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arXiv 2403.03047cond-mat.mes-hallcond-mat.quant-gasmath-phmath.MP
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三维Hofstadter模型中Weyl点的临界磁通量

Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model

Pierpaolo Fontana, Andrea Trombettoni

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13 pages + 7 pages appendix/references, 9 figures
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AI总结：

本研究针对立方晶格上沿对角线加各向同性磁场的三维Hofstadter模型，通过动量空间约化精确对角化发现临界磁通量$Φ_c(m)$，揭示其两侧能带结构差异，给出大$m$极限下的渐近值与相关数学猜想。

摘要：

我们研究了立方晶格上三维Hofstadter模型的能带结构，所加各向同性磁场沿立方体对角线方向，磁通量为$Φ=2 π\cdot m /n$，其中$m,n$为互质整数。利用动量空间中的约化精确对角化方法，我们证明，在固定$m$的情况下，存在一个与特定磁通量值相关的整数$n(m)$，我们将该磁通量记为$Φ_c(m) \equiv 2 π\cdot m/n(m)$，它划分出两种不同的区域。第一种区域对应磁通量$Φ<Φ_c(m)$，其特征为能带完全交叠；第二种区域对应磁通量$Φ>Φ_c(m)$，其特征为态密度中存在孤立的能带接触点，且第$m$条与第$(m+1)$条能带之间存在Weyl点。在Hasegawa规范下，第$(m+1)$条能带的最小值在临界磁通量$Φ_c(m)$处会从$k_z=0$突变到$k_z=π$。我们进一步论证，$Φ_c(m)$在$m$趋于无穷大时的极限存在且为有限值：$\lim_{m\to \infty} Φ_c(m) \equiv Φ_c$。我们估算得到$Φ_c/2π=0.1296(1)$。基于对整数$m\leq60$所确定的$n(m)$值，我们提出了一个关于大$m$极限下$Φ_c(m)$形式的数学猜想。利用该猜想得到的渐近临界磁通量为$Φ_c^{\rm (conj)}/2π=7/54$。

英文摘要：

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 $Φ_c^{\rm (conj)}/2π=7/54$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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