Thermodynamic curvature for a two-parameter spin model with frustration

Microscopic models of realistic thermodynamic systems usually involve a number of parameters, not all of equal macroscopic relevance. We examine a decorated $(1+3)$ Ising spin chain containing two microscopic parameters: a stiff parameter $K$ mediating the long-range interactions, and a sloppy $J$ operating within local spin groups. We show that $K$ dominates the macroscopic behavior, with varying $J$ having only a weak effect, except in regions where $J$ brings about transitions between phases through its conditioning of the local spin groups with which $K$ interacts. We calculate the heat capacity $C_H$, the magnetic susceptibility ${\chi }_T$, and the thermodynamic curvature $R$. For large $|J/K|$, we identify four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic, according to the signs of $K$ and $J$. We argue that for characterizing these phases, the strongest picture is offered by the thermodynamic geometric invariant $R$, proportional to the correlation length $\xi$. This picture has correspondences to other cases, such as fluids.

Figure 1

The decorated $(1+3)$ Ising chain. The $N$ lattice basis elements, each consisting of a single Ising spin and a three Ising spin plaquette, are enumerated by an index $i\in \{1,\cdots ,N\}$, with periodic boundary conditions $(N+1)\leftrightarrow 1$. The plaquette spins are enumerated by an index $\alpha \in \{1,2,3\}$. Spins within a plaquette interact with each other via a parameter $J$, and with the single neighboring Ising spins via a parameter $K$.

Figure 2

The ground-state spin configurations, as a function of $J$ and for $H\downarrow 0$, for (a) $K=-1$ and (b) $K=+1$. $S$ denotes the saturated phase, $F_A$ and $F_B$ the ferrimagnetic phases, and $AF$ the antiferromagnetic phase. These spin configurations repeat over the entire lattice. $J=1$ marks the phase boundary for $K=\pm 1$.

Figure 3

The heat capacity $C_H$, the magnetic susceptibility ${\chi }_T$, and the thermodynamic curvature ${\xi }_R=-R/2$ as functions of $J$ for several values of $T$, and for the three distinct cases $K=-1,0,+1$. Cases in the plateau regimes, with $J$ not near 1, are commensurate with the appropriate superspin chains. The dots in (c) show $\xi$ for $T=8/10$, in good agreement outside the transition regime with the corresponding ${\xi }_R$. There are negative values of ${\xi }_R$ in Fig. 3 near $J=1$, which are omitted on the log scale.

Figure 4

(a) ${\xi }_R=-R/2$ and $\xi$ for the saturated phase $S$ at zero magnetic field, with $\{J,K\}=\{-2,-1\}$. The agreement between ${\xi }_R$ and $\xi$ is excellent even to regimes with ${\xi }_R$ less than about a lattice constant. (b) the corresponding quantities for the ferrimagnetic phase $F_B$ with $\{J,K\}=\{+2,-1\}$. The agreement between ${\xi }_R$ and $\xi$ is likewise excellent, except when ${\xi }_R$ has value a fraction of a lattice site.