Brief description of the Project

The project will employ and develop tools from mathematical physics to study interacting quantum matter, focusing on its topological phases and its description through effective theories. Since the discovery of the integer and fractional quantum Hall effects and with the classification of topological insulators, the role of geometry and topology in condensed matter physics has been rapidly promoted to a fundamental paradigm, and it is nowadays a booming area of research. Mathematical physics, in particular the mathematical methods developed to treat quantum mechanics, provides the natural framework for the analysis of these phenomena. While we have a clear understanding of topological phases of matter in the 1-body approximation, the challenge for the near future is the comprehension of the interplay between topological effects and interactions among quantum particles. In typical condensed matter systems, the number of particles involved is huge, thus requiring the formulation of effective theories which describe their behavior in terms of fewer degrees of freedom. One of the most renowned examples is the cold Bose gas and its condensation, deeply related to the emergence of superfluidity and superconductivity in fermionic systems. From the point of view of mathematics, it appears that the techniques and effective theories used to describe many-body quantum systems will also play a crucial role in the analysis of topological phases. A clear mathematical description of these physical systems will in the long term also help in the development of new technologies, for example in quantum computation or solid-state memory devices.

Research lines

• Topology and transport in interacting and non-interacting quantum systems

• Spectral properties and effective dynamics for interacting many-body systems of bosons and fermions

Publications & preprints

• G. Basti, C. Boccato, S. Cenatiempo, and A. Deuchert. A new upper bound on the specific free energy of dilute Bose gases. Preprint at arXiv:2507.20877.

• G. Angelone, E. Ercolessi, P. Facchi, R. Maggi, G. Marmo, S. Pascazio, and F. V. Pepe. Electromagnetism: an intrinsic approach to Hadamard's method of descent. Preprint at arXiv:2506.05255.

• G. Angelone, M. Asorey, F. Ezquerro, and P. Facchi. Isospectrality and non-locality of generalized Dirac combs. Preprint at arXiv:2505.17920.

• C. Boccato, F. Pieroni, and D. Trevisan. On distances among Slater determinant states and determinantal point processes. Preprint at arXiv:2504.08971.

• N. Benedikter, C. Boccato, D. Monaco, and N. N. Nguyen. Derivation of Hartree-Fock dynamics and semiclassical commutator estimates for fermions in a magnetic field. Preprint at arXiv:2503.16001.

• W. Borrelli and U. Morellini. Global well-posedness in a Hartree-Fock model for graphene.  Preprint at arXiv:2411.14028. 

• M. Wesle, G. Marcelli, T. Miyao, D. Monaco, and S. Teufel. Exact linearity of the macroscopic Hall current response in infinitely extended gapped fermion systems. Preprint at arXiv:2411.06967. 

• M. Ferrero and D. Monaco. Effective quantum dynamics for magnetic fermions. Open Commun. Nonlinear Math. Phys. 4, 157–187 (2024). Preprint at arXiv:2406.15041.

• C. Boccato, J. Kerner, and M. Pechmann. Interacting many-particle systems in the random Kac–Luttinger model and proof of Bose–Einstein condensation. J. Math. Pures Appl. 189 (2024), 103594. Preprint at arXiv:2312.14357.

• V. Rossi and G. Panati. Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension. Preprint at arXiv:2407.14235.

• C. Boccato, A. Deuchert and D. Stocker. Upper Bound for the Grand Canonical Free Energy of the Bose Gas in the Gross–Pitaevskii Limit. SIAM J. Math. Anal. 56, Issue 2 (2024), 2611-266. Preprint at arXiv:2305.19173.

Events

• 2025/08/26-29. Seminar by Domenico Monaco @ Analytic and algebraic methods in physics XXII, Czech Technical University, Prague.

Effective dynamics for interacting magnetic fermions [abstract]

• 2025/06/16-20. PRIN 2022AKRC5P Workshop Topological Transport and Magnetic Matter @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

• 2025/02/26-30. Seminar by Gianluca Panati and contributed talks by Giuliano Angelone, Umberto Morellini and Gabriele Peluso @ Mathematics of Condensed Matter Systems, Politecnico di Milano.

G. Angelone: Isospectrality of generalized Dirac combs [abstract]

U. Morellini: Global well-posedness in a Hartree-Fock model for graphene [abstract]

G. Panati: Generalized Wannier bases and topological insulators: from periodic to non-periodic systems [abstract]

G. Peluso: Topological classification of chiral and particle-hole symmetric insulators in low dimension [abstract]

• 2025/02/09-14. Contributed talks by Giuliano Angelone, Umberto Morellini and Gabriele Peluso @ Mathematical Challenges in Quantum Mechanics, GSSI, L'Aquila (co-organized by Domenico Monaco).

G. Angelone: Classical echoes of quantum boundary conditions [abstract]

U. Morellini: The Pauli-Villars regularised free energy of Dirac’s vacuum in purely magnetic fields [abstract]

G. Peluso: Topological classification of chiral and particle-hole symmetric insulators in low dimension [abstract]

• 2024/10/09. Seminar by Giuliano Angelone @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

Boundary effects in quantum system [abstract]

• 2024/06/27. Seminar by Umberto Morellini @ Department of Mathematics, Università di Pisa.

The free energy of Dirac’s vacuum in purely magnetic fields [abstract]

• 2024/05/22. Seminar by Abdallah Maichine @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

Decomposition of operators commuting with magnetic translations and applications [abstract]

• 2024/03/13. Seminar by Jacob Shapiro @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

Classification of disordered insulators in 1D [abstract]

• 2024/03/06. Seminar by Giulia Basti @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

Upper bounds on the energy of dilute gases of hard-sphere bosons [abstract]

• 2024/03/04. Seminar by Joachim Kerner @ Department of Mathematics "Federigo Enriques", Università di Milano.

On Bose-Einstein Condensation in the Random Kac-Luttinger Model [abstract]

• 2024/02/28. Seminar by Horia Cornean @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

On the  self-consistent Landauer-Büttiker formalism [abstract]

• 2024/01/29-31. Workshop Physics and Topology @ Department of Physics, Sapienza Università di Roma.

• 2023/11/02. Seminar by Chiara Boccato @ Department of Mathematics "Guido Castelnuovo", Sapienza Università di Roma.

The free energy of the dilute Bose gas [abstract]