Theory of Hofstadter superconductors

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 $\mathrm{\Phi }=(p/q){\mathrm{\Phi }}_0$ comparable to the flux quantum ${\mathrm{\Phi }}_0=h/e$. 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 $\mathrm{\Phi }=(p/q){\mathrm{\Phi }}_0$ 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.

Figure 1

Brillouin zone and Fermi surfaces for a square lattice for (a) even and (b) odd $q$, and (c) the Hofstadter bands for the original Hofstadter tight binding model on a square lattice [1, 5, 104] for $q=3$. A single band in the absence of the magnetic field splits into $q$ bands, $q$-fold symmetric under translations by $\mathbf{Q}$. The red contours indicate the Fermi level that we take to be at $E=0$. The Brillouin zone is folded along the $k_x$ direction by a factor of $q$ relative to the Brillouin zone in the absence of the magnetic field. Due to the ${\widehat{T}}_1$ magnetic translation symmetry, the band structure repeats $q$ times along the $k_y$ direction. As a result, there are $q$ copies of a Fermi surface centered at momenta ${\mathbf{Q}}_{\ell }=\ell \mathbf{Q}$. The interactions are projected onto the single band that crosses the Fermi level.

Figure 2

Numerical solutions of uniform GL equations $|{\eta }_L|$ for (a) $q=5$ and (b) 7 as a function of ${\beta }_{22}$ treated as a variable parameter with ${\beta }_{23}=2, {\beta }_{12}=1.7$ for $q=5$ and ${\beta }_{12}=1.6$ for $q=7$, and all other ${\beta }_{MN}=1$. Numbers indicate the number of equal $|{\eta }_L|$ on each segment. For $q=5$, we find four first-order phase transitions between four kinds of phases: a ${\widehat{T}}_2$ symmetric phase with a single nonzero ${\eta }_L$ (blue); a ${\mathbb{Z}}_5$ symmetric phase with all $|{\eta }_L|$ equal (red); a phase with $\mathcal{I}$ symmetry with pairs of equal $|{\eta }_L|$ except for one (yellow); and a phase with no symmetries where none of the $|{\eta }_L|$ are equal (green). For $q=7$, we find a second-order phase transition between a ${\mathbb{Z}}_7$ symmetric phase with all $|{\eta }_L|$ equal (red) and a $\mathcal{I}$ symmetric phase with pairs of equal $|{\eta }_L|$ except for one (yellow).

Figure 3

Phase diagrams for the spin-polarized Hofstadter SC at $q=3$ in the ${\widehat{T}}_2{\widehat{T}}_1$ symmetric phase. (a) Full phase diagram in the space of $a_0, a_1$ and $b_1$, with $\vartheta =0$. The $C=-1$ and $C=-3$ phases are separated by a conical phase boundary. On the $b_1$ axis, the system is gapless and has a symmetry protected Bogoliubov Fermi surface (BFS). (b) Cut along $b_1=1$ with $\vartheta =0$. (c) Same as (b) but with $\vartheta =-\pi /6$. The phase boundary is a circle in this case.

Figure 4

BdG spectrum for $q=3$ with ${\mathrm{\Delta }}_0\left(\mathit{p}\right)=\sqrt{3}{(p_x+i{p}_y)}^3/p_F^3, p_F=10$ and ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=1$. We work in arbitrary units for the Fermi momentum and consider $p_F\ll \pi /a$. 6 Dirac nodes indicate a topological phase transition with Chern number changing from $C=-9$ to $C=-3$. (a) A cut of the BdG spectrum along the $\widehat{p}$ direction at the Dirac node. (b) A cut of the BdG spectrum along the Fermi momentum $p=p_F=10$ as a function of the angle $\theta$ of $\mathbf{p}=p_F(cos\theta ,sin\theta )$, which shows the presence of 6 Dirac touchings. Note also the unavoided crossings at nonzero energies indicative of the presence of the MTG symmetry $\tilde{T}$.

Figure 5

The phase diagrams for $q=5$ with ${\mathrm{\Delta }}_0=a_0(p_x+i{p}_y)/p_F,{\mathrm{\Delta }}_1=a_1(p_1+i{p}_2)/p_F+b_1,{\mathrm{\Delta }}_2=0, {\mathrm{\Delta }}_3\left(\mathbf{p}\right)=-{\mathrm{\Delta }}_2(-\mathbf{p})$ and ${\mathrm{\Delta }}_4\left(\mathbf{p}\right)=-{\mathrm{\Delta }}_1(-\mathbf{p})$. Cuts of the nested conical phase boundaries and Bogoliubov Fermi surface are shown along (a) $b_1=1$ and (b) $a_1=0$. (c) Bogoliubov Fermi surface ($E=0$) located at $\left|\mathit{p}\right|=10$ is shown in red for ${\mathrm{\Delta }}_0=0,{\mathrm{\Delta }}_1=1,\text{and}{\mathrm{\Delta }}_2=0$.