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arXiv:2609.37325

Boundary Condition dependent Universality Classes on a Hyperbolic Lattice

Anonymous, Debasish Banerjee, Arnab Kundu, Ritam Sinha

Published Sep 29, 2026• 0 citations
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Abstract

We study the ferromagnetic Ising model on finite hyperbolic tessellations of Euclidean AdS$_2$ with open and wired boundary conditions. Because a finite fraction of spins remains at the boundary as the lattice grows, these conditions select distinct thermodynamic behaviours. In the $\{5,4\}$ tessellation, Monte-Carlo simulations using efficient worm algorithms on open boundaries (OBC) yield a transition near $β_c J=0.632(3)$, with susceptibility data collapsing under scaling by the total number of spins $N$. The fitted finite-size exponents, $1/\barν \simeq 0.162$ and $γ/\barν \simeq 0.743$, differ substantially from the mean-field volume scaling. Wired boundaries (WBC), which correlate boundary spins, instead show a transition around $β_c J=0.348(4)$ to an ordered phase consistent with the mean-field exponents. While the mean-field criticality with WBC is consistent with previous studies and with the suppression of independent boundary fluctuations, the OBC exponents hint at the presence of a new universality class. Results from other tessellations support the robustness of the observed scaling with OBC. We discuss a possible route to interpolate between the different boundary conditions. Our findings show that boundary dynamics must be specified when characterizing critical behaviour on hyperbolic lattices.

DOI
10.48550/arxiv.2609.37325

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On the {5,4}\{5,4\}{5,4} hyperbolic lattice, the outermost generation holds 73\% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at Tc2=1.4782(15)J/kBT_{c2}=1.4782(15)J/k_{B}Tc2​=1.4782(15)J/kB​ without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at T∗≃1.67(3)T^{*}\simeq1.67(3)T∗≃1.67(3) inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between Tc2T_{c2}Tc2​ and T∗T^{*}T∗, the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that T∗T^{*}T∗ is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at Tpt=2.81(1)J/kBT_{pt}=2.81(1)J/k_{B}Tpt​=2.81(1)J/kB​, demonstrating that it remains accessible under an appropriate boundary perturbation.
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