Periodic Reporting for period 4 - UniCoSM (Universality in Condensed Matter and Statistical Mechanics)
Reporting period: 2021-09-01 to 2023-02-28
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 local 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 aimed 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 was 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 have been developed and tested in several model cases, of independent interest for current technological applications, such as: interacting spin and dimer models in two dimensions; interacting lattice models of nematic liquid crystals, of solids with dislocation defects, of Hall fluids, of topological insulators and of Weyl semimetals.
The new techniques developed by the UniCoSM project, sometimes unexpectedly, as compared to the original research plan, now open the way to the solution of some outstanding open problems in the field, such as: the non-perturbative construction of infrared lattice gauge theories in four dimensions, the proof of KPZ universal behavior for the fluctuations of the boundary of the Arctic circle in interacting dimer models, the systematic computation of the low-temperature magnetization in 3D quantum spin models, the rigorous confirmation of the Kosterlitz-Thouless-Halpering-Nelson-Young picture for two-dimensional melting, and the proof of the Lee-Huang-Yang correction to the ground state energy of the low-density hard-core Bose gas.
The UniCoSM project aimed 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 was 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 have been developed and tested in several model cases, of independent interest for current technological applications, such as: interacting spin and dimer models in two dimensions; interacting lattice models of nematic liquid crystals, of solids with dislocation defects, of Hall fluids, of topological insulators and of Weyl semimetals.
The new techniques developed by the UniCoSM project, sometimes unexpectedly, as compared to the original research plan, now open the way to the solution of some outstanding open problems in the field, such as: the non-perturbative construction of infrared lattice gauge theories in four dimensions, the proof of KPZ universal behavior for the fluctuations of the boundary of the Arctic circle in interacting dimer models, the systematic computation of the low-temperature magnetization in 3D quantum spin models, the rigorous confirmation of the Kosterlitz-Thouless-Halpering-Nelson-Young picture for two-dimensional melting, and the proof of the Lee-Huang-Yang correction to the ground state energy of the low-density hard-core Bose gas.
The research activity of the UniCoSM team focused on the study of several model systems: 2D magnets at the ferromagnetic-paramagnetic critical point (i.e. 2D Ising models with finite range interactions at the critical temperature); 3D ferromagnetic materials with weak crystalline anisotropies (i.e. 3D classical O(N) vector models); dense fluids of anisotropic molecules (i.e. 2D interacting/non-planar dimer models, and 3D models of anisotropic molecules with hard-core interactions); interacting quantum Hall fluids, topological insulators and Weyl semimetals (i.e. generalized Hubbard models on appropriate weighted and decorated lattices); solids with dislocations (i.e. the Ariza-Ortiz model); thin films with competing interactions (i.e. 2D Ising models with nearest-neighbor ferromagnetic and power-law anti-ferromagnetic interactions).
For all these cases, we succeeded in rigorously deriving and quantitatively characterizing the macroscopic behavior of the system and to exhibit new instances of universality, including: a construction of the scaling limit of 2D non-planar Ising models in domains with a boundary; the validity of the Kadanoff-Haldane scaling relations for non-planar dimer models; the existence of a uni-axial nematic phase in 3D systems of anisotropic hard plates; the first order nature of the isotropic to nematic phase transitions in models of elongated molecules with long-range interactions; the interaction independence of the quantum Hall conductivity in the Haldane-Hubbard model, and of the quadratic magneto-resistance coefficient (the `chiral anomaly') in interacting Weyl semimetals; the existence of long range translational order in the 3D Ariza-Ortiz model; the validity of spin-wave theory for the low-temperature magnetization of the 3D classical Heisenberg model.
The achievement of these results has been possible thanks to the development of new inter-methodological techniques, which allowed us to investigate and clarify some poorly understood technical aspects of the underlying theories, 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. In particular, we succeeded in combining multiscale analysis with the systematic use of lattice and asymptotic Ward Identities, in combining perturbation theory with reflection positivity, and in applying fermionic Renormalization Group to the study of boundary corrections and non-translationally invariant defect terms.
For all these cases, we succeeded in rigorously deriving and quantitatively characterizing the macroscopic behavior of the system and to exhibit new instances of universality, including: a construction of the scaling limit of 2D non-planar Ising models in domains with a boundary; the validity of the Kadanoff-Haldane scaling relations for non-planar dimer models; the existence of a uni-axial nematic phase in 3D systems of anisotropic hard plates; the first order nature of the isotropic to nematic phase transitions in models of elongated molecules with long-range interactions; the interaction independence of the quantum Hall conductivity in the Haldane-Hubbard model, and of the quadratic magneto-resistance coefficient (the `chiral anomaly') in interacting Weyl semimetals; the existence of long range translational order in the 3D Ariza-Ortiz model; the validity of spin-wave theory for the low-temperature magnetization of the 3D classical Heisenberg model.
The achievement of these results has been possible thanks to the development of new inter-methodological techniques, which allowed us to investigate and clarify some poorly understood technical aspects of the underlying theories, 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. In particular, we succeeded in combining multiscale analysis with the systematic use of lattice and asymptotic Ward Identities, in combining perturbation theory with reflection positivity, and in applying fermionic Renormalization Group to the study of boundary corrections and non-translationally invariant defect terms.
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 proofs of: the Kadanoff-Haldane scaling relations in interacting and non-planar dimer models, the universality of the magneto-resistance quadratic transport coefficient in Weyl semimetals, the existence of a uniaxial nematic phase in a 3D system of anisotropic molecules, the proof of the bulk-edge correspondence in interacting Hall fluids, the asymptotic nature of the low temperature expansion of the magnetization of the 3D classical Heisenberg model (via an elementary, non-multiscale, method).
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 a priori 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 four dimensions), some of which are already under investigation by the UniCoSM team. Moreover, we expect that they will help in creating new 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.
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 a priori 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 four dimensions), some of which are already under investigation by the UniCoSM team. Moreover, we expect that they will help in creating new 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.