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We numerically study quenches from a fully ordered state to the ferromagnetic regime of the chiral $mathbb{Z}_3$ clock model, where the physics can be understood in terms of sparse domain walls of six flavors. As in the previously studied models, the spread of entangled domain wall pairs generated by the quench lead to a linear growth of entropy with time, upto a time $ell/2v_g$ in size-$ell$ subsystems in the bulk where $v_g$ is the maximal group velocity of domain walls. In small subsystems located in the bulk, the entropy continues to further grow towards $ln 3$, as domain walls traverse the subsystem and increment the population of the two oppositely ordered states, restoring the $mathbb{Z}_3$ symmetry. The latter growth in entropy is seen also in small subsystems near an open boundary in a non-chiral clock model. In contrast to this, in the case of the chiral model, the entropy of small subsystems near an open boundary saturates. We rationalize the difference in behavior in terms of qualitatively different scattering properties of domain walls at the open boundary in the chiral model. We also present empirical results for entropy growth, correlation spread, and energies of longitudinal-field-induced bound states of domain wall pairs in the chiral model.
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In this paper we study the driven critical dynamics in the three-state quantum chiral clock model. This is motivated by a recent experiment, which verified the Kibble-Zurek mechanism and the finite-time scaling in a reconfigurable one-dimensional array of $^{87}$Rb atoms with programmable interactions. This experimental model shares the same universality class with the quantum chiral clock model and has been shown to possess a nontrivial non-integer dynamic exponent $z$. Besides the case of changing the transverse field as realized in the experiment, we also consider the driven dynamics under changing the longitudinal field. For both cases, we verify the finite-time scaling for a non-integer dynamic exponent $z$. Furthermore, we determine the critical exponents $beta$ and $delta$ numerically for the first time. We also investigate the dynamic scaling behavior including the thermal effects, which are inevitably involved in experiments. From a nonequilibrium dynamic point of view, our results strongly support that there is a direct continuous phase transition between the ordered phase and the disordered phase. Also, we show that the method based on the finite-time scaling theory provides a promising approach to determine the critical point and critical properties.
The Berezinskii-Kosterlitz-Thouless (BKT) transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method. A classical analog of the entanglement entropy $S( L, T )$ is calculated for $L times L$ square system up to $L = 129$, as a function of temperature $T$. The entropy exhibits a peak at $T = T^*_{~}( L )$, where the temperature depends on both $L$ and the boundary conditions. Applying the finite-size scaling to $T^*_{~}( L )$ and assuming presence of the BKT transitions, the two distinct phase-transition temperatures are estimated to be $T_1^{~} = 0.70$ and $T_2^{~} = 0.88$. The results are in agreement with earlier studies. It should be noted that no thermodynamic functions have been used in this study.
Entropy plays a key role in statistical physics of complex systems, which in general exhibit diverse aspects of emergence on different scales. However, it still remains not fully resolved how entropy varies with the coarse-graining level and the description scale. In this paper, we consider a Yule-type growth model, where each element is characterized by its size being either continuous or discrete. Entropy is then defined directly from the probability distribution of the states of all elements as well as from the size distribution of the system. Probing in detail their relations and time evolutions, we find that heterogeneity in addition to correlations between elements could induce loss of information during the coarse-graining procedure. It is also revealed that the expansion of the size space domain depends on the description level, leading to a difference between the continuous description and the discrete one.
We consider quantum quenches in an integrable quantum chain with tuneable-integrability-breaking interactions. In the case where these interactions are weak, we demonstrate that at intermediate times after the quench local observables relax to a prethermalized regime, which can be described by a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. We present explicit expressions for the approximately conserved charges characterizing this ensemble. We do not find evidence for a crossover from the prethermalized to a thermalized regime on the time scales accessible to us. Increasing the integrability-breaking interactions leads to a behaviour that is compatible with eventual thermalization.
We predict the emergence of turbulent scaling in the quench dynamics of the two-dimensional Heisenberg model for a wide range of initial conditions and model parameters. In the isotropic Heisenberg model, we find that the spin-spin correlation function exhibits universal scaling consistent with a turbulent energy cascade. When the spin rotational symmetry is broken by an easy-plane exchange anisotropy, we find a dual cascade of energy and an emergent conserved charge associated to transverse magnetization fluctuations. The scaling exponents are estimated analytically and agree with numerical simulations using phase space methods. We also define the space of initial conditions (as a function of energy, magnetization, and spin number $S$) that lead to a turbulent cascade. The universal character of the cascade, insensitive to microscopic details or initial conditions, suggests that turbulence in spin systems can be broadly realized in cold atom and solid-state experiments.
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