Interaction geometry and ground-state properties of sparse quantum lattice models

Abstract: We investigate how interaction geometry shapes the low-energy phases of sparse tunable long-range quantum models. We focus on a class of graphs whose degree grows logarithmically with system size, and show how symmetry and frustration in graph connectivity can drive, suppress, and reshape ground-state phase transitions. The central examples are power-of-$p$ graphs, where even and odd values of $p$ exhibit qualitatively distinct behaviour: even-$p$ graphs inherit the rich phase structure of the power-of-two model, while odd-$p$ graphs are governed by geometric frustration. Fibonacci graphs provide a contrasting case, lacking the discrete self-similarity of the power-of-$p$ family but exhibiting a direct geometric mapping between the short- and long-range limits. Across our models, we find that phase structure and criticality are governed by the same effective-geometry principle, unifying our framework for experimentally motivated long-range quantum systems.

• The paper introduces a geometry-driven framework that links tunable sparse interaction graphs to distinctive ground-state phenomena in quantum lattice models.
• It employs analytical and numerical techniques including exact diagonalization, Monte Carlo simulations, and Krylov methods to reveal phase transitions and gap closures.
• The study highlights potential applications in quantum simulation platforms and p-adic mapping for controlling non-local interactions in many-body systems.

Interaction Geometry and Ground-State Phenomena in Sparse Quantum Lattice Models

Sparse Long-Range Graphs: Structural Foundations

The study initiates with a generalized approach to tunable sparse long-range graphs, particularly focusing on families with logarithmic degree growth: power-of-$p$ and Fibonacci graphs. Each site in a system of $N$ nodes couples to $O(\log N)$ partners, yielding sparse, yet non-local, interaction structures. The power-of-$p$ graphs possess discrete scale self-similarity; bonds exist at distances $p^l$, enforcing recursive connectivity. In contrast, Fibonacci graphs break this symmetry, defining bonds at Fibonacci-separated distances, generating a less regular but equally sparse structure.

The geometric perspective is central: as the distance exponent $s$ is varied, the effective connectivity of these graphs morphs continuously, yielding distinct physical regimes and determining many-body ground-state structure. For power-of-$p$ graphs, $s$ deformation interpolates between local ring-like and “tambourine”-style connectivity, sparse all-to-all structures, and ultimately decoupled clusters.

Figure 1: Continuous deformation of interaction geometry for power-of-$p$ and Fibonacci graphs as the exponent $s$ is tuned.

Many-Body Hamiltonian Models and Analytical Approaches

Sparse graphs serve as the backbone for two quantum lattice models: antiferromagnetic transverse-field Ising (TFIM) and spinless fermion hopping models. Both exhibit interactions scaling as $N$0, with $N$1 modulating the dominance of short vs. long-range bonds. Analytical, exact diagonalization (ED), Monte-Carlo (MCMC) simulations, and Krylov/SQKD diagonalization techniques are employed for spectrum and ground-state analysis.

The TFIM Hamiltonian, $N$2 for spin-$N$3, fundamentally links classical frustration and quantum fluctuation. The fermionic Hamiltonian, $N$4, is quadratic and solved exactly via Fourier-Bogoliubov transformations.

Classical and Quantum Phase Structure on Sparse Graphs

Power-of-$N$5 Graphs

The classical ground-state phase structure for even $N$6 is dominated by recursive antiferromagnetic configurations. As $N$7 increases, criticality shifts: $N$8, the classical gap closing, approaches $N$9 as $O(\log N)$0 grows. This is verified numerically for $O(\log N)$1 via MCMC, and analytically by geometric recursion arguments (see Fig. 2, Fig. 8).

Figure 2: Classical phase diagrams for power-of-two, power-of-three, and Fibonacci graphs; critical transitions and gapless regimes are resolved with energy gap analysis.

Even-$O(\log N)$2 ground-states are constructed by iteratively doubling the system and concatenating globally flipped blocks (Fig. 3).

Figure 3: Recursive AFM construction for power-of-$O(\log N)$3 graphs: block concatenations yield ground-state patterns as system size is scaled.

Odd $O(\log N)$4 introduces geometric frustration: each shell ($O(\log N)$5) decomposes the ring into odd-length cycles, which prohibit full AFM satisfaction (Fig. 4). The AFM state frustrates exactly one bond per cycle, yielding a minimal frustration configuration that is proven to be globally optimal. For large positive $O(\log N)$6, degeneracies emerge as short-range costs vanish, leading to a gapless extensive manifold.

Figure 4: Minimal frustration for AFM patterns on power-of-three graphs; every odd cycle is frustrated exactly once.

Quantum phase diagrams retain the primary features of the classical landscape, but quantum fluctuations mix states and reshape transitions. For even $O(\log N)$7, rich universality classes emerge, including 2D Ising behavior at critical points. Odd $O(\log N)$8 (exemplified by $O(\log N)$9) remains robustly AFM for $p$0, transitioning to volume-law scaling as $p$1 increases (Fig. 5).

Figure 5: Quantum phase diagrams for sparse graphs show the interplay of AFM, paramagnetic, and Fibonacci AFM phases; entanglement scaling is annotated.

Fibonacci Graphs

Fibonacci graphs lack self-similarity, resulting in fundamentally mirrored geometric transitions across $p$2. For even Fibonacci $p$3, both nearest-neighbor and furthest-neighbor cycles are odd, ensuring unique AFM ground states and absence of classical phase transitions. Odd $p$4 instead exhibit a $p$5 transition between nearest-neighbor AFM and Fibonacci AFM phases, with gapless regimes at the transition point.

Effective interaction geometry remains ring-like for all limits, as verified through correlation analysis and geometric mapping (Fig. 7).

Figure 6: Effective geometry of Fibonacci graphs with variation in system size; dominant interaction bonds change the loop structure according to $p$6.

Exactly Solvable Fermionic Regime and Gap Scaling

The fermionic hopping model is solved for power-of-$p$7 graphs. Analytic gap closing is observed at $p$8, confirmed by numerical spectrum convergence for large $p$9 (Fig. 6). For positive $p^l$0, a second gapless regime emerges between $p^l$1 and $p^l$2, with gap scaling algebraically toward zero with increasing $p^l$3. Odd $p^l$4 never show analytic mode closing at $p^l$5, but numerical dispersion computation reveals gapless behavior for $p^l$6.

Figure 7: Spectral gap and coupling-function for PWR2 models: analytic gap closing at $p^l$7 and numerical gapless regime for $p^l$8.

Robustness and Sensitivity to System Size

Ground-state and spectral properties are highly sensitive to system size, particularly for $p^l$9 where dominant interaction cycles depend on $s$0. For Fibonacci graphs, varying $s$1 fragments the graph into disconnected loops or preserves ring structure, qualitatively impacting ground-state degeneracies (Fig. 7). For fermionic models, density-density correlators reveal pronounced peaks at graph-connected distances for positive $s$2, mirroring changes in loop connectivity and corroborating the geometric framework.

Conclusion

This work systematically demonstrates that sparse long-range quantum graphs exhibit ground-state phase behavior governed by their effective geometry. Recursive or frustrated connectivity in power-of-$s$3 graphs leads to two major qualitative classes, with power-of-two models showing maximal structural complexity. Fibonacci graphs, lacking self-similarity, manifest mirrored phase diagrams and transitions, providing a symmetric case for geometry-first analysis. Robust numerical and analytical treatment reveals strong sensitivity to system size, particularly for long-range regimes, where even a single-site shift can transform phase structure via changes in $s$4.

The implications for quantum simulation architectures are substantial: neutral-atom arrays and cavity-QED platforms can engineer these sparse geometries, enabling controlled exploration of distinct universality classes, degeneracies, and integrability properties. The $s$5-adic connection suggests future prospects for holographic mapping and tensor network investigations. Extending the geometry-driven framework beyond equilibrium, including dynamical phases and quantum information spreading, remains a promising research direction (2606.20387).