Critical phenomena in hyperbolic space

Mnasri, Karim; Jeevanesan, Bhilahari; Schmalian, Joerg

doi:10.1103/physrevb.92.134423

Cited by 6 publications

(8 citation statements)

References 31 publications

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“…So far, in this work, we analyzed the Ising model in 2D hyperbolic plane with Monte Carlo and high-temperature series expansion techniques. While our analysis confirms the mean-field nature of the phase-transition, our results are not compatible with the conjectured formulas for critical exponents given by the field theory calculations [8].…”

Section: The Ising Model In 3d Hyperbolic Spacecontrasting

confidence: 97%

“…From the finite size scaling analysis, we infer that the critical temperature is T c = 10.96 ± 0.01 and γ = 0.97 ± 0.02, β = 0.51 ± 0.04, which are close to the mean-field predictions. Comparing our results to those obtained by field theory (1/N ) computations [8], we see that the susceptibility exponent is not compatible with the field theory computations, who obtain γ = 2. On the other hand, the magnetization exponent, inferred from scaling relations, together with the field theory computations, agree.…”

Section: A Monte Carlo Analysiscontrasting

confidence: 90%

“…For instance, in contrast to the euclidean-space counterpart, the order-disorder phase-transition of the XY model on a 2D hyperbolic plane is driven by the fluctuations of topological defects, while the infrared properties of the spin-wave fluctuations are well-behaved [1]. Critical statistical mechanical systems show exponential decay of correlations [8] and new-phases with broken translational invariance can arise, which are absent in the flat-space counterparts. Mathematical proofs for the existence of such phases exist for models describing percolation [9][10][11] and ferromagnetic Ising models [12,13].…”

Section: Introductionmentioning

confidence: 99%

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Critical properties of the Ising model in hyperbolic space

2020

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The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two and three dimensional hyperbolic spaces using Monte Carlo and high and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the 'asymptotic' limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.

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Section: The Ising Model In 3d Hyperbolic Spacecontrasting

confidence: 97%

Section: A Monte Carlo Analysiscontrasting

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Critical properties of the Ising model in hyperbolic space

2020

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“…The real system can then be approximated and analyzed by introducing defects into the pristine hyperbolic idealization (see, e.g., [6] for a review). Other examples of this cross-fertilization include the control of infrared singularities in classical and quantum field theories in hyperbolic space [7], the anti-de Sitter/conformal field theory duality [8], phase transitions in curved spaces [9][10][11], and hyperbolic surface codes for quantum computation [12], among many others (see, e.g., [13] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Higher-dimensional Euclidean and non-Euclidean structures in planar circuit quantum electrodynamics

Saa,

Miranda,

Rouxinol

2021

Preprint

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We show that a recent proposal for simulating planar hyperbolic lattices with circuit quantum electrodynamics can be extended to accommodate also higher dimensional lattices in Euclidean and non-Euclidean spaces if one allows for circuits with more than three polygons at each vertex. The quantum dynamics of these circuits, which can be constructed with present-day technology, are governed by effective tight-binding Hamiltonians corresponding to higher-dimensional Kagomé-like structures (n-dimensional zeolites), which are well known to exhibit strong frustration and flat bands. We analyze the relevant spectra of these systems and derive an exact expression for the fraction of flat-band states. Our results expand considerably the range of non-Euclidean geometry realizations with circuit quantum electrodynamics.

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“…The role of a spatial hyperbolic geometry in disordered media is known to be of high relevance in the design of new metamaterials showing negative dielectric constant [34][35][36][37], as well as in the manipulation of complex light in optical metamaterials [38][39][40][41][42][43][44][45]. The hyperbolic space has recently been shown to influence critical phenomena [46]. The search for complex materials with zero or negative dielectric constant as been one of the road maps proposed for the amplification of the superconducting critical temperature to reach room temperature superconductivity [47][48][49][50][51][52][53].…”

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confidence: 99%

High-Temperature Superconductivity in a Hyperbolic Geometry of Complex Matter from Nanoscale to Mesoscopic Scale

Campi

Bianconi

2015

J Supercond Nov Magn

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While it was known that High Temperature Superconductivity appears in cuprates showing complex multiscale phase separation due to inhomogeneous charge density wave (CDW) order, the spatial distribution of CDW domains remained an open question for a long time, because of the lack of experimental probes able to visualize their spatial distribution at mesoscale, between atomic and macroscopic scale. Recently scanning micro X-ray Diffraction (SµXRD) revealed CDW crystalline electronic puddles with a complex fat-tailed spatial distribution of their size. In this work we have determined and mapped the anisotropy of the Charge Density Waves (CDW) puddles in HgBa 2 CuO 4+y (Hg1201) single crystal. We discuss the emergence of high temperature superconductivity in the interstitial space with hyperbolic geometry, that opens a new paradigm for quantum coherence at high temperature where negative dielectric function and interference between different pathways can help to raise the critical temperature. PACS numbers: 61.05.C-, 74.72.-h, 61.05.cf, 61.43 Organization of local lattice fluctuations and charge inhomogeneity determines intrinsic physical properties in new functional materials. This heterogeneity varies from the scale of microns to the atomic level [1-3]. Imaging structural fluctuations and inhomogeneity is an interdisciplinary fundamental issue for understanding function-structure relationship in complex materials and for the design of new functional materials.Nowadays, the improved X-ray optics and imaging techniques make possible to image the bulk structure at nanoscale and mesoscale that is the scale length between the atomic

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Critical phenomena in hyperbolic space

Cited by 6 publications

References 31 publications

Critical properties of the Ising model in hyperbolic space

Critical properties of the Ising model in hyperbolic space

Higher-dimensional Euclidean and non-Euclidean structures in planar circuit quantum electrodynamics

High-Temperature Superconductivity in a Hyperbolic Geometry of Complex Matter from Nanoscale to Mesoscopic Scale

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