Quantum quench dynamics of the sine-Gordon model in some solvable limits

In connection with the the thermalization problem in isolated quantum systems, we investigate the dynamics following a quantum quench of the sine-Gordon model in the Luther-Emery and the semiclassical limits. We consider the quench from the gapped to the gapless phase as well as reversed one. By obtaining analytic expressions for the one and two-point correlation functions of the order parameter operator at zero-temperature, the manifestations of integrability in the absence of thermalization in the sine-Gordon model are studied. It is thus shown that correlations in the long time regime after the quench are well described by a generalized Gibbs ensemble. We also consider the case where the system is initially in contact with a reservoir at finite temperature. The possible relevance of our results to current and future experiments with ultracold atomic systems is also critically considered.


I. INTRODUCTION
Recent experiments in the field of ultracold atomic gases have spurred much interest in understanding the thermalization dynamics of isolated quantum systems.  So far, the latter were considered mere idealizations of real systems, as most of the many particle systems of interest to quantum statistical mechanics, such as solids and quantum fluids, are strongly coupled to their environments. However, the creation of large ensembles of ultracold atoms with highly controllable properties, which remain fully quantum coherent for relatively long times (compared to the typical duration of an experiment), has completely changed this perception. This has also raised concerns about the mechanisms of thermalization in these systems, and even in some recent experiment [6] lack of thermalization has been observed.
The problem of thermalization in isolated quantum systems can be also posed as the study of the dynamics of the system following a quantum quench. That is to say, the study of the response of the system to a change of a control parameter of the Hamiltonian over a time-scale much shorter than any other relevant time scale of the system, so that the sudden approximation can be applied. Therefore, it is assumed that for t < 0 the Hamiltonian is H i and the system is in a given eigenstate of it, |Φ i . At t = 0 the Hamiltonian is changed to H f and thus for t > 0 the system evolves unitarily in isolation according to the dynamics dictated by H f . The quantum quench can be also used to describe the evolution of a system that has been prepared in a given state that is not an eigenstate of the Hamiltonian. Thus, the question naturally arises is whether a set of sufficiently interesting observables of the system reach some form of stationary state that can be described by a standard Gibbs ensemble. In such a case, we would speak of thermalization to a standard statistical ensemble (microcanonical, canonical, or grand-canonical). However, this may not be the case as it turns out that for several integrable models [11,14,15,19,26] the long-time behavior of certain observables can instead obtained from a generalized Gibbs ensemble characterized by a different temperature for each eigenmode of the Hamiltonian H f [11,14,[26][27][28]. The non-standard or generalized Gibbs ensembles follow from maximizing the von Neumann entropy with the constraints imposed by the set integrals of motion larger than the total energy and particle number. As to the situation concerning non-integrable systems, the issue of the thermalization dynamics is still not fully understood although some recent results [20,21,24] indicate that thermalization to a standard ensemble should eventually occur at sufficiently long times. However, the actual thermalization dynamics and how it depends to how close the system is to integrability still remain very poorly understood. Indeed, as far as one dimensional systems are concerned (for which the strong kinematic constrains usually lead to integrability being more ubiquitous than in higher dimensions), numerical simulations have found conflicting results concerning the existence of thermalization [12,13,23]. Indeed, in a recent work, Rigol and coworkers [29,30] has also pointed out that the statistics of the constituent particles may also play an important role in determining the thermalization dynamics.
Furthermore, in addition to studying the quench dynamics starting from a pure state, it can be also interesting to consider quantum quenches where the system is initially prepared in a thermal state by being at t < 0 in contact with an energy reservoir at temperature T . After the quench, the system is isolated from the reservoir and evolves unitarily according to the dynamics dictated by different Hamiltonian H f . This is the simplest kind of mixed initial state one can consider and it allows to analyze the effect on the quench dynamics of mixing (by means of the initial temperature) a fraction of the excited states with the ground state of H i .
In this work, we shall analyze the quench dynamics of the sine-Gordon model (sGM). This is an integrable field theory, but its exact solution is indeed quite difficult to deal with as the elementary excitations satisfy a rather non-standard algebra. Instead, we confine ourselves to two limits in which the model can written as a quadratic Hamiltonian. In one of these two limits, the so called Luther-Emery (LE) limit, using a trick called refermionization, [31,32] the model describes a system of one-dimensional massive (i.e. gapped) Dirac fermions. The fermions and their anti-particles (or holes, to use a solid-state Physics language) describe the solitonic and anti-solitonic excitations of the sGM, which, in the LE limit, happen to be non-interacting. In another limit, the model can be rather well approximated by a quadratic model of massive bosons. The latter describe a series of bound states of solitons and anti-solitons in the limit where interaction between them is strongly attractive. As explained elsewhere, [11,26] the quench dynamics of a quadratic Hamiltonian can be solved exactly via a time-dependent Bogoliubov transformation. We shall consider here two kinds of quenches, which correspond to the appearance and disappearance of the mass term (i.e. the gap) in the sGM. At zero temperature, we found correlations when the system is quenched from the gapped to the gapless phase at zero or low temperature exhibit an exponential decay with a correlation length/time fixed by the gap. This result is in agreement with the results of Calabrese and Cardy, [9,15] based on a mapping to a boundary conformal field theory. At high temperatures, however, the correlation length (time) is determined by the temperature. At intermediate temperatures, the system will exhibit a crossover between the zero-temperature (gap dominated) and the high-temperature (temperature dominated) regimes. On the other hand, correlations following a quench from the gapped to the gapless phase at the Luther-Emery and the semi-classical limit exhibit a somewhat different behavior, which may indicate a break-down of the semiclassical approximation or a qualitative change in the behavior of correlations as one moves away from the Luther-Emery limit.
This paper is organized as follows. In section II, we introduce the (quantum) sine-Gordon mode, its phases, and describe the problem of quantum quenches in this model. In section III, we consider quenches in the Luther-Emery limit at zero temperature, whereas in section IV we take up the quenches in semiclassical limit. In section V, we consider the effect of thermal fluctuations in the initial state assuming that contact with the energy reservoir is removed at the time when the system is quenched and therefore, it subsequently evolves in isolation. In section VI, we take up the issue of the long-time asymptotic behavior of correlations and expectation values, and show that it is given by a generalization of the Gibbs ensemble, as recently pointed out by Rigol and coworkers. [14] Finally in section VII we discuss some possible experimental consequences of this work. We provide a summary of the conclusions in section VIII. The results for the correlations at zero temperature that are reported in this manuscript have been available previously as a part of an unpublished preprint. [28] Since our preprint appeared other authors have also considered the thermalization dynamics in the sine-Gordon model. In particular (see this special issue), Sabio and Kehrein [33] used a flow equation method. Fioretto and Mussardo used [34] form factor methods to tackle quantum quenches in the sine-Gordon model, finding strong evidence that the long time behavior of a local operator is described by a generalized Gibbs ensemble. Furthermore, using the time-dependent renormalization group [35,36], Barmettler and coworkers [37] have investigated the quench dynamics of the XXZ model, which in the continuum limit reduces to the sine-Gordon model

II. THE SINE-GORDON MODEL
The sine-Gordon model is described by the following Hamiltonian: where : . . . : stands for normal order of the operators, [31,32,38] a 0 is a short-distance cut-off, and the phase and density fields, θ(x) and φ(x), are canonically conjugated in the sense that the obey This model can be regarded as a perturbation the Luttinger model (see e.g. Refs. 31, 32, 38 for a review), which still yields an integrable model. In equilibrium the model is known to exhibit two phases, which, according to the renormalization group analysis, [31,32] and for infinitesimal and positive values of the coupling in front of the cosine term, roughly correspond to K < 2 (gapped phase) and K ≥ 2 (gapless phase). In order to study the non-equilibrium (quench) dynamics, we will consider two different types of quenches: the quench from the gapless to the gapped phase and the reversed process, from the gapless to the gapped. In the first case, we assume that the dimensionless coupling g(t) is suddenly turned on, i.e. g(t) = g θ(−t). With this choice, H i = H sG (t ≤ 0) is a Hamiltonian whose ground state exhibits a frequency gap, m, to all excitations, whereas H f = H sG (t > 0) has gapless excitations. Conversely, in the second case, we consider that g(t) is suddenly turned off, i.e. g(t) = g θ(t). In this case, the ground state of H i is gapless whereas the Hamiltonian performing the time evolution, H f , has gapped excitations. However, although both H i and H f define integrable field theories, for a general choice of the parameters K and g, the quench dynamics cannot be analyzed, in general, by the elementary methods of Refs. [11,26]. Nevertheless, in two limits, the Luther-Emery (which corresponds to K = 1, see Sect. III) and in the semiclassical (that is, for K ≪ 1, Sect. IV) limits, it is possible to study the quench dynamics by methods similar to those of Ref. 26. However, as explained above, the statistics of the elementary excitations happens to be different in these two limits.

III. THE LUTHER-EMERY LIMIT
A. Introductory remarks Let us start by considering the sine-Gordon model, Eq. (3), for K = 1, which is the so-called Luther-Emery limit. It is convenient to introduce rescaled density and phase fields, which will be denoted as ϕ(x) = K −1/2 φ(x) and ϕ(x) = K 1/2 θ(x). Thus, the Hamiltonian of Eq. (1) becomes: where κ = 2 √ K. At the Luther-Emery limit κ = 2 (i.e. K = 1) and the model can be rewritten as a onedimensional model of massive Dirac fermions with mass by using the following bosonization formula for the Fermi field operators, [31,32] where s r = −s l = +1 and the chiral fields φ r (x) = ϕ(x) +φ(x), and φ l (x) = ϕ(x) −φ(x). For computational convenience, we choose the Majorana fermions in Eq. (4) to be η r = σ x and η l = iσ y , where σ x and σ y are the familiar Pauli matrices. In addition, we note that the gradient terms in Eq. (3) can be written as the kinetic energy of free massless Dirac fermions in one dimension: [31,32,38] As far as the cosine operator is concerned, the bosonization formula, Eq. (4), implies that: = Γ π : cos 2ϕ(x) : where Γ = iσ x σ y . This is almost the cosine term of the sGM in the LE limit (cf. Eq. 3), except for the presence of the operator Γ. However, we note that Γ 2 = 1 and that this operator also commutes with H 0 and with operator in the left hand-side of Eq. (6). The first property implies that the eigenvalues of Γ are ±1 whereas the second property and Γ can be diagonalized simultaneously. Upon choosing the eigenspace where Γ = −1, we obtain that is equivalent to Eq. (3) when κ = 2.
To gain some insight into the phases described by the sGM, let us first consider the Luther-Emery Hamiltonian, H LE in two (time-independent) situations: i) g(t) = 0 (the gapless free fermion phase, which coincides with the Luttinger model for K = 1 [31,32,38]), and ii) g(t) = g > 0, i.e. a time-independent constant (which corresponds to the gapped phase). In order to diagonalize the Hamiltonian, it is convenient to work in Fourier space and write the fermion field operator as: where α = r, l. The limit where the cut-off a 0 → 0 + should be formally taken at the end of the calculations, but in some cases we shall not do it in order to regularize certain short-distance divergences of the quantum sGM. It will also useful to introduce a spinor whose components are the right and left moving fields, and which will make the notation more compact: Thus the Hamiltonian for the gapless phase, H 0 , reads: where ω 0 (p) = vp is the fermion dispersion. However, the Hamiltonian of the gapped phase, corresponding to g(t) = g > 0, H LE , is not diagonal in terms of the right and left moving Fermi fields. In the compact spinor notation it reads: where being m = vg. Nevertheless, H LE can be rendered diagonal by means of the following unitary transformation: being tan 2θ(p) = m ω 0 (p) .
Thus the Hamiltonian of the gapped phase, in diagonal form, reads (after dropping an unimportant constant that amounts to the ground state energy): where ω(p) = ω 0 (p) 2 + m 2 . We associate ψ † v (p) (ψ † c (p)) with the creation operator for particles in the valence (conduction) band.
Before considering quantum quenches, let us briefly discuss some of the the properties of the ground states of the Hamiltonians H 0 and H LE . In what follows, these states will be denoted as |Φ 0 and |Φ , respectively. As mentioned above, the spectrum of H 0 is gapless, and the fermion occupancies in the ground state |Φ 0 are: That is, all single-particle levels with negative momentum are filled out. However, H LE has a gapped spectrum and, therefore, when constructing its ground state, |Φ , only the levels in the valence band (which have negative energy) are filled, whereas the levels in the conduction band remain empty: B. Quench from the gapped to the gapless phase The first situation we shall consider is when g(t) = g θ(−t) in Eq. (8), so that the spectrum of the Hamiltonian abruptly changes from gapped to gapless (i.e. quantum critical). In the following we denote The time evolution of ψ r,l for t > 0 is thus We first consider the zero temperature quench in this section, and postpone to Section V the discussion of the more complicated finite-temperature case. In the zero temperature case the initial state ρ i = |Φ Φ|. Notice that, in this state, Φ| cos 2φ(x)|Φ = Φ| cos 2ϕ(x)|Φ = Re Φ|e −2iϕ(x) |Φ = − ψ † r (x)ψ l (x) = 0 (the minus sign stems from the eigenvalue of the operator Γ = η r η l ), whereas in the ground state of H 0 , |Φ 0 , the expectation value of the same operator vanishes. Therefore, it behaves like an order parameter in equilibrium, and we can expect that it exhibits interesting dynamics out of equilibrium. Indeed, where, to evaluate the expectation value we have set T = 0 and therefore ψ † r (p)ψ l (p) = − 1 2 sin 2θ(p), as it follows from Eqs. (14,19,20). The above expression can be readily evaluated by recalling that sin 2θ(p) = m/ω(p), which yields, in the L → ∞ limit, where K 0 is the modified Bessel function. Thus we see that the 'order parameter' cos 2ϕ(0, t) decays exponentially at long times at T = 0. The decay rate is proportional to the gap between the ground state (the initial state) and the first excited state of the initial Hamiltonian H i = H. The existence of this gap means, in particular, that correlations in the initial state between degrees of freedom of the system are exponentially suppressed beyond a distance of the order ξ c ≈ v/m. Upon quenching, the evolution of the system is dictated by a critical Hamiltonian, H 0 , that is, a Hamiltonian describing excitations that propagate ballistically along 'light cones' corresponding to the 'trajectories' x ± vt. Thus, as discussed by Calabrese and Cardy [9,15] the correlation length scale characterizing the initial state translates into an exponential decay in time of the order parameter at long times. This exponential decay is also found (for the same type of quench) in the semiclassical limit of the sine-Gordon model (see Sect. IV below). Next we shall consider the (equal-time) two-point correlation function of the same operator, namely: When using the fermionic representation of e 2iϕ(x,t) , and upon expanding in Fourier modes we arrive at: Applying Wick's theorem, there are three different contractions of the above four fermion expectation value, which can be evaluated using Eqs. (14,19,20). This yields the following contractions: Hence, in the thermodynamic limit (L → ∞), we obtain: where in the limit a 0 → 0. Therefore, for |x| = 0, Let us examine the behavior of this correlation function in the asymptotic limit where |x| ≫ ξ c = v/m and 2vt ≫ ξ c . Since the Bessel functions decay exponentially for large values of their arguments, the leading term in G(x, t) depends on whether t < |x|/2v or t > |x|/2v. Thus, These results are in also agreement with those obtained using a mapping to a boundary conformal field theory by Calabrese and Cardy for general quantum quenches from a non-critical into a critical state. [9,15] C. Quench from the gapless to gapped phase We next consider the reversed situation to the one discussed in the previous subsection. In this case, we set g(t) = g θ(t) in Eq. (8), i.e. the initial state is critical and corresponds to the ground state of H i = H 0 , whereas the time evolution is performed according to H f = H LE . We shall consider the same correlation functions as in the previous subsection and therefore it is convenient in this case to obtain the time evolution of the operators ψ r (p) and ψ l (p), whose action on the initial state is known [cf. e.g. Eqs. (17,18)]. Once again, we first restrict ourselves to the T = 0 case and defer the discussion of finite temperature effects to Sect. V. We first note that the time-evolved Fermi operators can be related to the operators at t = 0 by means of the following (time-dependent) transformation: where f (p, t) = cos ω(p)t − i cos 2θ(p) sin ω(p)t and g(p, t) = i sin 2θ(p) sin ω(p)t. This transformation can be shown to respects the anti-commutation relations characteristic of Fermi statistics, and it is therefore a canonical transformation. Using Eqs. (34) to (35), we can now compute the decay of the order parameter operator e 2iϕ(x,t) . The calculation yields: In deriving the above expression we have used that f (−p, t) = f * (p, t) and g(−p, t) = g(p, t), which follows from cos 2θ(−p) = − cos 2θ(p) because cos 2θ(p) = ω 0 (p)/ω(p). Thus, setting Re [f * (p, t)g(p, t)] = − cos 2θ(p) sin 2θ(p) sin 2 ω(p)t and taking L → +∞, we find: where ci is the cosine integral function. The first term is a non-universal constant that depends on the short-distance cut-off a 0 introduced above (cf. Eq. 9). For long times this expression can be approximated by Hence we conclude that, when quenched from the critical (gapless) phase into the gapped phase, the order parameter exhibits an oscillatory decay towards a (non-universal) constant value, A(ma 0 ). Using similar methods the (equal-time) two-point correlation function, G(x, t) = e −2iϕ(x,t) e 2iϕ(0,t) , can be also evaluated. The resulting expression can be cast in a form identical to Eq.( 30) of subsect. III B. In the thermodynamic limit, we find that, in the present case, the function H(x, t) takes the form: We have been unable to obtain an closed analytical expression for this function at all times. However, in t → +∞ limit, in which case the term in the integrand proportional to cos 2ω(p)t oscillates very rapidly and therefore yields a vanishing contribution, an analytical expression can be obtained. Upon performing the momentum integral, we obtain the following result for large |x| (after taking the limit a 0 → 0): Hence, we obtain the following asymptotic behavior of the two-point correlation for t → ∞: This result is clearly different from the equilibrium behavior of the same correlation function in the gapped phase, where it decays exponentially to a constant. [31,32] Instead, when the system is quenched from the gapless into the gapped phase, we find that both the order parameter and the two-point correlation function (Eqs. 38 and 41) decay algebraically to constant (non-universal) values.

A. Introductory remarks
A good approximation to the sine-Gordon model (cf. Eq. 3) can be obtained in the limit where κ ≪ 1, which corresponds to the K ≪ 1 limit in the original notation of Eq. (1). In this limit, we can expand the cosine term in (3) about one of its minima, e.g. ϕ = 0. Retaining only the leading quadratic term yields the following quadratic Hamiltonian for the boson field ϕ(x): Within this approximation, the problem of studying a quantum quench in the sine-Gordon model becomes akin to the general problem studied in Ref. 26. To see this, let us first expand ϕ(x) in Fourier modes: where ω 0 (q) = v|q|; the b-operators introduced above obey the standard Heisenberg algebra: commuting otherwise. The first two terms in Eq. (43) are the so-called zero-modes, whose dynamics is only important at finite L. In what follows we restrict our attention to the thermodynamic limit (L → ∞) and therefore neglect the dynamics of those zero modes. Introducing (43) into (42), the Hamiltonian takes the general form: with the following identifications ω 0 (q) = v|q| and g(q, t) = m(q, t) = 2vg(t)κ 2 /|q|a 2−κ 2 /2 0 . As in the study of the Luther-Emery limit, we shall assume that g(t) = g θ(−t), which corresponds to a quench from the gapped to a gapless phase, [50], or g(t) = g θ(t), which corresponds to a quench for the gapless to the gapped phase. Following the procedure described in the Appendix of Ref. 26, the quench dynamics of this Hamiltonian can be solved by the following canonical transformation (indeed, the bosonic version of Eqs. 34,35) where Introducing m 2 = 4gv 2 κ 2 /a 2−κ 2 /2 0 , which is the the gap in the frequency spectrum of the gapped phase and setting m(q) = g(q) = m 2 /2ω 0 (q), the parameter β(q) satisfies: and the frequency: is the dispersion of the excitations in the gapped phase.
B. Quench from the gapped to the gapless phase Let us begin by discussing the situation where g(t) = g θ(−t). In this case, the initial state is the ground state of the following Hamiltonian (we omit the zero-mode part henceforth): where the operators a(q) and a † (q) are bosonic operators related to b(q) and b † (q) by means of the following canonical transformation: with β(q) satisfying Eq. (49). At t = 0 the Hamiltonian abruptly changes to H f = H 0 , which is diagonal in the b(q) and b † (q) basis, namely, In this case the evolution of the expectation value of the order parameter operator e −2iφ(x) = e −iκϕ(x) or its correlation functions can be obtained from the knowledge of the two-point (equal time) correlation function out of equilibrium for the boson field ϕ(x), i.e. F (x, t) = ϕ(x, t)ϕ(0, t) − ϕ 2 (0, t) , where the expectation value is taken over the ground state of H sc but the time evolution is dictated by H 0 . To compute this object, we first insert into the expectation value the Fourier expansion of ϕ(x), Eq. (43) and use Eq. (46). Thus, we arrive at: Using this result, let us consider the behavior of the order parameter following the quench. Taking into account that e −2iφ(0,t) = e −iκϕ(0,t) = e − κ 2 2 ϕ 2 (0,t) , we see that ϕ 2 (0, t) must be evaluated in closed form using Eq. (54) . Before performing any manipulation of this expression, it is convenient to subtract the constant ϕ 2 (0, 0) , which is formally infinite (i.e. it depends on the short distance cut-off, a 0 ). Thus, taking the thermodynamic limit where L → ∞, we obtain: Inserting the expressions for ω(p) and ω 0 (p) in the above equation, we obtain: where f (z) is defined as: being G 21 13 the Meijer G function. [39] Using the asymptotic expansion for this function, f (z) ≈ 1 − π|z| 2 , and hence the long-time behavior of the order parameter is: We next examine the behavior of the two-point correlation function of the same (order-parameter) operator, where we have defined F (x, t) = ϕ(x, t)ϕ(0, t) − ϕ 2 (0, t) . At zero temperature, with the help of Eq. (54), we find that where being a 0 is the short-distance cut-off. Evaluating the integrals: where f (z) has been defined in Eq. (57). Thus, asymptotically (for max{|x|/2v, t} ≫ m −1 ): where G(x, 0) describes the correlations in the initial (gapped ground) state, and exhibits the following asymptotic behavior: where B(a 0 ) is a non-universal prefactor. Thus we see that the asymptotic form of G(x, t) (Eq. 63), as well that of the order parameter, Eq. (58), have the same form as the results found limit the Luther-Emery limit, and also agree with the results of Calabrese and Cardy based on a mapping to boundary conformal field theory. [9,15] C. Quench from the gapless to the gapped phase In this case, the system finds itself initially in the ground state of H i = H 0 , and suddenly (at t = 0) the Hamiltonian is changed to H f = H sc . For this situation convenient to obtain the evolution of the observables from the time-dependent canonical transformation of Eq. (46), where β(q) and ω(q) are given by Eq. (49) and Eq. (50), respectively. In this case, As in the previous subsection, e −2iφ(x) = e − κ 2 2 ϕ 2 (0,t) , and using Eq. (65), we find that: Note, interestingly, that this result can be obtained from Eq. (55) by exchanging ω 0 (q) and ω(q). However, when evaluating the integral we find that ϕ 2 (0, t) = +∞, for all t > 0, due to the presence of infrared divergences that are not cured by the existence of a gap in the spectrum of H f = H sc . Thus, we conclude that e −2iφ(x) = e − κ 2 2 ϕ 2 (0,t) vanishes at all t > 0.
The above result for the evolution of the order parameter seems to indicate that the system apparently remains critical after the quench. This is conclusion is also supported by the behavior of the two-point correlation function of the operator e 2iφ(x,t) : Let G(x, t) = e 2iφ(x,t) e −2iφ(0,t) = e κ 2 F (x,t) , where F (x, t) = ϕ(x, t)ϕ(0, t) − ϕ 2 (0, t) . Using Eq. (46) and Eq. (43), we arrive at the following result (at zero temperature, and for L → +∞): To illustrate the above point about the apparent "criticality" of the (asymptotic) non-equilibrium state,we can analyze the behavior of the two-point correlation function, G(x, t), in two limiting cases, for t = 0 and t → +∞. At t = 0, the correlation function, as obtained from Eq. (68), reads: where A(a 0 ) depends on the short-distance cut-off a 0 . Thus, the correlations are power-law because the initial state is critical. In the limit where t → +∞, the part of the integral in Eq. (67) containing the term cos 2ω(q)t oscillates very rapidly and upon integration averages to zero. The remaining integral can be done with the help of tables, [39] yielding: where G 22 04 is a Meijer function Using the asymptotic behavior of the Meijer function, [39] we obtain: with B(a 0 ) being a non-universal constant. Thus, although initially the system is critical and therefore correlations at equilibrium decay as a power law with exponent 2κ 2 , when the system is quenched into a gapped phase (where equilibrium correlations exhibit an exponential decay characterized by a correlation length ξ c ≈ v/m), the correlations remain power-law, within the semiclassical approximation. The exponent turns out to be smaller, equal to κ 2 , which is half the exponent in the initial (gapless) state. In other words, within this approximation, it seems that the system keeps memory of its initial state, and behaves as if it was critical also after the quench. This behavior seems somewhat different from the results obtained for the same type of quench in the Luther-Emery limit, where both the order parameter and the correlations for t+∞ approach a constant value, A(ma 0 ) (unless the non-universal amplitude A(ma 0 ) = 0, which seems to require some fine-tuning). Whether the differences found here between the Luther-Emery and the semi-classical limits are due to a break-down of the quasi-classical approximation, which neglects the existence of solitons and anti-solitons in the spectrum of the sine-Gordon model, or to a qualitative change in the dynamics as one moves away from the Luther-Emery limit, it is not clear at the moment. To clarify this issue will require further investigation with more sophisticated methods.

V. DYNAMICS IN THE LUTHER-EMERY LIMIT AT FINITE TEMPERATURES
In this section we shall consider that the initial state of the system corresponds to thermal mixed state, which describes a sGM system in contact with an energy reservoir (i.e. the canonical ensemble) at a temperature T = β −1 . The state is thus mathematically described by a density operator ρ i = e −Hi/T /Z i . We shall assume that the coupling to the thermal bath is turned off at t = 0, and the system subsequently evolves unitarily in isolation according to H f . We shall consider only the Luther-Emery limit of the sGM, where exact results can be obtained at all temperatures within the sGM model. The latter is not true in the semiclassical limit discussed above because this approximation only captures the breather part of the spectrum and not the solitonic part. We shall therefore not consider finite temperature quenches in this limit here.

A. Quench from massive to massless
Consider first the quench from the gapped to the gapless phase. The initial (gapped) Hamiltonian H i = H LE is thus diagonal in the valence and conduction fermion basis and therefore immediately follows Here f F is the Fermi factor and · · · = Tr{e −βHi · · · }.
The final Hamiltonian H f = H 0 ; thus, using Eqs. (72)-(74) we can compute the finite temperature versions of Eqs. (27) to (29): Note that these averages automatically vanish if the fermion operators are evaluated at different values of p because of momentum conservation. The time evolution is again dictated by H f = H 0 as in the zero-temperature case. Let us next consider some interesting observables. We begin with the order parameter, which reads: This integral can be transformed into an infinite sum by expanding tanh βε/2 in powers of e −βε and integrating term by term. We thus obtain the following low temperature expansion: Since the function K 0 decays exponentially for large value of its argument, from this expression we see that when the temperature is decreased, less terms are needed to approximate the sum. In particular, at zero temperature (β → ∞) only the n = 0 term contributes and we recover the zero temperature result of Eq. (23). At finite but low temperatures, the asymptotic behavior at long times is an exponential decay where the characteristic time decay is fixed by the gap. However, at higher temperatures, more terms contribute to the sum whereas the alternating sign leads to some partial cancelations. As a result, we expect a faster decay in time of the order parameter. To further analyze the long time behavior in this regime, we shall use an identity of Bessel functions [see Eq. (A1)] which allows us to obtain the following high temperature expansion: Note that, for this expansion, the higher the temperature the smaller the number terms that needs to be retained to accurately approximate the sum. In particular, the infinite temperature limit, βm ≪ 1, only the l = 0 terms contributes and thus the decay in time is exponential, but now, the characteristic decay time τ c is now fixed by the inverse temperature: Thus, to summarize, the asymptotic behavior of the order parameter following a quench from the gap to the gapless phase is described by an exponential decay both at very low and very high temperatures. At very low temperatures, the characteristic decay time is given by the frequency gap, m, but as temperature of the initial state is increased, the characteristic time is determined by the temperature. The behavior for intermediate temperatures is a crossover between these two exponentially decaying behaviors. Moreover, it is also worth mentioning that the exponential decay at large temperatures is characteristic of a critical theory at finite temperatures. Indeed, in the sGM we expect that, as the temperature is raised well above the gap energy scale, m, the the properties of the system will become indistinguishable from those of a critical system. As for the the finite-temperature correlation function, it can be again cast in the same form as the zero temperature case, Eq. (30), with the function H(x, t; β) having the following t → +∞ limit: Using the same technique as for the order parameter, we evaluate this function as The reasoning is the same as before: at very low temperatures, only the term with n = 0 contributes, and we recover the zero temperature expression. However as the temperature increases more terms become important. We can use the dual expression Eq. (A2) which applied to Eq. (85) yields This function decays exponentially for long distances (only the l = 0 term contributes at high temperatures), with a characteristic correlation length that is fixed by the temperature of the initial state: Upon introducing this results into (the finite temperature equivalent of) Eq. (30), we arrive at an expression whose asymptotic behavior again depends on whether |x| > 2vt or |x| < 2vt (with correlation time/lengths given by m or T depending on the temperature range). Thus, the correlations at finite temperature also will exhibit the so-called 'light-cone' effect. [9,15] B. Quench from the gapless to the gapped phase We next consider that the system is quenched from the gapless phase intro the gapped phase. Thus we need to assume that the system was initially described by H i = H LE and in contact with an energy reservoir at temperature T = β −1 . The following expectation values (understood over the initial thermal ensemble) will be required in the calculations to follow: and hence Thus, the time evolution of the order parameter can be obtained and reads: As t → +∞ this function approaches a non-universal constant that depends on the energy cut-off a 0 and the temperature. At high temperatures, Thus, after the sudden quench at t = 0 from a high temperature state in the critical regime into the gapped phase, the order parameter shows an oscillatory decay towards a constant value. However, the exponent of the decaying law is different from the decaying exponent in the case of a quench from a zero (or low) temperature state. The whole picture is the following: the order parameter exhibits an oscillatory decaying low as t −3/2 for times smaller than a time scale fixed by the temperature, where there is a crossover to a t −1/2 behavior characteristic of low temperatures.
Concerning the two-point correlation function, it can be again recast as in Eq. (30) with the function H(x, t; β) being given by: At long times, the cosine within the integral oscillates very rapidly yielding a vanishing contribution. The remaining integral can be evaluated using the Cauchy theorem resulting in an infinite sum over positive odd Matsubara frequencies. The sum can be performed, yielding where Φ(x, y, z) is the Lerch function. [39] Likewise, the long distance behavior dominated by the lowest (Matsubara) frequency term: At long distances, this function decays exponentially, but two competing length scales appear: v/m and v β/π. The largest sets the characteristic length of the decay.

VI. LONG-TIME DYNAMICS AND THE GENERALIZED GIBBS ENSEMBLE
It was recently pointed out by Rigol and coworkers [14] that, at least for certain observables like the momentum distribution or the density, the asymptotic (long-time) behavior of an integrable system following a quantum quench can described by adopting the maximum entropy (also called 'subjective') approach to Statistical Mechanics, pioneered by Jaynes. [40,41] Within this approach, the equilibrium state of a system is described by a density matrix that extremizes the von-Neumann entropy, S = − Tr ρ ln ρ, subject to all possible constraints provided by the integrals of motion of the Hamiltonian of the system. In the case of an integrable system, if {I m } is a set of certain (but not all of the possible) independent integrals of motion of the system, this procedure leads to a 'generalized' Gibbs ensemble, described by the following density matrix: where Z gG = Tr e − m λmIm . The values of the Lagrange multipliers, λ m , must be determined from the condition that where ρ 0 describes the initial state of the system, and · · · gG stands for the average taken over the generalized Gibbs ensemble, Eq. (98). Although ρ i = |Φ(t = 0) Φ(t = 0)| in the case of a pure state, as was first used in Ref. 14, nothing prevent us from taking ρ 0 to be an arbitrary mixed state and in particular a thermal state characterized by an absolute temperature, T . In this case, the Lagrange multipliers will depend on T or any other parameter that defines the initial state. Rigol and coworkers numerically tested the above conjecture by studying the quench dynamics of a 1D lattice gas of hard-core bosons (see Ref. 14 for more details). One of us showed analytically [11] that correlations of Luttinger model also relax to averages taken over this ensemble. This result for the Luttinger model was extended to include finite temperature fluctuations in the initial state in Ref. 26. The question that naturally arises then is whether the family of integrable models studied in this work (see Eqs. (45), and their fermionic equivalences of Eq. (12) and (13)) relax in agreement with the mentioned conjecture. In other words, does the average O (t) at long times relax to the value O gG = Tr ρ gG O, for any of the correlation functions considered previously? In what follows we shall address this question by analyzing quantum quenches in the sGM at zero temperature. The generalization at finite temperatures should be straightforward, as discussed in Ref. 26 A. Quench from the gapped to the gapless phase in the Luther-Emery limit In this case, the type evolution of the system is performed by H 0 (cf. Eq. 11), which is diagonal in the operators n α (p) = : ψ † α (p)ψ α (p) : (α = r, l). Thus, the generalized Gibbs ensemble is defined by the following set of integrals of motion I m → I α (p) = n α (p). We see immediately that the fact that this ensemble is diagonal in n F α (p) means that the order parameter e −2iϕ(x) gG = ψ r (x)ψ l (x) gG = 0, which agrees with the t → +∞ limit of the order parameter, was shown in Sect III to exhibit an exponential decay to zero. However, the two-point correlator of e 2iϕ(x) has a non-vanishing limit for t → +∞. Thus, our main concern here will be the calculation of the correlation function: Since the ensemble is diagonal in the chirality index, α, as well as momentum, p, we evaluation of the above expression can be carried out by noting that: where the Lagrange multipliers λ(q) can be related to the values of the same expectation values in the initial states by imposing their conservation, that is, Hence, = δ p1,p4 δ p2,p3 sin 2 θ(p 1 ) sin 2 θ(p 2 ).
B. Quench from the gapless to the gapped phase in the Luther-Emery limit In this case the initial state is the gapless ground state of H 0 , Eq. (11), whereas the Hamiltonian that performs the time evolution has a gap in the spectrum and it is diagonal in the basis of the ψ v (p) and ψ c (p) Fermi operators (cf. Eq. 16). Therefore, the conserved quantities are The associated Lagrange multipliers (at zero temperature), λ v (p) and λ c (p) can be obtained upon equating I v,c (p) gG = Ψ(0)|I v,c (p)|Ψ(0) . This yields: where ϑ(p) denotes the step function. Using these expressions we next proceed to compute the expectation values of the following observables:

Order parameter
We start by computing the order parameter, and upon using Eqs. (112) and (113), where we have used that cos 2θ −p = − cos 2θ p ; A(ma 0 ) is the non-universal constant introduced in Sect. IV C. This result agrees with the one obtained in Sect. IV C for the order parameter in the limit t → +∞.

Two-point correlation function
We next consider the two-point correlator of the order parameter, namely The calculation of the average in this case is a bit more involved, but it can be performed by resorting to a factorization akin to Wick's theorem. This is applicable only in the thermodynamic limit, as it neglects terms in which the four momenta of the above expectation value are equal. These terms yield contributions of O(1/L) compared the others. When factorizing as dictated by Wick's theorem, the only non-vanishing terms are: the average over the generalized Gibbs ensemble of the four Fermi fields on the right hand-side of Eq. (116) can be computed and yields the following expression for the two-point correlation function (up to terms of O(1/L 2 )): The first term in r.h.s. of the above expression is just e 2iϕ(x) gG e −2iϕ(0) gG (cf. Eq. 115), whereas the first term in the right hand-side can be written as that is, it coincides with the t → +∞ limit of the second term in the right hand-side of Eq. 30 in Sect. III C (the function H(x, t) is defined in Eq. 39).
C. Quench from the gapless to the gapped phase in the semi-classical limit In this case Hamiltonian performing the time evolution is gapless (H 0 ) and thus diagonal in the b-operators. Hence, the conserved quantities are The Lagrange multipliers of the corresponding generalized Gibbs density matrix are fixed from the condition: where β(q) is defined by Eq. (49) Hence, using this result we next proceed to compute the order parameter and the two-point correlation function. We first note that the order parameter vanishes in the generalized Gibbs ensemble since e −2iφ(x) gG = e −2i φ 2 (0) gG and φ 2 (0) gG = κ 2 4 ϕ 2 (0) is divergent in the L → +∞ limit (see below). This agrees with the result found in Sect. IV B, where it was found that the order parameter decays exponentially in time.
Thus, in what follows we shall be concerned with the the two-point correlation function.

Two-point correlation function
Since e −2iφ(x) e 2iφ(0) gG = e − κ 2 2 C gG (x) , where C gG (x) = ϕ(x)ϕ(0) gG − ϕ 2 (0) gG . In order to obtain this correlator, we introduce the Fourier expansion of ϕ(x) (ignoring the zero-mode part), into the expectation value, and using (126) to evaluate the averages in the generalized Gibbs ensemble, we find that, in the thermodynamic limit, and therefore, where C(x, 0) ≡ C(x, t = 0) is defined in Eq. (61). Upon comparing the last result with Eq. (60) in the limit where t → +∞, we see they are identical.
D. Quench from the gapless to a gapped phase In this case the Hamiltonian that performs the time evolution is gapped, whereas the initial state is gapless. Thus, differently from the previous case, the Hamiltonian that performs the evolution is diagonal in the a-operators, and therefore, the conserved quantities are I(q) = a † (q)a(q). The corresponding Lagrange (at zero temperature) are fixed from the condition: where β(q) is given by Eq. (49). In order to obtain the one and two-point correlation functions of e 2iφ(x) = e 2iκϕ(x) we first need to write the ϕ(x) field in terms of the a-operators. Upon using the canonical transformation Eq. (52): Hence, since e −2iφ(x) gG = e −iκϕ(x) gG = e − κ 2 2 ϕ 2 (0) gG , and ϕ 2 (0) gG is logarithmically divergent in the thermodynamic limit (see expressions below), the find that e −iκϕ(x) gG = 0. This result is in agreement with the one found in Sect. IV C for the order parameter.

VII. RELEVANCE TO EXPERIMENTS
As we mentioned in the introduction, ultracold atomic systems are the ideal arena to study the quench dynamics of isolated quantum many-body systems. This is because they can be treated, to a large extent, as entirely isolated systems. Furthermore, since this work is concerned with the quench dynamics of a specific one dimensional model, the quantum sine-Gordon model, it is also worth emphasizing that the properties of these systems are highly tunable and, in particular, so is their effective dimensionality. Thus, there are already a number of experimental realizations of one-dimensional (1D) systems (see e.g. Refs. 42-44 and references therein), and in particular, there are also experiments where non-equilibrium dynamics has been probed in one-dimension, e.g. Refs. 6, 45 Thus, there may be a good chance that some of the results obtained above may be relevant to current or future quench experiments with ultracold atoms. However, since the sine-Gordon model considered in previous sections is nothing but an effective (low-energy) description of certain 1D systems, any comparison must be done with great care, as there is no fundamental reason why the low-energy effective theory should capture the essentials of the (highly non-equilibrium) quench dynamics. This is to be contrasted with the equilibrium dynamics, where renormalization group arguments show that the sine-Gordon model is indeed sufficient to describe the (universal) properties of certain 1D physical systems. There is in fact much evidence, both analytical and numerical, accumulated over the years, of the latter fact. However, we are not in a comparable situation in the case of non-equilibrium dynamics, and thus future studies should try to address this question more carefully.
With the above caveat, let us proceed to mention a few situations where the sGM is applicable, at least as a good description of the equilibrium state of a system that can be realized with ultracold atomic systems. There are basically two kinds of systems, depending on the interpretation of the order parameter operator, e −2iφ(x) . The fist instance is a 1D Bose gas moving in a periodic potential, where the the sine-Gordon model is the effective field-theory description the Mott insulator to superfluid transition (MI to SF) in 1D [31,46]. In this case, the order parameter field is the periodic component of the boson density. A quantum quench from the gapped (gapless) into the gapless (gapped) in this system can be realized by suddenly turning on (off) the periodic potential applied to the 1D gas. The evolution of the 1D density could be monitored by performing in-situ measurements, and the two-point correlations by measuring, at different times, the (instantaneous) structure factor using Bragg spectroscopy.
In the second instance, the order parameter field is interpreted as the (relative) phase of two [47] (or more [48,49]) 1D Bose gases coupled by Josephson coupling of two 1D Bose gases. Thus, in this setup a quench experiment [16] from the gapped (gapless) into the gapless (gapped) phase would correspond to suddenly switching on (off) the (Josephson) tunneling, which can be achieved by controlling the (optical or magnetic trapping) potentials that confine the atoms to 1D. This should be done with care, ensuring the atoms remain in the 1D regime both in the initial and final states, that is, that e.g. the potential trapping the atoms transversally is always sufficiently tight. The evolution of the relative phase can be monitored by analyzing the interference fringes at different times.

VIII. CONCLUSIONS
To sum up, we have investigated the time evolution of two-point correlation functions and the order parameter after a quantum quench of the relevant operator term in the sine-Gordon model. We considered two different kinds of quenches: a quench from a gapped phase to a gapless phase and viceversa. In addition to the initial pure state, we considered an initial mixed state coupled to an energy reservoir at finite temperature, that is suddenly disconnected at the same time that the quench is performed. In order to compute correlation functions, we studied two limits in which the Hamiltonian renders quadratic in terms of either fermion or boson operators, and the dynamics can be solved exactly: the Luther-Emery and the semiclassical limits. In the quench from the gaped to the gapless phase, the order parameter decays exponentially for long times to the ground state value, with a time scale fixed by the gap. In turns, the correlation function exhibit a light-cone effect: it decays exponentially with time with a time-scale fixed by the gap until it relaxes to a value that decreases exponentially with distance with correlation length fixed also by the mass. These results are valid for both, the Luther-Emery and the semiclassical limits, and agree with the results obtained in Ref. 9,15 . For an initial state at high temperature, these decays are also exponential (and the light-cone effect is preserved), but the characteristic time and length is set by the temperature. In between, a crossover connects these two limiting cases.
A general statement for the long time dynamics when the term that opens the gap is suddenly turned on is elusive, since in the semiclassical approximation the long times limit of the correlation function exhibit power-law decay with distance, and the order parameter vanishes, being these features characteristic of a critical state. On the other hand, the results in the Luther-Emery limit indicate a relaxation to a non-universal constant value for long times, signaling an ordered state. The latter behavior at the Luther-Emery limit is also present at finite temperature. Whether these differences are due to an artifact introduced by the semiclassical approximation or to a very special behavior that occurs at the solvable point needs further clarification.
We have shown that the long time behavior of correlation functions and the order parameter in the different types of quenches can be obtained from the generalized-Gibbs ensemble [14] in which the conservation of a certain set of independent integrals of motion is fixed as a constrain for the maximization of the statistical entropy. Finally, the relevance of the quantum quench dynamics in the sine-Gordon model for cold atomic gases is discussed. The superfluid-Mott insulator transition appears as the most appropriate scenario to observe the described effects.