# Finite-Time Universality in Nonequilibrium CFT

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## Abstract

Recently, remarkably simple exact results were presented about the dynamics of heat transport in the *local* Luttinger model for nonequilibrium initial states defined by position-dependent temperature profiles. We present mathematical details on how these results were obtained. We also give an alternative derivation using only algebraic relations involving the energy-momentum tensor which hold true in *any* unitary conformal field theory (CFT). This establishes a simple universal correspondence between initial temperature profiles and the resulting heat-wave propagation in CFT. We extend these results to larger classes of nonequilibrium states. It is proposed that such universal CFT relations provide benchmarks to identify nonuniversal properties of nonequilibrium dynamics in other models.

## Keywords

Nonequilibrium dynamics Conformal field theory Heat and charge transport Luttinger model## 1 Introduction

The study of heat, mass, charge, or spin transport in classical and quantum one-dimensional systems has a long history, see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Among problems that continue to make this an active field are questions concerning presence of diffusion, effects of integrability, interactions, or disorder, universality, and behaviors after quantum quenches, to mention a few. Studies of such questions were further spurred by experiments on ultracold atomic gases [11, 12] which recently triggered a rapid development of this field, see, e.g., [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Let us specifically mention the use of methods of conformal field theory (CFT) to gain better understanding of nonequilibrium steady states and transport in critical quantum 1*d* systems, see [24, 25, 26, 27] and references therein, [28] for an operator-algebraic approach, and [29, 30, 31, 32] that are particularly close to the context of the present paper.

*energy density*operator on a circle \(S^1\) with circumference

*L*parameterized by the coordinate \(x \in [-L/2, L/2]\). \(\mathcal {E}(x)\) together with the

*heat current*operator \(\mathcal {J}(x)\) satisfy the continuity equation

^{1}

*mixed*states of (1.3) and it was analyzed directly in real rather than imaginary time.

To study heat transport we were particularly interested in kink-like profiles \(1/\beta (x)\) interpolating between temperatures \(1/\beta _{\mathcal {L}}\) to the far left and \(1/\beta _{\mathcal {R}}\) to the far right, see Fig. 1. The *smooth temperature profile protocol* described above allows one to analytically compute the nontrivial behavior of the energy density and the heat current around the location of the kink at early times and their subsequent development into *heat waves* moving ballistically to the right and left. This should be contrasted with the results of the CFT description of the dynamics in the *partitioning protocol* employed in similar previous studies, see [27] and references therein. In such a description, argued to be valid after a transient time, the ballistic heat waves are compressed to simple jumps (shocks) without internal structure moving away from the contact point. In the smooth initial states that we consider, this happens only in the limit when *t* and *x* are sent to infinity at the same rate, as such a limit wipes out the nontrivial internal structure of the heat waves. The evolution of the energy density and the heat current obtained in [33] permits then to better understand the shortcomings of the partitioning protocol. It also sheds a new interesting light on transport in integrable systems and, in particular, on how its universal features [35] emerge at long times for a large class of nonequilibrium initial states, see [36] and also Sect. 4.3 below for a related discussion of charge transport in the Luttinger model. As a representative example, we plot \(\langle \mathcal {E}(x,t) \rangle _{\text {neq}}\) and \(\langle \mathcal {J}(x,t)\rangle _{\text {neq}}\) in Fig. 2 at four times for the Luttinger model with local interactions (defined in more detail below) starting from the kink-like temperature profile in Fig. 1. Note a peak and a dip in the energy density at time \(t=0\) in the region where the temperature changes and how this local shape evolves into heat waves. This is accompanied by a universal heat current building up in the region between the two heat-wave fronts. We note that, for local interactions, the wave fronts preserve their shapes in time. For the Luttinger model with nonlocal interactions, there are additional dispersive effects, which, however, eventually become unobservable in any finite region as the wave fronts leave such regions in finite time, see Fig. 1 in [33]. In the remainder of this paper we restrict our discussion to the local case.

*local Luttinger model*, the energy density is given formally by

^{2}

^{3}

*F*(

*x*) can be written in terms of the Schwarzian derivative

The method used in [33] was perturbative in a small parameter \(\epsilon \) measuring the distance to equilibrium in the initial state (i.e., the case \(\epsilon =0\) corresponds to the Gibbs state). This method is general, but generically one can only obtain useful low-order results. In the special case of *local interactions*, however, we were able to push the computations to all orders in \(\epsilon \), and, summing the resulting infinite series, we obtained the results in (1.7) and (1.8). In this paper we give a simpler derivation of these results extending them to *all* unitary CFT models, including models at finite *L*, and to other observables.

*“velocity” profile*\(\nu (x)\) taking

*c*of CFT (which is equal to 1 for the local Luttinger model) and with \(\beta (x)\) replaced by

^{4}

Finally, for CFTs with a double \(\text {U(1)}\) current algebra (e.g., the Luttinger model itself), we may also handle chemical-potential profiles \(\mu _{\pm }(x)\) in addition to temperature profiles \(1/\beta _{\pm }(x)\), possibly different for right and left movers. In this case, the functions \(F_{\pm }(x)\) pick up an additional term proportional to \(\mu _{\pm }(x)^2\). This is the largest class of nonequilibrium states that we consider. They are defined as in (1.3) but with \(\mathcal {G}\) in (4.40), see Sect. 4.3 for details.

^{5}We believe that this provides a useful benchmark for other models as follows. Typically, finite-time results are model-dependent, and more universal behavior is only obtained at long times [27]. As an example, we mention the nonlocal Luttinger model which exhibits finite-time dispersion effects depending on short-distance details of the interaction potential [33]. However, these effects have some qualitative features that are always present. We postulate that

*the CFT results have to be subtracted in order to identify the effects that come from the microscopic details*in the propagation of the heat or density waves emanating from the inhomogeneities of the initial state. In addition, such a subtraction should allow to identify the time scales when such model-dependent effects are important and when not. Moreover, it was argued in [19] that integrable systems come in two kinds: those that are purely ballistic and those with a nonzero diffusive contribution. For heat transport, in particular, one way to view this is through the thermal conductivity in the frequency domain

The plan of the rest of this paper is as follows. In Sect. 2, we sketch the original derivation of the result in (1.7) and (1.8) and explain the physical significance of the limit \(L\rightarrow \infty \) since this is also relevant for our more direct CFT argument. We also present special integrals whose exact evaluation was the key to this result and which, as we believe, are interesting in their own right. The reader may skip the second half of Sect. 2 without loss of continuity. The CFT derivation is given in Sect. 3. After collecting the results about Minkowski-space CFT that are needed in Sect. 3.1, we show in Sect. 3.2 how to use conformal transformations to straighten out position-dependent temperature profiles \(1/\beta (x)\) on a periodic interval. This allows one to exactly map the nonequilibrium expectations in CFT to the corresponding equilibrium ones. We make this mapping explicit for products of the components of the energy-momentum tensor. In Sect. 3.3, we study the thermodynamic limit of the finite-volume relations which allows one to treat temperature profiles on the infinite line with different asymptotic values on the left and right sides. As a byproduct, we calculate the thermal Drude weight. Section 4 is devoted to various generalizations. In Sect. 4.1, we briefly discuss the correlators of primary fields. In Sect. 4.2, we consider states with different temperature profiles for the right and left movers. They form a class of nonequilibrium states preserved by the Schrödinger-picture dynamics that lead to simple examples of generalized Gibbs states at long times. In Sect. 4.3, we extend the analysis to CFTs with a \(\text {U(1)}\) current algebra and states with temperature and chemical-potential profiles. For the Luttinger model we discuss how this implies universality of conductance for both the charge and axial currents, generalizing previous results in [36], see also [43]. Finally, in Sect. 5, we make contact with [29, 30, 31, 32], discussing the dynamics preserving states in (1.3) and the related Euclidian CFT description. We end with conclusions and directions for further developments in Sect. 6. The Appendix contains some mathematical details on the special integrals mentioned above.

## 2 Perturbative Derivation and Remarkable Integrals

*W*(

*x*) a function defined by this relation.

*W*(

*x*) becomes (say) 1 / 2 and \(-\,1/2\) to the far left and right, respectively, so that \(\bar{\beta }=(\beta _\mathcal {L}+\beta _\mathcal {R})/2\) and \(\epsilon =(\beta _\mathcal {L}-\beta _\mathcal {R})/\bar{\beta }\) in terms of the asymptotic values. On the other hand, for technical reasons, we used a model on a circle with circumference \(L<\infty \), which at first sight seems incompatible: a smooth function

*W*(

*x*) on the circle with a single kink is not possible, and there has to be at least one other opposite one. As an example, consider the periodic function

*t*, and then taking the limit \(L\rightarrow \infty \) [33] in which (2.2) turns into

*L*/

*v*, which is a time scale that becomes infinite in the limit \(L\rightarrow \infty \).

In the rest of this section we sketch the perturbative derivation of the result in (1.7) and (1.8), concentrating on remarkable integrals which were the key to this results. The readers mainly interested in our CFT derivation may pass directly to Sect. 3 without loss of continuity.

*n*-th order term is an \((n+1)\)-point correlation function for \(n=1,2,\ldots \) [33]. This method works, in principle, for

*any*model but, in practice, it is difficult to go beyond leading order \(n=1\). For a quasi-free bosonic model, to which the Luttinger model reduces, one can use general mathematical results [44, 45, 46] to replace the many-body computation by a much simpler one-particle one, and this makes it possible to extend the calculation to all orders [33].

The integrals in (2.7) are certainly nontrivial, but we found that they all can be computed exactly, giving the result in Lemma 2.1, and this was the key that led to (1.8).

## Lemma 2.1

A proof can be found in the Appendix.

It is remarkable that the result are even second order polynomials in the variables \(q_j\). It is clear from (2.6) that the constant term leads to contributions to \(F_n(x)\) which are proportional to \(W(x)^n\), whereas the terms with \(q_j^2\) and \(q_jq_k\), \(j\ne k\), lead to \(W(x)^{n-2}W''(x)\) and \(W(x)^{n-2}W'(x)^2\), respectively. Thus, the special form of the integrals in Lemma 2.1 implies that we get at most terms involving second derivatives of *W*(*x*). The explicit expression of these integrals allows one to compute \(F_n(x)\) exactly, and the result is simple enough to analytically sum the series in (2.5), which gives the result in (1.8) [33].

We describe this computation above since it allows one to interpret the argument in the next section as a partial proof of Lemma 2.1. Such a proof is only partial since all terms with \(q_j^2\) are identified with (say) \(q_1^2\), and all terms with \(q_jq_k\) for \(j\ne k\) are identified with \(q_1q_2\). This identification is also useful in order to explicitly derive (1.8) from Lemma 2.1, see Eq. (A4) in [33], which is implied by (2.9). Thus, the exact integrals in Lemma 2.1 contain more information than the result in (1.8). Since nontrivial integrals that can be computed exactly are rare and often not only have one but several applications in physics, we prove (2.9) in this paper. Moreover, since our derivation of (1.7) and (1.8) in the next section works even for finite *L*, it suggests interesting Riemann sum generalizations of the exact integrals in Lemma 2.1. We believe it would be interesting to work them out, but this is left for a future study.

We finally mention that our argument in the next section allows one to interpret the computation described above as a derivation of the conformal anomaly in CFT by a direct computation, see (3.7).

## 3 CFT Derivation

### 3.1 CFT in Minkowski Space

We consider the Minkowskian version of a unitary two-dimensional CFT where space is a circle \(S^1\) parameterized by the periodic coordinate *x* with the basic range \(-L/2\le x\le L/2\) and where \(t\in \mathbb {R}\) is time. We keep the propagation speed *v* in our equations to clearly indicate how it effects the (otherwise) universal law relating the temperature profiles to the heat-wave dynamics.

^{6}where, as before, \(x^\pm = x \pm vt\). They are distributions with values in the self-adjoint operators on the Hilbert space of states of the theory that satisfy the equal-time commutation relations

*L*-periodized \(\delta \)-function. The real number

*c*is the central charge of the theory. In terms of the Fourier modes,

^{7}\(K = \mathrm{e}^{2\varphi }\). The effective densities \(\tilde{\rho }_{\pm }\) act in a direct sum of bosonic Fock spaces that contains the interacting vacuum \(|\Psi \rangle \) and the Wick ordering in (3.5) is with respect to \(|\Psi \rangle \) [37, 38]. In the following arguments, the explicit form of the operators \(T_{\pm }(x)\) is not used.

*f*(

*x*) and \(f(x)+nL\) corresponding to the same diffeomorphism in \(\mathrm {Diff}_+(S^1)\). The operator-valued distributions \(T_{\pm }\) generate two commuting projective unitary representations \(U_{\pm }\) of \(\widetilde{\mathrm {Diff}}_+(S^1)\) on the Hilbert space of the theory such that for infinitesimal diffeomorphisms \(f(x)=x+\varepsilon \zeta (x)\) one has

*Sf*)(

*x*) given by (1.9). This was proven in [47] for the unitary highest-weight representations of the Virasoro algebra and carries over to the present context. The adjoint action of \(U_{\pm }(f)\) preserves \(T_{\mp }\).

### 3.2 Relating Nonequilibrium to Equilibrium Expectations

From the above, it is clear that the calculation of the time evolution of the nonequilibrium expectation values of the energy density and current operators is equivalent to computing \(\langle T_{\pm }(x^\mp )\rangle _{\text {neq}}\).

*f*such that

*f*given by (1.10) with the constant \(\beta _0\) determined by

*f*defines an element in \(\widetilde{\mathrm {Diff}}_+(S^1)\). Using this function

*f*, it follows from (1.3), (3.7), and (3.9) that

*f*, this reduces to

*U*(

*f*) straightens out the temperature profile.

*f*given by (1.10) and (3.12). In particular,

*y*, but they depend, in general, on \(v\beta _0\) and

*L*. By scaling, however, \((v\beta _0)^2 \big \langle T_{\pm }(y) \big \rangle _{\beta _0}\) depends only on \(v\beta _0/L\) but in a way dependent on the representation content of the CFT. As we shall see, what is universal, depending only on the central charge, is the \(L\rightarrow \infty \) limit of \((v\beta _0)^2 \big \langle T_{\pm }(y)\big \rangle _{\beta _0}\).

### 3.3 Thermodynamic Limit

^{8}Similarly, for fixed

*x*, \(\beta (x)\) will stabilize up to \(O(L^{-1})\) terms with any trace of the antikink gradually wiped out, and so does the function

*f*given by (1.10). The question about the large-

*L*limit of the nonequilibrium expectations in (3.17) then boils down to the one for the equilibrium expectations \(\big \langle \prod _{j} T_{r_j}(x_j) \big \rangle _{\beta _0}\), where, by rescaling, \(\beta _0\) may be set to its asymptotic value and the insertion points are allowed to have \(O(L^{-1})\) variations.

*L*and \(v\beta _0\), respectively. In a modular invariant CFT [40], they also have a dual representation

*L*/

*v*in the theory on the circle with circumference \(v\beta _0\). The components of the energy-momentum tensor with complex arguments on the right-hand side are defined by

*f*defined by (1.10). By scaling, the right-hand side is then independent of the choice of \(\beta _0\). In particular, (3.18) together with (3.24) show that in the limit \(L\rightarrow \infty \),

*F*given by the right-hand side of (1.8) multiplied by the central charge

*c*.

## 4 Generalizations

### 4.1 Other Correlators

*f*given by (1.10) and \(\beta _0\) by (3.12). Note that these are simpler relations than for the energy-momentum tensor components \(T_{\pm }\) since those fail to be primary fields with conformal weights (2, 0) and (0, 2) due to the Schwarzian derivative term in (3.7) reflecting the conformal anomaly. In a similar way as for observables built from the operators \(T_{\pm }\), the relations (4.3) hold also in the thermodynamic limit which may be controlled like before. Also, as before, \(\beta _0\) may be taken arbitrary in the infinite volume.

^{9}Their infinite-volume equilibrium 2-point correlation functions have the form

### 4.2 Temperature and Velocity Profiles

*U*(

*f*) replaced by \(U(f_+,f_-)= U_{+}(f_+)U_{-}(f_-)\) with two different diffeomorphisms \(f_{\pm }\). Choosing them as

^{10}

### 4.3 Temperature and Chemical-Potential Profiles

*j*satisfying

*K*is the Luttinger parameter). One may then consider states with both temperature and chemical-potential

^{11}profiles by taking

*k*WZW theory based on a compact Lie group [40] (e.g., \(\text {U(1)}\) or \(\text {SU(}N\text {)}\)) in which case \(\kappa ={k}/{2}\). It follows from [47, 50] that in these cases there exist, besides the two projective representations \(U_{\pm }\) of \(\widetilde{\mathrm {Diff}}_+(S^1)\) considered above for which

*f*as before and choose

*U*(

*f*) straightens out the chemical-potential and inverse-temperature profiles to \(\mu _0\) and \(\beta _0\), respectively,

^{12}

*F*and

*G*in (4.35) and (4.37) with

^{13}

## 5 Equilibrium Dynamics and Relation to Euclidian CFT

In this section, we discuss the relation between the articles [29, 30, 31, 32] and the present paper. In [29] it was argued that the kernel of the 1-particle density matrix in the ground state of a nonrelativistic high density Fermi gas in a trap may be described on mesoscopic scales by the 2-point function of the fermionic massless free field whose Fermi velocity varies in space. These results were generalized in [30, 31, 32] to certain nonrelativistic systems of interacting 1*d* bosons in traps. Despite similarities, there are several differences with the approach of the present paper. First, the arguments in [29] were based on the analysis of the ground-state Euclidian-time correlators in the presence of a trap and these were shown to correspond to Euclidian CFT correlators in an appropriate curved metric (in [32] also the coupling to a gauge field appeared implicitly). In this paper, we consider positive temperatures, but the correspondence of [29] generalizes to low-temperature states leading to the compactification of the Euclidian time direction in CFT, just as for homogeneous equilibria. Hence, for specific CFTs, the states in (1.3) may, indeed, be viewed as describing on mesoscopic scales the 1*d* nonrelativistic low-temperature matter in traps, with \(\beta (x)\) having the interpretation of the position-dependent Fermi velocity in appropriate units. Second, the argument of [29] was done for the equilibrium dynamics (although some nonequilibrium situations were also considered), whereas in the bulk of the present paper we study the dynamics generated by the homogeneous Hamiltonian that does not preserve the states in (1.3). Our considerations may, however, be generalized to dynamics induced by inhomogeneous Hamiltonians [56], in particular to the ones that preserve the states in (1.3). Third, we use the Minkowski version of CFT, whereas the papers [29, 30, 31, 32] employed the Euclidian CFT. That is usually considered as an innocent distinction handled by the Wick rotation. Indeed, for the primary fields, the correlators in the Euclidian theory in the metric considered in [29, 30, 31, 32] and with compactified time do agree, up to the Wick rotation, with the corresponding correlators in the states of (1.3) for which the time dependence is generated by the equilibrium dynamics. As shown below, however, that does not hold directly for the correlators of the energy-momentum tensor components which are of main interest in this paper. This points to the need of caution when one applies Euclidian techniques in the study of systems that are inhomogeneous in space or/and time.

*H*of (3.8) which results in the time dependence in the arguments of function

*f*.

*f*(

*x*) given by (1.10) and \(\beta _0\) by (3.12), and where \(\sigma (z,\bar{z})=-2\ln {f'(x)}\). In other words,

*h*of (5.4). Such a relation is well known for \(f(x)=x\). Its generalization to general

*f*follows from the identity

*d*fermions or bosons on mesoscopic scales in [29, 30, 31, 32], with the interpretation of \(v\beta (x)/\beta _0\) as the position-dependent Fermi velocity clearly reflected in the form of the metric in (5.4).

*f*(

*x*) one has [57], in the notation \(z^+=z,z^-=\bar{z}\),

## 6 Conclusions

We elaborated on the formula of [33] giving the full time evolution of the energy density and heat current from a nonequilibrium state with a preimposed temperature profile in the Luttinger model with local interactions. The formula was obtained in [33] by expanding the nonequilibrium state around the equilibrium to all orders. More details on the perturbative computation involving the exact calculation of complicated integrals, that may be interesting in its own right, were given. The main part of the paper was devoted to showing how the formula of [33], a result of the resummation of the perturbative series, may be obtained using Minkowskian conformal symmetries of the local Luttinger model. The idea was to use conformal transformations to map spatially inhomogeneous situations to homogeneous ones, straightening out a nonuniform temperature profile to a constant one. This led to a direct relation between nonequilibrium and equilibrium states, yielding the remarkable formula of [33] as a corollary. The CFT argument holds for a general class of unitary CFTs and could be applied to a wider class of nonequilibrium states that are preserved by the Schrödinger-picture evolution. The states in this class may be viewed as particular examples of simple generalized Gibbs states with local profiles, and they tend to ordinary generalized Gibbs states at long times, somewhat similarly as in the scenario recently advocated for integrable models where the evolution at certain length and time scales could be described by generalized hydrodynamics [14, 15, 16, 17, 18, 20, 21, 22, 23]. We obtained similar results also for CFTs with a \(\text {U(1)}\) current algebra (including the local Luttinger model itself) where we treated nonequilibrium states with temperature and chemical-potential profiles. Moreover, our results permit a more detailed analysis within CFT, compared to using the partitioning protocol studied before [27], of how a system starting in a state that looks like two different equilibria joint together evolves in time towards a nonequilibrium steady state described by a generalized Gibbs state.

As was discussed in Sect. 5, at least some families of the CFT nonequilibrium states that we studied in the present paper could be interpreted as providing a mesoscopic-scale description of dense nonrelativistic 1*d* matter in macroscopic traps [29, 30, 31, 32]. The dynamical correlators in such CFT states should similarly describe the corresponding nonrelativistic correlators at mesoscopic time scales both for after-quench and for equilibrium dynamics. The other way of arriving at the family of nonequilibrium CFT states that we considered is by reversing the logic of this paper. In the periodic-volume Minkowski CFT, the conformal symmetries (together with the gauge symmetries if a \(\text {U(1)}\) current algebra is present) are broken in the usual equilibria that are not preserved by the symmetries. Instead, the application of the symmetry transformations to the equilibrium states generates the family of nonequilibrium states that were studied here.

In the infinite volume, the states with kink-like profiles give access to the full counting statistics of the energy or charge transfers through the kinks, similarly to the states arising in the partitioning protocol [27]. Although such statistics in both approaches differ at finite times, they have the same long-time large deviations. This will be discussed elsewhere as it requires using different boundary conditions for finite volumes that allow one to avoid the duplication of kinks in the profiles. Finally, another interesting exercise, which was abundantly discussed in the similar context of quantum quenches [25], concerns the evolution of the entanglement entropy or negativity starting from states with profiles of the type consider here. By the replica trick, the latter may be extracted from nonequilibrium correlators of the twist primary fields in the replicated theory, to which our approach gives direct access. The analysis of the corresponding formulas is left for future research.

## Footnotes

- 1.
- 2.
- 3.
The symbols \(\mathcal {E}\),

*F*, and*f*here correspond to \(\mathcal {H}\),*G*, and*g*in [33]. - 4.
To avoid confusion, we stress that our subscripts ± refer to right \((+)\) and left (−) movers.

- 5.
Here we use the term “universal” as referring to the same form in different CFTs rather than to the independence of the microscopic details of the models.

- 6.
In the more standard notation for the energy-momentum tensor components in light-cone coordinates, \(T_+=T_{--}\), \(T_-=T_{++}\) and \(T_{+-}=0=T_{-+}\).

- 7.
- 8.
- 9.
The fermionic fields are represented as vertex operators related to the bosonic fields \(\tilde{\rho }_\pm \). Such operators require Wick ordering that provides their multiplicative renormalization, see, e.g., [38].

- 10.
This is proven using modular invariance for imaginary \(\nu _0\) and continuing analytically to real \(\nu _0\).

- 11.
The name is somewhat conventional. If \(\rho \) is charge density, rather than the particle density, then \(-\mu (x)\) would be the electric potential.

- 12.
The coefficients of linear response of currents \(\mathcal {J}\) and

*j*to small \(-\Delta \beta \) and \(\Delta (\beta \mu )\) would form a symmetric matrix that unveils a ballistic version of the Onsager reciprocal relations. - 13.
The result in [36] was more general in that it was for the Luttinger model with nonlocal interactions, but it was only for zero temperature states and \(\mu _+(x)=\mu _-(x)=\mu (x)\).

## Notes

### Acknowledgements

All the authors profited from valuable input and encouragement by Joel Lebowitz and Vieri Mastropietro. They gratefully acknowledge that this paper would not have seen the light of day without them. They would also like to thank Jouko Mickelsson and Herbert Spohn for helpful discussions. E.L. acknowledges support by VR Grant No. 2016-05167. P.M. is thankful for financial support from P. F. Lindström’s foundation (KTH travel scholarship VT-2017-0011 no. 4). The final version of the article has profited from pertinent comments of the anonymous referees who, in particular, drew our attention to the articles [29, 30, 31, 32].

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