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 =
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)
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.
While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter
Ms decays with imaginary time
τ as
Ms∝τ−β1/νz, where
β1 denotes the boundary order parameter exponent, and
ν and
z correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent
θ1, which is closely related to the critical initial slip behavior of
Ms characterized by the corresponding exponent
θ1′. In contrast to its positive bulk counterpart, the boundary initial-slip exponent
θ1′ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of
d-dimensional quantum phase transitions generally coincide with those of
(d+1)-dimensional classical phase transitions, we show that
θ1 does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.
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.
We study the critical behavior at the ordinary surface universality class of the three-dimensional
O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for
N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary transition.
We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find
yt1=0.718(2) and
yh1=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.