Abstract The classical dimer model on the cubic lattice hosts a columnar ordered phase and a disordered Coulomb phase, separated by a continuous phase transition that lies beyond the conventional Landau–Ginzburg–Wilson (LGW) paradigm. While its equilibrium critical properties have been extensively studied, the nonequilibrium critical dynamics of this model—particularly in the short-time regime—remains largely unexplored. In this work, we investigate the short-time critical dynamics near the transition using large-scale Monte Carlo simulations. By quenching the system from both ordered and disordered initial states with vanishing initial correlation length, we analyze the scaling behaviors of the order parameter and its time correlation function in the short-time stage. From these scaling behaviors, we accurately determine the critical temperature T c = 0.672(1) and the static critical exponent β / ν = 0.581(5) according to the scaling theory of the short-time dynamics. These results are in excellent agreement with previous equilibrium studies. Moreover, we extract the dynamic critical exponent z = 1.92(1) and, notably, find a negative critical initial slip exponent θ = − 1.052(5). This unusual negative value contrasts sharply with the positive θ typically observed in conventional critical dynamics. We attribute this anomalous behavior to the combined effects of the emergent SO(5) symmetry at criticality and the local U(1) gauge constraint (Gauss law), which enforces a conserved diffusive dynamics and enhances fluctuations in the short-time regime. Our results provide the first comprehensive characterization of nonequilibrium short-time criticality in the three-dimensional dimer model, shedding new light on the universal dynamical features of phase transitions beyond the LGW framework.
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 propose a scaling theory for the universal imaginary-time quantum critical dynamics for both short and long times. We discover that there exists a universal critical initial slip related to a small initial order parameter
M0. In this stage, the order parameter
M increases with the imaginary time $\ensuremath{\tau}$ as $M\ensuremath{\propto}{M}_{0}{\ensuremath{\tau}}^{\ensuremath{\theta}}$ with a universal initial-slip exponent $\ensuremath{\theta}$. For the one-dimensional transverse-field Ising model, we estimate $\ensuremath{\theta}$ to be
0.373, which is markedly distinct from its classical counterpart. Apart from the local order parameter, we also show that the entanglement entropy exhibits universal behavior in the short-time region. As the critical exponents in the early stage and in equilibrium are identical, we apply the short-time dynamics method to determine quantum critical properties. The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm and topological phase transitions.
The dynamic critical behaviour of the three-dimensional classical Heisenberg model was studied by using short-time Monte Carlo simulations. The dynamic critical exponents z , θ and θ g , as well as the static critical exponents ν and β, were estimated by following the time evolution of the z -component of the total magnetization and its moments of higher order. The anomalous dimension of the magnetization x 0 was also calculated from a relation obtained through finite size scaling in the non-equilibrium stage. Afterwards, the local persistence exponent θ l was obtained by following the power law behaviour of the local persistence probability when the system was quenched from high temperatures to the null one ( T = 0). The time evolution of P l ( t ) was also analysed for T >0 and the expected exponential stretch was observed.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at
(h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length
L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by
3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size
L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
Nonequilibrium short-time dynamics of first-order phase transition in a driven-disordered system at zero temperature is investigated. In a random-bond Ising model under external field, the largest discontinuous jump of order parameter shows a power-law evolution in short times: Δ M ( t ) ∼ t θ . The scaling exponent θ is equal to ( d − β/ν)/ z , where d is the dimensionality; β, ν and z are the critical exponents of the system. θ is found to be a universal exponent for any metastable relaxation in the short-time regime. This investigation suggests that the short-time dynamics is valid for the first-order phase transition in the driven disordered system and the critical phenomenon of the disordered system can be understood in the framework of nonequilibrium dynamics.
The classical cubic-lattice dimer model undergoes an unconventional transition between a columnar crystal and a dimer liquid, in the same universality class as the deconfined quantum critical point in spin-1/2 antiferromagnets but with very different microscopic physics and microscopic symmetries. Using Monte Carlo simulations, we show that this transition has emergent SO(5) symmetry relating quantities characterizing the two phases. While the low-temperature phase has a conventional order parameter, the defining property of the Coulomb liquid on the high-temperature side is deconfinement of monomers, and so SO(5) relates fundamentally different types of objects. Studying linear system sizes up to L=96, we find that this symmetry applies with an excellent precision, consistently improving with system size over this range. It is remarkable that SO(5) emerges in a system as basic as the cubic dimer model, with only simple discrete degrees of freedom. Our results are important evidence for the generality of the SO(5) symmetry that has been proposed for the noncompact CP^{1} field theory. We describe an interpretation for these results in terms of a consistent hypothesis for the renormalization-group flow structure, allowing for the possibility that SO(5) may ultimately be a near-symmetry rather than exact.