Interface localization in the 2D Ising model with a driven line
We study the effect of a one-dimensional driving field on the interface between two coexisting phases in a two dimensional model. This is done by considering an Ising model on a cylinder with Glauber dynamics in all sites and additional biased Kawasaki dynamics in the central ring. Based on the exact solution of the two-dimensional Ising model, we are able to compute the phase diagram of the driven model within a special limit of fast drive and slow spin flips in the central ring. The model is found to exhibit two phases where the interface is pinned to the central ring: one in which it fluctuates symmetrically around the central ring and another where it fluctuates asymmetrically. In addition, we find a phase where the interface is centered in the bulk of the system, either below or above the central ring of the cylinder. In the latter case, the symmetry breaking is ‘stronger’ than that found in equilibrium when considering a repulsive potential on the central ring. This equilibrium model is analyzed here by using a restricted solid-on-solid model.
pacs:05.50.+q, 05.70.Ln and 64.60.Cn
The properties of an interface separating two coexisting phases in two-dimensions has been studied extensively in the past. Many of these studies deal with the wetting transition between a ‘dry’ phase where the interface is attached to the boundary and a ‘wet’ phase where it is found in the bulk. This transition has been observed in various experimental settings (see review in ), as well as studied theoretically (see review in  and various studies in [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]). In most theoretical models considered, when the system is its wet phase, the interface performs a random walk in the bulk of the system in a potential that becomes flat in the thermodynamic limit. An interesting question, with significant experimental implications, is whether and how the interface can be confined to a certain region of the system or pinned to a specific line.
One of the prominent prototypical models studied in this context is the two-dimensional Ising model with Glauber (spin-flip) dynamics. The phase diagram of the equilibrium Ising model with cylindrical boundary conditions has been analyzed in the past using its exact solution [26, 3]. In the ordered phase and for boundary magnetic fields that are of opposite signs, the model exhibits an interface between a positive and a negative magnetic phases. When the weaker of the two boundary fields is of small enough amplitude, the model is in a ‘dry’ phase where the interface is attached to this boundary. As this boundary field increases, the model undergoes a second order wetting transition where the interface detaches from the boundary and moves to the bulk of the system. This transition has been studied analytically in more general geometries in [21, 27].
One natural way to confine the interface in the bulk is to modify the interaction strength of the model in a single line. This approach has been considered in [28, 29], by studying an Ising model on a cylinder where the interaction strength within the central ring of the cylinder, denoted by , is smaller than that in the bulk, denoted by . As often done in the study of wetting phenomena, the full two-dimensional model has been approximated in these studies by a one-dimensional restricted solid on solid (RSOS) model, which is more analytically tractable. The RSOS model describes the position of a sharp interface between pure positive and negative magnetic phases. The term ‘pure’ refers here to the absence of droplets of opposite sign in each phase. This description thus captures the low temperature behaviour of the full two dimensional Ising model. For and with boundary fields that are strong enough to induce a wet phase, it was found that the interface is pinned to the central ring and fluctuates symmetrically around it. Here we complement this analysis by considering also the case where . In this case we find a phase where the interface is repelled from the central ring and confined to one of its sides. Note that here and below we use the terms ‘dry’ and ‘wet’ to describe the effect of the boundary layers, as opposed to ‘pinned’ and ‘repelled’, which describe the effect of the central ring.
The aim of this paper is to study a different mechanism for confining an interface, obtained by exerting a local driving field, which results in a non-equilibrium steady-state. We study this mechanism by considering the example of the Ising model on a cylinder with Glauber dynamics (spin-flip) in the bulk and additional biased Kawasaki dynamics (spin-exchange) in the central ring of the cylinder.
In general, the addition of the biased Kawasaki dynamics to the equilibrium model results in a non-equilibrium steady-state measure which is unknown. However, as shown below, in the specific limit of fast drive and slow spin flips in the central ring of the cylinder, the phase diagram of the model can be studied analytically. Based on the approach presented in  and employed for the magnetization-conserving version of our model in , the dynamics of the magnetization in the central ring, , can be shown to correspond in this limit to an equilibrium random walk in a potential. This simplification is due to the fact that between every spin flip in the central ring, the bulk of the system above and below this ring relaxes to the steady-state of the two-dimensional Ising model. The effective potential, within which evolves, can be derived from the exact solution of the 2D Ising model, presented in . For the purpose of our analysis this solution, originally presented for the case of a single boundary field, is generalized below to the case of two boundary fields.
The steady-state distribution of the random walker yields the large deviation function of , from which the phase diagram of the model can be derived. In addition to the pinned phase and repelled phase observed in the equilibrium RSOS model, we find an asymmetrically-pinned phase where the sign-flip symmetry of is spontaneously broken. The interface is pinned in this case to the central ring but fluctuates asymmetrically around it.
One important difference between the equilibrium and non-equilibrium cases involves the nature of the repelled phase. In both cases, the repelled phase is composed of two equally probable states where the interface is either below or above the central ring. In equilibrium, the system switches between these two states on a time-scale which grows polynomially with the height and circumference of the cylinder. By contrast, in the driven case, this time scale is found to grow exponentially with the circumference of the cylinder. We thus refer to the latter phase as a ‘strongly repelled phase’. This strong repulsion is assumed to be due to long-range correlations cause by the driving field, which are known to exist generically in driven systems [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].
The asymmetrically-pinned phase has also been observed by Sadhu et al. who studied the same model considered here, only with conserving Kawasaky dynamics in the bulk . The original motivation for this work are experimental results on colloidal gas-liquid interface subjected to a shear flow parallel to the interface . These experiments have shown that the shear drive applied away from the interface strongly suppresses the fluctuations of the interface, making it smoother. As found below, the driving field, considered in , causes a spontaneous breaking of the sign-flip symmetry of , yielding a phase where the interface exhibits stronger fluctuations in one of the sides of the central ring. The analytic analysis of this conserving model was possible for strong drive, slow spin flips on the central ring and low temperature. In the present study, we analyze the non-conserving model in a similar limit but for finite temperatures. We are thus able to gain a better understanding of the fluctuations of the interface observed numerically in .
We complement the theoretical study by numerical simulation of the Ising model with a driven line. Interestingly, the two pinned phases and the strongly-repelled phase, found analytically for fast drive and slow spin-flips on the central ring, are found to exist significantly away from this limit.
The paper is organized as follows: In section 2 we define the Ising model with a driven line. In section 3.1 we discuss some of the properties of the equilibrium Ising model with , obtained from the exact solution of the 2D Ising model . These properties are used in the derivation of the non-equilibrium model and provide a context for our present work. The properties of the equilibrium model for are analyzed using a solid-on-solid model in section 3.2. The analysis of the non-equilibrium Ising model with a driven line is carried out in section 4.1. Section 4.2 presents results of Monte Carlo simulations which support some of the results of section 4.1. Concluding remarks are provided in section 5.
2 The Ising model on a cylinder with a driven line
The Ising-type model studied below is defined on a cylinder of size , illustrated in figure 1. A micro-configuration of the model is denoted by , where , and . Every spin in the system evolves by Glauber dynamics defined for spin as
where is the original spin-configuration obtained after flipping the spin in . Here denotes the local rate of the Glauber dynamics. These rates affect the steady-state of the model only when a driving field is present. In our analysis of the non-equilibrium steady-state of the model in section 4 we consider the limit where the dynamics on the central ring is slower, i.e. . The Hamiltonian in (1) is defined as
where denotes the coupling strength in the bulk, is the coupling strength within the central ring and are the boundary fields at the bottom and top of the cylinder, respectively (see figure 1). The periodic boundary condition is taken into account in (2) by considering .
In addition to the Glauber dynamics in (1), the model is driven out of equilibrium by Kawasaki dynamics on the central ring defined by the following rates:
where controls the asymmetry of the dynamics and is the rate of the Kawasaki dynamics. It is important to note that even for the model is out of equilibrium. The equilibrium Ising model is retained only when considering .
In section section 3 we discuss the properties of the equilibrium model (). In the case where the model reduces to the standard 2D Ising model which has been studied analytically by many authors. When the boundary fields are of opposite sign the model exhibits an interface between a plus phase and a minus phase. The model undergoes a second order wetting transition when is fixed and the boundary fields are increased ( or conversely when the boundary fields are fixed and is decreased). For , we study the model by considering the corresponding restricted solid on solid (RSOS) model, which is found to exhibit a phase where the interface is pinned to the central ring for and a phase where it is ‘weakly repelled’ from the central ring for . The terms weakly is clarified below. The non-equilibrium case is analyzed analytically in section 3.1 in the limit of fast driven and slow spin flips in the central ring, . The qualitative picture, which emerges from the analytic analysis, is confirmed numerically in section 4.2.
3 Analysis of the equilibrium Ising model
3.1 Wetting transition in the equilibrium model with
In this section we review the phase diagram of the equilibrium Ising model ( and ), as obtained using the exact solution of the model, presented in the treatise by McCoy and Wu . The authors have studied the Ising model on a cylinder with a single boundary field, e.g. for . In A we generalize their solution to the case of two boundary fields, where one finds a non-trivial phase diagram. This phase diagram can also be obtained based on the analysis of Abraham in . We chose, however, to use the solution in  since it readily yields an expression for the boundary magnetization (of the top and bottom rings), necessary for the analysis in section 4.1.
The partition function of the equilibrium Ising model on a cylinder is given by
Here is the well-known bulk-free-energy of the Ising model, is the free energy of each boundary in the absence of a magnetic field and is the leading order contribution in of the magnetic field to the free energy. These free energies are defined in terms of integrals over an angle , which result from the Fourier transform over the horizontal coordinate. More details are provided in section VI of  and in A. The bulk-free-energy is given by
where and is the solution with the larger magnitude of the following quadratic equation:
The boundary free energy in the absence of a boundary field is given in terms of by
The field-dependent boundary free energy, which is studied extensively below, is given by
In A we show that the free energy for and is given by
The fact that the term is simply a sum over those in (5) over the two fields, and , suggests that the two boundaries affect the system independently. However, there are coupling terms between the fields in the terms, which determine the overall magnetization of the system, i.e. the position of the interface.
In the analysis of the of the driven Ising model, defined in section 2, we use the expression for the boundary magnetization of the equilibrium model, obtained from the derivative of as
The properties of are analyzed in detail in Sec. VI.5 of  for the case of a single boundary field, obtained here by setting . In this case, below the Ising critical temperature, , the boundary magnetization, , exhibits a discontinuity at which signifies a sign flip of the bulk magnetization. This discontinuity is evident in figure 2, where is denoted by the thick solid line. McCoy and Wu have computed an additional solution for the boundary-magnetization using an analytic continuation of the integral in (3.1), which is given by
Here denotes either of the two solutions of the following quadratic equation:
This analytic continuation of , denoted by the two dashed lines in figure 2, is valid only in values of where is real, namely for , where
The subscript in signifies that is the magnetization when the system is in the dry phase and the interface is attached to the boundary. This interpretation is discussed below. Similarly is the field where the wetting transition occurs.
In , where the case of was studied, is interpreted as a metastable solution which forms a hysteresis loop of the boundary magnetization. The hysteresis loop is visible in figure 2 when combining the dashed and solid lines. When the system is on the dashed line the boundary field, , and bulk magnetization are of opposite signs. In this case there exists a phase with a lower free energy, where the bulk magnetization switches sign and the interface moves to the other boundary where . The transition between those two states involves creating an interface in the bulk of the system. The free energy barrier between the metastable state and the ground state can thus be interpreted as the free energy difference between the case where the interface is attached to a boundary with a field and the case where it is in the bulk. This barrier vanishes for , where the dashed and solid lines in figure 2 meet.
In the case of where there are two non-vanishing boundary fields, (10) implies that the magnetization in the top and bottom lines should still be given by either or with , respectively. However, the interpretation of each branch should be different in this case. When the two fields are of opposite sings, the state where the interface is attached to the field with the lower magnitude has the lower free energy out of the two dry states. We consider for concreteness the case where indeed and . Using the above interpretation of the free energy barrier of the metastable state, it is clear that for the state where the interface is attached to the lower boundary has a lower free energy than that of the wet state. In this case the magnetization of the lower boundary is given by . On the other hand when the state with the lowest free energy is that where the interface is found in the bulk. The boundary magnetization is given in this case by . One can therefore write the magnetization of the lower boundary as
Similarly, the expression for the upper-boundary magnetization can be obtained by exchanging and in the above equation.
Based on (15) we can draw the phase diagram of the Ising model on a cylinder with two boundary fields, shown for a fixed temperature below in figure 3. The thin solid line denote a second order wetting transition, between a state where the interfaces is attached to one of the boundaries and a state where it is found in the bulk of the system. The thick solid line denotes a first order transition where the interface shifts from one boundary to the other. It is important to note that the bulk magnetization changes smoothly across the second order transition line. Close to the transition line the interface is not in the middle of the cylinder, but closer to one of the boundaries (although still sites away from each). The exact position of the boundary can in principle be computed from the term in (10) which couples and explicitly.
One should note that the above interpretation of for the case of two boundary fields, from which (15) and the phase diagram given in figure 9 are derived, is done here on a heuristic level. A similar connection between and the wetting transition has already been mentioned in [44, 21]. A full mathematical proof of (15) based on the Pfaffian method employed by McCoy and Wu is beyond the scope of this paper.
It is also instructive to project the phase diagram of the model from the -space onto the plane, as shown in figure 4. The diagram shows the ordered-disordered second order transition at the bulk critical temperature and a wetting transition at . The wetting temperature, , has been derived in the past in . This transition has been studied analytically also for different geometries in [21, 27].
3.2 RSOS analysis for
In general, the solution of the equilibrium model with is unknown. However, as often done in the analysis of wetting phenomena, the qualitative behaviour of model can be analyzed by considering the corresponding restricted solid on solid (RSOS) model. Because RSOS models assume the two bulk phases to be pure (without droplets of phases of the opposite sign), they can be thought of as describing the low temperature behaviour of their corresponding two dimensional versions. In our case, the interface is described by an interface height variable, , such that all the spin above (below) are assumed to be of sign equal to that of (). The Hamiltonian of the RSOS model corresponding to the Hamiltonian in (1) is given by
and in addition whenever , which maintains the continuity of the interface. In this paper we are interested mainly in the dynamics of the central ring affects the wet phase. It is therefore sufficient to consider here the case where , in which the interface cannot be attached to the wall. The effect of the boundary fields can in principle be analyzed using the method described below.
Since the RSOS model is one-dimensional, its analysis can be done using the transfer matrix method. The transfer matrix corresponding to (16) is of dimension and is given by
where . The partition function of the model can be computed using the eigenvalues of , denoted by , as
In addition, we denote by the eigenvector corresponding to and by its components.
The properties of the different phases of the model can be understood by analyzing the probability distribution function (PDF) of , which is of course independent of . In order to compute , it is useful to define a basis of vectors, with , for which the ’th entry is and the rest are zero, i.e.
In terms of the PDF of is given to using the density matrix approach as
The PDF of is thus fully described by the eigenvectors and eigenvalues of . Analyzing the equation for with , one find an equation of the form
On the central lines the finite Laplacian operator in the LHS above is slightly modified, yielding
where is a real exponent and is a normalization factor. Inserting this form into (21) yields an expression for the eigenvalue, . Inserting this result and the ansatz into either (22) or (23) yields for the largest eigenvalue
and additional solutions that are smaller than by an amount that does not scale with . This gap between and the second largest implies that the properties of the system are thus fully controlled by the highest eigenvalue of . The fact that does not scale with implies that the density profile of decays in the bulk of the system as
The interface is thus pinned to the central ring for , with an exponentially small probability to be found in the bulk of the system. The interface fluctuates symmetrically around the central ring because it is energetically preferable for the interface to cross it. We verify this numerically in the full two-dimensional model and for a specific temperature in figure 5a. This phase is different from the asymmetrically pinned phase observed in the driven case below and in , where the interface fluctuates more into one side of the central ring.
When analyzing the RSOS model with , one has to consider a different solution for the eigenvectors of of the form
In general one expects to scale as in order to yield a smooth density profile. As a result, there is no finite gap between the largest eigenvalue and others, which implies that many of the wavevectors ( of them) contribute to the density profile. Expanding the above equation in small yields , which suggests that has the form with . Inserting this expansion yields for
and thus . Solution with may have a different form, but they do not contribute to the density profile due to the factor in (20). Using this result the probability of the profile to touch the central ring can be shown to scale as
for any finite . The interface is thus repelled from the central. This is clearly visible in figure 6 where is plotted for a specific temperature and several system-sizes.
In terms of temporal behaviour of the interface, (30) implies that as long as and scale similarly the interface is confined either above or below the central line and very rarely touches the central ring. The is shown to be the case for the full 2D Ising model in figure 5b, where is plotted for . When the interface is close to the central ring, there is an energetic barrier which scales as for creating a single crossing point. Expanding this crossing point has no energetic cost. There is, however, an entropic barrier which is expected to yield a crossing time between the two sides that grows polynomially with and . The analysis of the average crossing time, for several system-sizes is shown figure 7, which suggested that .
To conclude, modifying the interaction strength within the central ring was found yield two different wet phases. For the interface is strongly pinned to the central ring and fluctuates symmetrically around it, whereas for it is repelled from it but crosses it on a time-scale that grows polynomially with and . In the next section we study the effect of the drive on the interface and find that it leads to a different set of phases.
4 Analysis of the Ising model with a driven line
4.1 Phase diagram for fast drive and slow spin flips on the central ring
We now return to the Ising model with a driven line, illustrated in figure 1. In general the additional Kawasaki dynamics in (3) leads to a non-equilibrium steady-state, whose form is unknown. However, as explained below, one can derive the phase diagram of the model in the limit of fast drive and slow spin flips in the central ring. The different phases found in this study can be observed numerically also away from this limit, as shown in section 4.2.
The limit of fast drive is defined such that the relaxation time-scale of the conserving dynamics within the central ring is much faster than that of the boundary layers adjacent to that ring. Such a limit allows the conserving dynamics to reach a steady-state between every spin flip in the adjacent boundaries. The steady-state of the conserving dynamics is that of the Asymmetric Simple Exclusion Process (ASEP), which is uniform over all the spin-configurations with a fixed number of particles (which corresponds here to a a fixed magnetization in the central ring) [45, 46, 39]. This steady-state is known to be obtained on a time-scale of . On the other hand, since the dynamics in the adjacent rings is non-conserving, their magnetization relaxes on a time-scale of .
Similarly, the limit of slow spin flips in the central ring is defined such that the relaxation time-scale of the non-conserving dynamics in the central ring, given by , is much slower than the relaxation time of the adjacent layers. This enables the magnetization in these adjacent layers to reach the steady-state values of the 2D Ising model between every spin flip in the central ring. The limit we consider can thus be written as
Within these two limits the rate of every spin flip in the central ring is a function only of magnetization in the central ring, defined as
Summing over the rates of all the possible spin flips in the central ring, one obtains a master equation for , which corresponds to that of one-dimensional random walk in a potential. This can be formally written as
where are the rates of spin flips in the central ring for a given value of . These rates can be computed from the known steady-state properties of the 2D Ising model and of the ASEP, as discussed below. A similar approach has been employed in , where the rates were computed using a low temperature expansion of the Ising model. A finite temperature analysis was not possible in this case because the properties of the Ising model with a fixed magnetization (or equivalently fixed bulk magnetic field) are unknown. Equation (33) has a simple solution of the form
where increases in steps of . This can be written in the large limit as
where is the large deviation function of .
Equation (35) implies that depends only on the ratio between and . The first step in obtaining this ratio is to analyze the interaction between line and its neighbouring lines from above and below. Naively one would think that for a given magnetization, , the spins in line feel a magnetic field of strength . However, this turns out to be true only for high temperature, . In order to verify this, one has to consider the ratio between the rate of a spin flip in site and the that of the reverse process, given by
The fact that the steady-state measure of the central ring is uniform over all configuration implies that , which yields
The spins in line therefore feel an effective magnetic field which is in general different than . This field is plotted for various values of in figure 8. Figure 8a demonstrates the fact that . The fact that was noted also in , where the Ising interaction between two lines of spins was analyzed in the case where one line is strongly driven.
Since the spin flips in the driven line occur slowly, the systems above and below the driven line relax between every such spin flip to the steady-state of the 2D Ising model with boundary fields and , respectively 111The fact that the driven line exerts a field of strength and not was unaccounted for in . However, by repeating the derivation presented there, we verified that the use of does not change the qualitative results of that study.. Consequently, the magnetization in lines and is given on the time-scale in which is fixed by
Using an analysis similar to that of (4.1), the ratio between the rates of spin flips within the driven line are given by
This implies that apart for the average magnetic field from the two adjacent rings, and , every spin in the central ring feels a field of strength , exerted on it by the spins within the central ring.
The fact that the expression in (40) is independent of , enables us to derive from it the ratio between the rates of increase and decrease of by summing over all possible single spin flips, yielding
Inserting the expression for in (39) into the above equation yields an expression for which is fully determined by and is given by
Finally, the above result can be used in the solution of the large deviation function in (35), yielding
Note that the first two terms in the RHS above correspond to the entropy of the central ring. The following two integrals account for the interaction within the central ring, and with the two adjacent rings, respectively.
By studying the extrema of for different values of and one can obtain the phase diagram of the model. The resulting phase diagram is plotted in figure 9 for the relatively simple case where , in which the interface is never attached to the boundaries of the cylinder. Representing plots of in each phase are provided in figure 10.
The main features of the phase diagram are as follows: In the driven model, the central line become magnetized before the bulk of the system does, i.e. for . The system undergoes a second order phase transition between two phases where the bulk of the system is disordered and the central ring is either paramagnetic, , or ferromagnetic , . The plot of for this phase, referred here as the ordered line phase, is shown in figure 10a. Below the Ising critical point and for the system exhibits only one type of a wet phase, where the interface is repelled from the driven line, as shown in figure 10b. The sign-flip symmetry of is spontaneously broken in this phase, yielding two degenerate states, where the interface is either above of below the driven line. As suggested by the form , there is a barrier between those two degenerate states which increases with . This implies that the tunneling time between them increases exponentially with , as confirmed numerically in the next section. This is in contrast to the polynomial growth of the crossing-time found in the equilibrium repelled phase in section 3.2. In this respect the repelled phase observed here can be considered to be a ‘strongly-repelled’ phase.
As in equilibrium, decreasing below is generally expected to attract the interface to the driven line. However, in this case the effective repulsion due to the drive implies that one has to go significantly below in order to observe a pinned phase. For , one finds two phases where the interface is pinned to the central ring. For relatively high temperatures, we observe a phase similar to that observed in equilibrium, where the interface fluctuates symmetrically around the central ring, yielding (corresponding LDF shown in figure 10c). The transition between this phase and the disordered phase is of first order (thick solid line in figure 9). As the temperature is decreased the sign-flip symmetry of is spontaneously broken, yielding a phase where the interface found in one of the sides of the central ring but at a distance from it that does not scale with . The transition between the asymmetrically-pinned phase and the symmetrically-pinned phase is of second order. The plot of in the asymmetrically-pinned phase is shown in figure 10d. A reentrant second order transition back into the repelled phase is observed at lower temperature, where the field induced by the driven line is strong enough to unpin the interface, namely .
The asymmetrically-pinned phase was observed in , where Kawasaki dynamics was considered in the bulk and a low temperature limit was taken. The finite temperature fluctuation of the interface in this phase can thus be understood using the known boundary properties of the two-dimensional Ising model, analyzed extensively in .
4.2 Numerical results for the driven case
The phase diagram of the model presented in section 2 was computed in section 4.1 in the limit of , fast drive and slow spin flips on the central ring, . In this section we test numerically whether the results obtained in that limit yield a correct qualitative picture for finite and . For simplicity we consider here . The results appear not to depend much on the ratio between and and the asymmetry or the drive. We thus choose to study here the simple case where for all ’s and (totally asymmetric Kawasaki dynamics). On the other hand we find very different results for large and small . To test the prediction of the above analysis we considered relatively fast drive by taking .
We focus here on the three different wet phases, which appear for . We consider specifically . The three phases are clearly visible when observing the time evolution of the overall magnetization, defined as
Its behaviour in the repelled, symmetrically-pinned and asymmetrically-pinned phases is plotted respectively in Figs. 11a,11b and 11c for and in Figs. 11d,11e and 11f for . In the repelled phase, the system can be in two degenerate state where the magnetization is significantly different from . The transitions between these two states are clearly visible for , but cannot be observed for due to the limited simulation time. The exponential growth of the transition time is depicted by the grey triangles in figure 12, which shows the logarithm of the average time interval between the sign-flips of as function of . The symmetrically-pinned phase, where , appears in Figs. 11b and 11e to become more stable as the system-size increases. For system of size we were unable to observe the detachment of the interface. In the asymmetrically-pinned phase, shown in Figs. 11c and 11f, the system moves between two states where . As shown by the squares in figure 12, the crossing-time of the interface between those states grows exponentially with but at a smaller rate than that of the repelled phase.
The transitions between the various phases are examined in figure 13, which shows the histogram of the absolute value of the overall magnetization, , and of the magnetization on the driven line, . The histogram is measured for a relatively small system size, , for which the transition between the different metastable states could be observed. Above the Ising critical temperature () one finds a smooth transition into the ordered-line phase (OL) where increases whereas is still relatively close to . Below it we find the repelled phase (R) where both