Related papers: Canonical methods of constructing invariant tori b…
Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modelled by vector bundles over Brillouin tori,…
In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax…
The method of constructing of extended phase space for singular theories which permits the consideration of covariant gauges without the introducing of a ghost fields, is proposed. The extension of the phase space is carried out by the…
We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We…
We present a general approach to the bulk-boundary correspondence of noninvertible topological phases, including both topological and fracton orders. This is achieved by a novel bulk construction protocol where solvable $(d+1)$-dimensional…
We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we…
In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider St\"ackel…
An exact invariant is derived for $n$-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special {\it Ansatz\/} for the…
The Hamilton-Jacobi method of constrained systems is discussed. The equations of motion of a singular system with time dependent constraints are obtained as total differential equations in many variables. The integrability conditions for…
A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically…
We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $\o_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a R\"ussmann transversality…
One of the most demanding calculations is to generate random samples from a specified probability distribution (usually with an unknown normalizing prefactor) in a high-dimensional configuration space. One often has to resort to using a…
The diagonalization of the metrical Hamiltonian of a scalar field with an arbitrary coupling with a curvature in N-dimensional homogeneous isotropic space is performed. The energy spectrum of the corresponding quasiparticles is obtained.…
We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori…
We give a proof of the KAM theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is…
We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in `isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $\mathcal{H}=\mathcal{H}_0+\varepsilon…
Topological phases of matter are often understood and predicted with the help of crystal symmetries, although they don't rely on them to exist. In this chapter we review how topological phases have been recently shown to emerge in amorphous…
We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a…
In this article, using the idea of toric degeneration and the computation of the full potential function of Hirzebruch surface $F_2$, which is \emph{not} Fano, we produce a continuum of Lagrangian tori in $S^2 \times S^2$ which are…
Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"),…