Boundary criticality of the O(N) model in d = 3 critically revisited

Max A. Metlitski

paper

Boundary criticality of the O(N) model in d = 3 critically revisited

arXiv:2009.05119

Abstract

It is known that the classical $O (N)$ model in dimension $d > 3$ 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 $d = 3$ , it is less clear what happens to the extra-ordinary and special fixed points when $d = 3$ and $N \geq 2$ . Here we show that formally treating $N$ as a continuous parameter, there exists a critical value $N_{c} > 2$ separating two distinct regimes. For $N < N_{c}$ the extra-ordinary fixed point survives in $d = 3$ , albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of $lo g r$ . In particular, for $N = 2$ , starting in the surface phase with quasi-long-range order and approaching the bulk phase transition, the stiffness of the surface order parameter diverges logarithmically. For $N > N_{c}$ there is no fixed point with order parameter correlations decaying slower than power law; we discuss two scenarios for the evolution of the phase diagram past $N = N_{c}$ . Our findings appear to be consistent with recent Monte-Carlo studies of classical models with $N = 2$ and $N = 3$ . We also compare our results to numerical studies of boundary criticality in 2+1D quantum spin models.

v2. Added a discussion of classical Monte-Carlo results that have appeared after the first arXiv version. Interpretation of Monte-Carlo studies of quantum spin models updated. Normalization convention for boundary OPE coefficients changed to conform with recent CFT literature

Boundary criticality of the O(N) model in d = 3 critically revisited

Abstract

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