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

TL;DR

The paper provides an exact treatment of the coupled twin-diamond chain, mapping it to an effective Ising model via decoration-iteration and solving with a $4\times4$ transfer matrix. It identifies five zero-temperature phases, including two extensively degenerate frustrated sectors that produce entropy plateaus, and derives the full phase diagram with explicit ground-state energies and degeneracies. The study reveals dual pseudo-critical temperatures: two distinct low-temperature crossovers between ordered and frustrated sectors, manifested as sharp but continuous changes in entropy and magnetization, and finite peaks in specific heat and susceptibility. These results offer a clear, exactly solvable framework for understanding how internal frustration and competing local configurations yield dual pseudo-critical scales in one dimension, with potential relevance to materials like Cu$_2$(TeO$_3$)$_2$Br$_2$.

Abstract

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 pseudo-transition behavior. The coupled twin-diamond chain thus provides an exactly solvable setting in which competing local configurations and internal frustration lead to pronounced dual pseudo-critical features in one dimension.

Figures (6)

• Figure 1: (Top) Schematic representation of the spin-1/2 CTDC. Black spheres denote dimer spins ($S_{a,k}$ and $S_{b,k}$), while blue spheres represent nodal Ising spins. Blue (red) lines indicate the couplings between dimer and nodal spins, associated with the exchange interactions $J_{1}$ ($J_{0}$), respectively, whereas the black line corresponds to the intra-dimer exchange interaction ($J_{2}$). (Middle) Illustration of the decoration transformation. (Bottom) Resulting effective spin chain with nearest- and next-nearest-neighbor interactions.
• Figure 2: Zero-temperature phase diagram and low-temperature entropy density for the CTDC model. The phase diagram is shown in the ($J_{2}/J_{1},B/J_{1}$) plane for fixed $J_{0}=-0.2$, and equal gyromagnetic factors $g_{0}=g_{1}$, so that the Zeeman fields satisfy $h_{0}=h_{1}=g_{0}\mu_{\mathrm{B}}B$. The entropy plot at $k_{{\rm B}}T/J_{1}=0.05$ displays the corresponding finite-temperature structure.
• Figure 3: Low-temperature phase diagram of the CTDC for $g_{0}\ne g_{1}$ under a uniform external field $B$, assuming fixe $J_{0}/J_{1}=-0.1$. (a) Dimer spins have weaker Zeeman response, $g_{1}/g_{0}=0.7$. (b) Dimer spins have stronger Zeeman response, $g_{1}/g_{0}=1.3$. The entropy plot at $k_{{\rm B}}T/J_{1}=0.05$ displays the corresponding finite-temperature structure.
• Figure 4: Density plot of the entropy in the $(\mu_{{\rm B}}g_{0}B/J_{1},\,k_{{\rm B}}T/J_{1})$ plane, for a weak ferromagnetic coupling $J_{0}$. (a) $J_{0}/J_{1}=-0.01$, $J_{2}/J_{1}=1.2$ and $g_{1}/g_{0}=0.7$; (b) Similarly for $J_{0}/J_{1}=-0.01$, $J_{2}/J_{1}=2.4$ and $g_{1}/g_{0}=1.3$.
• Figure 5: (a) Entropy as functions of temperature for $J_{2}/J_{1}=1.24$, $J_{0}/J_{1}=-0.01$, $\mu_{{\rm B}}g_{0}B/J_{1}=1.1$, and $g_{1}/g_{0}=0.7$.(b) Entropy as functions of temperature for $J_{2}/J_{1}=2.48$, $J_{0}/J_{1}=-0.01$, $\mu_{{\rm B}}g_{0}B/J_{1}=1$, and $g_{1}/g_{0}=1.3$. (c) Specific heat for the parameters in panel (a), with an inset showing $C(T)$ on a logarithmic temperature scale.(d) Specific heat for the parameters in panel (b), with an inset showing $C(T)$ on a logarithmic temperature scale.