MajumdarGhosh — Spin-½ J₁–J₂ Chain at J₂/J₁ = 1/2

The Majumdar–Ghosh chain is the spin-½ J₁–J₂ Heisenberg chain locked to the special ratio $J_2/J_1 = 1/2$. At this point the ground state is exactly the product of nearest-neighbour singlets — one of the cleanest closed-form ground states in 1D quantum magnetism.

Hamiltonian

\[H = J \sum_i \mathbf{S}_i \cdot \mathbf{S}_{i+1} + \frac{J}{2} \sum_i \mathbf{S}_i \cdot \mathbf{S}_{i+2}, \qquad \mathbf{S}_i = \tfrac{1}{2}\boldsymbol{\sigma}_i, \qquad J > 0.\]

The next-nearest-neighbour coupling is locked to $J/2$ by the model definition; only $J$ is a free parameter.

Exact dimer ground state

At $J_2/J_1 = 1/2$ the ground state is the product of nearest- neighbour singlets, with two inequivalent dimer coverings (even/odd):

\[|\psi_0^{\pm}\rangle = \prod_i |\text{singlet}\rangle_{(i, i+1)}.\]

Each singlet contributes $\langle \mathbf{S}\cdot\mathbf{S}\rangle = -3/4$ and adjacent dimers are orthogonal, so all $\mathbf{S}_i\cdot\mathbf{S}_{i+2}$ matrix elements on the dimer state vanish. The size-independent ground-state energy density is therefore

\[\boxed{\;\frac{E_0}{N} = -\frac{3J}{8}.\;}\]

The dimer state is an exact eigenstate for both periodic (even $N$) and open boundary conditions; the ground state is two-fold degenerate on both.

Excitation gap

Both stored values are exposed through the MassGap quantity with a method kwarg. Note that the SS $J/4$ value exceeds the actual DMRG gap ($0.25 > 0.234$), so it must be read as a bound on a specific excitation sector (likely the local-triplet sector on the trimer-projector decomposition) rather than on the absolute spectral gap. Rigorous absolute-gap bounds (Caspers–Magnus 1982; Magnus 1991: $\Delta \geq 0.117\,J$) are weaker.

Coverage Matrix

Quick-look code

using QAtlas

m = MajumdarGhosh(; J=1.0)

QAtlas.fetch(m, GroundStateEnergyDensity(), Infinite())            # -3/8
QAtlas.fetch(m, GroundStateEnergyDensity(), PBC(8))                # -3/8
QAtlas.fetch(m, MassGap(), Infinite())                             # 0.234 (default; White–Affleck DMRG)
QAtlas.fetch(m, MassGap(), Infinite(); method=:numerical)          # 0.234
QAtlas.fetch(m, MassGap(), Infinite(); method=:trimer_bound)       # 1/4   (Shastry–Sutherland trimer-sector bound)
QAtlas.fetch(m, MassGap(), Infinite(); method=:lower_bound)        # 1/4   (legacy alias of :trimer_bound; emits deprecation @warn)

References

• C. K. Majumdar, D. K. Ghosh, "On Next-Nearest-Neighbor Interaction in Linear Chain. I/II", J. Math. Phys. 10, 1388 (1969) — exact dimer ground state at $J_2/J_1 = 1/2$.
• B. S. Shastry, B. Sutherland, "Excitation spectrum of a dimerized next-neighbour antiferromagnetic chain", J. Phys. C 14, L765 (1981) — analytical lower bound $\Delta \geq J/4$.
• S. R. White, I. Affleck, "Dimerization and incommensurate spiral spin correlations in the zigzag spin chain", Phys. Rev. B 54, 9862 (1996) — DMRG gap $\Delta \approx 0.234\,J$.

Related

• Heisenberg1D — the $J_2 = 0$ parent chain (gapless, Bethe-ansatz ground state $e_0 = J(1/4 - \ln 2)$).
• XXZ1D — anisotropic generalisation of the nearest-neighbour Heisenberg point.

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Verified hubs

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