Abstract
The ground-states of the spin-S antiferromagnetic chain H_{{text {AF}}} with a projection-based interaction and the spin-1/2 XXZ-chain H_{{text {XXZ}}} at anisotropy parameter Delta =cosh (lambda ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar Q-state Potts model at sqrt{Q}= 2S+1 =2cosh (lambda ). The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for H_{{text {AF}}} at all S> 1/2, and Néel order for H_{{text {XXZ}}} at lambda >0. The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for Q>4. Altogether, the quantum manifestation of the change between Q=4 and Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.
Highlights
The focus of this work is the structure of the ground-states in two families of antiferromagnetic quantum spin chains, each of which includes the spin-1/2 Heisenberg anti-ferromagnet as a special case
In the infinite volume limit, with the exception of their common root, in both cases the systems exhibit symmetry breaking at the level of ground-states
Henri Poincare other case, the Hamiltonian is frustration free and the symmetry breaking is expressed in long-range Neel order
Summary
The focus of this work is the structure of the ground-states in two families of antiferromagnetic quantum spin chains, each of which includes the spin-1/2 Heisenberg anti-ferromagnet as a special case. The results presented here are based on non-perturbative structural arguments They may be worth presenting since in the models considered such arguments allow full characterization of the conditions under which the symmetry breaking occurs, as well as other qualitative features of the model’s ground-states. The two ground-states · even and · odd are distinct, each invariant under the 2-step shift, each being the 1-step shift of the other Their translation symmetry breaking is manifested in energy oscillations, namely, for every n ∈ N. In [20], the loop representation of the critical Q-state Potts model on the square lattice with Q > 4 was proved to have two distinct infinite-volume measures under which the probability of having large loops is decaying exponentially fast (see [28] for the case of large Q). Let us note that under the dimerization scenario, which is established for its full range (S > 1/2), other physically interesting features follow: 1. Spectral gap: As was argued already in [6, Theorem 7.1], the exponential decay of truncated correlations (1.16) in the t-direction implies a nonvanishing spectral gap in the excitation spectrum above the even and odd ground-states
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