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I review the concept of a disorder operator, introduced originally by Kadanoff in the context of the two-dimensional Ising model. Disorder operators acquire an expectation value in the disordered phase of the classical spin system. This concept has had applications and implications to many areas of physics ranging from quantum spin chains to gauge theories to topological phases of matter. In this paper I describe the role that disorder operators play in our understanding of ordered, disordered and topological phases of matter. The role of disorder operators, and their generalizations, and their connection with dualities in different systems, as well as with majorana fermions and parafermions, is discussed in detail. Their role in recent fermion–boson and boson–boson dualities is briefly discussed.

In this article we study imaginary Gaussian multiplicative chaos—namely a family of random generalized functions which can formally be written as e iX(x), where X is a log-correlated real-valued Gaussian field on R d , that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the so-called Random Field Ising model (rfim) provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilizing self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and that account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of rfim-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith’s invisible hand can fail badly at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria from being reached. As a theoretical challenge, the study of so-called “detailed-balance” violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.

We establish that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting \"free energy density\" which coincides with the (nonrigorous) Bethe prediction of statistical physics. Using this result, we characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs without the use of the second moment method employed in previous works on these questions. As a consequence, we show that for both the hard-core model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has nonuniqueness on the d-regular tree. Together with results of Jerrum–Sinclair, Weitz, and Sinclair–Srivastava–Thurley, this gives an almost complete classification of the computational complexity of homogeneous two-spin systems on bounded-degree graphs.

We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin–spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on Zd, d⩾3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in d⩾3 satisfies the conjectured strong spatial mixing property in the entire high temperature regime.

The task of finding optimal protocols that minimize the energetic cost of thermodynamic processes of long yet finite duration τ is a pressing one. We approach this problem here in a rigorous and systematic fashion by means of the adiabatic perturbation theory of closed Hamiltonian quantum systems. Our main finding is a 1/ τ 2 scaling of the excess work for large τ in gapped systems. This result is at odds with the asymptotic 1/ τ prediction of the geometric approach to optimization, which is predicated on the slow evolution of open systems close to canonical equilibrium. In contrast, our approach does not lead to an obvious geometric interpretation. Furthermore, as the thermodynamic work does not depend on how an isolated quantum system is split into a system of interest and its environment, our results imply the failure of the geometric approach prediction even for open systems. Additionally, we provide alternative optimization procedures, both for slowly-varying processes described by adiabatic perturbation theory and for weakly-varying processes described by linear response theory. Our findings are benchmarked and confirmed through the application to the driven transverse-field Ising chain.

Single-time and two-time correlators are computed exactly in the 1D Glauber-Ising model after a quench to zero temperature and on a periodic chain of finite length N, using a simple analytical continuation technique. Besides the general confirmation of finite-size scaling in non-equilibrium dynamics, this allows for testing the scaling behaviour of the plateau height C∞(2), to which the two-time auto-correlator converges when deep in the finite-size regime.

An extension of the Ising spin configurations to continuous functions is used for an exact representation of the random field Ising model’s order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.

Social insects such as honey bees exhibit complex behavioral patterns, and their distributed behavioral coordination enables decision-making at the colony level. It has, therefore, been proposed that a high-level description of their collective behavior might share commonalities with the dynamics of neural processes in brains. Here, we investigated this proposal by focusing on the possibility that brains are poised at the edge of a critical phase transition and that such a state is enabling increased computational power and adaptability. We applied mathematical tools developed in computational neuroscience to a dataset of bee movement trajectories that were recorded within the hive during the course of many days. We found that certain characteristics of the activity of the bee hive system are consistent with the Ising model when it operates at a critical temperature, and that the system’s behavioral dynamics share features with the human brain in the resting state.

Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using time-evolving block decimation, we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations.