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arXiv:2501.06287 · SciPost Physics

Boundary operator expansion and extraordinary phase transition in the tricritical O(N) model

Xinyu Sun, Shao-Kai Jian

Published Jun 27, 2025• 5 citations
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Abstract

We study the boundary extraordinary transition of a three-dimensional (3D) tricritical O(N) O ( N ) model. We first compute the mean-field Green’s function with a general coupling of |\vec \phi|^{2n} | ϕ → | 2 n (with n=3 n = 3 corresponding to the tricritical model) at the extraordinary phase transition. Then, using layer susceptibility, we obtain the boundary operator expansion for the transverse and longitudinal modes within the \epsilon=3 - d ϵ = 3 − d expansion. Based on these results, we demonstrate that the tricritical point exhibits an extraordinary transition characterized by an ordered boundary for any N N . This provides the first nontrivial example of continuous symmetry breaking in 2D in the context of boundary criticality.

DOI
10.21468/scipostphys.18.6.210

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We study the boundary extraordinary transition of a three-dimensional (3D) tricritical O(N) O ( N ) model. We first compute the mean-field Green’s function with a general coupling of |\vec \phi|^{2n} | ϕ → | 2 n (with n=3 n = 3 corresponding to the tricritical model) at the extraordinary phase transition. Then, using layer susceptibility, we obtain the boundary operator expansion for the transverse and longitudinal modes within the \epsilon=3 - d ϵ = 3 − d expansion. Based on these results, we demonstrate that the tricritical point exhibits an extraordinary transition characterized by an ordered boundary for any N N . This provides the first nontrivial example of continuous symmetry breaking in 2D in the context of boundary criticality.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4) y s = 0.86 ( 4 ) and \eta_{\parallel}=-0.33(1) η ∥ = − 0.33 ( 1 ) , which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.
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.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized for the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into the higher-dimensional boundary conformal field theories. In this work, we discover a direct special transition from the ordinary phase into an extraordinary phase on the surface of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmically decaying spin correlations in the 3D classical O(2) and O(3) models. The anomalous dimension of the surface antiferromagnetic order in the extraordinary phase η∥=−0.29(1)\eta_{\parallel}=-0.29(1)η∥​=−0.29(1) is independent of the surface coupling strength. The special transition point also has a new set of critical exponents, ys=0.82(4)y_{s}=0.82(4)ys​=0.82(4) and η∥=−0.304(5)\eta_{\parallel}=-0.304(5)η∥​=−0.304(5), which are distinct from the special transition of the classical O(3) model and indicates a new surface universality class of the 3D O(3) Wilson-Fisher theory.
We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.
Employing Wilson's renormalization group scheme, we investigate the critical behaviour of a modified Ginzburg-Landau model with a nonlocal mode-coupling interaction in the quartic term. Carrying out the calculations at one-loop order, we obtain the critical exponents in the leading order of , where ρ is an exponent occurring in the nonlocal interaction term and d is the space dimension. Interestingly, the correlation exponent η is found to be non-zero at one-loop order and the expansion corresponds to an expansion about the tricritical mean-field theory in three dimensions, unlike the conventional theory. The ensuing critical exponents are in good agreement with experimental values for samples close to tricriticality. Our analysis indicates that tricriticality is a feature only in three dimensions.
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.