Central charges of aperiodic holographic tensor network models (1911.03485v4)

Abstract: Central to the AdS/CFT correspondence is a precise relationship between the curvature of an anti-de Sitter (AdS) spacetime and the central charge of the dual conformal field theory (CFT) on its boundary. Our work shows that such a relationship can also be established for tensor network models of AdS/CFT based on regular bulk geometries, leading to an analytical form of the maximal central charges exhibited by the boundary states. We identify a class of tensors based on Majorana dimer states that saturate these bounds in the large curvature limit, while also realizing perfect and block-perfect holographic quantum error correcting codes. Furthermore, the renormalization group description of the resulting model is shown to be analogous to the strong disorder renormalization group, thus giving the first example of an exact quantum error correcting code that gives rise to a well-understood critical system. These systems exhibit a large range of fractional central charges, tunable by the choice of bulk tiling. Our approach thus provides a precise physical interpretation of tensor network models on regular hyperbolic geometries and establishes quantitative connections to a wide range of existing models.

Central Charges of Aperiodic Holographic Tensor Network Models

The paper "Central charges of aperiodic holographic tensor network models" explores the intricate relationship between a holographic duality approach using tensor network models and AdS/CFT correspondence. The authors aim to establish that this relationship also holds in models based on regular bulk geometries, presenting an analytical derivation of maximal central charges observable in boundary states. Their focus on a new class of tensors formulated from Majorana dimer states reveals how these states saturate central charge bounds in conditions where curvature is significant. They further elucidate how these tensor network models form exact quantum error correcting codes and establish the relationship of this renormalization group description to strong disorder renormalization groups (SDRG).

Summary of Key Concepts and Results

1. AdS/CFT Correspondence and Central Charges: At the core of the paper is the substantial grounding of AdS/CFT correspondence, particularly how the characteristic curvature of AdS spaces correlates with central charges in associated conformal field theories (CFTs). Historically, this has been significant for theorizing how holographic duals of physical systems can be described.
2. Tensor Network Models: Tensor networks have become prominent in discretizing aspects of holography, an approach that coincides with the Ryu-Takayanagi formula for entanglement entropy. The authors employ a measure via hyperbolic tiling methods to approximate critical boundary states, a methodological pivot from generally continuous forms of holography.
3. Majorana Dimer States: Tensors based on Majorana dimers are introduced, which exemplify how such models can achieve analytical maximal central charges. This includes what is termed 'block-perfect' codes, representing a perfect holographic quantum error correcting code.
4. Numerical and Analytical Results: The paper showcases how suitable tensor networks on regular tilings can be computed efficiently and how these lead to boundary states distinguished by high fractional central charges. The numerical illustrations expose how geodesic lengths relate to entanglement across different vertex inflations within these models, confirming theoretical results.
5. Implications of Critical Systems: Through analysis, the paper suggests that these models represent a discretized version of aperiodic critical systems, a form that breaks translational invariance but exhibits disorder on all scales. This translates into deep connections with critical systems prominent in condensed matter physics, offering a novel exploration area within discrete holographic models.

Practical and Theoretical Implications

The implications of this research predominantly reside in better understanding models that exhibit holographic duality and quantum error correction mechanisms. The values derived from understanding central charges aid in illuminating properties of stable and efficient quantum systems, providing insights into constructing models for quantum computing. Moreover, the theoretical implications extend to entwining various branches of physics, from string theory and gravity to condensed matter systems, reinforcing the paper’s multifaceted approach toward analyzing holographic principles.

Future Directions

Looking ahead, the research paves the way for investigating diverse tensor network formulations that can be adapted to model physical phenomena in a more refined, accurate manner. Exploring other configurations beyond Majorana dimers may reveal additional configurations where holographic theories prove robust. Additionally, broadening the connections between SDRG and holographic systems can deepen our understanding of the entanglement properties and symmetry-breaking processes observed in critical systems. This interconnection between tensor network models and various quantum mechanical systems represents a fertile ground for future exploration and application within theoretical physics.