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 $M^{2}_{s}\propto R^{(1+η_{s})/r_{s}}[\log(LR^{1/η_{s}})]^{-q}$ , where $R$ is the driving rate, $L$ the system size, $η_{s}$ the surface anomalous dimension, $r_{s}$ 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.
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.
The celebrated Kibble-Zurek mechanism (KZM) describes the scaling of physical quantities when external parameters sweep through a critical point. Boundaries are ubiquitous in real systems, and critical behaviors near the boundary have attracted extensive research. Different boundary universality classes, including ordinary, special, extraordinary, and surface transitions, have been identified. However, the driven critical dynamics near boundaries remains unexplored. Here, we systematically investigate the driven critical dynamics in various boundary universality classes of the Ising model in both two and three dimensions, and discover a wealth of dynamic scaling behaviors. We find that for heating dynamics in all boundary universality classes, as well as for cooling dynamics in special, extraordinary, and surface transitions, the dynamic scaling behaviors of the order parameter can be described by a normal generalization of the KZM, called boundary finite-time scaling (BFTS). In contrast, for cooling dynamics in ordinary transition, we discover an abnormal logarithmic scaling on the driving rate. Moreover, for the special transition, in addition to temperature driving, we also consider the driven dynamics by driving the surface couplings. For increasing the surface coupling across the special transition point along the line of the ordinary transition, the prerequisite of the KZM, which requires that the correlation length/time in the initial state to be short-ranged, breaks down. We develop a generalized BFTS for a nonequilibrium initial state characterized by the waiting time, or the ``age'', of the boundary. Possible generalizations are also discussed.
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.
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.
Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance r as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with d the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the O(N) system. In this logarithmic universality, g(r) decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional XY model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of g(r,L) has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with L as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the r-dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.
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 .
The Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide a foundational framework for driven critical dynamics, yet their predictive power has been largely confined to local observables. Here, we establish a universal finite-time scaling theory for the nonequilibrium dynamics of quantum entanglement. Using unbiased quantum Monte Carlo simulations, we investigate the corner entanglement entropy of (2+1)-dimensional interacting Dirac fermions driven from ordered phases toward a quantum critical point. We find that the corner entanglement accurately obeys a universal driven scaling governed by the driving rate and system size, persisting whether the initial ordered state is fully gapped or hosts gapless Goldstone modes. Crucially, this dynamical entanglement exhibits a logarithmic dependence on the driving rate, from which the universal corner coefficient of the underlying conformal field theory can be robustly extracted far from equilibrium. These results generalize the KZM from local observables to the intrinsic nonlocal quantum information measures, offering a practical blueprint for characterizing quantum criticality and entanglement on programmable quantum simulators.