QM3 Quantum Matter meets Maths

Planned sessions

Turbulent hydrodynamics in strongly correlated Kagome metals

A current challenge in condensed matter physics is the realization of strongly correlated, viscous electron fluids. These fluids are not amenable to the perturbative methods of Fermi liquid theory, but can be described by holography, that is, by mapping them onto a weakly curved gravitational theory via gauge/gravity duality. The canonical system considered for realizations has been graphene, which possesses Dirac dispersions at low energies as well as significant Coulomb interactions between the electrons. In this work, we show that Kagome systems with electron fillings adjusted to the Dirac nodes of their band structure provide a much more compelling platform for realizations of viscous electron fluids, including non-linear effects such as turbulence. In particular, we find that in stoichiometric Scandium (Sc) Herbertsmithite, the fine-structure constant, which measures the effective Coulomb interaction and hence reflects the strength of the correlations, is enhanced by a factor of about 3.2 as compared to graphene, due to orbital hybridization. We employ holography to estimate the ratio of the shear viscosity over the entropy density in Sc-Herbertsmithite, and find it about three times smaller than in graphene. These findings put, for the first time, the turbulent flow regime described by holography within the reach of experiments.

Reference

Quantum many-body scars: a new form of weak ergodicity breaking in constrained quantum systems

Recent experiments on large chains of Rydberg atoms [1] have demonstrated the possibility of realising one-dimensional, kinetically constrained quantum systems. It was found that such systems exhibit surprising signatures of non-ergodic dynamics, such as robust periodic revivals in global quenches from certain initial states. This weak form of ergodicity breaking has been interpreted as a manifestation of "quantum many-body scars" [2], i.e., the many-body analogue of unstable classical periodic orbits of a single particle in a chaotic stadium billiard. Scarred many-body eigenstates have been shown to exhibit a range of unusual properties which violate the Eigenstate Thermalisation Hypothesis, such as equidistant energy separation, anomalous expectation values of local observables and subthermal entanglement entropy. I will demonstrate that these properties can be understood using a tractable model based on a single particle hopping on the Hilbert space graph, which formally captures the idea that scarred eigenstates form a representation of a large $\operatorname{SU}(2)$ spin that is embedded in a thermalising many-body system. I will show that this picture allows to construct a more general family of scarred models where the fundamental degree of freedom is a quantum clock [3]. These results suggest that scarred many-body bands give rise to a new universality class of constrained quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Topological theory of non-Hermitian photonic systems

Recently, topological materials and topological effects have elicited a great interest in the photonics community [1]. While condensed-matter phenomena are traditionally described by Hermitian operators, the same is not true in the context of macroscopic electrodynamics where a dissipative response is the rule, not the exception. In this talk, I will discuss how to determine the topological phases of dissipative (non-Hermitian) photonic structures from first principles using a gauge-independent Green function [2, 3]. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. I apply the developed theory to nonreciprocal electromagnetic continua and photonic crystals, with lossy and or gainy elements. Furthermore, I discuss the validity of the bulk-edge correspondence in the non-Hermitian case.

[1] L. Lu, J. D. Joannopoulos, M. Soljačić, "Topological photonics", Nat. Photonics, 8, 821, (2014).

[2] M. G. Silveirinha, "Topological theory of non-Hermitian photonic systems", Phys. Rev. B, 99, 125155, 2019.

[3] F. R. Prudêncio, M. G. Silveirinha, First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals, arXiv:2003.01539

The insulating state of matter: a geometrical theory

The insulating versus conducting behavior of condensed matter is commonly addressed
in terms of electronic excitations and/or conductivity. At variance with such wisdom, W. Kohn hinted in 1964 that the insulating state of matter reflects a peculiar organization of the electrons in their ground state, and does not require an energy gap.
Kohn’s “theory of the insulating state” got a fresh restart in 1999; at the root of these
developments is the modern theory of polarization, developed in the early 1990s, and
based on a geometrical concept (Berry phase). Since insulators and metals polarize
in a qualitatively different way, quantum geometry also discriminates insulators
from conductors. A common geometrical “marker”, based on the quantum metric,
caracterizes all insulators (band insulators, Anderson insulators, Mott insulators,
quantum Hall insulators. . . ); such marker diverges in conductors.

Four-dimensional semimetals with tensor monopoles: from surface states to topological responses

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. A prominent example is provided by the chiral magnetic effect in three-dimensional Weyl semimetals, which stems from the chiral anomaly. Here, we reveal a distinct quantum effect, coined "parity magnetic effect", which is induced by the parity anomaly in a four-dimensional topological semimetal. Upon preserving time-reversal symmetry, the spectrum of our model is doubly degenerate and the nodal (Dirac) points behave like Z2 monopoles. When time-reversal symmetry is broken, while preserving the sublattice (chiral) symmetry, our system supports spin- 3/2 quasiparticles and the corresponding Dirac-like cones host tensor monopoles characterized by a Z number, the Dixmier-Douady invariant. In both cases, the semimetal exhibits topologically protected Fermi arcs on its boundary. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.

The goal of QM3 is to discuss recent topics in quantum matter from a mathematical perspective while building bridges between the physics and mathematics communities.

New QM3 site: https://qm3.tecnico.ulisboa.pt/

Organizers: Bruno Mera, João Pimentel Nunes, José Mourão, Pedro Ribeiro, Roger Picken, Vitor Rocha Vieira