Hofstadter超导体理论
Theory of Hofstadter Superconductors
摘要：
我们研究平均场状态导致的电子配对的时间反演破碎分形霍夫斯塔德带，这出现在二维晶格中的单位单元陷阱磁通量$Φ= (p/q)Φ_0$可比的通量量子$Φ_0 = h/e$。它建立了由电荷2 e配对场提供的磁平移群（MTG）的不可约表示的维数和简并度具有不同于由单粒子Bloch态提供的性质，特别是被证明依赖于分母$q$的奇偶性。我们探索这种对称性分析，制定一个Ginzburg-Landau理论描述的Hofstadter超导体的热力学性质，在任意合理的通量$Φ= (p/q)Φ_0$的多组分序参量，描述了有限的动量配对的电子在不同的费米表面补丁。这种唯象理论导致了丰富的相图，其特征在于不同的对称性破缺模式的MTG，这可以被解释为不同类别的涡旋晶格。鉴定了一类$\mathbb{Z}_q$对称的Hofstadter SC，其中MTG分解为$\mathbb{Z}_q$亚组。我们研究了这种$\mathbb{Z}_q$对称的Hofstadter SC的拓扑性质，并证明了$q$的奇偶性决定了陈数的奇偶性.我们确定了实现Bogoliubov费米表面的奇偶校验和MTG对称性的存在下，建立一个新的拓扑不变量捕捉这种电荷中性的无隙激发的存在条件。我们的研究结果，这可能承担相关的描述重入超导电性的莫尔系统在霍夫施塔特政权，建立霍夫施塔特SC作为一个肥沃的设置，探索对称性破缺和拓扑秩序。
英文摘要：
We study mean-field states resulting from the pairing of electrons in time-reversal broken fractal Hofstadter bands, which arise in two-dimensional lattices where the unit cell traps magnetic flux $Φ= (p/q)Φ_0$ comparable to the flux quantum $Φ_0 = h/e$. It is established that the dimension and degeneracy of the irreducible representations of the magnetic translation group (MTG) furnished by the charge 2e pairing fields have different properties from those furnished by single particle Bloch states, and in particular are shown to depend on the parity of the denominator $q$. We explore this symmetry analysis to formulate a Ginzburg-Landau theory describing the thermodynamic properties of Hofstadter superconductors at arbitrary rational flux $Φ= (p/q)Φ_0$ in terms of a multicomponent order parameter that describes the finite momentum pairing of electrons across different Fermi surface patches. This phenomenological theory leads to a rich phase diagram characterized by different symmetry breaking patterns of the MTG, which can be interpreted as distinct classes of vortex lattices. A class of $\mathbb{Z}_q$-symmetric Hofstadter SCs is identified, in which the MTG breaks down to a $\mathbb{Z}_q$ subgroup. We study the topological properties of such $\mathbb{Z}_q$-symmetric Hofstadter SCs and show that the parity of the Chern numbers is fixed by the parity of $q$. We identify the conditions for the realization of Bogoliubov Fermi surfaces in the presence of parity and MTG symmetries, establishing a novel topological invariant capturing the existence of such charge-neutral gapless excitations. Our findings, which could bear relevance to the description of re-entrant superconductivity in moiré systems in the Hofstadter regime, establish Hofstadter SC as a fertile setting to explore symmetry broken and topological orders.
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