Exact Haldane mapping for all S and super universality in spin chains

The low energy dynamics of the anti-ferromagnetic Heisenberg spin S chain in the semiclassical limit S → ∞ is known to map onto the O(3) nonlinear σ model with a θ term in 1+1 dimension. Guided by the underlying dual symmetry of the spin chain, as well as the recently established topological significance of “dangling edge spins,” we report an exact mapping onto the O(3) model that avoids the conventional large S approximation altogether. Our new methodology demonstrates all the super universal features of the θ angle concept that previously arose in the theory of the quantum Hall effect. It explains why Haldane’s original ideas remarkably yield the correct answer in spite of the fundamental complications that generally exist in the idea of semiclassical expansions. In 1983, Haldane proposed that the low energy dynamics of the anti-ferromagnetic Heisenberg spin S chain can be taken from the O(3) nonlinear σ-model (NLSM) in 1 + 1 dimension and in the presence of the θ term. [1] At least within the limitations of a large S approximation the parameter θ was found to take on the values 0 or π only, dependent on whether S is integral or half-integral respectively. Since the standard O(3) model with θ = 0 is known to display a mass-gap, [2] Haldane concluded that the integral spin chain is always gapped. This is unlike the S = 1/2 chain, for example, which from the Bethe ansatz solution is known to display gapless excitations [3]. Based on numerical work it is now generally accepted that the uniform integral spin chain is always gapped whereas the half integral spin chain is generally gapless . It has recently been pointed out, however, that a universal topological feature of the θ angle concept has historically been overlooked. [4] The θ vacuum quite generally displays massless chiral edge excitations [5] that have important consequences for the theory on the strong coupling side. Within the grassmannian U(M +N)/U(M)×U(N) nonlinear sigma model approach to localization and interaction phenomena, for example, it was shown that the massless edge excitations are directly related to the existence of robust topological quantum numbers that explain the stability and precision of the quantum Hall effect. [6] This has led to the idea of super universality of quantum Hall physics that, unlike the common belief in the field, is independent of the details of the theory such as the number of field components or, for that matter, replica limit M,N → 0. [6] Based on the Haldane mapping it is next natural to expect that the super universality statement can be extended to also include quantum spin liquids. Indeed, renormalization group studies have clearly indicated that the dimerised S = 1/2 spin chain displays all the basic features of the quantum Hall effect [4] in much the same manner as what has recently been observed,for example, in the exactly solvable largeN limit of the CP model. [7] The dynamics of the dangling edge spins of the chain thereby plays a role that is in many ways the same as that of the massless chiral edge excitations of the θ vacuum. Unfortunately, a generalized Haldane mapping that includes the dangling spins at the edges of the spin S chain has sofar been obtained only for the semiclassical S = ∞ limit. [4] Given the extensive literature on the subject of quantum spin chains, it is somewhat surprising to know that not a single attempt has been reported that would in principle resolve this longstanding drawback and extend the Haldane mapping to include finite values of S.