrational $\alpha $ $α$ . It turns out that when $\alpha $ $α$ is irrational, the magnetic hull associated with the flux operator forms a Cantor set. On the other hand, for rational $\alpha $ $α$ this set coincides with the two-point compactification of $\mathbb {Z}$ $Z$ . This characterization, along with the use of the Pimsner-Voiculescu exact sequence, is the main ingredient for the computation of the K-theory. Once the K-theory is known, with the use of the index theory one can deduce the bulk-interface correspondence for tight-binding Hamiltonians subjected to an Iwatsuka magnetic field. Notably, it occurs that the topological quantization of the interface currents remains independent of the slope $\alpha $ $α$ (as physically expected).