Final Report Summary - COMBOS (Collective phenomena in quantum and classical many body systems)
The CoMBoS project focused on the collective equilibrium properties of many body systems, in particular on exotic phenomena such as non-Fermi liquid phases in interacting Fermi systems, stripes and periodic patterns
formation in dipolar systems, Bose-Einstein Condensation in homogeneous Bose gases and the emergence of conformal symmetry in 2D systems at criticality. While apparently very different, these problems share common underlying features, such as the spontaneous emergence of ordered structures and hidden symmetries at very large length scales, induced by the "constructive interference" of self-similar driving mechanisms acting in a multiscale fashion on very different length scales, starting from the microscopic scale, up to the macroscopic one. In the context of these problems, the project aimed at developing new methods based on the combination of advanced mathematical techniques such as Renormalization Group (RG), cluster expansion, localization and functional estimates, Reflection Positivity (RP), as a mean to unveil the hidden connections between these problems, and to allow further progresses in the study of collective phenomena in the classical and quantum realms.
The project produced several remarkable results on interacting Fermi and Bose gases, quantum spins, pattern formation and 2D systems at criticality. While focused on the development of mathematical methods for the rigorous foundation of these systems, the CoMBoS project, with its accomplishments, allowed to settle certain experimental observations, which for a while did not have a clear theoretical explanation: I refer here, in particular, to the universality of the optical conductivity in single-layer graphene.
The achievements of the CoMBos project have been accompanied by the development of some novel techniques, such as: (1) the systematic implementation of exact lattice Ward Identities within a rigorous RG scheme; (2) the combination of RP estimates with localization estimates, coarse graining and functional inequalities; (3) the combination of RG methods with discrete holomorphicity.
Here are some of the problems in mathematical condensed matter that the CoMBoS project solved, thus opening the way to further developments:
INTERACTING GRAPHENE. We proved universality of the conductivity in the optical frequency range, in a fully non-perturbative fashion, thus solving a dispute present in the physics literature about the role of interactions on the value of the optical conductivity. We introduced and studied a fundamental lattice gauge theory model of graphene, we identified its dominant quantum instabilities at low temperatures and proposed a mechanism for the spontaneous generation of a Peierls-Kekule' distortion pattern.
QUANTUM SPIN SYSTEMS We proved the validity of the spin wave picture in the low temperature limit for the 3D quantum Heisenberg ferromagnet, thus solving a problem open since 80 years, i.e. since the fundamental works of F. Bloch on collective spin excitations in quantum magnets.
PATTERNED STATES We proved the emergence of periodic striped states in several models characterized by a competition between attractive short-range and repulsive long-range forces. We proved the existence of nematic order in a system of elongated molecules with repulsive interactions, thus rigorously confirming after 65 years the excluded volume mechanism proposed by L. Onsager.
CRITICAL SYSTEMS. We constructed the scaling limit for the energy sector of interacting 2D Ising models at criticality. We studied the height fluctuations for interacting dimers, establishing their convergence to those of the gaussian free field.
BOSE GASES We studied the 2D Bose gas and analyzed the stability of the condensate, which we proved for length scales smaller than the Ginzburg scale. We studied the effect of fast rotation on 3D Bose-Einstein condensates, establishing an explicit formula for the vortex distribution in rotating Bose gases.
formation in dipolar systems, Bose-Einstein Condensation in homogeneous Bose gases and the emergence of conformal symmetry in 2D systems at criticality. While apparently very different, these problems share common underlying features, such as the spontaneous emergence of ordered structures and hidden symmetries at very large length scales, induced by the "constructive interference" of self-similar driving mechanisms acting in a multiscale fashion on very different length scales, starting from the microscopic scale, up to the macroscopic one. In the context of these problems, the project aimed at developing new methods based on the combination of advanced mathematical techniques such as Renormalization Group (RG), cluster expansion, localization and functional estimates, Reflection Positivity (RP), as a mean to unveil the hidden connections between these problems, and to allow further progresses in the study of collective phenomena in the classical and quantum realms.
The project produced several remarkable results on interacting Fermi and Bose gases, quantum spins, pattern formation and 2D systems at criticality. While focused on the development of mathematical methods for the rigorous foundation of these systems, the CoMBoS project, with its accomplishments, allowed to settle certain experimental observations, which for a while did not have a clear theoretical explanation: I refer here, in particular, to the universality of the optical conductivity in single-layer graphene.
The achievements of the CoMBos project have been accompanied by the development of some novel techniques, such as: (1) the systematic implementation of exact lattice Ward Identities within a rigorous RG scheme; (2) the combination of RP estimates with localization estimates, coarse graining and functional inequalities; (3) the combination of RG methods with discrete holomorphicity.
Here are some of the problems in mathematical condensed matter that the CoMBoS project solved, thus opening the way to further developments:
INTERACTING GRAPHENE. We proved universality of the conductivity in the optical frequency range, in a fully non-perturbative fashion, thus solving a dispute present in the physics literature about the role of interactions on the value of the optical conductivity. We introduced and studied a fundamental lattice gauge theory model of graphene, we identified its dominant quantum instabilities at low temperatures and proposed a mechanism for the spontaneous generation of a Peierls-Kekule' distortion pattern.
QUANTUM SPIN SYSTEMS We proved the validity of the spin wave picture in the low temperature limit for the 3D quantum Heisenberg ferromagnet, thus solving a problem open since 80 years, i.e. since the fundamental works of F. Bloch on collective spin excitations in quantum magnets.
PATTERNED STATES We proved the emergence of periodic striped states in several models characterized by a competition between attractive short-range and repulsive long-range forces. We proved the existence of nematic order in a system of elongated molecules with repulsive interactions, thus rigorously confirming after 65 years the excluded volume mechanism proposed by L. Onsager.
CRITICAL SYSTEMS. We constructed the scaling limit for the energy sector of interacting 2D Ising models at criticality. We studied the height fluctuations for interacting dimers, establishing their convergence to those of the gaussian free field.
BOSE GASES We studied the 2D Bose gas and analyzed the stability of the condensate, which we proved for length scales smaller than the Ginzburg scale. We studied the effect of fast rotation on 3D Bose-Einstein condensates, establishing an explicit formula for the vortex distribution in rotating Bose gases.