Holography on tessellations of hyperbolic space

Asaduzzaman, Muhammad; Catterall, Simon; Nelson, Roice; Unmuth-Yockey, Judah

doi:10.1103/physrevd.102.034511

Cited by 42 publications

(39 citation statements)

References 11 publications

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“…Given these developments, one also anticipates implementations of hyperbolic lattices using other metamaterial platforms such as photonic crystals ( 8 , 9 ). The above concrete realizations of hyperbolic lattices in the laboratory open up vistas for the exploration of quantum mechanics in (negatively) curved space, with possibly far-reaching implications for fundamental physics in the areas of string theory ( 10 – 12 ), quantum gravity ( 13 – 15 ), and quantum information ( 16 – 21 ). In the long-wavelength limit, the Hamiltonian of a quantum particle on a hyperbolic lattice reduces to the well-known Laplace–Beltrami operator on the Poincaré disk ( 22 , 23 ), whose spectrum is well understood.…”

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confidence: 99%

Automorphic Bloch theorems for hyperbolic lattices

Maciejko

Rayan

2022

Proc. Natl. Acad. Sci. U.S.A.

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Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.

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confidence: 99%

Automorphic Bloch theorems for hyperbolic lattices

Maciejko

Rayan

2022

Proc. Natl. Acad. Sci. U.S.A.

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“…Just as in the original Poincaré disk model, one can tessellate a timeslice of AdS spacetime; the resulting fractal pattern in the bulk of AdS breaks the continuous symmetries of the global conformal group 23 , 34 – 36 on the boundary. In 45 , it was shown that the holographic dictionary survives the truncation associated with approximating a continuous Poincaré disk model of AdS with a regular tessellation; moreover, as described in 26 , a tensor-network realization that imitates a tessellated Poincaré disk can be described using tensors at every vertex of the bulk. Conversely, edge tensors can be added to a tensor-network manifold that mimics the tessellated Poincaré disk, as in hyMERA.…”

Section: Quasiperiodic Boundaries and Variational Optimizationmentioning

confidence: 99%

Conformal properties of hyperinvariant tensor networks

Steinberg

Prior

2022

Sci Rep

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Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. Although this new class of tensor network shows much potential for simulating conformal field theories arising from hyperbolic bulk manifolds with quasiperiodic boundaries, many issues are unresolved. In this manuscript we analyze the challenges related to optimizing tensors in a hyMERA with respect to some quasiperiodic critical spin chain, and compare with standard approaches in MERA. Additionally, we show two new sets of tensor decompositions which exhibit different properties from the original construction, implying that the multitensor constraints are neither unique, nor difficult to find, and that a generalization of the analytical tensor forms used up until now may exist. Lastly, we perform randomized trials using a descending superoperator with several of the investigated tensor decompositions, and find that the constraints imposed on the spectra of local descending superoperators in hyMERA are compatible with the operator spectra of several minimial model CFTs.

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“…1 Details of the lattice construction can be 1 Regular two-dimensional hyperbolic geometry can be represented by Schläfli symbol, {p, q} where p represents the p-sided polygon used as a building block to create the lattice and q represents the coordination number of each vertex found in Ref. [4]. The use of the triangle symmetry group for the lattice construction was also emphasized by Brower et al in Ref.…”

Section: The Model and Bulk Phase Structurementioning

confidence: 99%

“…For example: in previous work [4], we studied a model of a massive free scalar field propagating on tessellations of two and three-dimensional hyperbolic space. Despite strong lattice artifacts associated with finite lattice spacing and finite volume, the boundary lattice theory displays the usual features of conformality, exhibiting power-law fall-off of boundary-to-boundary correlators with boundary distance, where the inferred scaling dimensions match precisely with continuum analysis.…”

Section: Introductionmentioning

confidence: 99%

Holography for Ising spins on the hyperbolic plane

Asaduzzaman,

Catterall,

Hubisz

et al. 2021

Preprint

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Motivated by the AdS/CFT correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary susceptibility which yields a scaling exponent consistent with the scaling dimension extracted from the boundary correlation function. This observation provides evidence that the connection between continuum boundary conformal symmetry and isometries of the bulk hyperbolic space survives for simple interacting field theories even when the bulk is approximated by a discrete tessellation.

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Holography on tessellations of hyperbolic space

Cited by 42 publications

References 11 publications

Automorphic Bloch theorems for hyperbolic lattices

Automorphic Bloch theorems for hyperbolic lattices

Conformal properties of hyperinvariant tensor networks

Holography for Ising spins on the hyperbolic plane

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