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arXiv:2506.06103

Dimerization in $O(n)$-invariant quantum spin chains

Jakob E. Björnberg, Kieran Ryan

Published Jun 6, 2025• 0 citations
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Abstract

We establish dimerization in $O(n)$-invariant quantum spin chains with large enough $n$, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to $\mathbb{Z}$ and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop $O(n)$ models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.

DOI
10.48550/arxiv.2506.06103

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Dimerization in O(n)O(n)O(n)-invariant quantum spin chains
We establish dimerization in O(n)O(n)O(n)-invariant quantum spin chains with large enough nnn, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to Z\mathbb{Z}Z and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop O(n)O(n)O(n) models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.
Abstract The ground-states of the spin-Santiferromagnetic chain HAFH_{{\text {AF}}}HAF​ HAF with a projection-based interaction and the spin-1/2 XXZ-chain HXXZ H_{{\text {XXZ}}}HXXZ​ HXXZ at anisotropy parameter Δ=cosh⁡(λ)\Delta =\cosh (\lambda ) Δ=cosh(λ) Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planarQ-state Potts model at Q=2S+1=2cosh⁡(λ) \sqrt{Q}= 2S+1 =2\cosh (\lambda )Q​=2S+1=2cosh(λ) Q=2S+1=2cosh(λ) . 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 HAFH_{{\text {AF}}}HAF​ HAF at all S>1/2S> 1/2S>1/2 S>1/2 , and Néel order for HXXZ H_{{\text {XXZ}}} HXXZ​ HXXZ at λ>0\lambda >0λ>0 λ>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>4Q>4Q>4 Q>4 . Altogether, the quantum manifestation of the change between Q=4Q=4Q=4 Q=4 and Q>4Q>4Q>4 Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.
The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system’s boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.
We prove that for the spin-1/2 Heisenberg XXZ chain in the range Δ∈[−1,1/2]Δ\in[-1,1/2]Δ∈[−1,1/2], the ground state on the torus of length LLL converges to an infinite volume ground state ⟨⋅⟩\langle\cdot\rangle⟨⋅⟩ as L→∞L\to\inftyL→∞, and that the spin-spin correlation ⟨S0(1)Sx(1)⟩\langle S_0^{(1)}S_x^{(1)}\rangle⟨S0(1)​Sx(1)​⟩ decays polynomially fast in xxx. In the range Δ∈[−1,0]Δ\in[-1,0]Δ∈[−1,0] we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that LLL and β=1/Tβ=1/Tβ=1/T can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that ⟨S0(1)Sx(1)(t)⟩\langle S_0^{(1)}S_x^{(1)}(t)\rangle⟨S0(1)​Sx(1)​(t)⟩ decays to zero in ∣(x,t)∣|(x,t)|∣(x,t)∣, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points Δ=0,−1Δ=0,-1Δ=0,−1), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.
Abstract We consider quantum spins with S≥1S\ge 1S≥1 S ≥ 1 , and two-body interactions with O(2S+1)O(2S+1)O(2S+1) O ( 2 S + 1 ) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.