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arXiv:2111.03613 · Physical Review Letters

Boundary Criticality of the 3D O( N ) Model: From Normal to Extraordinary

Francesco Parisen Toldin, Max A. Metlitski

Published May 27, 2022• 53 citations
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Abstract

It was recently realized that the three-dimensional O(N) model possesses an extraordinary boundary universality class for a finite range of N≥2. For a given N, the existence and universal properties of this class are predicted to be controlled by certain amplitudes of the normal universality class, where one applies an explicit symmetry breaking field to the boundary. In this Letter, we study the normal universality class for N=2, 3 using Monte Carlo simulations on an improved lattice model and extract these universal amplitudes. Our results are in good agreement with direct Monte Carlo studies of the extraordinary universality class serving as a nontrivial quantitative check of the connection between the normal and extraordinary classes.

DOI
10.1103/physrevlett.128.215701

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It was recently realized that the three-dimensional O(N) model possesses an extraordinary boundary universality class for a finite range of N≥2. For a given N, the existence and universal properties of this class are predicted to be controlled by certain amplitudes of the normal universality class, where one applies an explicit symmetry breaking field to the boundary. In this Letter, we study the normal universality class for N=2, 3 using Monte Carlo simulations on an improved lattice model and extract these universal amplitudes. Our results are in good agreement with direct Monte Carlo studies of the extraordinary universality class serving as a nontrivial quantitative check of the connection between the normal and extraordinary classes.
This paper studies the critical behavior of the 3d classical O (N) ( N ) model with a boundary. Recently, one of us established that upon treating N N as a continuous variable, there exists a critical value N_c > 2 N c > 2 such that for 2 \leq N < N_c 2 ≤ N < N c the model exhibits a new extraordinary-log boundary universality class, if the symmetry preserving interactions on the boundary are enhanced. N_c N c is determined by a ratio of universal amplitudes in the normal universality class, where instead a symmetry breaking field is applied on the boundary. We study the normal universality class using the numerical conformal bootstrap. We find truncated solutions to the crossing equation that indicate N_c \approx 5 N c ≈ 5 . Additionally, we use semi-definite programming to place rigorous bounds on the boundary CFT data of interest to conclude that N_c > 3 N c > 3 , under a certain positivity assumption which we check in various perturbative limits.
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 .
Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .
It was recently found that the classical 3d O (N) ( N ) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as \langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q} ⟨ S → ( x ) ⋅ S → ( 0 ) ⟩ ∼ 1 ( log x ) q . This universality class exists for a range 2 ≤ N < N_c 2 ≤ N < N c and Monte-Carlo simulations and conformal bootstrap indicate N_c > 3 N c > 3 . In this work, we extend this result to the 3d O (N) ( N ) model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N ≥2 N ≥ 2 . We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = ∞ N = ∞ is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N 1 / N corrections. We study the “central charge” a a for the O(N) O ( N ) model in the boundary and interface geometries and provide a non-trivial detailed check of an a a -theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O (N)
We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision Monte Carlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naïve arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models.