Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation

We consider quantum spins with $$S\ge 1$$ S ≥ 1 , and two-body interactions with $$O(2S+1)$$ 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.

Section: A Family Of Quantum Spin Chains With O(n)-invariant Interactionssupporting

confidence: 88%

Section: Our Results About Dimerization Let Us Introduce the Operatorsmentioning

confidence: 99%

Section: 3mentioning

confidence: 98%

“…. , S, and related to the standard eigenbasis of the third spin matrix S (3) , satisfying S (3) |α = α|α , as follows: for α = 0 take e 0 = i S |0 , and for α > 0 define…”

Section: A Family Of Quantum Spin Chains With O(n)-invariant Interactionsmentioning

confidence: 99%

“…We use it to define a "contour model" in Sect. 3 where contours are shown to have small weights. This allows to use the method of cluster expansion and prove dimerization in Sect.…”

Section: Introductionmentioning

confidence: 97%

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Dimerization in Quantum Spin Chains with O(n) Symmetry

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On the Six-Vertex Model’s Free Energy

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Krachun

et al. 2022

Self Cite

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $$\Delta <1$$ Δ < 1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $$a=b=1$$ a = b = 1 and $$c\ge 1$$ c ≥ 1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

Macroscopic Loops in the Bose Gas, Spin O(N) and Related Models

Quitmann

Taggi

2023

We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model,random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in $${\mathbb {Z}}^d$$ Z d , $$d \ge 3$$ d ≥ 3 , and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate $${\mathbb {Z}}^d$$ Z d by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.