We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum case and for scalar modes, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass squared. We solve the Klein-Gordon equation both analytically and numerically for finite cutoff. We then numerically calculate the BF bound for both cases. The results agree and are independent of the specific tiling. We also propose a model for a hyperbolic electric circuit and find again numerical agreement with the modified BF bound. This circuit is readily accessible in the laboratory, allowing for the experimental verification of our results.

No abstract

As a new step towards defining complexity for quantum field theories, we consider Nielsen's geometric approach to operator complexity for the SU (N ) group. We develop a tractable large N limit which leads to regular geometries on the manifold of unitaries. To achieve this, we introduce a particular basis for the su(N ) algebra and define a maximally anisotropic metric with polynomial penalty factors. We implement the Euler-Arnold approach to identify incompressible inviscid hydrodynamics on the two-torus as a novel effective theory for the evaluation of operator complexity of large qudits. Moreover, our cost function captures two essential properties of holographic complexity measures: ergodicity and conjugate points. We quantify these by numerically computing the sectional curvatures of SU (N ) for finite large N . We find a predominance of negatively curved directions, implying classically chaotic trajectories. Moreover, the non-vanishing proportion of positively curved directions implies the existence of conjugate points, as required to bound the growth of holographic complexity with time.

We propose a new example of discrete holography that provides a new step towards establishing the AdS/CFT duality for discrete spaces. A class of boundary Hamiltonians is obtained in a natural way from regular tilings of the hyperbolic Poincaré disk, via an inflation rule that allows to construct the tiling using concentric layers of tiles. The models in this class are aperiodic spin chains, whose sequences of couplings are obtained from the bulk inflation rule. We explicitly choose the aperiodic XXZ spin chain with spin 1/2 degrees of freedom as an example. The properties of this model are studied by using strong disorder renormalization group techniques, which provide a tensor network construction for the ground state of this spin chain. This can be regarded as discrete bulk reconstruction. Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic Hamiltonian. For both approaches, a logarithmic growth of the entanglement entropy in the subsystem size is identified. The coefficients, i.e. the effective central charges, depend on the bulk discretization parameters in both cases, albeit in a different way.

We propose a covariant definition of the fractional Klein-Gordon equation with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional Klein-Gordon equation on AdS2+1, computing the explicit kernel representation of the fractional Laplace-Beltrami operator as well as the two-point propagator of the fractional Klein-Gordon equation. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.