bibkey: fabbrimarcelligiuliani2026conductivity
authors: Simone Fabbri; Giovanna Marcelli; Alessandro Giuliani
year: 2026
title: "The conductivity matrix at the topological phase transition"
doi: null
url: https://arxiv.org/abs/2609.03832v1
claim: "The paper studies the conductivity matrix of two-dimensional lattice fermion models at a topological phase transition, under a gap assumption on the Bloch bands for nonzero mass; for its pi/2-flux Hofstadter example it supports the gap assumption only numerically, stating that for every m in [-2,2] minus {0} the fourth band stays above and the third band below the Fermi energy over the Brillouin zone, and expects an analytic proof."
strata_touched:
  - D5/S3/SpectralTopology/HalfFluxHofstadterMassGap
license: citation-only
triage: anchor

Fabbri, Marcelli and Giuliani, the conductivity matrix at the topological phase transition

S. Fabbri, G. Marcelli and A. Giuliani, arXiv:2609.03832v1 [math-ph] (3 September 2026). Quotations are from the v1 TeX source, with the macros \v{k}, \l, \mbb{B} written as $\vec{k}$, $\lambda$, $\mathbb{B}$ and cross-references omitted.

The bands:

we denote by ${\lambda }_1(\vec{k};m)\le \dots \le {\lambda }_M(\vec{k};m)$ the eigenvalues of ${\hat{H}}^0(\vec{k};m)$ (i.e., its energy bands), repeated according to multiplicity.

The gap assumption (item 1 of the second assumption): for an index $n$ with ${\lambda }_{\pm }={\lambda }_{n+1},{\lambda }_n$ and ${\mu }_{\pm }(m)$ the minimum and maximum of ${\lambda }_{\pm }(\cdot ;m)$ over $\mathbb{B}$,

${\mu }_{+}(m)>{\mu }_{+}(0)={\mu }_{-}(0)>{\mu }_{-}(m)\forall m\in [-m_0,m_0]\setminus \{0\}$

The $\pi /2$-flux Hofstadter example, with $d\in \mathbb{R}$:

$${\hat{H}}^0(\vec{k})=-\left(\begin{array}{cccc}d& 1-i{e}^{2i{k}_1}& 1-i{e}^{2i{k}_2}& 0\\ 1+i{e}^{-2i{k}_1}& 0& 0& -i+e^{2i{k}_2}\\ 1+i{e}^{-2i{k}_2}& 0& 0& 1+i{e}^{2i{k}_1}\\ 0& i+e^{-2i{k}_2}& 1-i{e}^{-2i{k}_1}& -d\end{array}\right),$$

with ${\vec{k}}_F:=(\frac{\pi }{4},\frac{\pi }{4})$ and

Letting the mass parameter be $m:=d-2$ and ${\hat{H}}^0(\vec{k};m):={\hat{H}}^0(\vec{k}){|}_{d=2+m}$, we then set the Fermi energy $\mu ={\lambda }_3({\vec{k}}_F;m){|}_{m=0}={\lambda }_4({\vec{k}}_F;m){|}_{m=0}$.

In the MathematicaNotebook1.nb sheet we provide numerical evidence of the validity of Assumption 2.1, and specifically of the fact that for all $m\in [-2,2]\setminus \{0\}$, ${\min }_{\vec{k}\in \mathbb{B}}({\lambda }_4(\vec{k};m)-\mu )>0$ and ${\max }_{\vec{k}\in \mathbb{B}}({\lambda }_3(\vec{k};m)-\mu )<0$. We expect that, with some additional technical effort, this can be fully proved analytically, but we prefer not to belabor the details of such a proof here.

Verified locator

Source: https://arxiv.org/abs/2609.03832v1

• URL: https://arxiv.org/abs/2609.03832v1 (the e-print is a gzipped tar archive; member FMG_v1.tex, 2259 lines, md5 8e03f8b9d7f104d07ebd07e00baad267):
  • the Brillouin zone, Eq. def:Brillouin, l. 328–332;
  • the bands, l. 354;
  • the gap assumption, Assumption assum:H, l. 356–392, inequality eqn:54 at l. 364;
  • the Bloch matrix, Eq. eqn:Hofst_Hamiltonian, l. 663–671;
  • the Fermi point, l. 674;
  • the Fermi energy and the numerical expectation, l. 686–690.
• The author list and the date are from the arXiv metadata; Crossref lists no journal version.