Abstract
The study of critical quantum many-body systems through conformal field theory (CFT) is one of the pillars of modern quantum physics. Certain CFTs are also understood to be dual to higher-dimensional theories of gravity via the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. To reproduce various features of AdS/CFT, a large number of discrete models based on tensor networks have been proposed. Some recent models, most notably including toy models of holographic quantum error correction, are constructed on regular time-slice discretizations of AdS. In this work, we show that the symmetries of these models are well suited for approximating CFT states, as their geometry enforces a discrete subgroup of conformal symmetries. Based on these symmetries, we introduce the notion of a quasiperiodic conformal field theory (qCFT), a critical theory less restrictive than a full CFT and with characteristic multi-scale quasiperiodicity. We discuss holographic code states and their renormalization group flow as specific implementations of a qCFT with fractional central charges and argue that their behavior generalizes to a large class of existing and future models. Beyond approximating CFT properties, we show that these can be best understood as belonging to a paradigm of discrete holography.
Highlights
Quantum field theories constitute a central cornerstone of modern theoretical physics, describing a large part of physical phenomena from condensed matter to high-energy settings
We found that tensor networks on regular {n, k} tilings with equivalent tensors on all sites lead to boundary states that respect a discrete subset of conformal symmetries, invariance under local and global scale transformations, as well as an approximate translation invariance
States that are invariant under all discrete quasiperiodic conformal field theory (qCFT) transformations are associated with ground states, while excited states need only respect the asymptotic symmetries at small scales
Summary
Quantum field theories constitute a central cornerstone of modern theoretical physics, describing a large part of physical phenomena from condensed matter to high-energy settings. Further characterizing the resulting symmetries, we argue that these states realize instances of what we call a quasiperiodic conformal field theory (qCFT) with inherently discrete structure that exhibits discrete analogues of scale and translation invariance found in a more constrained continuum CFT. These boundary transformations follow from invariance transformations of the hyperbolic disk that preserve regular hyperbolic tilings, characterized by Fuchsian groups. Using examples from previous literature in which tensor network models with these symmetries have already been constructed, we illustrate the range of realizable qCFTs models
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