Four interacting spins: addition of angular momenta, spin-spin correlation functions, and entanglement

TL;DR

This paper analyzes four spin-1/2 particles on a ring with competing Heisenberg couplings $J$ and $J_2=\alpha J$ using the addition of angular momenta to obtain an exact spectrum. It reveals ground-state evolution with $\alpha$, characterized by level crossings and distinct spin-spin correlation patterns, and shows that entanglement entropy depends crucially on how the system is partitioned into subsystems. By examining multiple bipartitions, the authors illustrate rich entanglement structures tied to the underlying ground states and identify signatures analogous to finite-size quantum critical behavior, including the Majumdar-Ghosh point at $\alpha_{\rm MG}=1/2$. The discussion extends to larger systems, highlighting finite-size scaling and connections to real materials, with implications for quantum information processing in small spin networks.

Abstract

We study four spins on a ring coupled through competing Heisenberg interactions between nearest neighbors, $J$, and next-nearest neighbors, $J_2\equivαJ>0$. The spectrum is obtained in a simple way by using the rules for addition of 4 angular momenta. This allows us to follow the evolution of the ground state with $α$, characterized by level crossings and by analyses of spin-spin correlation functions. Further insight is obtained by examining the entanglement between different parts of the system: we observe that the entanglement entropy is strongly dependent on how the system is partitioned.

Figures (5)

• Figure 1: A semiclassical representation of the 4 spins on a ring; $J$ and $J_2$ are the exchange coupling constants between spins on nearest (dotted lines) and next-nearest neighbor sites (full lines), respectively.
• Figure 2: Energy eigenvalues for the 4-site $J$-$J_2$ Heisenberg model as a function of $\alpha\equiv J_2/J$, with $J_2>0$. The curves are labelled by a simplified notation, $|S_Ts_{13}s_{24}\rangle$, since for each $S_T$ they are degenerate in $M$; $S_T$ is the total spin quantum number, and $s_{13}$ and $s_{24}$ are the quantum numbers specifying the partial sums. Still within this notation, we recall that the states $|110\rangle$ and $|101\rangle$ are degenerate. The energies of some states display a discontinuity at $\alpha=0$, which have been purposely smoothed for easier identification.
• Figure 3: Correlation functions for a 4-site ring in the $|S_TMs_{13}s_{24}\rangle$ states as functions of the distance between spins: (a) $|0000\rangle$, (b) $|0011\rangle$, and (c) $|2M11\rangle$; in the latter case, the quintuplet, they are degenerate in $M=-2,-1,0,1, 2$.
• Figure 4: Entanglement entropies as functions of $\alpha$: (a) $S(13)$, which measures entanglement between spins on different sublattices, (b) $S(12)$, which measures entanglement between dimers, and (c) $S(1)$, which measures entanglement between a single spin and the remaining ones. In all panels, the entropy depends on $|M|$ in the region $-1/4 < \alpha < 0$.
• Figure 5: (a) The phase diagram for the 4-site ring, where the kets follow the notation of Fig. \ref{['fig:Evsa4']}, and FM corresponds to the quintuplets $|2M11\rangle$; see text. (b) The phase diagram obtained from different numerical approaches allowing for extrapolations to the thermodynamic limit Tonegawa89Okamoto92Zinke09Majumdar69. IC stands for incommensurate spiral phase, FM for the saturated ferromagnetic phase, SL for a spin liquid, and D for dimerized; see text.