Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913601603764224 |
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| author | Fontana, Pierpaolo Trombettoni, Andrea |
| author_facet | Fontana, Pierpaolo Trombettoni, Andrea |
| contents | We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux $Φ=2 π\cdot m /n$, where $m,n$ are co-prime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed $m$, there exists an integer $n(m)$ associated with a specific value of the magnetic flux, that we denote by $Φ_c(m) \equiv 2 π\cdot m/n(m)$, separating two different regimes. The first one, for fluxes $Φ<Φ_c(m)$, is characterized by complete band overlaps, while the second one, for $Φ>Φ_c(m)$, features isolated band touching points in the density of states and Weyl points between the $m$- and the $(m+1)$-th bands. In the Hasegawa gauge, the minimum of the $(m+1)$-th band abruptly moves at the critical flux $Φ_c(m)$ from $k_z=0$ to $k_z=π$. We then argue that the limit for large $m$ of $Φ_c(m)$ exists and it is finite: $\lim_{m\to \infty} Φ_c(m) \equiv Φ_c$. Our estimate is $Φ_c/2π=0.1296(1)$. Based on the values of $n(m)$ determined for integers $m\leq60$, we propose a mathematical conjecture for the form of $Φ_c(m)$ to be used in the large-$m$ limit. The asymptotic critical flux obtained using this conjecture is $Φ_c^{\rm (conj)}/2π=7/54$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model Fontana, Pierpaolo Trombettoni, Andrea Mesoscale and Nanoscale Physics Quantum Gases Mathematical Physics We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux $Φ=2 π\cdot m /n$, where $m,n$ are co-prime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed $m$, there exists an integer $n(m)$ associated with a specific value of the magnetic flux, that we denote by $Φ_c(m) \equiv 2 π\cdot m/n(m)$, separating two different regimes. The first one, for fluxes $Φ<Φ_c(m)$, is characterized by complete band overlaps, while the second one, for $Φ>Φ_c(m)$, features isolated band touching points in the density of states and Weyl points between the $m$- and the $(m+1)$-th bands. In the Hasegawa gauge, the minimum of the $(m+1)$-th band abruptly moves at the critical flux $Φ_c(m)$ from $k_z=0$ to $k_z=π$. We then argue that the limit for large $m$ of $Φ_c(m)$ exists and it is finite: $\lim_{m\to \infty} Φ_c(m) \equiv Φ_c$. Our estimate is $Φ_c/2π=0.1296(1)$. Based on the values of $n(m)$ determined for integers $m\leq60$, we propose a mathematical conjecture for the form of $Φ_c(m)$ to be used in the large-$m$ limit. The asymptotic critical flux obtained using this conjecture is $Φ_c^{\rm (conj)}/2π=7/54$. |
| title | Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model |
| topic | Mesoscale and Nanoscale Physics Quantum Gases Mathematical Physics |
| url | https://arxiv.org/abs/2403.03047 |