Holography for Ising spins on the hyperbolic plane

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

doi:10.1103/physrevd.106.054506

Cited by 9 publications

(12 citation statements)

References 14 publications

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“…This includes a generalization of previous results for entanglement entropy in aperiodic chains to a larger class of modulations [77,78]. We obtain a piecewise linear behavior of the entanglement entropy (57), with logarithmic enveloping functions (58). The proof we provide for these results is applicable to a given class of {p, q} inflation rules.…”

Section: Introductionmentioning

confidence: 60%

“…In particular, if we take M 1 = M 2 = M σ , where M σ is the substitution matrix of the inflation rule generating the original sequence, it is straightforward to verify that the entanglement entropy (57) reduces to the result of [77], which does not depend on the parity of n(L). A more detailed investigation of (57) shows that the dependence of S A on L occurs through n(L), which is a floor function. This leads to a piecewise linear behavior of the entanglement entropy, typical of aperiodic systems [77,78].…”

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

“…Nevertheless, it is possible to make contact between these two classes of systems through the following observation. The piecewise curve (57) has breaking points corresponding to L = Λ k , where Λ k is defined in (42)- (43). The sets of points {Λ 2k−1 } and {Λ 2k } uniquely determine two distinct logarithmic envelopes of (57) with equal coefficients, but different additive constant.…”

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

“…In Fig. 9 we show the entanglement entropy (57) with the envelopes (58) for our prime example of a 2cycle bond distribution attractor {6, 5}. In the next subsection we consider the aperiodicities that satisfy the requirements discussed above and are generated by σ {p,q} for some {p, q} and discuss how the corresponding c eff depends on p and q.…”

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

“…Aperiodic spin chains have attracted a lot of attention after the discovery of quasicrystals [47] and are very well-known in condensed matter theory [48][49][50][51][52][53]. Specifically, we start from regular hyperbolic tilings, which are canonical discretizations of hyperbolic space and have previously been considered in many studies of discrete hyperbolic geometry [15,19,21,32,[40][41][42][54][55][56][57][58][59]. These tilings are characterized by their Schläfli symbol {p, q}, denoting a tiling where q regular p-gons meet at each vertex.…”

Section: Introductionmentioning

confidence: 99%

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Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings

et al. 2022

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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.

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Section: Introductionmentioning

confidence: 60%

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

Section: Entanglement Entropy In Aperiodic Singlet Phasesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings

et al. 2022

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Statistical mechanics approach to the holographic renormalization group: Bethe lattice Ising model and p-adic AdS/CFT

Okunishi,

Takayanagi

2023

Progress of Theoretical and Experimental Physics

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The Bethe lattice Ising model —a classical model of statistical mechanics for the phase transition— provides a novel and intuitive understanding of the prototypical relationship between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary spin correlations by introducing the relation between the lattice network and an effective Poincare metric on a unit disk. We compare the Bethe lattice model in the high-temperature region with a scalar field in AdS2, and then discuss its more direct connection to the p-adic AdS/CFT. In addition, we find that the phase transition in the interior induces a crossover behavior of boundary spin correlations, depending on the depth of the corresponding correlation path.

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