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Universality in Condensed Matter and Statistical Mechanics

Periodic Reporting for period 3 - UniCoSM (Universality in Condensed Matter and Statistical Mechanics)

Reporting period: 2020-03-01 to 2021-08-31

Universality is a central concept in several branches of mathematics and physics, at the core of our understanding of the collective behavior of large systems of interacting particles. In this context, universality refers to the exact independence of certain macroscopic observables from the microscopic details of the system. Remarkable examples include: the correlations among density fluctuations at the liquid-vapor critical point of a simple liquid, or among the magnetization in the ferromagnetic phase of a magnet with broken continuous symmetry, and the universal value of the Hall conductivity in interacting or disordered quantum many-body systems. Notwithstanding the existence of sophisticated, approximated, theories that allow us to explain the conceptual reasons behind the universality phenomenon, a fundamental understanding is still missing. There are several examples of systems for which we are still unable to quantitatively predict the collective behavior of the system (e.g. the phase diagram of a superconducting material or the conductivity coefficients of a two-dimensional electron gas in a magnetic field) starting from its well known microscopic structure. This lack of understanding severely limits our ability in engineering materials with the desired electronic, magnetic, mechanical or `topological' properties.

The UniCoSM project aims at developing new mathematical tools for the theoretical treatment of large systems of interacting particles, both in the classical and in the quantum realms. The idea is to combine ideas and techniques arisen in the last years in different branches of mathematical physics, such as the use of lattice Ward Identities in a constructive renormalization group treatment of fermionic systems, the use of discrete harmonicity in the study of scaling limits of lattice fields, reflection positivity, and functional localization estimates. The effective combination of these sophisticated tools will be developed and tested in a few model cases, of independent interest for current technological applications, such as interacting spin systems in two and three dimensions, or interacting lattice models for carbon nanotubes, for graphene and for topological semimetals. The mid-term goal is to rigorously derive and quantitatively characterize the macroscopic behavior of these systems and to exhibit in these contexts new instances of universality. Our research will mostly focus on the study of some poorly understood technical aspects, such as the role of boundary corrections and the loss of translational invariance in multiscale analysis, and the phenomenon of continuous symmetry breaking in systems with a non-abelian symmetry. The long term goal is to enhance our understanding of the macroscopic properties of interacting many particle systems, including `exotic' ones, such as superconducting materials, quantum Hall fluids and Bose-Einstein condensates.
In the first half of the project, the research activity of the UniCoSM team focused on the study of several model systems: two dimensional magnets at the ferromagnetic-paramagnetic critical point (technically, 2D Ising models with finite range interactions at the critical temperature); dense fluids of anisotropic molecules (technically, 2D interacting dimer models and 3D anistropic hard plates with discrete orientations); interacting quantum Hall fluids (technically, the 2D Haldane-Hubbard model); interacting 3D Weyl semimetals; atomistic models of solids with dislocations (technically, the 3D Ariza-Ortiz model). For all these cases, we succeeded in rigorously characterizing the macroscopic behavior of the system, including: a construction of the macroscopic correlation functions and the computation of their critical exponents in 2D interacting Ising and dimer models, as well as the rigorous derivation of certain `universal scaling relations' relating them (`Haldane relations'); a proof of the existence of a uni-axial nematic phase in 3D systems of anisotropic hard plates; a proof of the universality of the quantum Hall conductivity in the Haldane-Hubbard model and construction of its topological phase diagram; a proof of of the universality of the quadratic magnetoresistance coefficient in interacting Weyl semimetals; a proof of long range translational order in the 3D Ariza-Ortiz model.
All the results mentioned above represent a substantial progress in the understanding of the phase diagram or scaling limit of the aforementioned interacting many body systems. In some cases, our result are the first rigorous ones of their kind (we refer here, e.g. to the construction of the scaling limit and to the proof of the scaling relations in interacting dimer models, or to the proof of universality of the magnetoresistance in Weyl semimetals, or to the proof of existence of a uniaxial nematic phase in a 3D system of anisotropic molecules). Their proofs required the development of novel techniques, such as the use of multiscale methods in the presence of boundary terms or of potentials breaking the translation invariance of the system, the effective combination of Ward Identities with apriori decay bounds on correlations, or the introduction of cluster expansion and duality techniques in atomistic models for solids. These new techniques have the potential to allow us to attack new problems, which remained unsolved so far (such as the bulk-edge correspondence in the interacting Landau Hamiltonian, or the understanding of long range orientational order in 2D atomistic models for solids, or the construction of infrared lattice QED in d=3+1), and to create bridges among different, even though contiguous, communities (such as the one in mathematical statistical mechanics and those in probability, or in applied variational analysis and material science, or in lattice quantum field theory). In its second half, the UniCoSM project plans to further generalize the results on the model systems investigated so far, and to attack and solve new challenging problems, including the localization/delocalization properties of interacting electron systems in a quasi-random potential, and the phenomenon of low-temperature symmetry breaking in systems with a non-abelian continuous symmetry.
A modulated graphene sheet