Generalized Exact Holographic Mapping with Wavelets

TL;DR

This paper introduces a generalized holographic mapping using wavelets that preserves Hamiltonian forms and clarifies the emergence of bulk geometry, enhancing the connection between scale invariance, holography, and wavelet theory.

Contribution

It develops a new wavelet-based holographic mapping that maintains Hamiltonian structure and elucidates the geometric origin of the dual space.

Findings

Number of wavelet moments correlates with AdS space radius.

Mapping preserves a broad class of lattice Hamiltonians.

Proposed modifications for systems with Fermi pockets.

Abstract

The idea of renormalization and scale invariance is pervasive across disciplines. It has not only drawn numerous surprising connections between physical systems under the guise of holographic duality, but has also inspired the development of wavelet theory now widely used in signal processing. Synergizing on these two developments, we describe in this paper a generalized exact holographic mapping that maps a generic N-dimensional lattice system to a N+1-dimensional holographic dual, with the emergent dimension representing scale. In previous works, this was achieved via the iterations of the simplest of all unitary mappings, the Haar mapping, which fails to preserve the form of most Hamiltonians. By taking advantage of the full generality of biorthogonal wavelets, our new generalized holographic mapping framework is able to preserve the form of a large class of lattice Hamiltonians. By explicitly separating features that are fundamentally associated with the physical system from those that are basis-specific, we also obtain a clearer understanding of how the resultant bulk geometry arises. For instance, the number of nonvanishing moments of the high pass wavelet filter is revealed to be proportional to the radius of the dual Anti deSitter (AdS) space geometry. We conclude by proposing modifications to the mapping for systems with generic Fermi pockets.