Séminaires du LPTMC

Quantum chaos and complexity in Krylov space

The dynamics of a quantum system is confined to a subspace of the state or operator space, the Krylov space, which is often smaller than the full Hilbert/operator space. Krylov space methods therefore provide a compact and computationally efficient description of quantum evolution, which is particularly useful for nonequilibrium phenomena in many-body systems with large Hilbert spaces. In this talk, I will discuss a notion of complexity defined in Krylov space and its use as a probe of operator growth, quantum chaos, information scrambling, and state evolution in both closed and open quantum systems. Illustrative examples include quantum spin chains, Sachdev–Ye–Kitaev models, and random matrix theory. The talk is based on selected topics from the review article arXiv:2405.09628 (Physics Reports) and more recent developments.

Is the stability matrix of the FRG flow around a fixed point self-adjoint?

Abstract: In unitary conformal field theories, it has been shown that the spectrum of scaling dimensions is real and bounded from above. FRG analysis always reveals a discrete spectrum with real parts bounded from above, but complex eigenvalues are often obtained. The aim of this talk, which relates to ongoing work, is to analyse the extent to which these complex values are physical or spurious. We will review the main characteristics of the spectrum obtained in the Derivative Expansion of the FRG. We will then analyse whether the results obtained in unitary conformal field theories are consistent with these findings. Finally, we will prove that the LPA equation for any scalar theory with a real fixed point potential has a stability matrix operator which, when combined with an appropriate scalar product, is self-adjoint. This implies that, for eigenperturbations with bounded norm, have eigenvalues which form a discrete and real spectrum in LPA approximation (for real fixed points). Finally, we will discuss the extent to which this result can be generalised to more complex approximations.

Nonequilibrium nucleation theory for nonconserved fields: from active matter to population dynamics

Classical nucleation theory (CNT) describes the formation of a stable phase from a metastable one. In equilibrium systems, it quantifies the free-energy competition between a favorable bulk gain and an unfavorable interfacial cost. For systems without detailed balance, the corresponding nonequilibrium nucleation theory (NNT) was so far developed only for cases with a conserved order parameter, such as active fluid-fluid phase separation. Here we construct the NNT for systems with a (single, scalar) nonconserved order parameter. Unlike in the conserved case, the nucleation barrier controlling (noise-driven) droplet growth is profoundly altered by deviations in the interfacial density profile from the one arising during (deterministic) droplet relaxation. The barrier can nonetheless be analysed by carefully defining the reaction coordinate (droplet radius) to project out those deviations. We give explicit NNT predictions for models drawn from population dynamics and active matter, finding excellent agreement with numerical studies.

Quantum geometry: a white cane for unconventional and topological superconductivity ?

In this talk, I will present a recent preprint on the quantum geometry of superconductors [1]. I will begin by introducing quantum geometry as a general consequence of the U(1) gauge invariance of quantum mechanics, leading to different incarnations throughout physics (and beyond). I will then discuss the specific example of quantum geometry in the band theory of crystals, along with its physical interpretations. I will then summarize my PhD research, where I studied the influence of the normal state quantum geometry on the superconducting phase. From this discussion, I will then motivate the study of the quantum geometry of the superconducting state itself, notably regarding the limitations of classification schemes in topological superconductivity and experimental studies of unconventional superconductivity. I will then present the results of Ref. [1], where we determine sufficient conditions for the quantum geometry of the superconducting phase To exactly separate into a normal-state contribution and a "pairing" quantum geometry, driven by superconductivity. We then determine explicit expressions for the pairing quantum geometry of all pairings, including non-unitary triplet pairings. These results will hopefully serve as a foundation for geometry-based design rules for superconductors, and contribute to the experimental exploration of unconventional superconductivity.

[1] F. Simon, T. Bernat & A.M. Black-Schaffer, Composite quantum geometry of superconductors (2026). Arxiv. doi:10.48550/arXiv.2608.02434

Conformal symmetry as a guiding principle for approximation schemes in the functional renormalization group

Interest in conformal symmetry dates back at least fifty years, to the work of Migdal and others, who showed that it constrains the form of three-point correlation functions and connects naturally with critical phenomena. Since then, conformal symmetry has become a powerful tool for computing physical observables and, in some cases, for solving quantum field theories exactly, most notably in two dimensions.

More recently, conformal symmetry has begun to receive attention within the functional renormalization group (FRG). The study of its interplay with standard FRG approximation schemes, and in particular with the derivative expansion (DE), is even more recent and remains largely unexplored. In this talk, we explain how conformal symmetry is realized in the FRG and examine the consequences of employing the derivative expansion (DE). We then propose a framework for incorporating conformal symmetry systematically into the derivative expansion.