Global surfaces of section for Reeb flows in dimension three and beyond

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written for Proceedings of the ICM, Rio 2018; in v3 we made minor additional corrections, updated references, added a reference to work of Lu on the Conley Conjectur

Competing Adiabatic Thouless Pumps in Enlarged Parameter Spaces

The transfer of conserved charges through insulating matter via smooth deformations of the Hamiltonian is known as quantum adiabatic, or Thouless, pumping. Central to this phenomenon are Hamiltonians whose insulating gap is controlled by a multi-dimensional (usually two-dimensional) parameter space in which paths can be defined for adiabatic changes in the Hamiltonian, i.e., without closing the gap. Here, we extend the concept of Thouless pumps of band insulators by considering a larger, three-dimensional parameter space. We show that the connectivity of this parameter space is crucial for defining quantum pumps, demonstrating that, as opposed to the conventional two-dimensional case, pumped quantities depend not only on the initial and final points of Hamiltonian evolution but also on the class of the chosen path and preserved symmetries. As such, we distinguish the scenarios of closed/open paths of Hamiltonian evolution, finding that different closed cycles can lead to the pumping of different quantum numbers, and that different open paths may point to distinct scenarios for surface physics. As explicit examples, we consider models similar to simple models used to describe topological insulators, but with doubled degrees of freedom compared to a minimal topological insulator model. The extra fermionic flavors from doubling allow for extra gapping terms/adiabatic parameters - besides the usual topological mass which preserves the topology-protecting discrete symmetries - generating an enlarged adiabatic parameter-space. We consider cases in one and three \emph{spatial} dimensions, and our results in three dimensions may be realized in the context of crystalline topological insulators, as we briefly discuss.Comment: 21 pages, 7 Figure

Emergent SU(N) symmetry in disordered SO(N) spin chains

Strongly disordered spin chains invariant under the SO(N) group are shown to display random-singlet phases with emergent SU(N) symmetry without fine tuning. The phases with emergent SU(N) symmetry are of two kinds: one has a ground state formed of randomly distributed singlets of strongly bound pairs of SO(N) spins (a `mesonic' phase), while the other has a ground state composed of singlets made out of strongly bound integer multiples of N SO(N) spins (a `baryonic' phase). The established mechanism is general and we put forward the cases of $\mathrm{N}=2,3,4$ and $6$ as prime candidates for experimental realizations in material compounds and cold-atoms systems. We display universal temperature scaling and critical exponents for susceptibilities distinguishing these phases and characterizing the enlarging of the microscopic symmetries at low energies.Comment: 5 pages, 2 figures, Contribution to the Topical Issue "Recent Advances in the Theory of Disordered Systems", edited by Ferenc Igl\'oi and Heiko Riege

Test of the Additivity Principle for Current Fluctuations in a Model of Heat Conduction

The additivity principle allows to compute the current distribution in many one-dimensional (1D) nonequilibrium systems. Using simulations, we confirm this conjecture in the 1D Kipnis-Marchioro-Presutti model of heat conduction for a wide current interval. The current distribution shows both Gaussian and non-Gaussian regimes, and obeys the Gallavotti-Cohen fluctuation theorem. We verify the existence of a well-defined temperature profile associated to a given current fluctuation. This profile is independent of the sign of the current, and this symmetry extends to higher-order profiles and spatial correlations. We also show that finite-time joint fluctuations of the current and the profile are described by the additivity functional. These results suggest the additivity hypothesis as a general and powerful tool to compute current distributions in many nonequilibrium systems.Comment: 4 pages, 4 figure

Highly-symmetric random one-dimensional spin models

The interplay of disorder and interactions is a challenging topic of condensed matter physics, where correlations are crucial and exotic phases develop. In one spatial dimension, a particularly successful method to analyze such problems is the strong-disorder renormalization group (SDRG). This method, which is asymptotically exact in the limit of large disorder, has been successfully employed in the study of several phases of random magnetic chains. Here we develop an SDRG scheme capable to provide in-depth information on a large class of strongly disordered one-dimensional magnetic chains with a global invariance under a generic continuous group. Our methodology can be applied to any Lie-algebra valued spin Hamiltonian, in any representation. As examples, we focus on the physically relevant cases of SO(N) and Sp(N) magnetism, showing the existence of different randomness-dominated phases. These phases display emergent SU(N) symmetry at low energies and fall in two distinct classes, with meson-like or baryon-like characteristics. Our methodology is here explained in detail and helps to shed light on a general mechanism for symmetry emergence in disordered systems.Comment: 26 pages, 12 figure

Effective response theory for zero energy Majorana bound states in three spatial dimensions

We propose a gravitational response theory for point defects (hedgehogs) binding Majorana zero modes in (3+1)-dimensional superconductors. Starting in 4+1 dimensions, where the point defect is extended into a line, a coupling of the bulk defect texture with the gravitational field is introduced. Diffeomorphism invariance then leads to an $SU(2)_2$ Kac-Moody current running along the defect line. The $SU(2)_2$ Kac-Moody algebra accounts for the non-Abelian nature of the zero modes in 3+1 dimensions. It is then shown to also encode the angular momentum density which permeates throughout the bulk between hedgehog-anti-hedgehog pairs.Comment: 7 pages, 3 figure