This project aims to construct a rigorous, computable, and testable discrete holographic duality model within the CDGM framework. Based on the discretization of the {5,4} tiling of the two-dimensional hyperbolic plane, the bulk network is exactly mapped to a multi-scale entanglement renormalization ansatz (MERA) tensor network on the boundary, with explicit constructions of isometry tensors and disentanglers. Anticipated results include: the entanglement entropy of a boundary region satisfies the discrete RT formula with a relative error less than 0.5% in the thermodynamic limit; the boundary two-point correlation function exhibits power-law decay with a scaling exponent consistent with a free scalar conformal field theory. This preregistration establishes a timestamped record of the research plan; full derivations and numerical details will be presented in subsequent publications.
This project aims to construct a rigorous, computable, and testable discrete holographic duality model within the CDGM framework. Based on the discretization of the {5,4} tiling of the two-dimensional hyperbolic plane, the bulk network is exactly mapped to a multi-scale entanglement renormalization ansatz (MERA) tensor network on the boundary, with explicit constructions of isometry tensors and disentanglers. Anticipated results include: the entanglement entropy of a boundary region satisfies the discrete RT formula with a relative error less than 0.5% in the thermodynamic limit; the boundary two-point correlation function exhibits power-law decay with a scaling exponent consistent with a free scalar conformal field theory. This preregistration establishes a timestamped record of the research plan; full derivations and numerical details will be presented in subsequent publications.
Standard Ryu-Takayanagi (RT) bounds and Multiscale Entanglement Renormalization Ansatz (MERA) networks assume a costless, unitary mapping between continuous bulk geometry and discrete boundary information. In this Letter, we show that projecting continuous bulk fields into discrete boundary states introduces a mandatory topological phase-space friction governed by the Euler-Mascheroni constant γ. Normalizing this harmonic discretization residue across the geometric capacity of a two-dimensional boundary yields the exact holographic delamination invariant, Λ2=exp(γ/π2)≈1.060228. By demonstrating that discrete MERA layer scaling is the exact mathematical dual to the logarithmic AdS radial measure, we prove this additive topological penalty exponentiates into a strict multiplicative scaling of thermodynamic work. We construct non-unitary dissipative MERA disentangler operators (Υ^) that enforce a factorization bound on accessible boundary entropy Sacc. Acting as a covariant topological scalar, this invariant preserves the Equivalence Principle while establishing a fundamental thermodynamic and semantic limit on holographic state projection.
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error-correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
Modelling strongly correlated systems is an incredibly hard problem, but a very fruitful one when achieved. The main reason for its difficulty is that perturbation theory approaches do not work for certain interesting coupling regimes. However it is believed that a holographic duality would map models with strong coupling strengths into weakly coupled bulk regimes where our current tools do work. Interest in applications to condensed matter systems has been growing throughout the past decade Though questions remain about how to do it and if the duality is valid in a relevant regime.Recently, it has been proposed that the multi-scale entanglement renormalisation ansatz (MERA) m an efficient representation for ground states of local Hamiltonians on a lattice m realises at least some features of the AdS/CFT correspondence. For example, an one dimensional (1D) quantum lattice model described by the MERA has its description stored in the renormalisation group (RG) flow of the ground state, detailing the state at different length scales. When implemented, this flow is encoded in a two dimensional (2D) hyperbolic tensor network where the extra dimension is the aforementioned length scale of the lattice system. Specifically the ansatz is based on a real space RG transformation, known as entanglement renormalisation, that removes local entanglement before coarse-graining the state.To look at this I studied the MERA tensor network by adding additional degrees of freedom on each bond - a connection between separate tensors in the network - to obtain a physical geometry in 2D. The focus was to keep the bulk theory description in the tensor network regime to allow straightforward applications of numerical techniques so that this procedure can be applied to any quantum field theory regardless of coupling strength. To achieve this, myself and collaborators introduced the new degrees of freedom by inserting tensors into the original MERA network, qliftingq it to the bulk MERA.n We also extended the formalism to gauge fields by analysing the symmetric MERA - a MERA constructed out of tensors symmetric under the action of some group G.By making use of symmetric tensors and symmetric MERA the formalism naturally extends to give rise to emergent gauge fields in the bulk two dimensional state. This occurred due to symmetric tensor networks decomposing into magnetic and degeneracy components via Schur's lemma and restrictions on the magnetic components to give rise to what we call the gauge degrees of freedom. Using this it was possible to analyse gauge properties of the bulk, including: Wilson loops, entanglement between gauge and degeneracy degrees of freedom, and local gauge symmetries in the bulk. Finally we also demonstrate that it is possible to construct a Hamiltonian for the 2D bulk MERA which is local, constructed from vertex and loop operators, and symmetric under this gauge symmetry.As this project was motivated from a numerical perspective a major component of the work done was to generate a code for manipulating tensor networks. For tensor networks without symmetries this is simple, however for symmetric tensor networks this becomes quite complicated. Therefore this work also included the development of a library to numerically manipulate symmetric tensor networks, including anyonic symmetries such as the quantum SU(2) symmetry. Doing this we gained access to a complete family of unitary minimal model conformal field theories with which we could numerically test the correspondence.With the critical models tested I found that this procedure could replicate a number of properties predicted by the holographic dictionary. These included: global on-site symmetries corresponding to local on-site symmetries, the Ryu-Takayanagi formula relating entropy on the boundary to geometry in the bulk, and a correspondence between bulk and boundary operators.n
This paper reinterprets holographic duality as a geometric manifestation of information compression in the hierarchical structure of discrete constraint networks.The bulk constraint network forms a hierarchy through local constraint satisfaction,and the effective theory on the boundary emerges as the projection of constraintpropagation patterns onto the boundary subset. Tensor networks — particularlythe multi-scale entanglement renormalization ansatz — provide a computable andconcrete implementation of this correspondence. It is demonstrated that the intrinsic redundancy of the constraint network naturally yields an area-law bound forentropy, and its error-correcting code structure explains the microscopic origin ofsubregion duality and bulk locality. Furthermore, the dynamical evolution of bulkgeometry corresponds to an endogenous reconstruction process of the constraintnetwork, and the gravitational effects perceived by boundary observers are projection phenomena arising from the network’s pursuit of global consistency. Thisframework grounds the holographic principle in information theory and discretedynamics, and suggests possible generalizations to non-AdS geometries.
Discrete models of holographic dualities, typically modeled by tensor networks on hyperbolic tilings, produce quantum states with a characteristic quasiperiodic disorder not present in continuum holography. In this work, we study the behavior of XXZ spin chains with such symmetries, showing that lessons learned from previous non-interacting (matchgate) tensor networks generalize to more generic Hamiltonians under holographic disorder: While the disorder breaks translation invariance, site-averaged correlations and entanglement of the disorder-free critical phase are preserved at a plateau of nonzero disorder even at large system sizes. In particular, we show numerically that the entanglement entropy curves in this disordered phase follow the expected scaling of a conformal field theory (CFT) in the continuum limit. This property is shown to be non-generic for other types of quasiperiodic disorder, only appearing when our boundary disorder ansatz is described by a "dual" bulk hyperbolic tiling. Our results therefore suggest the existence of a whole class of critical phases whose symmetries are derived from models of discrete holography.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.