Magnetic order and novel quantum criticality in the strongly interacting quasicrystals

TL;DR

This work studies the Hubbard model on two-dimensional quasicrystals (Penrose and Thue-Morse) to uncover how aperiodic geometry shapes quantum criticality under strong electronic interactions. Using sign-problem-free projector quantum Monte Carlo (PQMC) with a novel boundary-construction strategy and finite-size scaling, the authors show the Penrose lattice hosts Néel order for any $U>0$ due to a divergent DOS at the Fermi level, while the Thue-Morse lattice requires a finite critical interaction $U_c \u223c 2.32 t$ to onset magnetic order. Finite-size scaling yields a quantum critical point on the Thue-Morse lattice with exponents $ u=0.94(1)$, $eta=0.72(1)$, and $z=1.51(2)$, distinct from the conventional $(2+1)$D $O(3)$ universality class, signaling a novel universality class arising from the interplay of correlations and aperiodic geometry. The results additionally identify altermagnetic order on the Thue-Morse lattice, highlighting the unique magnetic phenomena possible in quasicrystals and motivating future field-theoretical formulations of quantum criticality in aperiodic environments.

Abstract

We present the sign-problem-free quantum Monte Carlo study of the half-filled Hubbard model on two-dimensional quasicrystals, revealing how specific aperiodic geometries fundamentally dictate quantum criticality. By comparing the Penrose and Thue-Morse quasicrystals, we demonstrate that the nature of the magnetic phase transition is controlled by the electronic density of states (DOS): while the singular DOS of the Penrose tiling induces magnetic order at infinitesimal interaction strengths, the Thue-Morse lattice requires a finite critical interaction to drive the transition. Crucially, through a novel boundary construction strategy and rigorous finite-size scaling, we identify a quantum critical point on the Thue-Morse quasicrystal with critical exponents ($ν\approx 0.94$, $β\approx 0.72$ and $z\approx 1.51$) that deviate significantly from the conventional $(2+1)$D Heisenberg $O(3)$ class. These findings establish the existence of a novel universality class driven by the interplay between electronic correlations and aperiodic geometry, challenging standard paradigms of magnetic criticality in two dimensions.

Figures (5)

• Figure 1: (a) Schematic representation of the two-dimensional Thue-Morse quasicrystal. (b) DOS for the Thue-Morse tight-binding model. Calculations were performed using the KPM on a supercell of $8,388,608$ sites. The DOS remains finite at the Fermi level ($E=0$). (c) Schematic representation of the two-dimensional Penrose quasicrystal. (d) DOS for the Penrose tight-binding model for a cluster of $N_s = 6,738,718$ sites. In contrast to the Thue-Morse case, a divergent peak is observed at the Fermi level.
• Figure 2: The Neel order on Penrose Quasicrystals. (a) Variation of the Binder ratio $U_L$ with $U$. The colors of the curves represent different system sizes $L$. In the range $U > 0$, $U_L$ systematically increases with increasing $L$ and gradually saturates. (b) Extrapolation of the squared order parameter $M_2$ with $1/L$. Different symbols correspond to different $U$ values, and the dashed line denotes the power function fit, where $f=ax^b+c$. Inset: Variation of $M_2$ in the thermodynamic limit with $U/t$, indicating the immediate emergence of non-zero magnetization for $U > 0$.
• Figure 3: (a) Finite-size scaling of the squared staggered magnetization $M_2$ plotted against the inverse system size $1/L$ for various interaction strengths $U$. The dash lines represent power function fit to the thermodynamic limit ($1/L \to 0$). Inset: The extrapolated $M_2$ in the thermodynamic limit, $M_{2}^{\infty}$, as a function of $U$. It shows a continuous onset of magnetic order at the critical point $U_c$. (b) The Binder ratio $U_L$ as a function of $U/t$ for different system sizes $L$. The crossing point of these curves identifies the critical interaction strength $U_c$.
• Figure 4: (a) Finite-size scaling data collapse of the squared staggered magnetization $M_{2}$. The rescaled $M_{2} L^{\frac{2\beta}{\nu}}$ is plotted against the scaling variable $(U-U_{c})L^{1/\nu}$. Using the critical exponents $\nu=0.94(1)$ and $\frac{2\beta}{\nu}=1.54(1)$, the data for different system sizes collapse onto a single universal curve. (b) Log-log plot of $M_{2}$ versus system size $L$ at the critical point $U_c = 2.32(1)$. The dash line represents a linear fit, yielding a slope of $\frac{2\beta}{\nu}=1.56(2)$, which is consistent with the result from the collapse analysis in (a). (c) Non-equilibrium finite-size scaling data collapse of $M_2$. Using the critical exponents $z=1.51(2)$ and $\frac{2\beta}{\nu}=1.53(2)$, the data for different system sizes collapse onto a single universal curve.
• Figure S1: Imaginary-time critical relaxation dynamics starting from the fully polarized Néel state on the Thue-Morse quasicrystal with $U=U_c$. (a) $U_L$ as a function of the projection imaginary time $\tau$ for different numbers of lattice sites $N_s$. (b) Data collapse for panel (a), with $z = 1.50(1)$. (c) Power-law decay of $M_2$ with $N_s=1800$ in the short-time stage, $M_2 \propto \tau^{-\frac{2\beta}{\nu z}}$, yielding $\frac{2\beta}{\nu z} = 1.01(3)$.