Reentrant Topological Phases Achieves Universal Class Invariance in Moire-Modulated SSH Model

Moiré physics presents exciting possibilities for progress in phase transition studies, yet the behaviour of reentrant phase transitions , where a system returns to a previous phase after transitioning , remains largely unexplored. Guo-Qing Zhang, L. F. Quezada, and Shi-Hai Dong, from Huzhou University and the Instituto Politécnico Nacional, investigate this phenomenon within the framework of the moiré-modulated extended Su-Schrieffer-Heeger model. Their research clarifies the sequence of reentrant transitions and confirms the consistency of universality class, offering analytical derivations for simplified scenarios and numerical calculations for more complex systems. This work establishes a crucial link between zero-energy edge modes, entanglement spectra, and changes in topological invariants, ultimately enhancing understanding of universal characteristics and bulk-boundary correspondence in one-dimensional condensed matter systems influenced by moiré patterns.

A wealth of opportunities exists for significantly advancing the field of quantum phase transitions, yet the properties of reentrant phase transitions driven by moiré strength remain poorly understood. This research investigates the reentrant sequence of phase transitions and the invariance of the universality class within the moiré-modulated extended Su-Schrieffer-Heeger (SSH) model. Considering a simplified scenario with intercell hopping set to zero, analytical derivations of the Hamiltonian parameters’ renormalization relations explain the observed reentrant phenomenon. For the general case, numerical phase boundaries are calculated in the thermodynamic limit, establishing a connection between zero-energy edge modes and entanglement.

Topological Phases and Bulk-Boundary Correspondence

Topological phases of matter are characterised by robust edge states that persist even with disorder or perturbations, and can be classified using topology and algebraic geometry. Examples include topological insulators, superconductors, and semimetals, with the bulk-boundary correspondence relating bulk material properties to surface states. Topological invariants are used to classify these phases, with the quantum Hall effect serving as a key experimental realisation of topological order. Chern numbers function as topological invariants for two-dimensional systems, and non-trivial topology can arise from symmetry-protected mechanisms.

Fractional quantum Hall states exhibit non-Abelian statistics, while topological phases can exist at finite temperatures and with interactions. The quantum spin Hall effect is a topologically protected state with gapless edge modes, and topological insulators possess robust surface states with a gapped bulk. The Haldane model demonstrates topological order in two dimensions, and symmetry-protected topological phases require specific symmetries for their existence. The quantum anomalous Hall effect represents a state with quantized transverse conductance without magnetic fields, highlighting the diverse manifestations of topological order.

Moiré Physics Reveals Reentrant Phase Transitions

Recent work has advanced understanding of phase transitions through moiré physics, but the behaviour of reentrant phase transitions driven by moiré strength remained unclear. Scientists investigated the reentrant sequence of phase transitions within a moiré-modulated extended Su-Schrieffer-Heeger (SSH) model, focusing on the invariance of the universality class. Analytical derivations of renormalization relations for Hamiltonian parameters, under the condition of zero intercell hopping, successfully explained the observed reentrant phenomenon. Numerical phase boundaries were then calculated in the thermodynamic limit, establishing a correlation between zero-energy edge modes and the entanglement spectrum, demonstrating a bulk-boundary correspondence through their degeneracy.

This signifies a connection between the system’s bulk properties and edge behaviour, crucial for topological phases. The research also established a relationship between the central charge, derived from entanglement entropy, and the change in winding number during the phase transition, providing a novel characterisation method. This breakthrough delivers a detailed understanding of universal characteristics and bulk-boundary correspondence for moiré-induced reentrant phase transitions in one-dimensional condensed-matter systems. The team demonstrated that the universality class remains invariant across the reentrant sequence, as determined by critical exponents obtained through finite-size analysis of the winding number and energy gap. Finite-size analysis solidified the robustness of the observed universality, and the moiré pattern universally induces reentrant topological phase transition sequences with winding numbers exceeding one, while maintaining the invariant universality class. The study employed rigorous numerical simulations of the extended SSH model, modulating the intracell hopping with a moiré pattern created by superposing two commensurate periods.

Moiré Strength Dictates Topological Phase Reentrance

This work details a comprehensive investigation into reentrant topological phase transitions within a moiré-modulated extended Su-Schrieffer-Heeger model, a one-dimensional condensed matter system. Researchers analytically derived renormalization relations for simplified conditions, explaining the reentrant phenomenon observed when varying moiré strength, and numerically calculated phase boundaries for more general cases. The study establishes a clear correspondence between bulk properties, specifically the winding number, and boundary characteristics, demonstrated through the degeneracy of both entanglement spectra and zero-energy edge modes. Calculations of central charge from entanglement entropy consistently align with changes in the winding number, further validating the connection between bulk and boundary properties. The authors acknowledge limitations stemming from the complexity of fully analytical solutions for all parameter values, particularly with non-zero intercell hopping. Future research could focus on experimentally realising this model using ultracold atoms in synthetic momentum lattices, with the potential to probe topological phases via quench dynamics and explore the feasibility of implementing long-range hopping terms.