At the bulk upper critical dimension, marginally irrelevant interactions generate multiplicative logarithmic corrections to mean-field scaling. While these corrections are well understood for bulk observables, their consequences for boundary criticality, particularly for finite-size scaling, remain much less explored. Here we combine large-scale quantum Monte Carlo simulations with boundary renormalization-group analysis to study a (3 + 1)D O(3) quantum critical point. After verifying the known logarithmically modified bulk finite-size scaling, including the correlation-length scaling governed by the logarithmic finite-size exponent \hat{\coppa}, we tune the surface coupling to identify ordinary, special, and extraordinary boundary regimes. For the ordinary and special transitions, we derive logarithmic correction exponents and \hat{\coppa}-dependent finite-size scaling forms for boundary correlations, including results that have not been systematically established before. These predictions are quantitatively supported by Monte Carlo data. In the extraordinary regime, we find long-range surface magnetic order and a logarithmically enhanced surface-bulk correlation.
We investigate the quantum phase transition in an
S=1/2 dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks
O(N) symmetry with
N=3 in
D=3+1 dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization
ms and the N\'eel temperature
TN. We present a scaling ansatz for
TN, including logarithmic corrections, which agrees with our data and indicates exact linearity with
ms, implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted
N-independent leading and subleading logarithmic corrections in the size dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods, and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as
TlCuCl3.
We investigate nonequilibrium driven dynamics across the special surface phase transition in the three-dimensional classical Heisenberg model with open boundaries, where tuning the surface coupling gives access to an extraordinary-log boundary critical state characterized by logarithmic, rather than power-law, decay of correlations. Using Monte Carlo simulations, we realize four driving protocols: temperature heating and cooling across the special transition, and surface-coupling ramps from the ordinary and extraordinary-log critical states into the special point. For temperature-driven protocols, the surface order parameter obeys a generalization of the finite-time scaling (FTS) and the Kibble-Zurek mechanism. The central finding emerges when the system is driven from the extraordinary-log critical state: the large-rate scaling relation acquires a logarithmic correction and takes the novel form
Ms2∝R(1+ηs)/rs[log(LR1/ηs)]−q , where
R is the driving rate,
L the system size,
ηs the surface anomalous dimension,
rs the scaling dimension of
R, and
q the exponent governing the logarithmic boundary criticality. We demonstrate that this form follows from the general FTS framework by incorporating the logarithmic initial-state memory, and we achieve excellent data collapse over a wide range of system sizes and driving rates. Our results establish that extraordinary-log initial states alter nonequilibrium critical scaling, extending boundary FTS beyond conventional power-law initial conditions.
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) is independent of the surface coupling strength. The special transition point also has a new set of critical exponents,
ys=0.82(4) and
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 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.
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.
Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the
O(n) models on the simple-cubic lattice with
n=1, 2, and 3, i.e., the Ising,
XY, and Heisenberg models. For the critical couplings we find
Kc(n=2)=0.4541659 (10) and
Kc(n=3)=0.693003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be
yh1(o)=0.7374(15), 0.781 (2), and 0.813 (2) for
n=1, 2, and 3, respectively. Then we vary the surface coupling
K1 and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are
yt1(s)=0.715 (1) and
yh1(s)=1.636 (1) for the Ising model, and
yt1(s)=0.608 (4) and
yh1(s)=1.675 (1) for the
XY model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.