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arXiv:2301.05728 · SciPost Physics

A plane defect in the 3d O(N) model

Abijith Krishnan, Max A. Metlitski

Published Sep 12, 2023• 34 citations
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Abstract

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)

DOI
10.21468/scipostphys.15.3.090

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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)
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 .
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.
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.
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.
Criticality on Rényi Defects at (2+1)DDD O(3) Quantum Critical Points
At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)DDD quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index nnn, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.
It is known that the classical O(N) O ( N ) model in dimension d > 3 d > 3 at its bulk critical point admits three boundary universality classes: the ordinary, the extra-ordinary and the special. For the ordinary transition the bulk and the boundary order simultaneously; the extra-ordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in d = 3 d = 3 , it is less clear what happens to the extra-ordinary and special fixed points when d = 3 d = 3 and N \ge 2 N ≥ 2 . Here we show that formally treating N N as a continuous parameter, there exists a critical value N_c > 2 N c > 2 separating two distinct regimes. For 2 \leq N < N_c 2 ≤ N < N c the extra-ordinary fixed point survives in d = 3 d = 3 , albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of \log r log r . For N > N_c N > N c there is no fixed point with order parameter correlations decaying slower than power law. We discuss several scenarios for the evolution of the phase diagram past N = N_c N = N c . Our findings appear to be consistent with recent Monte Carlo studies of classical models with N = 2 N = 2 and N = 3 N =