arXiv subjects

J. E. Björnberg

Publications and source records attributed to J. E. Björnberg.

5 recordsLinked to original sources

arXiv2026

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

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.

math-ph↗

arXiv2015

The free energy in a class of quantum spin systems and interchange processes

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin $S=\tfrac12$ the model is the Heisenberg ferromagnet, for general spin $S\in\tfrac12\mathbb{N}$ it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when $S=\tfrac12$). The critical temperature is shown to coincide (as a function of $S$) with that of the $q=2S+1$ state classical Potts model, and the phase transition is discontinuous when $S\geq1$.

math-ph↗

arXiv2015

Large cycles in random permutations related to the Heisenberg model

We study the weighted version of the interchange process where a permutation receives weight $θ^{\#\mathrm{cycles}}$. For $θ=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $θ>1$, that large cycles appear at `low temperature'.

math.PR↗

arXiv2014

Sharpness versus robustness of the percolation transition in 2D contact processes

We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in [17] for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of 'robust critical behaviour' with power law cluster size distribution for a range of parameter values, as suggested in [17].

math.PR↗

arXiv2009

The phase transition of the quantum Ising model is sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called `random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.

math-ph↗