Heisenberg Chain

Overview

The spin-1/2 antiferromagnetic Heisenberg chain is the paradigmatic model of quantum magnetism. Unlike the TFIM, it is not mappable to free fermions; its exact solution requires the Bethe ansatz (1931), one of the earliest and most celebrated applications of integrability in condensed matter physics.

\[H = J \sum_{i} \mathbf{S}_i \cdot \mathbf{S}_{i+1}\]

where $\mathbf{S}_i = \tfrac{1}{2}\boldsymbol{\sigma}_i$ are spin-1/2 operators and $J > 0$ is the antiferromagnetic exchange coupling.

Parameters: Exchange coupling $J$ (default 1.0; $J > 0$ AFM, $J < 0$ FM).

Key physics: The 1D Heisenberg chain is gapless with algebraically decaying spin correlations. At low energies it is described by a $c = 1$ conformal field theory (free boson / Luttinger liquid), in contrast to the $c = 1/2$ Ising CFT governing the TFIM. The ground state is a spin singlet ($S_{\text{tot}} = 0$) for any finite even $N$ with AFM coupling.

Dimer Spectrum (N = 2, OBC)

Statement

For two spin-1/2 sites coupled by $H = J\,\mathbf{S}_1 \cdot \mathbf{S}_2$, the Hilbert space $\mathbb{C}^2 \otimes \mathbb{C}^2$ decomposes into a singlet ($S_{\text{tot}} = 0$) and a triplet ($S_{\text{tot}} = 1$):

\[\text{Spectrum} = \left\{-\frac{3J}{4},\; \frac{J}{4},\; \frac{J}{4},\; \frac{J}{4}\right\}\]

The singlet–triplet gap is $\Delta = J$.

Derivation

Rewriting the Hamiltonian via the total-spin identity

\[\mathbf{S}_1 \cdot \mathbf{S}_2 = \frac{1}{2}\!\left[S_{\text{tot}}(S_{\text{tot}}+1) - \frac{3}{2}\right]\]

immediately gives $E_s = -3J/4$ for $S_{\text{tot}} = 0$ and $E_t = J/4$ for $S_{\text{tot}} = 1$. Full details are in Heisenberg Dimer: Singlet-Triplet.

References

• A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, 1994), Section 2 – textbook derivation.
• Any quantum mechanics textbook (e.g., Sakurai, Ch. 4).

QAtlas API

# Full 4-state spectrum of the dimer
λ = QAtlas.fetch(Heisenberg1D(), ExactSpectrum(); N=2, J=1.0, bc=:OBC)
# → [-0.75, 0.25, 0.25, 0.25]

Verification

4-Site PBC Ring (N = 4)

Statement

For $N = 4$ spins on a periodic ring with Hamiltonian $H = J \sum_{i=1}^{4} \mathbf{S}_i \cdot \mathbf{S}_{i+1}$ ($\mathbf{S}_5 \equiv \mathbf{S}_1$), the $2^4 = 16$ eigenvalues are:

The ground state $E_0 = -2J$ is a unique singlet. The ferromagnetic sector ($|\!\uparrow\uparrow\uparrow\uparrow\rangle$ etc.) sits at $E = +J$, since each of the 4 bonds contributes $S^z_i S^z_{i+1} = 1/4$.

Derivation

The 4-site ring can be solved by direct diagonalization of the $16 \times 16$ Hamiltonian or by exploiting the full SU(2) symmetry and translational invariance. The result is also obtainable as a special case of the Bethe ansatz for $N = 4$.

References

• A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, 1994), Section 2.3.
• H. Bethe, Z. Physik 71, 205 (1931) – original Bethe ansatz (applicable to any $N$).

QAtlas API

# Full 16-state spectrum of the 4-site PBC ring
λ = QAtlas.fetch(Heisenberg1D(), ExactSpectrum(); N=4, J=1.0, bc=:PBC)
# → [-2.0, -1.0, -1.0, -1.0, 0.0, ..., 1.0, 1.0, 1.0, 1.0, 1.0]

Verification

Bethe Ansatz Ground-State Energy Density

Statement

In the thermodynamic limit ($N \to \infty$, PBC), the ground-state energy per site of the spin-1/2 AFM Heisenberg chain is

\[e_0 = J\!\left(\frac{1}{4} - \ln 2\right) \approx -0.4431\,J\]

This exact result was first obtained by Hulthen (1938) from the Bethe ansatz solution.

Derivation

The Bethe ansatz parameterizes eigenstates in the $M$-down-spin sector by a set of rapidities $\{\lambda_j\}$ satisfying the Bethe equations. In the thermodynamic limit, the ground-state rapidity distribution satisfies a linear integral equation whose solution yields the energy through integration. The full calculation is given in Bethe Ansatz: Heisenberg $e_0$.

Finite-size corrections

For a PBC chain of $N$ sites:

\[\frac{E_0(N)}{N} = e_0 + O\!\left(\frac{1}{N^2}\right)\]

with logarithmic corrections also present. Due to the negative sign of the leading finite-size correction (PBC), $E_0(N)/N < e_0$ for finite $N$.

References

• H. Bethe, "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette", Z. Physik 71, 205 (1931) – original Bethe ansatz.
• L. Hulthen, "Uber das Austauschproblem eines Kristalles", Ark. Mat. Astron. Fys. 26A, No. 11, 1 (1938) – first evaluation of $e_0 = 1/4 - \ln 2$.

QAtlas API

# Exact thermodynamic-limit ground-state energy per site
e₀ = QAtlas.fetch(Heisenberg1D(), GroundStateEnergyDensity(); J=1.0)
# → -0.44314718055994530

Verification

Connections

• Universality: The low-energy physics of the Heisenberg chain is described by a $c = 1$ Luttinger liquid (free boson CFT), not the $c = 1/2$ Ising CFT. This is a fundamentally different universality class from the TFIM / IsingSquare.
• Entanglement verification: The Heisenberg chain provides a $c = 1$ test case for the Calabrese–Cardy formula, complementing the $c = 1/2$ test from the TFIM.
• Cross-verification: The Bethe ansatz $e_0$ is cross-checked against finite-size ED in testuniversalitycross_check.jl, constituting an independent physical verification from two different theoretical lines (Bethe ansatz vs exact diagonalization).