The transition from a gapless liquid to a gapped dimerized ground state that occurs in the frustrated antiferromagnetic Majumdar-Ghosh (or ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg) model is revisited from the point of view of entanglement. We study the evolution of entanglement spectra, a ``projected subspace'' block entropy, and concurrence in the Schmidt vectors through the transition. The standard tool of Schmidt decomposition along with the existence of the unique Majumdar-Ghosh (MG) point where the ground states are degenerate and known exactly suggest the projection into two orthogonal subspaces that is useful even away from this point. Of these, one is a dominant five-dimensional subspace containing the complete state at the MG point and the other contributes marginally, albeit with increasing weight as the number of spins is increased. We find that the marginally contributing subspace has a minimum von Neumann entropy in the vicinity of the dimerization transition. Entanglement content between pairs of spins in the Schmidt vectors, studied via concurrence, shows that those belonging to the dominant five-dimensional subspace display a clear progress towards dimerization as the MG point is approached. In contrast, study of the Schmidt vectors in the marginally contributing subspace, as well as in the projection of the ground state in this space, display pair concurrence which decreases on both sublattices as the MG point is approached. The robustness of these observations indicate their possible usefulness in the study of models that have similar transitions, and have hitherto been difficult to study using standard entanglement signatures.
The von Neumann entanglement entropy of exact valence-bond ground states is studied in two frustrated one-dimensional spin chains: the spin-1/2 Majumdar-Ghosh (MG) model and the spin-3/2 J1-J2-J3 chain in its fully dimerized (FD) and partially dimerized (PD) phases. Using matrix-product-state representations, the entropy is computed as a function of system size for three complementary bipartitions - half-chain, single-site, and pairwise - under both open and periodic boundary conditions. In all cases, the entropy saturates to a finite constant in the thermodynamic limit, confirming area-law behavior. The saturation values, extracted via finite-size scaling, are directly related to the underlying virtual-spin bond structure. The MG model and FD phase exhibit similar entanglement behavior, differing primarily in saturation magnitude determined by spin value and bond multiplicity, and both display even-odd oscillations and exponential convergence with system size. In contrast, the PD phase shows qualitatively distinct signatures, including multiple half-chain saturation values depending on the bond type at the cut, asymmetric edge contributions in the single-site entropy, and a multi-band structure in the pairwise entropy reflecting the coexistence of single- and double-singlet bonds. These results establish entanglement entropy as a robust signature of frustrated bond architecture, enabling clear distinction among dimerized phases with different spin magnitude, bond multiplicity, and dimerization patterns.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where
J2 is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where
J3 refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing
J2, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing
J3. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive
J2. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
We compute concurrence, a measure of bipartite entanglement, of the first excited state of the
1-D Heisenberg frustrated
J1-
J2 spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength
α=J2/J1≈0.241, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with
8 sites to
16 sites, and the value of
α at which the change in entanglement is observed asymptotically tends to a value
αc≈0.24116. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the
1-D Heisenberg frustrated
J1-
J2 spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like
2-D Heisenberg
J1-
J2 model and Shastry-Sutherland model and detect quantum phase transitions.
We study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.