Scaling behavior of Ising systems at first-order transitions

In a finite system of size $L$ in the low-temperature phase the variation of the magnetization $m$ with the external field $h$ is perfectly smooth: rather than an infinitely steep variation with $h$, $m$ has a large but finite slope. In the case of neutral boundary conditions preserving the $\mathbb{Z}_{2}$ inversion symmetry, like PBC and OBC, a FSS theory can be developed in the case of a field-driven FOT for a ferromagnetic system below the critical temperature [17, 10].

For $h\rightarrow 0^{\pm}$ and $L\gg\xi$, where $\xi$ is the correlation length of the low-temperature phase, the probability distribution of the magnetization $P_{L}(m)$ is a sum of two gaussians, centered at $m_{0}$ and $-m_{0}$. For $h\neq 0$ the probability distribution is again a sum of two gaussians, but now centered around the shifted values $\pm m_{0}+h\chi_{m}$, where $\chi_{m}$ is the magnetic susceptibility: in this case the weights of the two peaks are different, i.e. one of the phases is favored by $h$ according to its sign.

From $P_{L}(m)$ we can obtain the average magnetization

For small $h$, the relevant scaling variable in the 2D static case is

so the appropriate universal behavior is observed when $h\rightarrow 0$, $L\rightarrow\infty$ at fixed $r_{1}$. In particular, when the FSS limit is considered the magnetization per site $m$ becomes

If the value of $r_{1}$ is finite then $m\neq|m_{0}|$, indicating that both free-energy minima contribute to equilibrium properties, a sign of the fact that the system is always in the coexistence region.

The double-gaussian approximation is reasonable only near $m=\pm m_{0}$, while it gives an underestimate of the real value of $P_{L}(m)$ in the interval $-m_{0}\ll m\ll m_{0}$. In this region the probability distribution is dominated by configuration corresponding to the two-phase coexistence in the system. The correct value of $P_{L}(m)$ for $m\approx 0$ can be obtained by considering interface contributions in the free energy $\mathcal{F}_{L}(m)$ and then writing

where $Z_{L}$ is the partition function. Let us denote with

the free energy difference between the configuration characterized by the phase coexistence, i.e. $m\approx 0$, and the configuration characterized by the presence of a single phase, i.e. $m\approx m_{0}$. Then the interface contributions are [17]

in the $d$-dimensional case. These differences are computed taking into account the typical configurations of the system in the coexistence region, as we shall see in the next sections.

It is worth mentioning that in these specific cases there is no shift of the transition, due to inversion symmetry. If boundary conditions that explicitly break this symmetry are considered, the transition point is shifted to a non-trivial value $h(L)\propto L^{-1}$ [18].

We study the equilibrium scaling behavior of finite systems at magnetic FOTs, when the boundary conditions at two opposite sides of the system generate an interface. For the 2D Ising model, these boundary conditions are realised by considering OFBC along the $x$-direction, i.e. a column of positive spins (on the right) favors the phase with positive magnetization while a column of negative spins (on the left) favors the other one, and PBC in the $y$-direction.

We analyze the equilibrium scaling behavior of the average magnetization $\langle M\rangle$ in terms of the size $L$. As for PBC, these boundary conditions preserve the inversion symmetry $h\rightarrow-h$ and then we expect that the correct scaling variable for small $h$ is $r_{1}=hL^{2}$ in the 2D case.

We perform various Monte Carlo (MC) simulations with different fixed value of $r_{1}$ and fixed $T=0.8T_{c}$, using a standard Metropolis single-spin flip algorithm. If the scaling variable is correct we should observe that

where $g_{m}$ is a scaling function.

Data are shown in fig. 1 and we clearly see that the scaling is optimal. We can interpret this in terms of the interface generated by the boundary conditions: in the coexistence region the typical configuration has the phase with positive magnetization on the right side of the lattice, and negative magnetization on the left side, as in fig. 2.

Let us denote with $X_{i}$ the location of the interface along the $x$-axis and with $A_{\pm}$ the areas of positive/negative magnetization regions. Setting $x=0$ in the center of the lattice, we have that $X_{i}\in[-L/2,L/2]$. Furthermore

We are able to write the magnetization as a function of $A_{\pm}$, indeed

We can relate $m(a_{+})$ to the coordinate $X_{i}$: if we suppose $X_{i}>0$, i.e. that the interface is located in the right half of the lattice, then

therefore

Thus, considering also eq.(17), it follows that $X_{i}=\widetilde{g}_{m}(r_{1})$, where $\widetilde{g}_{m}$ is another scaling function such that $\widetilde{g}_{m}\propto g_{m}$.

The connection between $r_{1}$ and $X_{i}$ is represented by this proportionality: if we move towards positive values of $r_{1}$ the interface is moved to the left; conversely, if we move towards negative values of $r_{1}$ the interface will be moved to the right. In this sense the interface follows the scaling variable $r_{1}$.

The coexistence region for the 2D Ising model is represented by the line $h=0$, $T<T_{c}$ in the relative phase diagram. Close to this segment, physical observables show a scaling behavior in terms of $h$ and $L$ that depends on the boundary conditions. To extend FSS to the dynamic case it is necessary to identify the time scale of the dynamics. Recalling that in the static case $h\propto L^{-2}$ at fixed $r_{1}$, we note that $h\rightarrow 0$ in the scaling limit $L\rightarrow\infty$. This means that the system is in the coexistence region in this limit, thus as a relevant time scale we can consider the one that controls the large-time dynamic behavior for $h=0$.

If the considered boundary conditions are symmetric, for $T<T_{c}$ the largest relaxation times are associated with flips of the magnetization. In the PBC case the typical configuration is characterized by the presence of two interfaces separating the coexistent phases, while configurations with spherical droplets are unstable in this region [19, 20]. Since the time needed to observe a complete reversal of the magnetization is proportional to $e^{2\beta\kappa L}$ [21, 22], the correct time scale is

where $\alpha$ is an appropriate exponent and the factor of two in the exponent is due to the presence of two interfaces separating the phases.

The arguments used in the PBC case can be applied also for OBC; however these boundary conditions give rise to different configurations in the lattice. In particular, the typical configurations in the coexistence region are characterized by the presence of spherical domains or by the presence of a single planar interface. These two types of configurations are characterized by different values of the magnetization, and there is a critical value that separates them. In the 2D case, this value is [17]

where $M_{0}=\pm 1$ is the value of the magnetization in one of the pure phases of the system. We clearly see that, since $M_{0}\approx m_{0}$, in the region $\Lambda=[-0.841m_{0},0.841m_{0}]$ configurations characterized by the presence of a single interface should be energetically preferred.

Assuming that the relevant mechanism for the generation of configurations with two coexisting phases is the creation of domain walls parallel to the lattice axis, we therefore expect that the correct time scale for OBC is

and we define

as our dynamic scaling variable.

The behavior in this case is expected to be ruled by the same mechanism of the PBC case, but with different time scales since different configurations are promoted. For this reason, in the next section we briefly summarize what is know for PBC, in order to extend the DFSS also to the OBC case.

In the PBC case, physical observables such as $m_{r}(t,r_{1},L)$ and $T_{f}(\mu,r_{1},L)$ are expected to show a scaling behavior in terms of the scaling variables we have identified. In particular we expect

These scaling function can be exactly predicted by modeling the dynamics of the system as a simple two-level dynamics, provided that time scales of the order of $\tau(L)$ are considered. Indeed, in this case what we observe for a single dynamic history is that the system oscillates between the two Ising phases, characterized by the values of the spontaneous magnetization equal to $\pm m_{0}$. The fact that the time scales are of the order of $\tau(L)$ allows us to consider the flip of the magnetization essentially instantaneous.

The dynamics, assuming it is Markovian, is completely parametrized by the rates

being $P(\cdot)$ the probability of such a transition from a phase to the other one.
Considering different dynamic realizations of the process, it can also be shown that the average renormalized magnetization is given by

being the rates related by

Since in the considered limit the flips are instantaneous, $T_{f}(\mu,r_{1},L)$ is expected to be independent of $\mu$, i.e. $T_{f}(\mu,r_{1},L)\approx\tau_{f}(r_{1},L)$. Moreover, the rate $I_{+}$ and the FPT are related via

This allows us to write, in the scaling limit

where

and $g_{\tau}(r_{1})$ is a scaling function satisfying

These equations tell us two things: the first is that the time scale $\tau(L)$ can be evaluated once the mean FPT is known at fixed $\mu$ and $r_{1}$, while the second is that the behavior of the average renormalized magnetization is exponential in a proper variable, which is nothing but the time $t$ rescaled with the mean FPT.

This theory was succesfully analyzed numerically in Ref. [23]; however, we expect that the DFSS applies also in the OBC case, since these boundaries do not favor any specific phase of the system. In particular, we expect that the dynamics is again exponential, with a time scale that is a half of the one characterizing the PBC case. Moreover, we argue that the behavior of the average renormalized magnetization is again exponential, but in principle with a different scaling function.

To see if our predictions are correct, we perform MC simulations for different values of $L\in[20,40]$ and $r_{1}$, using again the Metropolis algorithm to implement a purely relaxational dynamics at fixed $T=0.9T_{c}$.

The mean FPT is independent of $\mu$ for $L\rightarrow\infty$, thus we can fix $\mu=0.9$ as our reference value.

We first observe what is the typical configuration in the coexistence region, i.e. when $\langle M(t)\rangle\approx 0$. An example is reported in fig. 3: we can see that opposite phases are separated by a single (approximately) planar interface, without spherical domains. Therefore on the basis of what obtained with MC simulations we can neglect spherical droplets and consider eq. (24) as the time scale for OBC.

Moreover, by looking at the MC histories, we observe that $M(t)$ oscillates between two phases that are characterized by a value of the magnetization which is slightly different from the one given by eq. (3) at the given $\beta=1/T$. This value $\tilde{m}_{0}$ turns out to be smaller than $m_{0}$, in particular $\tilde{m}_{0}\approx 0.81$ while $m_{0}\approx 0.895$ by using eq. (3). This is due to finite-size effects, since we are using values of $L$ up to $L=40$ in our simulations, while eq. (3) holds in the thermodynamic limit. Moreover, the fact that now OBC are considered renders more pronounced this discrepancy: the PBC case [23] is obviously different, since they minimize the effects due to the finiteness of the lattice.

We want to check the size dependence of this time scale. Numerically we compute $T_{f}(\mu=0.9,r_{1},L)$ for different values of $r_{1}$, precisely $r_{1}=5$, $r_{1}=7.5$ and $r_{1}=10$. Initially we fit these data to the unbiased ansatz

i.e. not taking into account the power-law corrections. If eq. (24) holds we should find a value $a\approx\kappa\beta$, which for the chosen value of $T$ is $\kappa\beta\approx 0.189514$.

The results of the fits are reported in Table 1.

Concerning $a$, we observe a trend with the size $L$ of the system: when lower values of $L$ are discarded $a$ grows, except for the case $r_{1}=5$. Furthermore, it seems that there is a dependence also on the value of $r_{1}$: as it increases $a$ becomes smaller. All these values of $a$ are not distant from the theoretical value $\beta\kappa$, but they are not compatible with each other within the errors. Despite this, we can conclude that the analysis is consistent with an exponentially slow dynamics, with an exponent close to the expected value.

To check if there are significant power-law corrections to this exponential behavior we can fix $a=\beta\kappa$ in eq. (24) and try to make a fit of the form

Parameters for different values of $r_{1}$ are reported in Table 2. We observe again a trend with the size $L$: as the latter increases also $\alpha$ increases. It seems that for $\alpha$ these is a dependence on the value of $r_{1}$, as happens for the estimated exponent $a$. Moreover, big values of $\chi^{2}/\mathrm{ndof}$ suggest that there are significant corrections to scaling: therefore, it is difficult to give a final reliable estimate of $\alpha$. Despite that, if we look at fig. 4 is clear that the dynamics is consistent with a mainly exponential behavior with power-law corrections, even if here additional corrections to scaling are not considered.

Finally, we analyze the exponential nature of the renormalized magnetization. We use as time scale the estimated one for $\mu=0.9$ and consider the previous three values of $r_{1}$, i.e. $r_{1}=5,\;7.5,\;10$. The observed scaling behavior, reported in the figs. 5, 6, 7, is quite good, since the data collapse on a single curve. This confirms the exponential behavior of $m_{r}$ given by eq. (32) (black lines in the figures), with typical time scale given by $T_{f}(0.9,r_{1},L)$.

The equilibrium dynamic exponent $z$ of the relaxational dynamics is strictly related to the equilibrium large-$L$ behavior of the autocorrelation time $\tau$ of observables near the transition, i.e. $\tau\sim L^{z}$. We expect that at FOTs it depends on the boundary conditions. The case of PBC has already been analyzed [19], in particular $\tau$ is expected to exponentially increase with increasing $L$, i.e. $\tau\sim e^{\sigma L}$ where $\sigma$ is the interfacial free energy density and $z\rightarrow\infty$: this is related to the exponential increasing of the tunneling time between the coexisting phases at FOTs.

Here we consider OFBC: they are symmetric boundary coditions, as well as PBC, but promote explicitly the formation of an interface in the lattice. We thus expect that the behavior of the autocorrelation time drastically changes in this case: the dynamics near the transition is related to this interface, which moves within the lattice. This gives rise to a scaling law which is not an exponential but a power-law, i.e.

defining the equilibrium dynamic exponent of the Metropolis dynamics.

This behavior is analyzed for temperature-driven FOTs [12], in particular for the 2D Potts model at $T=T_{c}$ with $q=20$ and mixed boundary conditions: in this case the boundaries favor on one side the low-$T$ phase and at the opposite side the high-$T$ phase. Numerical results support such a power-law behavior for the integrated autocorrelation time. However we can not extend a priori this result to our case, since the type of transition is different (it is a field-driven FOT).

We numerically estimate $z$ by equilibrium MC simulations for various linear dimension $L\in[15,100]$. Details on the estimators used to estimate the autocorrelation time of the magnetization are reported in Appendix A.

In order to obtain an estimate of $z$ we make a log-log fit of the form

since the log-log plot of the data is a straight line, as shown in fig. 8.

In order to obtain a better estimate of the exponent we try to keep only some values of $L$, choosing only data with $L>L_{min}$. We observe that the values of the exponent obtained in this way are compatible within the errors. The results of the fit are reported in Table 3.

Given the values of $z$ in Table 3 we can conclude by considering

as our final estimate of the equilibrium dynamic exponent in the case of OFBC on a square 2D Ising lattice.

It is necessary to make some general comments on the value of $z$: its value should be intrinsically related to the interface dynamics within the lattice, thus we could try to extend the obtained result to other finite systems at magnetic FOTs with boundary conditions that favor the formation of such an interface. Furthermore, it should extend to the whole class of purely relaxational dynamics, including also the heat bath upgrading. As a consequence, different classes of dynamics may lead to other values of the dynamic exponent $z$.