Quantum Spin Chains and Haldane Physics

Antiferromagnetic spin chains separate three related statements that are often conflated: the uniform isotropic integer-spin Heisenberg chain has a Haldane gap, half-integer chains obey a Lieb–Schultz–Mattis obstruction under specified symmetries, and odd-integer Haldane phases can be symmetry-protected topological phases with fractionalized edges. Each statement has different assumptions and diagnostics; anisotropy, frustration, or additional interactions can produce other integer-spin phases or critical points.

Required background. Antiferromagnets, Sigma Models, and Theta Terms supplies the $\theta=2\pi S$ continuum distinction; Non-Abelian Bosonization and Spin Sectors supplies the SU(2)$_1$ description of a critical half-integer chain. Helpful background. Interacting SPT Diagnostics supplies projective-edge and entanglement criteria.

Integer and half-integer chains

For the translation-invariant Heisenberg chain

$$H=J\sum_j\mathbf S_j\cdot\mathbf S_{j+1}, \qquad J>0,$$

the sigma-model angle is $\theta=2\pi S$. Integer spin corresponds to $\theta=0$ modulo $2\pi$ and a massive bulk; half-integer spin corresponds to $\theta=\pi$ and destructive interference of instantons. Haldane predicted exponential correlations and a nonzero gap for integer $S$, in contrast to the critical spin-$1/2$ chain Haldane 1983, pp. 1153–1156.

The lattice statement for half-odd-integer spin per unit cell is more general. With translation and spin-rotation symmetry, a short-range one-dimensional system cannot have a unique, symmetric, gapped ground state. It may be gapless, as the nearest-neighbor spin-$1/2$ chain is, or gapped with ground-state degeneracy from broken translation. The theorem is an obstruction, not a universal prediction of criticality. Modern quasi-adiabatic proofs make the locality assumptions precise Hastings 2004, §§ II–III.

The AKLT representative

The spin-1 Affleck–Kennedy–Lieb–Tasaki Hamiltonian is

$$H_{\mathrm{AKLT}}=\sum_j \left[\mathbf S_j\cdot\mathbf S_{j+1} +\frac13(\mathbf S_j\cdot\mathbf S_{j+1})^2+\frac23\right].$$

Each spin 1 is represented by two symmetrized spin-$1/2$ constituents, and neighboring constituents form singlets. On a periodic chain this gives a unique gapped valence-bond ground state; on an open chain one spin-$1/2$ remains at each edge. The displayed parent Hamiltonian is a sum of twice the projectors onto bond spin 2, so its positive normalization leaves the valence-bond state at zero energy Affleck et al. 1987, pp. 799–802.

The edges transform projectively under SO(3), time reversal, or the dihedral subgroup of $\pi$ rotations. As long as an appropriate protecting symmetry is preserved and the bulk gap stays open, an odd-integer Haldane phase cannot be deformed into a product state. Two copies can combine their edge projective representations, giving the interacting $\mathbb Z_2$ distinction between odd and even integer chains Pollmann et al. 2010, §§ II–IV.

Diagnostics and their limits

The nonlocal string correlator

$$O_{\mathrm{string}}^z= \lim_{|i-j|\to\infty} \left\langle S_i^z \exp\!\left(i\pi\sum_{k=i+1}^{j-1}S_k^z\right) S_j^z\right\rangle$$

is nonzero in the AKLT state and throughout much of the spin-1 Haldane phase. It reveals hidden order in that basis, but it is not by itself a complete SPT invariant: its value can depend on symmetry and choice of operator, and some trivial or symmetry-broken states can support related strings.

Stronger diagnostics include projective symmetry action on Schmidt states, protected degeneracy in the entanglement spectrum, robust edge representations, and quantized many-body response where defined. Edge spins alone also require care: a boundary can bind an extrinsic impurity. Their protection is established by how symmetry acts and by stability against local symmetric boundary perturbations.

An integer-spin gap is not synonymous with a nontrivial Haldane SPT. A large single-ion anisotropy $D\sum_j(S_j^z)^2$ produces a trivial product-like phase, separated from the spin-1 Haldane phase by a bulk transition when protecting symmetry is maintained. Breaking all protecting symmetries can connect the phases without closing the gap.

Exercises

1. Show that each AKLT bond term is a projector onto total spin 2.

Solution

For two spin-1 sites, $x=\mathbf S_i\cdot\mathbf S_j=[S_{\mathrm{tot}}(S_{\mathrm{tot}}+1)-4]/2$ takes values $-2,-1,1$ for total spin $0,1,2$. The polynomial $x+x^2/3+2/3$ gives $0,0,2$ respectively. Thus it equals $2P_{S_{\mathrm{tot}}=2}$.

2. Why does the half-integer obstruction allow a dimerized gapped phase?

Solution

Spontaneous dimerization doubles the unit cell and produces at least two translation-related ground states. The phase is gapped but not unique and translation symmetric in a single pure ground state, so it satisfies rather than violates the obstruction.

References

• Affleck, I., T. Kennedy, E. H. Lieb, and H. Tasaki. “Rigorous Results on Valence-Bond Ground States in Antiferromagnets.” Physical Review Letters 59 (1987): 799–802. DOI.
• Haldane, F. D. M. “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State.” Physical Review Letters 50 (1983): 1153–1156. DOI.
• Hastings, M. B. “Lieb–Schultz–Mattis in Higher Dimensions.” Physical Review B 69 (2004): 104431. DOI.
• Pollmann, F., A. M. Turner, E. Berg, and M. Oshikawa. “Entanglement Spectrum of a Topological Phase in One Dimension.” Physical Review B 81 (2010): 064439. DOI.