Spectral mechanism and nearly reducible transfer matrices for pseudotransitions in one-dimensional systems

TL;DR

This paper introduces a spectral framework explaining pseudotransitions in one-dimensional systems, arising from nearly block-diagonal transfer matrices, with applications to models like the Ising chain and nanowire chains.

Contribution

It develops a general spectral approach to understand pseudotransitions caused by nearly reducible transfer matrices, with exact solutions and symmetry-based reductions for specific models.

Findings

Pseudotransitions are linked to eigenvalue competition in nearly block-diagonal transfer matrices.

Residual entropy at interfaces remains bounded, indicating sharp but analytic anomalies.

Effective low-rank matrices accurately capture crossover behaviors in complex models.

Abstract

While true phase transitions are forbidden in one-dimensional systems with short-range interactions, several models have recently been shown to exhibit sharp yet analytic thermodynamic anomalies that mimic thermal phase transitions. We show that this behavior arises from transfer matrices that are mathematically irreducible but possess a nearly block-diagonal structure due to the weak contribution of off-diagonal Boltzmann weights in the low-temperature regime. This results in weakly coupled competing sectors whose eigenvalue competition produces abrupt crossovers without nonanalyticity, a mechanism we term nearly block-diagonal irreducible. A key thermodynamic signature of such pseudotransitions is that the residual entropy at the interface remains bounded between the residual entropies of the competing sectors. We develop a general spectral framework to describe this behavior and apply it to two representative models: the Ising chain with internal degeneracy (Doniach model) and a hexagonal nanowire chain with mixed spin-1/2 and spin-1 components. In the first case, we derive exact expressions for the pseudo-critical temperature and residual entropy. In the second, we reduce the full $1458\times1458$ transfer matrix via symmetry decomposition and construct a low-rank effective matrix that accurately captures the crossover between quasi-ferromagnetic and quasi-core-ferromagnetic regimes. Our results demonstrate that pseudotransitions can be understood as spectral phenomena emerging from irreducible but functionally decoupled structures within the transfer matrix.