Extraordinary-log surface phase transition in the three-dimensional $XY$ model

Abstract

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance $r$ as $g(r)\sim r^{2-d-\eta }$, with $d$ the spatial dimension and $\eta$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of O($N$) system. In this logarithmic universality, $g(r)$ decays in a power of logarithmic distance as $g(r)\sim (\ln r{)}^{-\hat{\eta }}$, dramatically different from the standard scenario. We explore the three-dimensional $XY$ model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with $L$ as $g(L)\sim (\ln L{)}^{-{\hat{\eta }}^{\prime }}$ as well as the $r$-dependent term $g(r)\sim (\ln r{)}^{-\hat{\eta }}$, with ${\hat{\eta }}^{\prime }\approx \hat{\eta }-1$. The critical exponent ${\hat{\eta }}^{\prime }$, characterizing the height of the plateau, obeys the scaling relation ${\hat{\eta }}^{\prime }=(N-1)/(2\pi \alpha )$ with the RG parameter $\alpha$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.

arXiv comment6+11 pages, 4+4 figures

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• A plane defect in the 3d O$(N)$ model
• Surface criticality of antiferromagnetic Potts model
• Quantum phase transition between topologically distinct quantum critical points
• Extraordinary-log Universality of Critical Phenomena in Plane Defects
• Surface critical properties of the three-dimensional clock model
• Special Transition and Extraordinary Phase on the Surface of a Two-Dimensional Quantum Heisenberg Antiferromagnet
• Tensor network Monte Carlo simulations for the two-dimensional random-bond Ising model
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• Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge
• Sublattice extraordinary-log phase and new special point of the antiferromagnetic Potts model
• BKT transitions of the XY and six-state clock models on the various two-dimensional lattices
• The ordinary surface universality class of the three-dimensional O($N$) model
• Measuring the Boundary Gapless State and Criticality via Disorder Operator
• Extraordinary surface critical behavior induced by symmetry-protected topological state of a two-dimensional quantum magnet
• Universal finite-size scaling in the extraordinary-log boundary phase of three-dimensional $O(N)$ model
• Boundary operator expansion and extraordinary phase transition in the tricritical O(N) model
• Boundary phase transitions of two-dimensional quantum critical XXZ model
• Boundary criticality for the Gross-Neveu-Yukawa models
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• Accurate boundary bootstrap for the three-dimensional O($N$) normal universality class
• Deep Boundary Perturbations at a Quantum Critical Point
• Long-Range Order in a Strictly Short-Range Quasi-2D XY Model: When Critical Fluctuations Matter