Quasi-Solid and Supersolid from Quasiperiodic Long-Range Interactions

TL;DR

This work demonstrates that purely interaction-driven quasiperiodicity can stabilize novel quantum phases in a one-dimensional hard-core boson system without external disorder or on-site potentials. By engineering Fibonacci-modulated long-range interactions $V_{ij} = V \cos(\pi \alpha i)\cos(\pi \alpha j)$ with $\alpha = (\sqrt{5}-1)/2$ and using large-scale quantum Monte Carlo with the worm algorithm, the authors map a phase diagram featuring incompressible quasi-solid lobes at fillings $\langle n \rangle = 1/\alpha$ and $1-1/\alpha$, and an intermediate quasi-supersolid region where density order coexists with finite superfluid density. Finite-size scaling confirms the robustness of these states in the thermodynamic limit, with Bragg-like peaks at Fibonacci-related wavevectors $q^*$ and multiple harmonics reflecting the underlying quasiperiodic spectrum. The results provide a new mechanism for translational symmetry breaking with quantum coherence and point to experimental platforms—such as Rydberg arrays, cavity QED, and programmable quantum simulators—that can realize Fibonacci-like interaction kernels to explore quasi-solid and quasi-supersolid phases.

Abstract

We investigate hard-core bosons in one dimension with quasiperiodic long-range interactions defined by V_ij = V0 cos(pi * alpha * i) cos(pi * alpha * j), where alpha = (sqrt(5) - 1)/2 is the inverse golden ratio. Large-scale quantum Monte Carlo simulations reveal incompressible density plateaus at incommensurate fillings tied to Fibonacci ratios. These plateaus feature emergent nonuniform density profiles and robust long-range correlations, as captured by the structure factor. Depending on filling and interaction strength, the system realizes either a quasi-solid phase with suppressed superfluidity, a quasi-supersolid phase where density order coexists with finite superfluid density, or a superfluid phase. Our results demonstrate that purely interaction-induced quasiperiodicity, without external potential or disorder, can stabilize novel quantum phases that simultaneously break translational symmetry and sustain quantum coherence.

Figures (7)

• Figure 1: Ground-state phase diagram of the system in the $\mu/V$–$t/V$ plane for $L=233$. Two incompressible quasi-solid (qS) lobes emerge near fillings $\langle n \rangle =0.618 \approx 1/\alpha$ and $\langle n \rangle =0.382 \approx 1 - 1/\alpha$, where $\alpha = (1+\sqrt{5})/2$ is the golden ratio. These phases exhibit vanishing compressibility and superfluid density, along with sharp incommensurate peaks in $S(q)$, indicating Fibonacci-induced spatial order. The light blue region marks the quasi-supersolid (qSS) regime, where incommensurate density modulations coexist with finite superfluid density.
• Figure 2: Average filling $\langle n \rangle$ (green circles), superfluid density $\rho_s$ (red squares), and structure factor at the dominant incommensurate peak $S(q^*)$ (purple upward trangules) as functions of chemical potential $\mu/V$ for $L=233$ at $t/V=0.5$. Two prominent incompressible plateaus appear at irrational fillings $\langle n \rangle = 0.618$ and $\langle n \rangle =0.382$, corresponding to the golden-ratio fractions $1/\alpha$ and $1 - 1/\alpha$ with $\alpha = (1+\sqrt{5})/2$. These densities are naturally commensurate with the underlying Fibonacci-modulated interaction pattern and host quasi-solid order, characterized by vanishing $\rho_s$ and enhanced $S(q^*)$. Between these plateaus, finite $\rho_s$ coexists with nonzero $S(q^*)$, signaling the emergence of a quasi-supersolid phase stabilized purely by quasiperiodic long-range interactions. Error bars are within symbols if not seen in the figure.
• Figure 3: (a) Site-resolved average density profile $\langle n_x \rangle -\langle n \rangle$ along a one-dimensional chain of length $L=233$, with total particle number $N=144$, corresponding to filling $\langle n \rangle=0.618 \approx 1/\alpha$, where $\alpha = (1+\sqrt{5})/2$ is the golden ratio at $t/V=1/3$. The quasiperiodic long-range interaction follows a Fibonacci modulation pattern, leading to a quasi-solid phase characterized by incommensurate density modulations, vanishing superfluid density, and zero compressibility. (b) Zoom-in of sites 1–54 of the same system. The circle radii are proportional to the local density at each site, emphasizing the incommensurate spatial modulation.
• Figure 4: (a) Density–density correlation function $C(x)$ at $t/V=1/3$ for three representative fillings on a system of size $L=233$: $\langle n \rangle=0.618$ (red circles), $\langle n \rangle=0.694$ (blue squares), and $\langle n \rangle =0.780$ (green upward triangles). Red data correspond to the inverse golden ratio $\langle n \rangle =0.618 \approx \alpha^{-1}$, at which quasiperiodic modulation is most pronounced. Fitting curves are superimposed on the data. (b) Corresponding structure factor $S(q)$ for each filling, displaying prominent peaks at incommensurate wavevectors $q_1 = 2.398$ and $q_2 = 3.882$ that reflect the dominant Fourier components of the Fibonacci pattern. Inset: finite-size scaling of the dominant peak height $S(q^*)$ at $q^*=2.398$ for $\langle n \rangle= 0.618$ with the fitting function as $S(q^*)(L)= 0.126991 + 14.4886\,\left(\frac{1}{L}\right)^{1.45515}$, which confirms the persistence of quasi-long-range order in the thermodynamic limit.
• Figure 5: (a) Average filling $\langle n \rangle$, superfluid density $\rho_s$, and structure factor $S(q^*)$ as functions of the chemical potential $\mu/V$ for $t/V = 2$ at system size $L = 233$. The system is predominantly superfluid, exhibiting finite $\rho_s$, while $S(q^*)$ remains nearly zero across most of the chemical potential range. A weak enhancement of $S(q^*)$ ($\max S(q^*) \approx 0.035$) occurs near the golden-ratio fillings $\langle n \rangle = 0.382$ and $\langle n \rangle = 0.618$, indicating the quasiperiodic order. (b) Corresponding data for $t/V = 1/3$. In this regime, well-defined incompressible plateaus appear at $\langle n \rangle= 0.382$ and $\langle n \rangle = 0.618$, accompanied by enhanced $S(q^*)$, marking the emergence of stable quasi-solid phases. Between the plateaus, $\rho_s$ remains finite while $S(q^*)$ stays nonzero, signifying a quasi-supersolid region where phase coherence coexists with incommensurate density modulation.