Conformal Quasicrystals and Holography (1805.02665v2)

Abstract: 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.

Overview of Conformal Quasicrystals and Holography

The paper "Conformal Quasicrystals and Holography" by Latham Boyle, Madeline Dickens, and Felix Flicker, presents an innovative exploration into the discrete geometry underpinning holographic tensor network models. The paper unveils a novel structural concept, the conformal quasicrystal (CQC), which serves as a discrete model of conformal geometry at the boundary of hyperbolic space tessellations. This discovery has potential implications across theoretical physics, including fields such as quantum gravity, condensed matter physics, and quantum information theory.

Key Findings and Methodologies

Recognizing the absence of discrete boundary geometry in regular tessellations of hyperbolic spaces within holographic models, the authors introduce CQCs as a natural setting for boundary degrees of freedom. They define a class of one-dimensional CQCs, developing a systematic construction and addressing higher-dimensional examples like the Penrose tiling.

The authors employ a methodological approach involving vertex completion procedures applied to regular $\{p,q\}$ tessellations in hyperbolic geometry. This process generates a series of increasingly larger tile sets, leading to a limit which defines the CQC at the boundary. This construction ensures a discrete symmetry that aligns with conformal geometry, highlighting the infinite discrete subgroup symmetry preservation.

Numerical Insights and Contradictory Claims

The paper demonstrates that the largest eigenvalue associated with the inflation rule matrices for CQCs, $\lambda_{+}(p,q)$, is always an irrational Pisot-Vijayaraghavan (PV) number when the tessellation condition $\frac{1}{p} + \frac{1}{q} < \frac{1}{2}$ is satisfied. This numerical insight supports the notion that CQCs exhibit a crystalline form in their diffraction patterns despite their quasiperiodic ordering.

The framework contrasts the discrete boundary with periodic lattice assumptions previously utilized in condensed matter physics, suggesting a more fundamental discrete geometry inherent in holographic constructions. This position highlights a bold claim that the natural geometry for boundary conditions in holography is non-periodic and quasicrystalline.

Practical and Theoretical Implications

From a practical standpoint, CQCs offer key insights into the simulation of critical systems like conformal field theories (CFTs) and condensed matter systems at criticality. The emergent discrete conformal symmetry may result in more efficient simulations, potentially allowing more accurate analyses of near-critical phenomena.

Theoretically, the introduction of CQCs suggests new pathways to discretize CFTs, leveraging exact scale symmetry rather than translation symmetry, thus advancing the understanding of phase transitions. Furthermore, CQCs might form the basis for defining discrete versions of holographic dualities, thereby contributing significantly to quantum gravity research.

Speculation on Future Developments

Looking ahead, extending the concept of CQCs to higher dimensions, such as $\{3,5,3\}$ regular honeycombs in three-dimensional hyperbolic space, appears promising. Such endeavors may yield further insights into the relationship between discrete holographic networks and quasicrystalline structures.

Additionally, quasi-MERA tensor networks based on CQCs could revolutionize the description of quantum critical states, enhancing computational techniques and algorithms within quantum many-body systems. Such innovations hold the potential to reshape numerical capabilities in scale-invariant systems.

In conclusion, the paper by Boyle et al. provides a compelling contribution to the discourse surrounding holographic dualities, proposing the conformal quasicrystal as a crucial element in understanding the discrete boundary geometries involved. This work lays the groundwork for a transformative approach to the paper of quantum systems, suggesting significant advancements in both theoretical frameworks and practical methodologies.