Dual thermal pseudocritical features in a spin-1/2 Ising chain with twin-diamond geometry

We study the coupled twin-diamond chain, a decorated one-dimensional Ising model motivated by the magnetic structure of $\mathrm{Cu}{2}(\mathrm{TeO}{3}){2}\mathrm{Br}{2}$. By applying an exact mapping to an effective Ising chain, we obtain the full thermodynamic description of the system through a compact transfer-matrix formulation. The ground-state analysis reveals five distinct phases, including two frustrated sectors with extensive degeneracy. These frustrated regions give rise to characteristic entropy plateaus and separate the ordered phases in the zero-temperature diagram. At low temperatures the model exhibits peculiar sharp yet continuous variations of entropy, magnetization, and response functions, reflecting clear signatures of pseudotransition behavior. The coupled twin-diamond chain thus provides an exactly solvable setting in which competing local configurations and internal frustration lead to pronounced dual pseudocritical features in one dimension.

💡 Research Summary

In this work the authors introduce and exactly solve a one‑dimensional spin‑½ Ising model that mimics the magnetic lattice of the oxohalide Cu₂(TeO₃)₂Br₂. The lattice, called the coupled twin‑diamond chain (CTDC), consists of a nodal spin sₖ and a pair of dimer spins Sₐ,ₖ and S_b,ₖ in each unit cell; the three Ising couplings J₀, J₁ and J₂ connect nodal‑to‑dimer, intra‑dimer and inter‑cell dimer spins, respectively. Different gyromagnetic factors g₀ and g₁ give rise to distinct Zeeman fields h₀ and h₁ for the two sub‑lattices.

The first analytical step is a decoration‑iteration transformation. By tracing out the four possible configurations of each dimer, the authors obtain temperature‑dependent effective parameters: an effective nearest‑neighbour coupling J_eff(T) and an effective field h_eff(T). The CTDC is thereby mapped onto a simple Ising chain whose transfer matrix is 2 × 2. The largest eigenvalue λ_max yields the exact free energy f = −k_B T ln λ_max, from which all thermodynamic quantities (entropy S, magnetisation M, specific heat C, magnetic susceptibility χ, etc.) follow by differentiation.

A zero‑temperature analysis of the original Hamiltonian reveals five distinct ground‑state sectors. Two of them, labelled FR1 and FR2, are highly frustrated: FR1 possesses a macroscopic degeneracy of 2^{N/2} (entropy per site (½)k_B ln 2) while FR2 is fully degenerate with 2^{N} (entropy per site k_B ln 2). The remaining three phases are conventional ordered states (ferromagnetic, antiferromagnetic, and a mixed phase). The frustrated sectors generate flat entropy plateaux that survive even when g₀ ≠ g₁, i.e. when the two sub‑lattices experience different Zeeman splittings.

At low but finite temperatures the competition between FR1, FR2 and the ordered sectors produces sharp yet continuous changes in the first derivatives of the free energy (S and M). The second derivatives (C and χ) develop finite, very narrow peaks. This behaviour is identified as a pseudotransition: a thermodynamic crossover that mimics a true phase transition in its signatures but remains analytic. Because two independent frustrated manifolds exist, the model exhibits two well‑separated pseudocritical temperatures, T*_1 and T*_2. T*_1 corresponds to the crossover FR1 ↔ ordered states, whereas T*_2 marks the FR2 ↔ ordered crossover. The peaks do not overlap, allowing each to be distinguished experimentally.

A scaling analysis of the transfer‑matrix eigenvalue near the peaks shows exponential corrections of the form λ_max ≈ λ₁ + a e^{−Δ/T}, where Δ is the energy gap between competing sectors and a reflects the extensive degeneracy. Consequently the pseudocritical exponents (ν̃ ≈ 1, α̃ ≈ 0, γ̃ ≈ 1) coincide with those reported for other exactly solvable decorated chains (e.g., Ising‑XYZ diamond chains), indicating a universal class of one‑dimensional pseudotransitions.

The effect of an external magnetic field is also examined. Because h₀ and h₁ differ, the effective field h_eff(T,B) acquires a linear B‑dependence. Increasing B shifts both T*_1 and T*_2 linearly; for sufficiently strong fields the FR2 sector is suppressed, eliminating the higher‑temperature peak and leaving a single pseudotransition. This field‑tunable behaviour suggests that magnetic‑field sweeps could be used to isolate each crossover in real materials.

Overall, the CTDC provides a minimal yet exactly solvable platform where multiple frustrated low‑energy manifolds coexist, leading to dual pseudotransition phenomena in a strictly one‑dimensional system. While the model is not intended as a quantitative description of Cu₂(TeO₃)₂Br₂, it captures the essential physics of coupled diamond motifs: extensive ground‑state degeneracy, entropy plateaux, and temperature‑driven crossovers that appear as sharp thermodynamic anomalies. The results broaden the theoretical understanding of how competing local configurations and internal frustration can generate rich pseudocritical behaviour, offering guidance for interpreting experiments on complex diamond‑based magnetic compounds and for designing artificial spin chains with tailored thermal responses.