Highly Entangled Quantum Spin Chains on Fermat's Spiral

Abstract

We investigate the entanglement entropy (EE) and spectral gap properties of highly entangled spin chains arranged along a Hamiltonian path on a two-dimensional (2D) lattice with geometries reminiscent of Fermat's spiral. Interpreting the interactions along the spin chain as the strongly anisotropic limit of a 2D model, with couplings oriented along different directions in different quadrants, we construct an exactly solvable ground state (GS) that exhibits volume scaling of EE across bipartition through the center in any direction. This provides another mechanism for realizing 2D GSs with local interactions that violate the entanglement area law. As in the previously studied coupled-chains paradigm, the new construction features an entanglement phase transition, but with distinct scaling at the critical point and in the weakly entangled phase, and a faster closing of the spectral gap in the highly entangled phase. The corresponding tensor network representation uses lower-rank tensors while preserving a global geometry similar to that of coupled-chains model. Finally, the Fermat-spiral layout naturally generalizes to two highly entangled 1D chains coupled by a quantum junction at the center of the 2D system.