Entropy production for complex Langevin equations

TL;DR

This paper investigates entropy production in complex-valued Langevin equations modeling nonlinear oscillator networks with non-Hermitian dissipation, providing explicit thermodynamic expressions and analyzing steady states and energy flows.

Contribution

It introduces a stochastic thermodynamics framework for complex Langevin systems with non-Hermitian terms, deriving explicit entropy production formulas and exploring effects of asymmetric coupling.

Findings

Explicit entropy production rates for complex Langevin networks.

Steady states characterized by constant entropy production and energy flows.

Asymmetric coupling influences entropy production significantly.

Abstract

We study irreversible processes for nonlinear oscillators networks described by complex-valued Langevin equations that account for coupling to different thermo-chemical baths. Dissipation is introduced via non-Hermitian terms in the Hamiltonian of the model. We apply the stochastic thermodynamics formalism to compute explicit expressions for the entropy production rates. We discuss in particular the non-equilibrium steady states of the network characterised by a constant production rate of entropy and flows of energy and particle currents. For two specific examples, a one-dimensional chain and a dimer, numerical calculations are presented. The role of asymmetric coupling among the oscillators on the entropy production is illustrated.

Figures

Equations

$${\dot{\psi }}_m=F_m+{\xi }_m.$$

$$⟨{\xi }_m(t){\xi }_n^{\ast }(t^{\prime })⟩=D_m{\delta }_{mn}\delta (t-t^{\prime }).$$

$${\partial }_m\equiv \frac{\partial }{\partial {\psi }_m}=\frac{1}{2}\left(\frac{\partial }{\partial x_m}-i\frac{\partial }{\partial y_m}\right),$$

$$\dot{P}=\sum _m\left[-{\partial }_m(F_{m}P)-{\partial }_m^{\ast }(F_m^{\ast }P)+2D_m{\partial }_m{\partial }_m^{\ast }P\right].$$

$${\mathcal{J}}_m^I$$

$${\mathcal{J}}_m^R$$

$$\dot{P}=\sum _m\left(-{\partial }_m{\mathcal{J}}_m-{\partial }_m^{\ast }{\mathcal{J}}_m^{\ast }\right).$$

$$S=-⟨\log P⟩\equiv -\int P\log Pdx$$

$$\dot{S}=\int \sum _m({\partial }_m{\mathcal{J}}_m+{\partial }_m^{\ast }{\mathcal{J}}_m^{\ast })\ln Pdx.$$

$$\dot{S}=-2\mathrm{Re}\int \sum _m{\mathcal{J}}_m^I\frac{{\partial }_{m}P}{P}dx.$$

$$\frac{{\partial }_{m}P}{P}={\partial }_m\ln P=\frac{1}{D_m}(F_m^{I\ast }-{\mathcal{J}}_m^{I\ast }/P)$$

$$\dot{S}=-2\mathrm{Re}\int \sum _m{\mathcal{J}}_m^I\frac{F_m^I}{D_m}dx+\int \sum _m\frac{|{\mathcal{J}}_m^I{|}^2}{D_{m}P}dx.$$

$$\Phi =\sum _m{\Phi }_m=\sum _m\left[2\frac{⟨|F_m^I{|}^2⟩}{D_m}+2\mathrm{Re}⟨{\partial }_m{F}_m^I⟩\right]$$

$${\dot{P}}^s=\sum _m\left[-{\partial }_m{\mathcal{J}}_m^s-{\partial }_m^{\ast }{\mathcal{J}}_m^{\ast s}\right]\equiv 0.$$

$${\Lambda }_m\equiv \frac{{\mathcal{J}}_m}{P}-\frac{{\mathcal{J}}_m^s}{P^s}=-D_m{\partial }_m^{\ast }\ln \frac{P}{P^s}.$$

$${\Pi }^a$$

$${\Pi }^{na}$$

$${\mathcal{H}}^R$$

$${\mathcal{H}}^I$$

$${\dot{u}}_1$$

$${\dot{u}}_1=-{\Phi }_{1}T-\frac{2}{\alpha }\mathrm{Re}⟨F_1^I{F}_1^{\ast R}⟩=-{\Phi }_{1}T-j_1^q,$$

$$j_m^q-j_{m-1}^q$$

$${\mathcal{H}}^R=\sum _{m=1}^L\left(|{\psi }_m{|}^4+{\psi }_m^{\ast }{\psi }_{m+1}+{\psi }_{m+1}^{\ast }{\psi }_m\right)$$

$${\mathcal{H}}^I$$

$$\mathcal{H}$$

$${\dot{\psi }}_1$$

$${\dot{\psi }}_2$$

$$j_{12}^p=2\mathrm{Im}⟨A_{12}{\psi }_1^{\ast }{\psi }_2⟩,$$

$$j_{12}^E=2\mathrm{Re}⟨A_{12}{\psi }_1^{\ast }{\dot{\psi }}_2⟩.$$

$${\mathcal{H}}^I$$

$${\mathcal{H}}^R$$

Full text

I Introduction

Simple oscillator models allow one to tackle fundamental problems of non-equilibrium statistical mechanics Bonetto et al. (2000); Lepri et al. (2003); Dhar (2008); Basile et al. (2007) and to study energy transport in systems that are ubiquitous in physics, chemistry, biology and nanosciences Balandin and Nika (2012); Lepri (2016). Examples include, but are not limited to, the dynamics of spin systems Slavin and Tiberkevich (2009); Borlenghi et al. (2015a), Bose-Einstein condensates, lasers, mechanical oscillators Kevrekidis (2009) and photosynthetic reactions Iubini et al. (2015).

A central issue is to identify the conditions under which a network of oscillators reaches thermal equilibrium, or is driven in a non-equilibrium steady state characterised by the propagation of coupled currents. One basic observable characterizing the state is the entropy production, whose calculation in terms of the microscopic variables is the object of the present paper. In particular, we address this issue using the language of Stochastic Thermodynamics (ST) Crooks (1999); Evans and Searles (2002); Seifert (2005); Esposito and Mukamel (2006); Deffner and Lutz (2011); Esposito (2012); Seifert (2012). Within the ST framework, the out of equilibrium dynamics is described combining the Langevin and associated Fokker-Planck (FP) equations or a (quantum or classical) master equation Tomé (2006); Tomé and de Oliveira (2010, 2012); Van den Broeck and Esposito (2010); Esposito (2012); Seifert (2012). Those allow one to define the evolution of probability over the phase space and to derive consistent expressions for thermodynamic forces/flows and for entropy production for states arbitrarily far from equilibrium.

Another issue that can be considered is the presence of asymmetric couplings in the system Hamiltonian. Physical systems that can be described by asymmetrically coupled oscillators include magnetic materials with asymmetric exchange coupling Cheong and Mostovoy (2007), synthetic lattice gauge fields Celi et al. (2014), transport in topological insulators Rivas and Martin-Delgado (2016) and parametrically driven oscillators Salerno and Carusotto (2014). Here we discuss how, in a network of coupled oscillators, detailed balance can be broken either by the presence of thermal baths at different temperatures and chemical potential or by an anti-Hermitian coupling among the oscillators. The use of anti-Hermitian Hamiltonians to describe phenomenologically irreversibility both in classical and quantum systems has been widely investigated Dekker (1981); Rajeev (2007); Rotter (2009). Here we move a step forward by quantifying irreversibility in those systems using the ST language.

Although our formulation is completely general, we shall mostly refer to the dynamics of coupled nonlinear oscillators in the form of the discrete nonlinear Schrödinger equation (DNLS) Eilbeck et al. (1985); Kevrekidis et al. (2001); Eilbeck and Johansson (2003) whose off-equilibrium properties have received a certain attention recently Iubini et al. (2012, 2013); Borlenghi et al. (2015a); Kulkarni et al. (2015); Mendl and Spohn (2015). The spin-Josephson effect Borlenghi et al. (2015b), the connection between gauge invariance and thermal transport Borlenghi (2016) and heat/spin rectification Ren and Zhu (2013); Borlenghi et al. (2014a, b) are a few of the effects within the DNLS field that can be captured by the ST formalism. One appealing feature of this class of models is the presence of two conserved quantities, namely energy and norm Rasmussen et al. (2000); Iubini et al. (2012, 2013) that give rise to coupled transport effects between the associated currents Iubini et al. (2012). This constitutes a further element of novelty that has not yet been considered in the existing literature.

The remainder of the paper is organised as follows. In Sec. I we describe the dynamics of a network of complex Langevin equations, and we introduce the associated Fokker-Planck (FP) equation. In Sec. II we derive the entropy flow and entropy production for this system, and in Sec. III we identify the adiabatic and non-adiabatic components of entropy production. In Sec. IV we show the link between heat and entropy flows and report simulations for the specific case of a DNLS chain with boundary thermostats. In Sec. V we discuss example of the dimer, the simplest realisation of the DNLS consisting of only two coupled oscillators. We present some numerical simulations that elucidate its off-equilibrium dynamics. Finally, in Sec. V we conclude the work and summarize the main results.

II Stochastic network model

Let us consider a network, where the dynamics of each of the $m=1,...,M$ nodes is described by the following Langevin equations (see the sketch in Fig.1(a))

[TABLE]

Here the dot indicates time derivative and $\psi_{m}=\sqrt{p_{m}(t)}e^{i\phi_{m}(t)}$ is a complex oscillator amplitude. The force $F_{m}$ is an arbitrary function of the $\psi$s and their complex conjugate. We assume that both the coupling between the $\psi$s and the local forcing and damping are contained in the definition of $F$. The white noises $\xi_{m}$, which model the stochastic baths, are complex Gaussian random processes with zero average and correlation

[TABLE]

Here $D_{m}=\alpha_{m}T_{m}$ is the diffusion constant, with $\alpha_{m}$ the damping rate and $T_{m}$ the temperature of bath $m$. Eq.(1) is a general model that describes a multitude of systems encountered in physics, chemistry and biology.

Throughout the paper we adopt the following conventions: we set the Boltzmann constant $k_{B}$ equals to one. Vectors and matrices are written in plain text, while their component are denoted by the $m$ and $n$ subscripts.

We define the Wirtinger derivatives as

[TABLE]

with $\psi_{m}=x_{m}+iy_{m}$ and $\partial_{m}^{*}=\frac{\partial}{\partial\psi_{m}^{*}}$ its complex conjugated. The variables $(\psi_{m},i\psi_{m}^{*})$ are canonically conjugate. The total forces $F_{m}=F_{m}^{I}+F_{m}^{R}$ are the sum of dissipative (or irreversible, $I$) and conservative (or reversible, $R$) components. Those are given by the derivatives $F_{m}^{I/R}=i\partial_{m}^{*}\mathcal{H}^{I/R}$ of anti-Hermitian/Hermitian Hamiltonians $\mathcal{H}^{I/R}$, respectively. The latter have opposite parity under the time reversal transformation and the total Hamiltonian $\mathcal{H}$ is defined as $\mathcal{H}=\mathcal{H}^{I}+\mathcal{H}^{R}$.

The Fokker-Planck (FP) equation associated to Eq.(1) reads Haken (1969); Xü-Bing and Hong-Yi (2010)

[TABLE]

Eq.(4) gives the evolution of the probability $P$ to find the system in the configuration $(\psi_{1},...,\psi_{M},\psi_{1}^{*},...,\psi_{M}^{*})$ at time $t$. Following Refs.Spinney and Ford (2012); Landi et al. (2013), we define the irreversible and reversible probability currents

[TABLE]

with $\mathcal{J}_{m}=\mathcal{J}_{m}^{I}+\mathcal{J}_{m}^{R}$ and $\mathcal{J}_{m}^{*}$ its complex conjugated. In terms of those currents the FP equation Eq.(4) assumes the form of a continuity equation:

[TABLE]

The steady state corresponds to $\dot{P}=0$, while thermal equilibrium corresponds to $\mathcal{J}_{m}=\mathcal{J}_{m}^{*}=0$.

The average of an arbitrary function $f$ of the observables is expressed by means of $P$ as $\left\langle f\right\rangle=\int fPdx$, where $dx=(\frac{i}{2})^{N}\prod_{m=1}^{N}d\psi_{m}\wedge d\psi_{m}^{*}$ is the phase space volume element. Note that this average is equivalent to ensemble-average of Eq.(1) over different realisations of the stochastic processes. As usual Tomé (2006); Esposito (2012), we consider the case where the probability currents and the thermodynamical forces vanish at infinity, so that the cross terms in the integration by part can be discarded.

III Entropy flow and entropy production

The entropy flow $\Phi$ and entropy production $\Pi$ are obtained starting from the definition of phase space entropy

[TABLE]

and computing its time derivative by means of Eq.(4):

[TABLE]

Upon integrating by parts, using Eqs.(5), (6) and (7) and assuming that the reversible forces have zero divergence Tomé (2006); Tomé and de Oliveira (2010), Eq.(9) becomes

[TABLE]

From Eqs. (5) one has

[TABLE]

and $\partial_{m}^{*}\ln P$ its complex conjugate. Substituting this into Eq.(10) gives

[TABLE]

The two terms in Eq.(12) correspond respectively to minus the entropy flow $\Phi$ from the system to the environment and entropy production $\Pi$. Note in particular that $\Phi$ has the usual form of products between probability fluxes $\mathcal{J}_{m}$ and thermodynamical forces $F_{m}^{*}/D_{m}$ and that $\Pi$ is positive-definite. In non-equilibrium stationary states, one has $\dot{S}=-\Phi+\Pi=0$, so that $\Phi=\Pi$. These quantities are both zero only at thermal equilibrium. Upon using Eq.(5), integrating by parts and substituting the integrals over $P$ with ensemble average, the total entropy flow becomes

[TABLE]

where $\Phi_{m}$ is the entropy flow on site $m$. Note that Eq.(13) is the generalization of the expression given in Ref.Tomé (2006) to the case where forces are complex-valued.

Before concluding the section, let us briefly discuss the more general case of non-stationary conditions. To this aim, it is useful to separate the entropy production into adiabatic and non-adiabatic components, which correspond respectively to steady and non steady states [Van den Broeck and Esposito, 2010]. Upon indicating with superscript $s$ the steady state probability $P^{s}$ and fluxes $\mathcal{J}^{s}$, one writes the steady state FP equation as

[TABLE]

By using Eqs.(5) and (6), it is convenient to define the following quantity

[TABLE]

Eq.(15) defines the discrepancy between a stationary and non-stationary state. By inserting $\Lambda_{m}$ into the definition of entropy production Eq.(10) and integrating by parts, one can show that the latter splits into the sum $\Pi=\Pi^{a}+\Pi^{na}$ of two parts which are respectively the adiabatic and non adiabatic components:

[TABLE]

The adiabatic component corresponds to non-equilibrium steady state, obtained for example connecting the system to baths at different constant temperature. On the other hand, the non-adiabatic component corresponds to non stationary states, obtained by applying a time dependent driving to the system.

IV Steady state heat flow

Let us return to the stationary case and consider the relation between the entropy flux $\Phi$ derived in the previous section and the heat flow. For clarity, we specialize to the relevant case of DNLS oscillators in contact with boundary reservoirs Iubini et al. (2013). In particular, we consider the geometry sketched in Fig.1(b), where the first site of the chain is in contact with a reservoir on the left at temperature $T$ and chemical potential $\mu$ and with the rest of the chain on the right. This setup is described by the following Hamiltonians Borlenghi et al. (2015a)

[TABLE]

where $h_{m}$ is the local energy yielding the conservative forces $F_{m}^{R}\delta_{mj}=i\partial_{m}^{*}h_{j}$. Analogously, the irreversible forces are $F_{m}^{I}\delta_{m1}=i\partial^{*}_{m}\mathcal{H}^{I}$. Let us now evaluate the variation of the local internal energy $u_{1}=\langle h_{1}-\mu p_{1}\rangle$ on a stationary state.

[TABLE]

where $\alpha\neq 0$ is assumed. Upon substituting the dissipative forces and using the anti-hermitianity of $\mathcal{H}^{I}$, one has

[TABLE]

By inserting the equations of motion, Eq. (1), the above equation becomes

[TABLE]

assuming that $\langle F_{m}^{I}\xi_{j}\rangle=\alpha T\langle\partial_{m}F_{m}^{I}\rangle$ as in Refs. Tomé (2006); Tomé and de Oliveira (2010), one gets

[TABLE]

where $j^{q}_{1}$ is the heat flux on site $1$. Indeed, for a lattice site $m$ in contact with the reservoir, we have

[TABLE]

The first term of the right-hand side corresponds to the energy flow difference $j^{h}_{m}-j^{h}_{m-1}$ while the second term is the particle flow difference $j_{m}^{p}-j_{m-1}^{p}$ Borlenghi et al. (2015a) multiplied by the chemical potential. Therefore, we consistently obtain the relation $j_{m}^{q}=j_{m}^{h}-\mu j_{m}^{p}$ Lepri (2016). Finally, since $\dot{u}_{1}=0$ on a stationary state and $j_{0}^{q}=0$, we recover the basic thermodynamical relation $j^{q}_{1}=-\Phi_{1}T$.

The consistency of Eq. (23) has been tested numerically on a chain of $L$ DNLS oscillators in contact with two boundary heat baths. The system Hamiltonian can be explicitly written as

[TABLE]

and the heat baths are implemented as in Eq. 1 with $D_{1}=\alpha T_{1}$ and $D_{2}=\alpha T_{2}$. Assuming fixed boundary conditions $(\psi_{0}=\psi_{L+1}=0)$, the dissipative Hamiltonian reads Iubini et al. (2013); Borlenghi et al. (2015a)

[TABLE]

Fig. 2 shows the heat-flux balance between the boundary currents $\Phi_{j}T_{j}$ $(j=1,L)$ and the average bulk flux $\overline{j^{q}}=1/L\sum_{m}j^{q}_{m}$ near a nonequilibrium stationary state with different boundary temperatures. When the stationary state is reached, a relation analogous to Eq. (23) holds separately at the rightmost boundary. This regime corresponds to a linear temperature profile along the chain (see the inset) and a flat profile of $j_{m}^{q}$ (data not shown).

V dynamics of a dimer

For a better physical insight, and to appreciate the role of coupling on transport, we now discuss the simplest realisation of the DNLS consisting of only two coupled oscillators $L=2$ (see Fig.1(c) for a cartoon). The system is described by the non-Hermitian Hamiltonian

[TABLE]

The quantities $\omega_{m}(p_{m})=\omega^{0}_{m}+Qp_{m}$ and $\alpha\omega_{m}(p_{m})$, $m=1,2$ are respectively the non-linear frequency and damping with $Q$ the nonlinearity coefficient, while $\mu_{m}$ is the chemical potential. For simplicity we do not write the explicit dependence of the frequencies on the powers. The coupled equations of motion, given by $\dot{\psi_{m}}=i\partial_{m}^{*}\mathcal{H}+\xi_{m}$, $m=1,2$ read

[TABLE]

From the previous section, one has the following expressions for particle and energy currents:

[TABLE]

When the two reservoirs have different temperatures and/or chemical potentials or an asymmetric coupling, the system reaches a non-equilibrium steady state where the currents are constant. Thermal equilibrium, which corresponds to the case where the currents are zero is obtained where both baths have the same temperature and chemical potentials and the coupling is symmetric, $A_{12}=A_{21}\equiv A$. Note that if the coupling is symmetric, one has $j_{12}^{p/E}=-j_{21}^{p/E}$. However, for an asymmetric coupling those currents are different and transport is described by the net currents $j_{net}^{p/E}=j_{12}^{p/E}-j_{21}^{p/E}$.

As discussed previously, the $I$ and $R$ components of the thermodynamical forces $F_{m}^{I/R}=i\partial_{m}^{*}\mathcal{H}^{I/R}$, are the ones that change (resp. do not change) sign upon the time reversal operation $\mathcal{H}^{I/R}(t)\rightarrow\mathcal{H}^{I/R*}(-t)$. To separate the Hamiltonian in $I/R$ parts, it is convenient to split the coupling between the oscillators as $A=B+C$, respectively into Hermitian and anti-Hermitian parts. A straightforward calculation gives

[TABLE]

and the thermodynamical forces read

[TABLE]

Note that one has the same decomposition if the coupling matrix $A$ is real, but in this case $B$ and $C$ are respectively its symmetric and anti-symmetric components. One can see here that the presence of anti-Hermitian (or anti symmetric) components adds extra terms in both the irreversible and reversible forces.

Following Eq.(13), the entropy production for the dimer finally reads:

[TABLE]

with $\partial_{m}F_{m}=-\alpha(\mu_{m}+\omega_{m})$.

We turn now to numerical simulations of Eqs.(28) and (29). In the following, the parameters $\alpha=0.02$, $\omega_{1}^{0}=\omega_{2}^{0}=1$ where used. At first, we have calculated the observables for a system with symmetric coupling $A_{12}=A_{21}\equiv A=0.1$, keeping $T_{1}=0.2$ and varying $T_{2}$ between $0.2$ and $2.7$ model units. Fig.3 shows the observables as a function of $\Delta T=T_{1}-T_{2}$. One can see that both the entropy production and the currents increase linearly at low temperature and then saturate. This behavior is similar to what has been observed in several systems previously studied, such as the spin-caloritronics diode and artificial spin chains Borlenghi et al. (2014a, b, 2015b, 2015a). It is due to the fact that at increasing temperature, thermal fluctuation hinder synchronisation between the oscillators thus reducing the currents.

The power difference $\Delta p=p_{1}-p_{2}$ decrease linearly as a function of $\Delta T$, since $p_{1}$ remains constant and $p_{2}$ is proportional to the temperature $T_{2}$.

Next, we focus on the effect of chemical potential difference on transport. In Fig.(4) the observables as a function of $\Delta\mu=\mu_{1}-\mu_{2}$ are reported. The simulations where performed keeping $T_{1}=T_{2}=0.1$ and $\mu_{2}=0.01$ fixed and varying $\mu_{1}$ between $0.01$ and $0.05$. One can observe that both $\Phi$ and $\Delta p$ grows quadratically, while the currents increase linearly as a function of $\Delta\mu$. Note in particular that no saturation is observed in this case.

Finally, let us discuss the case in which the model is brought outside equilibrium by an asymmetric coupling. Fig. 5 displays the observables at constant temperature $T=0.2$ as a function of the asymmetry $\Delta A=A_{12}-A_{21}$ of the coupling. One can see in panel a) that the entropy flow increases quadratically with the coupling, while the other observables are linear in $\Delta A$. Note also that the observables, and in particular the entropy production and the energy current, are much larger than in the case of symmetric coupling and temperature difference, showing that the asymmetric coupling is a very efficient means to drive the system out of equilibrium.

VI conclusions

In summary, we considered assembly of coupled nonlinear oscillators coupled to Langevin baths. Within the ST approach we compute explicit expressions for the entropy production rate and demonstrated their concrete use for specific model cases: the DNLS chain and dimer. In the case of the chain, we showed how the approach to the steady state can be studied by monitoring $\Phi$. For the dimer, we emphasized the role of asymmetry in the coupling as a means to effectively drive the system out of equilibrium. The asymmetry reflects the presence of anti-Hermitian components in the Hamiltonian.

The role of non-Hermitian Hamiltonians in classical and quantum oscillators has been long investigated Dekker (1981). Recently, the differences between the Lindblad and non-Hermitian formulation of open quantum systems have been clarified Zloshchastiev and Sergi (2014). The present work can serve to elucidate how anti-Hermitian components contribute to drive out of equilibrium this kind of systems. Generalising these results to the case of multiplicative noise should allow to treat genuinely quantum systems and provide a connection with the formalism of quantum state diffusion equations Gisin and Percival (1992).

The importance of the dimer is that it is simplest object that can be investigated, and yet it exhibits a rich dynamics due to the fact that it has two conserved quantities with associated currents. In magnetic system and in particular in spin valve structures, the dipolar interaction between layers introduces naturally an asymmetric coupling Naletov et al. (2011), and further investigation is needed to understand coupled transport in those systems. Most of the times these setup can be described by simple dimer models as the one treated in this paper Slavin and Tiberkevich (2009).

Generally speaking, the off-equilibrium observables are of importance to quantify irreversibility in a multitude of physical systems. Possible applications include the description of transport in mechanical oscillators Salerno and Carusotto (2014), synthetic gauge fields Celi et al. (2014) and topological insulators Rivas and Martin-Delgado (2016). Similar expression for entropy productions have also been obtained in the context of granular media Gradenigo et al. (2012).

We remark that the role of asymmetric coupling in the dynamics of oscillator network has attracted a certain attention in recent years, especially in connection with synchronisation phenomena and the dynamics of neural network Yamada (2002); Blasius (2005); Belykh et al. (2006); Zeitler et al. (2009); Cantos et al. (2016). Our work moves a step forward by addressing the off-equilibrium thermodynamics of those type of systems using a very general approach.

We mention also that a possible mechanism to create an asymmetric or complex coupling consists in forcing parametrically the coupled oscillators in such a way that the forcing has a fixed phase.

Acknowledgements.

We thank V. Tosatti and M. Polettini for illuminating discussions. This research was supported by the Stiftelsen Olle Engkvist Byggmästare.