37 research outputs found
A nodal domain theorem for integrable billiards in two dimensions
Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, , of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger equation (Helmholtz
equation) is separable admit trivial expressions for the number of domains.
Here, we discover that for all separable and non-separable integrable
billiards, satisfies certain difference equations. This has been
possible because the eigenfunctions can be classified in families labelled by
the same value of , given a particular , for a set of quantum
numbers, . Further, we observe that the patterns in a family are similar
and the algebraic representation of the geometrical nodal patterns is found.
Instances of this representation are explained in detail to understand the
beauty of the patterns. This paper therefore presents a mathematical connection
between integrable systems and difference equations.Comment: 13 pages, 5 figure
Nodal domains of the equilateral triangle billiard
We characterise the eigenfunctions of an equilateral triangle billiard in
terms of its nodal domains. The number of nodal domains has a quadratic form in
terms of the quantum numbers, with a non-trivial number-theoretic factor. The
patterns of the eigenfunctions follow a group-theoretic connection in a way
that makes them predictable as one goes from one state to another. Extensive
numerical investigations bring out the distribution functions of the mode
number and signed areas. The statistics of the boundary intersections is also
treated analytically. Finally, the distribution functions of the nodal loop
count and the nodal counting function are shown to contain information about
the classical periodic orbits using the semiclassical trace formula. We believe
that the results belong generically to non-separable systems, thus extending
the previous works which are concentrated on separable and chaotic systems.Comment: 26 pages, 13 figure
Enhanced thermal Hall effect in the square-lattice N\'eel state
Recent experiments on several cuprate compounds have identified an enhanced
thermal Hall response in the pseudogap phase. Most strikingly, this enhancement
persists even in the undoped system, which challenges our understanding of the
insulating parent compounds. To explain these surprising observations, we study
the quantum phase transition of a square-lattice antiferromagnet from a
confining N\'eel state to a state with coexisting N\'eel and semion topological
order. The transition is driven by an applied magnetic field and involves no
change in the symmetry of the state. The critical point is described by a
strongly-coupled conformal field theory with an emergent global
symmetry. The field theory has four different formulations in terms of
or gauge theories, which are all related by dualities; we relate all
four theories to the lattice degrees of freedom. We show how proximity of the
confining N\'eel state to the critical point can explain the enhanced thermal
Hall effect seen in experiment.Comment: 8+5 pages, 4+1 figure