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

TL;DR

The paper identifies nearly block-diagonal irreducible transfer matrices as the mechanism behind pseudotransitions in one-dimensional systems, where weak cross-sector couplings lift degeneracies without producing true nonanalyticities. It develops a general spectral framework centered on the minimal gap between leading eigenvalues and applies it to two models: the Doniach Ising chain with internal degeneracy and a mixed spin-1/2 spin-1 hexagonal nanowire. For the Doniach model it derives exact pseudo-critical temperatures and residual entropy at the interface, while for the nanowire it constructs a low-rank effective matrix to capture the crossover between quasi-ferromagnetic and quasi-core-ferromagnetic regimes. The results show that pseudotransitions are driven by spectral proximity and constrained coupling within irreducible transfer matrices, with the interface entropy lying between the adjacent sector entropies, offering a diagnostic for identifying such sharp but analytic crossovers in 1D systems.

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.

Figures (5)

• Figure 1: Schematic representation of the two leading eigenvalues$\lambda_{1}$ and $\lambda_{2}$ as a function of control parameter $x$ at fixed temperature $T_{p}$. The minimal gap $2w_{0}$ appears at $x=x_{p}$, where $w_{a}=w_{b}$ when $w_{0}\to0$.
• Figure 2: Surface plot of $\zeta(h,T)$ in the $h-T$ plane for $m_{\text{eff}}=2$ and $J=1$. The blue curve corresponds to the condition $\zeta_{h}=0$ [\ref{['eq:T_p']}], while the red curve follows $\zeta_{T}=0$ [\ref{['eq:T_p*-cond']}].
• Figure 3: Schematic representation of mixed spin-$1/2$ (red) and spin-1 (blue) hexagonal nanowire. Shell spins interact via $J_{s}$, core-shell via $J_{1}$, and core-core via $J_{c}$.
• Figure 4: Pseudo-critical temperature $T_{p}$ as a function of $D$, assuming zero magnetic field and fixed $J_{1}=J_{s}=J_{c}=1$. The curves correspond to different pseudotransition conditions: \ref{['eq:gen-Tp*']} (blue curve), \ref{['eq:gen-xp']} (red curve), the 0th-roder correction \ref{['eq:0th-cor-cond']} (pink curve), and 1st-order correction \ref{['eq:1st-corr-con']} (black curve). The dashed line shows the MFA result from Mendes.
• Figure 5: Leading eigenvalues of the transfer matrix as a function of $D$ at fixed temperature $T=0.5$. Solid curves represent exact numerical results: the red curve shows $\hat{\lambda}_{1}$, the blue curve shows $\hat{\lambda}_{2}$, and the orange curve corresponds to $\hat{\lambda}_{1'}$. Dashed lines indicate the first-order approximations $\hat{w}_{a}$ and $\hat{w}_{b}$ from the truncated matrix analysis.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3