We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum, our results carry over to the three-dimensional case.

Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math...

We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due...

In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion.

A new construction of BPS monodromies for 4d \({\mathcal {N}}=2\) theories of class \({\mathcal {S}}\) is introduced. A novel feature of this construction is its manifest invariance under Kontsevich–Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV...

Motivated by a recent result of Ciesielski and Jasiński we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first...

We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is \(\textsf {QMA}_{{\textsf {EXP}}}\)-complete, even for systems of low local dimension (\(\approx 40\)). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can...

In Theorem 1.4 (iii) of the original article, we stated that $$\begin{aligned} \sqrt{N} (F_N - F(\beta )) \Rightarrow \mathcal {N}(0, \alpha _2) \end{aligned}$$ (1)in the ferromagnetic regime \(J > 1\) and \(\beta > \frac{1}{2J}\). The proof was based on Theorem 1.5 (iii), which we proved in the paper, and a known random matrix theory result, given in the second part of (1.19...

Given a compact surface \(\mathcal {M}\) with a smooth area form \(\omega \), we consider an open and dense subset of the set of smooth closed 1-forms on \(\mathcal {M}\) with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate...

We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on...

We consider the problem of deciding if a set of quantum one-qudit gates \(\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G\) is universal, i.e. if \({<}\mathcal {S}{>}\) is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in \(\mathcal {S}\) we assign the orthogonal matrix \(\mathrm {Ad}_g\) that is image of g under the adjoint...

We describe the construction of a geometric invariant characterising initial data for the Kerr–Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr–Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr–Newman spacetime in terms of Killing spinors. The space...

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic many-body models: part 1—main results and algebra, arXiv:1609.01745, 2016, The small field parabolic flow for...

A holomorphic family of closed operators with a rank one perturbation given by the function \(x^{\frac{m}{2}}\) is studied. The operators can be used in a toy model of renormalization group.

Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free \(\mathbb {Z}^d\) actions on Cantor sets admit “small cocycles.” These represent classes in \(H^1\) that are mapped to small vectors in \(\mathbb {R}^d\) by the Ruelle–Sullivan (RS) map. We show that there exist \(\mathbb {Z}^2\) actions where no such...

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.

Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying...

In this work, we argue that the \(\alpha '\rightarrow 0\) limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus. An important technical input from tropical geometry is the use of tropical theta functions with characteristics to...

We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum.

The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the...

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein–Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric...

The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with \(\delta \)-interactions supported on closed curves in \(\mathbb {R}^3\). We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive...

The paper is devoted to operators given formally by the expression $$\begin{aligned} -\partial _x^2+\left( \alpha -\frac{1}{4}\right) \frac{1}{x^{2}}. \end{aligned}$$This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real \(\alpha \), or closed operator for complex \(\alpha \), we find that this homogeneity can be...

We introduce and study a Markov field on the edges of a graph \(\mathcal {G}\) in dimension \(d\ge 2\) whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a...

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on \({\mathbb{R}^3}\). In particular, we are interested in those operators \({\mathcal{D}_B}\) for which the associated magnetic field \({B}\) is given by pulling back a two-form \({\beta}\) from the sphere \({\mathbb{S}^2}\) to \({\mathbb{R}^3}\) using a combination of the...