Electronic instabilities in Penrose quasicrystals: Competition, coexistence, and collaboration of order

Quasicrystals lack translational symmetry, but can still exhibit long-range order, promoting them to candidates for unconventional physics beyond the paradigm of crystals. Here, we apply a real-space functional renormalization group approach to the prototypical quasicrystalline Penrose tiling Hubbard model treating competing electronic instabilities in an unbiased, beyond-mean-field fashion. Our work reveals a delicate interplay between charge and spin degrees of freedom in quasicrystals. Depending on the range of interactions and hopping amplitudes, we unveil a rich phase diagram including antiferromagnetic orderings, charge density waves, and subleading, superconducting pairing tendencies. For certain parameter regimes, we find a competition of phases, which is also common in crystals, but additionally encounter phases coexisting in a spatially separated fashion and ordering tendencies which mutually collaborate to enhance their strength. We therefore establish that quasicrystalline structures open up a route towards this rich ordering behavior uncommon to crystals and that an unbiased, beyond-mean-field approach is essential to describe this physics of quasicrystals correctly.

Figure 1

Illustration of quasicrystal lattices. Penrose tiling, iterated four times, with the vertex model (left) and the center model (right). Nearest neighbors are marked as red bonds. The rhombi referred to in the main text are shown in gray for the center model. The lattice sites are marked as blue dots. The hoppings in the Hamiltonian are set to $t_0$ only if a red line connects the two sites in question.

Figure 2

Vertex model at $\mu =0.0$: Competition of order. We employ Eq. (7) for the interaction. The upper left plot visualizes the charge order parameter at $U^{\prime }=1.5$ and $U=1$. The lower left plot visualizes the spin order parameter (or magnetization pattern) at $U^{\prime }=0.5$ and $U=2.0$. On the right side, the critical scales depending on the nearest-neighbor interaction are shown for $U=3.0$ in the vicinity of the phase transition, to highlight the suppression of the critical scale upon approaching the phase transition. We find a competition of the two phases at the transition, similar to the behavior prototypically found in conventional crystals.

Figure 3

Center model at $\mu =2.0$: Coexistence of order. Gap values of the decoupled effective interaction at ${\mathrm{\Lambda }}_c$ for $U=1$ and $U^{\prime }=1.015$. Leading ordering strengths of the charge and spin ordering channels have approximately the same magnitude. The calculated gap magnitudes are shown in a red-blue color scheme for the spin gap, which is equivalent to the magnetization. In the violet-lime color scheme, the charge gap is shown. The two faces have no spatial overlap of magnitudes above ${10}^{-3}\max \left(\mathrm{\Delta }\right)$. The coexistence of the two phases is found to be a general feature of this phase transition.

Figure 4

Center model with exponentially decaying hopping amplitudes at $\mu =0.99$: Collaboration of order. In the upper left, the maximal eigenvalues of the pairing and charge channel relative to the one of the spin channel are shown for $U=0.3$. The expected ordering pattern for the spin channel is visualized in the upper right plot, the one for the charge channel in the lower left plot, and the one for the pairing channel in the lower right plot. We observe a slow decay of the relative maximal eigenvalues, pointing at a possible unconventional ground state. The ordering patterns of the three channels have spatial overlap, and therefore there is no spatial evasion of order.

Figure 5

Gap magnitudes of SDW and CDW in the center Penrose model at $\mu =2.0$, $U=1.0$, $U^{\prime }=1.015$, and $T={10}^{-3}$ without fixed particle number. It compares well to the gap predicted by the fixed particle number calculation. The calculated gap magnitudes are shown in a red-blue color scheme for the spin gap, which is equivalent to the magnetization. In the violet-lime color scheme, the charge gap is shown. The two faces have no spatial overlap of magnitudes above ${10}^{-3}\max \left(\mathrm{\Delta }\right)$.

Figure 6

Average on-site and nearest-neighbor interaction depending on the distance defined by the real-space positions to the center in the vertex-type Penrose model.

Figure 7

Noninteracting DoS for the three types of Penrose models used. Left: the vertex-type Penrose model with $21106$ sites with the $\delta$-like peak at $\mu =0$. Middle: the center-type model with hoppings defined using the graph; the main peak is at $\mu =2.0$. Right: the center-type model with exponential decaying hoppings, with three main peaks. In both center-type models, the lattice contains $20800$ sites.

Figure 8

Phase diagram of the vertex Penrose model employing Eq. (C2). We find a SDW (star marker) and a CDW (triangular marker) phase. The critical scales are given by the color code.

Figure 9

The upper left plot shows the charge order parameter at $U^{\prime }=1.5$ and $U=1$. The lower left plot visualizes the spin order parameter (or magnetization pattern) at $U^{\prime }=0.5$ and $U=2.0$. The middle plot shows the critical scales depending on the nearest-neighbor interaction for $U=3.0$ in the vicinity of the phase transition, to highlight the suppression of the critical scale upon approaching the phase transition. The right plot shows the phase diagram of the vertex model with a SDW (star marker) and a CDW (triangular marker) phase; the phase diagrams for the pure case and the one employing an envelope differ only very slightly.

Figure 10

Phase diagram of the center model at different chemical potentials: $\mu =1.9$ (upper left), $\mu =2.0$ (upper middle), $\mu =2.1$ (upper right), $\mu =2.3$ (lower left), $\mu =2.35$ (lower middle), and $\mu =2.4$ (lower right). $U$ and $U^{\prime }$ are varied at $T={10}^{-3}$; nearest neighbors are included in the calculation.

Figure 11

Ordering predictions for some chosen data points at each chemical potential: $\mu =1.9$ (upper left), $\mu =2.0$ (upper middle), $\mu =2.1$ (upper right), $\mu =2.3$ (lower left), $\mu =2.35$ (lower middle), and $\mu =2.4$ (lower right), from FRG plus postproduction mean-field theory at chosen data points. $U$ and $U^{\prime }$ are varied at $T={10}^{-3}$; nearest neighbors are included in the calculation.

Figure 12

Critical scales depending on the interaction strength in the center model with exponential hoppings at $T={10}^{-3}$ including all sites with a distance $\le 3a$ in the calculation.

Figure 13

Orderings at varying chemical potentials and interaction strengths, where the left column show orderings for $U=0.3$ and the right one shows orderings for $U=3.0$, and the chemical potential is shown from the lowest ($\mu =0.83$) (top row) to the highest ($\mu =1.23$) (bottom row), for the physical center model. Calculations are performed at $T={10}^{-3}$ using a frequency cutoff including all sites with a distance $\le 3a$ in the calculation. The color map is diverging, and thus blue and red mark different signs; we normalized the gap to $\pm 1$ and applied a logarithmic scale.