https://www.researchsquare.com/article/rs-5433512/v1
Phys. Rev. Research (2026) https://doi.org/10.1103/b3py-tyv8
Ultrametricity is a fundamental mathematical concept that describes a particular metric space in which every triplet of points in the space forms an isosceles triangle. The ultrametric space differs from the usual Archimedean metric, where three points are allowed from any triangle.
Ultrametricity is the topology of hierarchical architectures. Examples can be found in taxonomy, where phylogenetic trees are ultrametric, mathematics with p-adic numbers, geography for measuring landscape complexity, and physics, where complex systems have intrinsically an ultrametric structure.
The Noble Prize Giorgio Parisi demonstrated this within the theory of spin glasses, where the overlap between spins exhibits ultrametricity, with the mathematical solution given by the full replica symmetry breaking.
An experimental demonstration of this is still lacking due to the difficulty of finding measurable physical observables.
In 2015, we introduced random lasers as photonic counterparts of spin glasses, and we demonstrated the replica symmetry breaking by directly measuring the overlap between spins, known as the order parameter in the description of glass phase transitions.
In the work, we clearly show the hierarchical organization of the overlap matrix reproducing the Parisi Ansatz, and we experimentally prove the ultrametric nature of the replica states.
For the first time, we measure the distance between any three replicas forming a triangle, and we report the growth of the distribution of isosceles tringles when the system enters the glassy regime. This is an unambiguous way to demonstrate ultrametricity and has been previously done only in numerical simulations.
In addition, from the hierarchical structure of the spin states, illustrated as dendrograms, and the distances between replicas, we attain the first topological energy landscape of a complex system from experiments.
The great potentiality of our research is the ability to access measurable spins from emission spectra and to quantify the overlap parameter. Random lasers are photonic spin glasses, as they manifest a clear phase transition from a paramagnetic ordered state to a glassy disordered one by increasing the system’s energy. Thanks to this powerful asset, we demonstrate the ultrametricity of the replica space. We report the experimental energy landscape with a topology that changes from a flat large basin to the coexistence of many metastable minima and the braking of ergodicity in the glassy state.
