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Daniel Ueltschi

Publications and source records attributed to Daniel Ueltschi.

At least 19 recordsLinked to original sources
arXiv2025

Symmetry breaking and thermal phase transition of the spin-1 quantum magnet with SU(3) symmetry on the simple cubic lattice

Using a combined analysis from Poisson-Dirichlet and symmetry-breaking calculations as well as quantum Monte Carlo simulations, we examine the ordered phase and the thermal phase transition of the three-dimensional spin-1 quantum magnet on the simple cubic lattice with bilinear and biquadratic interactions and SU(3) internal symmetry. We obtain exact results for the order parameter distribution function that compare well to the quantum Monte Carlo data. Furthermore, based on a detailed finite-size analysis, we provide evidence that the thermal melting transition at the SU(3) point is either weakly first-order in this system or, if continuous, it falls beyond the unitary-bounds of conformal field theory.

cond-mat.str-el↗
arXiv2024

Kac-Ward solution of the 2D classical and 1D quantum Ising models

We give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants, and (ii) the one-dimensional quantum Ising model. We use the method of Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni--Duminil-Copin--Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model.

math-ph↗
arXiv2024

A two-table theorem for a disordered Chinese restaurant process

We investigate a disordered variant of Pitman's Chinese restaurant process where tables carry i.i.d. weights. Incoming customers choose to sit at an occupied table with a probability proportional to the product of its occupancy and its weight, or they sit at an unoccupied table with a probability proportional to a parameter $θ>0$. This is a system out of equilibrium where the proportion of customers at any given table converges to zero almost surely. We show that for weight distributions in any of the three extreme value classes, Weibull, Gumbel or Fréchet, the proportion of customers sitting at the largest table converges to one in probability, but not almost surely, and the proportion of customers sitting at either of the largest two tables converges to one almost surely.

math.PR↗
arXiv2022

Poisson-Dirichlet distributions and weakly first-order spin-nematic phase transitions

We provide a quantitative characterization of generic weakly first-order thermal phase transitions out of planar spin-nematic states in three-dimensional spin-one quantum magnets, based on calculations using Poisson-Dirichlet distributions (PD) within a universal loop model formulation, combined with large-scale quantum Monte Carlo calculations. In contrast to earlier claims, the thermal melting of the nematic state is not continuous, instead a weakly first-order transition is identified from both thermal properties and the distribution of the nematic order parameter. Furthermore, based on PD calculations, we obtain exact results for the order parameter distribution and Binder cumulants at the discontinuous melting transition. Our findings establish the thermal melting of planar spin-nematic states as a generic platform for quantitative approaches to weakly first-order phase transitions in quantum systems with a continuous SU(2) internal symmetry.

cond-mat.str-el↗
arXiv2022

Universal behaviour of 3D loop soup models

These notes describe several loop soup models and their {\it universal behaviour} in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, long-range order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a {\it Poisson-Dirichlet distribution}. This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.

cond-mat.stat-mech↗
arXiv2022

Loop correlations in random wire models

We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson-Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson-Dirichlet counterpart.

math-ph↗
arXiv2021

Dimerization in quantum spin chains with $O(n)$ symmetry

We consider quantum spins with $S\geq1$, and two-body interactions with $O(2S+1)$ 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.

math-ph↗
arXiv2019

Quantum spins and random loops on the complete graph

We present a systematic analysis of quantum Heisenberg-, XY- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson--Dirichlet distributions.

math-ph↗
arXiv2019

Characterising random partitions by random colouring

Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $1/2$.

math.PR↗
arXiv2018

Critical temperature of Heisenberg models on regular trees, via random loops

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.

math-ph↗
arXiv2018

A direct proof of dimerization in a family of SU(n)-invariant quantum spin chains

We study the family of spin-S quantum spin chains with a nearest neighbor interaction given by the negative of the singlet projection operator. Using a random loop representation of the partition function in the limit of zero temperature and standard techniques of classical statistical mechanics, we prove dimerization for all sufficiently large values of S.

math-ph↗
arXiv2018

Singularity analysis for heavy-tailed random variables

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindelöf integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws $p(k) = c\exp( -k^α)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^β)$, $β>2$; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem.

math.PR↗
arXiv2017

Critical parameter of random loop model on trees

We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process our results improve on earlier work by Angel and by Hammond, in that we determine the second-order term of the critical parameter.

math.PR↗
arXiv2017

An improved tree-graph bound

This report discusses the improved bound of the cluster expansion, recently proposed by Procacci and Yuhjtman (Lett. Math. Phys. 107, 31, 2017). Brydges and Helmuth noticed the relevance of Kruskal's algorithm, which allows to streamline and simplify the argument. These clarifications have proved useful in arXiv:1705.04385.

math-ph↗
arXiv2017

Decay of correlations in 2D quantum systems with continuous symmetry

We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan-Spencer-Koma-Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg-, Hubbard-, and t-J models, and to certain models of random loops.

math-ph↗
arXiv2016

Correlation inequalities for classical and quantum XY models

We review correlation inequalities of truncated functions for the classical and quantum XY models. A consequence is that the critical temperature of the XY model is necessarily smaller than that of the Ising model, in both the classical and quantum cases. We also discuss an explicit lower bound on the critical temperature of the quantum XY model.

math-ph↗
arXiv2016

Correlation inequalities for the quantum XY model

We show the positivity or negativity of truncated correlation functions in the quantum XY model with spin 1/2 (at any temperature) and spin 1 (in the ground state). These Griffiths-Ginibre inequalities of the second kind generalise an earlier result of Gallavotti.

math-ph↗