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arXiv:2610.23221

Inducing Extraordinary-log Criticality through a Quantum Spin Chain

Grigorios Makris, Francesco Parisen Toldin, Stefan Weßel

Published Oct 16, 2026• 0 citations
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Abstract

We demonstrate the emergence of extraordinary-log criticality in a quantum many-body system by coupling a spin-$1/2$ Heisenberg chain to a two-dimensional quantum critical Heisenberg bilayer. Under the quantum-to-classical mapping, the chain acts as a plane defect in a three-dimensional critical environment for which the extraordinary-log universality class has recently been predicted. Using large-scale sign-problem-free quantum Monte Carlo simulations, we directly probe the renormalization-group flow of the effective defect coupling through the spin stiffness. We observe the characteristic logarithmic growth with linear system size $L$, corresponding to a stable weak-coupling fixed point and a running coupling $g(L)\sim 1/\log L$. Furthermore, the extracted step-scaling function quantitatively follows the universal renormalization-group flow predicted for the extraordinary-log universality class, including the leading and subleading universal coefficients. Our results establish extraordinary-log criticality in a genuine quantum many-body setting and show that its universal behavior persists despite the topological $θ$-term present in the low-energy field theory of the isolated spin chain.

DOI
10.48550/arxiv.2610.23221

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We demonstrate the emergence of extraordinary-log criticality in a quantum many-body system by coupling a spin-1/21/21/2 Heisenberg chain to a two-dimensional quantum critical Heisenberg bilayer. Under the quantum-to-classical mapping, the chain acts as a plane defect in a three-dimensional critical environment for which the extraordinary-log universality class has recently been predicted. Using large-scale sign-problem-free quantum Monte Carlo simulations, we directly probe the renormalization-group flow of the effective defect coupling through the spin stiffness. We observe the characteristic logarithmic growth with linear system size LLL, corresponding to a stable weak-coupling fixed point and a running coupling g(L)∼1/log⁡Lg(L)\sim 1/\log Lg(L)∼1/logL. Furthermore, the extracted step-scaling function quantitatively follows the universal renormalization-group flow predicted for the extraordinary-log universality class, including the leading and subleading universal coefficients. Our results establish extraordinary-log criticality in a genuine quantum many-body setting and show that its universal behavior persists despite the topological θθθ-term present in the low-energy field theory of the isolated spin chain.
The recent discovery of the extraordinary-log (E-Log) criticality is a celebrated achievement in modern critical theory and calls for generalization. Using large-scale Monte Carlo simulations, we study the critical phenomena of plane defects in three- and four-dimensional O(n) critical systems. In three dimensions, we provide the first numerical proof for the E-Log criticality of plane defects. In particular, for n=2, the critical exponent q[over ^] of two-point correlation and the renormalization-group parameter α of helicity modulus conform to the scaling relation q[over ^]=(n-1)/(2πα), whereas the results for n≥3 violate this scaling relation. In four dimensions, it is strikingly found that the E-Log criticality also emerges in the plane defect. These findings have numerous potential realizations and would boost the ongoing advancement of conformal field theory.
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Recent advances in boundary critical phenomena have led to the discovery of a new surface universality class in the three-dimensional O ( N ) model. The newly found “extraordinary-log” phase can be realized on a two-dimensional surface for N < N c , with N c > 3 , and on a plane defect embedded into a three-dimensional system, for any N . One of the key features of the extraordinary-log phase is the presence of logarithmic violations of standard finite-size scaling. In this work we study finite-size scaling in the extraordinary-log universality class by means of Monte Carlo simulations of an improved lattice model. We simulate the model with open boundary conditions, realizing the extraordinary-log phase on the surface for N = 2 , 3 , as well as with fully periodic boundary conditions and in the presence of a plane defect for N = 2 , 3 , 4 . In line with theory predictions, renormalization-group invariant observables studied here exhibit a logarithmic dependence on the size of the system. We numerically access not only the leading term in the β function governing these logarithmic violations, but also the subleading term, which controls the evolution of the boundary phase diagram as a function of N .
There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents yt=0.58(1){y}_{t}=0.58(1)yt​=0.58(1) and yh=1.690(1){y}_{h}=1.690(1)yh​=1.690(1), respectively, which are close to recent estimates from models with discrete spin variables. The existence of extraordinary-log universality is evidenced by the critical exponent $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}=0.58(2)$ from two-point correlation and the renormalization-group parameter $\ensuremath{\alpha}=0.28(1)$ from superfluid stiffness, which obey the scaling relation of extraordinary-log critical theory and recover the logarithmic finite-size scaling of critical superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results bridge recent observations of surface critical behavior in the classical statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701 (2021); Hu et al., Phys. Rev. Lett. 127, 120603 (2021); Parisen Toldin et al., Phys. Rev. Lett. 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun etetet al.al.al., arXiv:2205.00878 (2022)].
We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic XXZ model with easy-plane anisotropy, with particular emphasis on the surface critical behaviors of the (2+1)-dimensional quantum critical points of the model that belong to the classical three-dimensional O(2) universality class, for both S=1/2S=1/2S=1/2 and S=1S=1S=1 spins using quantum Monte Carlo simulations. We find completely different surface behaviors on two different surfaces of geometrical settings: the dangling-ladder surface, which is exposed by cutting a row of weak bonds, and the dangling-chain surface, which is formed by cutting a row of strong bonds along the direction perpendicular to the strong bonds of a periodic system. Similar to the Heisenberg limit, we find an ordinary transition on the dangling-ladder surface for both S=1S=1S=1 and S=1/2S=1/2S=1/2 spin systems. However, the dangling-chain surface shows much richer surface behaviors than in the Heisenberg limit. For the S=1/2S=1/2S=1/2 easy-plane model, at the bulk critical point, we provide evidence supporting an extraordinary surface transition with a long-range order established by effective long-range interactions due to bulk critical fluctuations. The possibility that the state is an extraordinary-log state seems unlikely. For the S=1S=1S=1 system, we find surface behaviors similar to that of the three-dimensional classical XY model with sufficiently enhanced surface coupling, suggesting an extraordinary-log state at the bulk critical point.
Quantum Heisenberg spin chains with random couplings and spin sizes are studied using a real-space renormalization group technique. These systems belong to a new universality class of disordered quantum spin systems realized in, e.g., ${\mathrm{Sr}}_{3}\mathrm{Cu}{\mathrm{Pt}}_{1\ensuremath{-}x}{\mathrm{Ir}}_{x}{\mathrm{O}}_{6}$. The low-energy fixed point is characterized by the formation of weakly coupled large effective spins. At low temperature TTT the entropy obeys a power law ${T}^{\ensuremath{\delta}}$ ($\ensuremath{\delta}\ensuremath{\approx}0.44$), and the susceptibility follows a Curie-like behavior.