45 research outputs found
Time Quasilattices in Dissipative Dynamical Systems
We establish the existence of `time quasilattices' as stable trajectories in
dissipative dynamical systems. These tilings of the time axis, with two unit
cells of different durations, can be generated as cuts through a periodic
lattice spanned by two orthogonal directions of time. We show that there are
precisely two admissible time quasilattices, which we term the infinite Pell
and Clapeyron words, reached by a generalization of the period-doubling
cascade. Finite Pell and Clapeyron words of increasing length provide
systematic periodic approximations to time quasilattices which can be verified
experimentally. The results apply to all systems featuring the universal
sequence of periodic windows. We provide examples of discrete-time maps, and
periodically-driven continuous-time dynamical systems. We identify quantum
many-body systems in which time quasilattices develop rigidity via the
interaction of many degrees of freedom, thus constituting dissipative discrete
`time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron
Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333.
Submission to SciPos
Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
We consider infinite sequences of superstable orbits (cascades) generated by
systematic substitutions of letters in the symbolic dynamics of one-dimensional
nonlinear systems in the logistic map universality class. We identify the
conditions under which the topological entropy of successive words converges as
a double exponential onto the accumulation point, and find the convergence
rates analytically for selected cascades. Numerical tests of the convergence of
the control parameter reveal a tendency to quantitatively universal
double-exponential convergence. Taking a specific physical example, we consider
cascades of stable orbits described by symbolic sequences with the symmetries
of quasilattices. We show that all quasilattices can be realised as stable
trajectories in nonlinear dynamical systems, extending previous results in
which two were identified.Comment: This version: updated figures and added discussion of generalised
time quasilattices. 17 pages, 4 figure
Charge Ordering Geometries in Uniaxially-Strained NbSe
Recent STM experiments reveal niobium diselenide to support domains of
striped (1Q) charge order side-by-side with its better-known triangular (3Q)
phase, suggesting that small variations in local strain may induce a quantum
phase transition between the two. We use a theoretical model of the charge
order in NbSe, based on a strong momentum- and orbital-dependent
electron-phonon coupling, to study the effect of uniaxial strain. We find that
as little as anisotropic shift in phonon energies breaks the threefold
symmetry in favor of a 1Q state, in agreement with the experimental results.
The altered symmetries change the transition into the ordered state from
weakly-first-order in the 3Q case, to second order in the 1Q regime. Modeling
the pseudogap phase of NbSe as the range of temperatures above the onset of
long-range order in which phase coherence is destroyed by local phonon
fluctuations, we find a shortening of the local ordering wavevector with
increasing temperature, complementing recent X-ray diffraction observations
within the low-temperature phase.Comment: 5 pages, 3 figure
Charge Order in NbSe
We develop in detail a model of the charge order in NbSe deriving from a
strong electron-phonon coupling dependent on the ingoing and outgoing electron
momenta as well as the electronic orbitals scattered between. Including both
dependencies allows us to reproduce the full range of available experimental
observations on this material. The stability of both experimentally-observed
charge-ordered geometries (1Q and 3Q) is studied within this model as a
function of temperature and uniaxial strain. It is found that a small amount of
bulk strain suffices to stabilize the unidirectional order, and that in both
ordering geometries, lattice fluctuations arising from the strong
electron-phonon coupling act to suppress the onset temperature of charge order,
giving a pseudogap regime characterized by local order and strong phase
fluctuations.Comment: 18 pages, 22 figure
Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling
The decades-long search for a shape that tiles the plane only aperiodically
under translations and rotations recently ended with the discovery of the
`spectre' aperiodic monotile. In this setting we study the dimer model, in
which dimers are placed along tile edges such that each vertex meets precisely
one dimer. The complexity of the tiling combines with the dimer constraint to
allow an exact solution to the model. The partition function is
where is the
number of `Mystic' tiles. We exactly solve the quantum dimer (Rokhsar Kivelson)
model in the same setting by identifying an eigenbasis at all interaction
strengths . We find that test monomers, once created, can be infinitely
separated at zero energy cost for all , constituting a deconfined phase in
a 2+1D bipartite quantum dimer model.Comment: 7 pages, 4 figures, 1 tabl
Conformal Quasicrystals and Holography
Recent studies of holographic tensor network models defined on regular
tessellations of hyperbolic space have not yet addressed the underlying
discrete geometry of the boundary. We show that the boundary degrees of freedom
naturally live on a novel structure, a conformal quasicrystal, that provides a
discrete model of conformal geometry. We introduce and construct a class of
one-dimensional conformal quasicrystals, and discuss a higher-dimensional
example (related to the Penrose tiling). Our construction permits
discretizations of conformal field theories that preserve an infinite discrete
subgroup of the global conformal group at the cost of lattice periodicity.Comment: v1: 8 pages, 4 figures; v2: 9 pages, 4 figures, expanded Introduction
and Discussion, added references, matches version to be published in PR
Realizing Hopf Insulators in Dipolar Spin Systems
The Hopf insulator represents a topological state of matter that exists
outside the conventional ten-fold way classification of topological insulators.
Its topology is protected by a linking number invariant, which arises from the
unique topology of knots in three dimensions. We predict that three-dimensional
arrays of driven, dipolar-interacting spins are a natural platform to
experimentally realize the Hopf insulator. In particular, we demonstrate that
certain terms within the dipolar interaction elegantly generate the requisite
non-trivial topology, and that Floquet engineering can be used to optimize
dipolar Hopf insulators with large gaps. Moreover, we show that the Hopf
insulator's unconventional topology gives rise to a rich spectrum of edge mode
behaviors, which can be directly probed in experiments. Finally, we present a
detailed blueprint for realizing the Hopf insulator in lattice-trapped
ultracold dipolar molecules; focusing on the example of KRb, we
provide quantitative evidence for near-term experimental feasibility.Comment: 6 + 7 pages, 3 figure