7 research outputs found
Synthetic Helical Liquids with Ultracold Atoms in Optical Lattices
We discuss a platform for the synthetic realization of key physical
properties of helical Tomonaga Luttinger liquids (HTLLs) with ultracold
fermionic atoms in one-dimensional optical lattices. The HTLL is a strongly
correlated metallic state where spin polarization and propagation direction of
the itinerant particles are locked to each other. We propose an unconventional
one-dimensional Fermi-Hubbard model which, at quarter filling, resembles the
HTLL in the long wavelength limit, as we demonstrate with a combination of
analytical (bosonization) and numerical (density matrix renormalization group)
methods. An experimentally feasible scheme is provided for the realization of
this model with ultracold fermionic atoms in optical lattices. Finally, we
discuss how the robustness of the HTLL against back-scattering and
imperfections, well known from its realization at the edge of two-dimensional
topological insulators, is reflected in the synthetic one-dimensional scenario
proposed here
Discrete deep structure
The discrete scale space representation L of f is continuous in scale t. A computational investigation of L however must rely on a finite number of sampled scales. There are multiple approaches to sampling L differing in accuracy, runtime complexity and memory usage. One apparent approach is given by the definition of L via discrete convolution with a scale space kernel. The scale space kernel is of infinite domain and must be truncated in order to compute an individual scale, thus introducing truncation errors. A periodic boundary condition for f further complicates the computation. In this case, circular convolution with a Laplacian kernel provides for an elegant but still computationally complex solution. Applied in its eigenspace however, the circular convolution operator reduces to a simple and much less complex scaling transformation. This paper details how to efficiently decompose a scale of L and its derivative t L into a sum of eigenimages of the Laplacian circ ular convolution operator and provides a simple solution of the discretized diffusion equation, enabling for fast and accurate sampling of L