Memory Effects in the Nonequilibrium Critical Behavior of the Two-Dimensional XY Model in the Low-Temperature Berezinskii Phase

The Monte Carlo study of nonequilibrium memory effects in the two-dimensional pure and structurally disordered XY model in the low-temperature Berezinskii phase has been performed. Features of the correlation between memory and aging effects have been demonstrated. A qualitatively novel phenomenon for memory effects has been revealed: dynamic curves of the autocorrelation function in the thermal cycling time interval tend to the dynamic curves with the initial temperature. A unique implementation of memory effects has been demonstrated at both cooling and heating cycling of the system under the condition that cooling and heating temperatures are in the low-temperature Berezinskii phase. The influence of structural disorder on memory effects has been analyzed. It has been found that they are enhanced in the structurally disordered system owing to the enhancement of aging effects.

The current status of the statistical physics of phase transitions is determined primarily by the investigation of the nonequilibrium critical behavior of macroscopic bulk and low-dimensional systems [1–6]. This is due to their anomalously slow dynamics, which is characterized by the appearance of aging and memory effects, as well as by the violation of the fluctuation–dissipation theorem [2, 4]. Nonequilibrium aging effects are manifested in the slowing down of relaxation and correlation processes in the system with an increase in the age of a sample [1]. Aging effects in the critical relaxation of disordered statistical systems were studied in [6–8]. Memory effects in systems with slow dynamics are manifested in the return of time dependences of the characteristic of nonequilibrium relaxation to the initial dependences and states after a short-term action on the system [4]. The temperature action in the studies of memory effects in critical phenomena is chosen in the form of thermal cycling variation of the state of the system [9–11].

The reduction of the dimension of the system and passage to systems with continuous degeneracy are accompanied by the significant enhancement of fluctuation effects [12]. The two-dimensional XY model is particularly remarkable among the two-dimensional classical spin models of statistical mechanics [12, 13]. The Berezinskii–Kosterlitz–Thouless (BKT) topological phase transition occurs at the temperature T_BKT in this model [14–18]. The low-temperature Berezinskii phase at T < T_BKT is characterized by the appearance of a quasi-long-range order and by a continuous set of fixed points of the renormalization-group transformation [13, 15]; consequently, each temperature in the low-temperature phase up to the phase transition point T_BKT is an “analog” of the critical point. The nature of nonequilibrium relaxation processes in the two-dimensional XY model significantly depends on its initial state [19–21]: relaxation at evolution from the high- and low-temperature initial states is predominantly vortex and spin-wave, respectively.

Because of the continuous set of pseudocritical points in the low-temperature phase, the investigation of nonequilibrium processes in the two-dimensional XY model [4] is of particular interest in view of the possible manifestation of slow dynamics in the broad temperature range of $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}$ rather than only near the critical point T_c, as in the other classical models of statistical physics [6, 7, 9–11]. The predicted and observed low-temperature coarsening in systems with the long-range order at T < T_c [22] is characterized by a noticeably faster decrease in two-time characteristics—autocorrelation and response functions—with increasing observation time and by much weaker aging effects than those in these systems at T = T_c and the more so in the two-dimensional XY model with very slow dynamics and long-term aging effects at $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}$. These properties lead to hardly observable long-term aging effects in systems with the long-range order at cooling temperatures beyond the critical region. This feature of the low-temperature Berezinskii phase allowed the study of nonequilibrium aging effects and the violation of the fluctuation–dissipation theorem in the two-dimensional XY model taking into account the temperature factor [19–21]. The effect of structural disorder results in the pinning of vortices on structural defects in the two-dimensional XY model. The pinning of vortices in models with mobile defects can also lead to the inverse action on structural disorder, more precisely, to the nonequilibrium vortex annealing of structural disorder [23].

The appearance of memory effects at thermal cycling in the nonequilibrium critical behavior was studied for a number of classical models of statistical physics in [9–11]. However, these studies of a system with a single critical point T_c were limited to the action of cooling, which temporarily reduces the temperature of the system with respect to the critical temperature T_c because no memory effects appear in such systems under heating. In this work, we study memory effects in the nonequilibrium behavior of the two-dimensional pure and structurally disordered XY model at the relaxation of a system from the initial high-temperature state under both cooling and heating with respect to the temperature T in the low-temperature phase, which is a specific feature of this model. The choice of the initial state is related to the implementation of canonical aging in the nonequilibrium dynamics of the two-dimensional XY model under evolution from the high-temperature initial state [19, 21]. The relaxation of the system from the initial low-temperature state is accompanied by super- and subaging phenomena [20]; for this reason, the choice of such initial state significantly distorts the general picture of nonequilibrium memory effects and is not considered in this work.

Memory effects in the slow dynamics of the system are manifested primarily as the return of the relaxation characteristics to the initial values after a short-term external action; thermal cycling with both cooling and heating is used in this work as such external action. The main feature of the manifestation of memory effects in the two-dimensional XY model is the transition of processes of nonequilibrium relaxation of the system from one critical point to another. Correspondingly, it is expected that the system moving off the initial fixed point in the thermal cycling time interval approaches another fixed point corresponding to the thermal cycling temperature if it is in the low-temperature Berezinskii phase. Such a manifestation of nonequilibrium effects is inherent only in the two-dimensional XY model because of the existence of the low-temperature Berezinskii phase.

It is expected that nonequilibrium aging and memory effects will be connected to each other, which can lead to the slowing down of the recovery of the critical relaxation of the system after the action on the system. For this reason, memory effects are studied in this work by analyzing the two-time spin–spin autocorrelation function given by the expression [20]

$$\begin{array}{*{20}{l}} {C\left( {t,{{t}_{{\text{w}}}}} \right) = \left[ {\left\langle {\frac{1}{{p{{L}^{2}}}}\sum\limits_i \,{{p}_{i}}{{{\mathbf{S}}}_{i}}(t){{{\mathbf{S}}}_{i}}({{t}_{{\text{w}}}})} \right\rangle } \right],} \end{array}$$

(1)

where the angle and square brackets stand for statistical averaging and averaging over the different configurations of defects in the structurally disordered system.

Slow dynamic effects in the two-time dependence C(t, t_w) are mainly manifested in the aging regime at the characteristic times t – t_w ~ t_w [19–21]. For this reason, the beginning time t₁ of thermal cycling was chosen equal to the waiting time t₁ – t_w = t_w, and the thermal cycling duration was chosen t₂ – t₁ = t_w, 2t_w, and 3t_w. Thermal cycling was carried out both by reducing the temperature T of the system by ΔT = 0.5T under cooling and by heating by ΔT = T; i.e., the temperature of the system is changed in the thermal cycling time interval by a factor of 2. It is noteworthy that the manifestation of memory effects is expected only when the cooling, T, and thermal cycling, T + ΔT, temperatures are in the low-temperature Berezinskii phase with $T,\;T + \Delta T\;\leqslant \;{{T}_{{{\text{BKT}}}}}$.

In this work, the Hamiltonian $H[{\mathbf{S}}]$ of the two-dimensional structurally disordered XY model was chosen in the form

$$H\left[ {\mathbf{S}} \right] = - \frac{J}{2}\sum\limits_{\langle i,j\rangle } \,{{p}_{i}}{{p}_{j}}{{{\mathbf{S}}}_{i}}{{{\mathbf{S}}}_{j}},$$

(2)

where ${{{\mathbf{S}}}_{i}}$ is the classical planar spin with the unit length associated with the ith site of the two-dimensional square lattice with the linear size L and p_i is the occupation number of the ith site, i.e., p_i = 1 and 0 for the site with the spin and defect of the structure, respectively. Defects were distributed over the lattice uniformly with the probability 1 – p, where p is the spin concentration. Summation in Eq. (2) is performed over all pairs $\langle i,j\rangle$ of nearest neighbors. The temperature T was measured in units of the exchange integral $J > 0$.

The nonequilibrium critical behavior of the two-dimensional XY model from the initial high-temperature state is simulated in this work using the Metropolis algorithm, which implements the dynamics corresponding to the critical relaxation dynamics with the nonconserved order parameter [19]. Times were measured in units of the Monte Carlo step per spin (MCS/s); the Monte Carlo step includes $N = p{{L}^{2}}$ rotations of spins per unit time. We chose the spin concentrations p = 1.0 (pure system), 0.9, and 0.8 (structurally disordered system). For these spin concentrations, the temperatures T_BKT(p) of the BKT phase transitions were calculated quite accurately and were used in [21] to study in detail aging effects in the two-dimensional XY model.

The simulation of the system was started from the initial high-temperature state with the magnetization ${{m}_{0}} \ll 1$, which was prepared at the temperature ${{T}_{0}} \gg {{T}_{{{\text{BKT}}}}}(p)$. The temperatures of the system T/T_BKT(p) = 1.0, 0.5, 0.25, and 0.125 were taken in the range $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}(p)$ and waiting times were taken t_w = 100, 400, 1000, and 2000 MCS/s. The chosen observation time t – t_w = 10 000 MCS/s allowed us to study two-time dependences C(t, t_w) for the critical relaxation of the two-dimensional XY model for the main dynamic regimes t – t_w $\ll$ t_w and t – t_w $\gg$ t_w, between which the dynamic crossover region t – t_w ~ t_w is observed. The linear size of the system L = 256 was chosen much larger than the correlation length $\xi (t) \ll L$ at the considered times t, which makes it possible to reveal the asymptotic properties of the critical relaxation of the system [21]. The statistical averaging of dynamic dependences for the pure system was performed over 3000 statistical configurations. When simulating the structurally disordered system, averaging was performed over 1000 configurations of the distribution of the structural disorder in the system and over 10 statistical configurations for each impurity configuration with the spin concentration p.

Figure 1 shows the two-time dynamic dependences C(t, t_w) for the critical relaxation of the two-dimensional XY model with different spin concentrations p calculated taking into account thermal cycling processes. They demonstrate features of the effect of the reduction of the temperature of the system T = T_BKT(p) to the temperature T_BKT(p)/2 (Figs. 1a1–1a3) and its increase from T = T_BKT(p)/2 to the temperature T_BKT(p) (Figs. 1b1, 1b2). In particular, the cooling of the system first leads to an increase in C(t, t_w) compared to the value at T_BKT(p) and to successive approach to the C(t, t_w) values for the temperature T_BKT(p)/2 with increasing cooling time interval ${{t}_{2}}{\kern 1pt} - {\kern 1pt} {{t}_{1}}$ in units of t_w. Beginning with the time t₂ at which the temperature of the system returns to T_BKT(p), the autocorrelation function C(t, t_w) decreases and approaches the curve at T_BKT(p). On the contrary, the heating of the system results in the decrease in C(t, t_w) compared to the value at T_BKT(p)/2 and C(t, t_w) approaches C(t, t_w) at the temperature T_BKT(p). Beginning with the time t₂ at which the temperature of the system returns to T_BKT(p)/2, the autocorrelation function C(t, t_w) increases and tends to the curve at T_BKT(p)/2.

Fig. 1.

(Color online) Two-time autocorrelation function C(t, t_w) of thermal cycling critical relaxation in the two-dimensional XY model for spin concentrations p = 1.0 (pure system), 0.9, and 0.8 (structurally disordered system) for the waiting time t_w = 100 MCS/s (a1–a3) at cooling processes from the temperature T = T_BKT(p) to T_BKT(p)/2; (b1, b2) at heating processes from the temperature T = T_BKT(p)/2 to T_BKT(p); (b3) at heating processes from the temperature T = T_BKT(p = 1.0) to 2T_BKT(p = 1.0); and (c1–c3) C(t, t_w) with cut thermal cycling time intervals. For comparison, C(t, t_w) curves for systems without thermal cycling at temperatures T and 0.5T are shown. The comparison of the results presented in (b1), (b2) and (b3) demonstrates the disappearance of memory effects when the thermal cycling temperature leaves the low-temperature phase T + ΔT > T_BKT(p).

To reveal memory effects and to compare the dependences C(t, t_w) for the system with and without thermal cycling, temperature variation time intervals should be “cut” in C(t, t_w) (Figs. 1a, 1b) [9–11]. Such two-time dependences C(t, t_w) with cut thermal cycling time intervals are presented in Figs. 1c1–1c3 for the case of the cooling of the system from the temperature T = T_BKT(p) to T = T_BKT(p)/2. The presented results demonstrate features of the manifestation of memory effects: the autocorrelation function obtained from C(t, t_w) by subtracting the thermal cycling time interval ${{t}_{2}} - {{t}_{1}}$ from the observation time $t - {{t}_{w}}$ tends to return to the value at the cooling time ${{t}_{1}}$ and the C(t, t_w) curves for the thermally cycled system smoothly return to the initial curves at the temperature $T$ for the system without thermal cycling, which is accompanied by a transient temperature process. Figures 1c1–1c3 demonstrate that the C(t, t_w) curves for the pure model with p = 1.0 return to the curve at the cooling time ${{t}_{1}}$ for all considered cooling intervals, whereas the memory of the state at the cooling time ${{t}_{1}}$ with an increase in the defect concentration is restored only after shorter cooling time intervals $[{{t}_{w}},3{{t}_{w}}]$ and $[{{t}_{w}},2{{t}_{w}}]$ for the models with p = 0.9 and 0.8, respectively. For longer cooling time intervals, much longer observation times t – t_w are necessary for C(t, t_w) of the structurally disordered system to return to the value at the cooling time ${{t}_{1}}$. Thus, memory effects in the nonequilibrium critical behavior of the two-dimensional XY model are enhanced with increasing defect concentration: in this case, such a system not only “remembers” its state at the cooling time ${{t}_{1}}$ for a longer time but also requires a longer time to “return” to this state.

If the state of the two-dimensional XY model under heating enters the high-temperature phase with T > T_BKT(p), the autocorrelation function in this temperature range decreases exponentially with the time $C(t,{{t}_{w}})$ ~ $\exp [ - (t - {{t}_{{\text{w}}}}){\text{/}}{{t}_{{{\text{cor}}}}}]$, where ${{t}_{{{\text{cor}}}}}$ is the correlation time, and thermal cycling does not induce memory effects in C(t, t_w); i.e., it does not return to the state at the heating time ${{t}_{1}}$ and to C(t, t_w) at the initial temperature $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}(p)$. This is demonstrated in Fig. 1b3 for the calculated dependence C(t, t_w) in the pure model heated from the initial temperature T = T_BKT(p) to the temperature 2T_BKT.

To analyze the effect of the waiting time t_w and the cooling time interval on the appearance of memory effects, the two-time C(t, t_w) curves at different t_w values are plotted together in Figs. 2a1–2a3. According to the presented results, an increase in t_w slows down the transient relaxation process. At the chosen observation time t – t_w = 10 000 MCS/s, the C(t, t_w) curves completely coincide only for short waiting times t_w = 100 and 400 MCS/s. At long waiting times t_w, the C(t, t_w) curves with and without thermal cycling also approach each other but the chosen observation time t – t_w = 10 000 MCS/s is insufficient for their complete coincidence. We note that this slowing down is inherent in the approach of dynamic C(t, t_w) curves. The duration of the transient process in which the sharp drop is observed in C(t, t_w) increases insignificantly with t_w. On the logarithmic scale in the observation time t – t_w used in Figs. 2a1–2a3, this transient process becomes more abrupt; namely, the value ΔC(t, t_w) of the sharp decrease in the autocorrelation function C(t, t_w) increases. The expansion of the thermal cycling time interval (in units of t_w) also slows down the transient relaxation process and, thereby, the approach of C(t, t_w) curves with and without thermal cycling. This phenomenon is due to the influence of aging effects [19, 21] on the memory effects: the time required for the system to return to its initial state increases with the waiting time t_w.

Fig. 2.

(Color online) Two-time autocorrelation function C(t, t_w) of thermal cycling critical relaxation in the two-dimensional XY model with cut time intervals of cooling of the system (a1–a3) for different waiting times t_w at the spin concentration p = 0.9 and the initial temperatures of the system T = (a1) 1.0T_BKT(p), (a2) 0.5T_BKT(p), and (a3) 0.25T_BKT(p) and (b1–b3) for different spin concentrations p at the waiting time t_w = 400 MCS/s and the initial temperatures T = (b1) 1.0T_BKT(p), (b2) 0.5T_BKT(p), and (b3) 0.25T_BKT(p). For comparison, C(t, t_w) curves for systems without thermal cycling are shown.

In order to examine the effect of the temperature of the system T on memory effects, we compared the    two-time autocorrelation functions C(t, t_w) (Figs. 2a1–2a3) under cooling from different initial temperatures T. As seen in Figs. 2a1–2a3, the decrease in the temperature T slows down transient processes. As a result, the C(t, t_w) curves with and without thermal cycling approach each other at much longer observation times t – t_w. However, the duration of the transient process with the abrupt drop in C(t, t_w) decreases significantly with decreasing temperature T. This transient process becomes sharper with decreasing T owing to an increase in the value ΔC(t, t_w) of the sharp decrease in C(t, t_w). These features are due to the slowing down of relaxation processes in the two-dimensional XY model with decreasing temperature T in the low-temperature Berezinskii phase T < T_BKT(p), which was mentioned previously in [19, 21].

To separate the effect of the structural disorder on memory effects in the critical dynamics of the two-dimensional XY model, two-time autocorrelation functions C(t, t_w) for systems with different spin concentrations p are presented in Figs. 2b1–2b3. It is seen that the increase in the defect concentration (decrease in p) and the decrease in the temperature T in the low-temperature Berezinskii phase with $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}(p)$ strongly slow down relaxation processes, which enhances memory effects; i.e., the memory of the state at the cooling time ${{t}_{1}}$ is restored completely after longer cooling time intervals for structurally disordered models compared to the pure model. Furthermore, the increase in the defect concentration is accompanied by the slowing down of transient relaxation processes occurring in the system at times longer than the thermal cycling time interval. This slows down the approach of dynamic C(t, t_w) curves for systems with and without thermal cycling.

We separate segments corresponding to the longest considered cooling time interval 3t_w in the two-time dependence C(t, t_w) and compare the resulting curves in the cooling time interval with the calculated dependence C(t, t_w) at the cooling temperature 0.5T without the thermal cycling of the system for observation times beginning with the cooling time ${{t}_{1}}$. The resulting dynamic dependences for the autocorrelation function C(t, t_w) are presented in Fig. 3. It is seen in Figs. 3a1–3a3 that the curves corresponding to the dynamic dependences C(t, t_w) in the cooling time interval with the expansion of the cooling time interval, which is reached by the increase in the waiting times from t_w = 100 MCS/s to t_w = 2000 MCS/s, asymptotically approach the unperturbed dependences C(t, t_w) at the cooling temperature 0.5T. An important feature is that the decrease in the temperature of the system T accelerates the approach of C(t, t_w) curves in the cooling time interval to the dependences C(t, t_w) at a temperature of 0.5T, which are not perturbed by thermal cycling. The reason is the decrease in the temperature perturbation $\Delta T$ with a decrease in the temperature of the system T. With a further expansion of the cooling time interval, in fact, the waiting time t_w, one can expect the overlap of these curves and the implementation of the cooling time dynamics described by unperturbed dependences C(t, t_w). This effect is a specificity of the low-temperature Berezinskii phase distinguishing it from other statistical systems [9–11], where the system under the variation of the temperature does not leave the critical point but passes from one critical state to another. The results presented in Figs. 3b1–3b3 for different spin concentrations p demonstrate an important feature: the increase in the defect concentration in the structure leads to the strong slowing down of correlation processes in the system, which is manifested in the slowing down of a decrease in C(t, t_w) with increasing t_w at the cooling temperature 0.5T; as a result, the time of approach of the C(t, t_w) curves increases with increasing t_w. This is due to the enhancement of aging effects in structurally disordered systems compared to pure systems, as mentioned in [19, 21].

Fig. 3.

(Color online) Two-time autocorrelation function C(t, t_w) in a cooling time interval of 3t_w in comparison with C(t, t_w) at a cooling temperature of T/2 (a1–a3) at different waiting times t_w for the pure system with the initial temperatures T = (a1) 1.0T_BKT(p), (a2) 0.5T_BKT(p), and (a3) 0.25T_BKT(p) and (b1–b3) for systems with different spin concentrations p at the waiting time t_w = 100 MCS/s and the initial temperatures T = (b1) 1.0T_BKT(p), (b2) 0.5T_BKT(p), and (b3) 0.25T_BKT(p).

To summarize, the numerical study of the two-time autocorrelation function has revealed for the first time nonequilibrium memory effects in the two-dimensional XY model in the low-temperature Berezinskii phase at temperatures $T\;\leqslant \;{{T}_{{{\text{BKT}}}}}(p)$ with respect to thermal cycling processes. Their features fundamentally different from similar phenomena in other statistical systems are due to the continuous set of fixed points in the low-temperature phase. Under the temperature action, the system does not leave a critical state but passes to another critical state of the low-temperature phase, which ensures the stability of memory effects because the system remembers for a longer time the initial nonequilibrium state before thermal cycling. Features of the correlation between memory and aging effects have been revealed. Structural defects are decisive for the enhancement of memory and aging effects. The results obtained in this work significantly expand the understanding of slow dynamic effect in the nonequilibrium critical behavior of the two-dimensional XY model.