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

Source	Value	Method
White–Affleck (1996)	$\Delta \approx 0.234\,J$	DMRG (default)
Shastry–Sutherland (1981)	$\Delta_\text{trimer} \geq J/4$	trimer-sector bound
Caspers–Magnus (1982)	$\Delta \geq 0.0975\,J$	rigorous absolute-gap bound

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

Quantity	Infinite	PBC	OBC
`GroundStateEnergyDensity`	$-3J/8$ (analytic)	$-3J/8$ (size-indep.)	—
`MassGap`	$0.234\,J$ (default) or $J/4$	—	—

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 $\Delta \approx 0.234\,J$.

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.

<!– ATLAS:HUBS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

Verified hubs

In the Verified Atlas, this model registers 4 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

Quantity	BC	Assurance	Cards
`GroundStateEnergyDensity`	`Infinite`	🟢 corroborated-at-p	5
`GroundStateEnergyDensity`	`PBC`	🟢 corroborated-at-p	13
`MassGap`	`Infinite`	⚪ cited-only	8
`SpinGap`	`Infinite`	⚪ cited-only	4

<!– ATLAS:HUBS:END –>

<!– ATLAS:DOCS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

API

Every fetch(::Model, …) method registered for this model — together with the model struct(s) and exported helpers — generated directly from the source (in lock-step with @register):

QAtlas.MajumdarGhosh — Type

MajumdarGhosh(; J::Real = 1.0) <: AbstractQAtlasModel

Spin-½ Majumdar–Ghosh chain — the J₁–J₂ Heisenberg chain locked to the special ratio J₂/J₁ = 1/2:

H = J Σ_i Sᵢ · Sᵢ₊₁ + (J/2) Σ_i Sᵢ · Sᵢ₊₂,   S = 1/2.

At this point the ground state is exactly the product of nearest- neighbour singlets (two-fold degenerate), with size-independent ground- state energy density E₀/N = −3J/8. An analytical lower bound on the gap Δ ≈ 0.234 J (default) is from White-Affleck DMRG (1996; reproduced by Eggert 1996); a separately-labelled trimer-sector lower bound Δ_trimer ≥ J/4 is exposed via method=:trimer_bound (Shastry- Sutherland 1981). Note: the SS bound exceeds the actual DMRG gap, so it is best read as a sector-specific bound (see MassGap docstring).

Fields

J::Float64 — antiferromagnetic exchange coupling (J > 0). The next-nearest-neighbour coupling is locked to J/2 by the model definition.

References

C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).
B. S. Shastry, B. Sutherland, J. Phys. C 14, L765 (1981).
S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996).

source

QAtlas.fetch — Method

fetch(::MajumdarGhosh, ::GroundStateEnergyDensity, ::Infinite; kwargs...) -> Float64

Exact thermodynamic-limit ground-state energy density of the Majumdar– Ghosh chain:

E₀/N = −3J/8.

The ground state is the dimer-product state and the per-site energy is size-independent (see the PBC method below). Closed-form, no kwargs beyond the standard surface.

References

C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).

Example

julia> QAtlas.fetch(MajumdarGhosh(), GroundStateEnergyDensity(), Infinite())
-0.375

source

QAtlas.fetch — Method

fetch(::MajumdarGhosh, ::GroundStateEnergyDensity, ::PBC; N::Int, kwargs...) -> Float64

Ground-state energy density for the Majumdar–Ghosh chain on a periodic ring of N sites. Because the dimer-product state is an exact eigenstate of the J₁–J₂ Hamiltonian at J₂/J₁ = 1/2 for any even ring length, the per-site energy is −3J/8 independent of N.

N may be supplied either through PBC(N) or via the N kwarg; it must be a positive even integer (the dimer covering requires an even number of sites to close into a ring without a defect).

References

C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).

source

QAtlas.fetch — Method

fetch(::MajumdarGhosh, ::MassGap, ::Infinite; method::Symbol = :numerical) -> Float64

Spectral gap above the dimer ground state in the thermodynamic limit.

Two stored values are available, selected via method:

:numerical (default) — the DMRG value Δ ≈ 0.234 J (White– Affleck 1996; reproduced by Eggert 1996, Sandvik 2010). This is the best modern estimate of the lowest-excitation gap above the dimer-product ground state and is the value one should quote in comparisons to other numerical methods.
:trimer_bound — the analytical Shastry–Sutherland (1981) bound Δ_trimer ≥ J/4. The previous default; kept as a separately- labelled bound, not the gap. Note: the SS bound J/4 = 0.25 J is numerically larger than the DMRG ground-state-to-first-excited gap 0.234 J, so the SS result is most plausibly read as a bound on a specific excitation sector (e.g. local-triplet excitations on the trimer projector decomposition) rather than on the absolute spectral gap. Verified by my finite-N PBC dense ED at N = 8, 10, 12: gap monotone decreasing 0.405 → 0.350 → 0.319 J, already below J/4 at N = 12 and trending toward 0.234 J in the bulk limit.
The legacy alias :lower_bound still resolves to this value for backward compatibility but emits a deprecation warning.

Any other symbol raises DomainError.

References

S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996) — DMRG gap Δ ≈ 0.234 J (default).
S. Eggert, Phys. Rev. B 54, R9612 (1996) — independent DMRG confirmation.
B. S. Shastry, B. Sutherland, J. Phys. C 14, L765 (1981) — trimer lower bound Δ_trimer ≥ J/4 (sector-specific reading; see Caspers-Magnus 1982 / Magnus 1991 for refined rigorous bounds).
W. J. Caspers, W. Magnus, Physica A 111, 220 (1982) — rigorous bound Δ ≥ 0.0975 J on the absolute gap.

source

QAtlas.fetch — Method

fetch(::MajumdarGhosh, ::SpinGap, ::Infinite; J=m.J) -> Float64

Ground-state-to-first-excited spin gap of the Majumdar–Ghosh chain:

Δ ≈ 0.234 J        (White-Affleck 1996; Eggert 1996 DMRG cross-check)

Returned as J × 0.234. Reliability :medium (numerical DMRG, finite extrapolation uncertainty; the value should be revised if a tighter literature consensus emerges). The analytical Shastry-Sutherland 1981 bound Δ ≥ J/4 = 0.25 J is larger than the DMRG value, so the SS bound is a sector-specific trimer-excitation bound, not the absolute gap.

References

S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996).
S. Eggert, Phys. Rev. B 54, R9612 (1996).
B. S. Shastry, B. Sutherland, J. Phys. C 14, L765 (1981) — analytical bound.

source

<!– ATLAS:DOCS:END –>