Athermality of generalized Gibbs ensembles

1 Introduction

Isolated quantum systems evolving unitarily from non-equilibrium initial states generically thermalize [1–7]. This means that, at long times, the reduced density matrix of a finite subsystem embedded in a thermodynamically large system approaches the reduced density matrix of a thermal ensemble, with the temperature uniquely fixed by the energy density of the initial state. As a consequence, although the full system remains at all times in a pure state characterized by zero von Neumann entropy, finite subsystems behave locally as if they were in thermal equilibrium. In this sense, entanglement with the rest of the system effectively acts as a thermal bath for local observables [8–11]. This mechanism provides the modern understanding of thermalization in isolated quantum many-body systems.

From a dynamical perspective, one of the most intriguing aspects of quantum integrable models [12] is their failure to thermalize under quantum unitary dynamics. In these systems, the existence of an extensive number of mutually compatible conservation laws strongly suppresses the scrambling of local quantum information, providing a natural mechanism for local memory retention. Importantly, despite the fragile nature of integrability, this behavior persists over long prethermal timescales [13–27] even in experimentally realistic settings [28–30]. For this reason, at long times an integrable system does not thermalize to the standard Gibbs ensemble (GE). Instead, it relaxes to a special statistical ensemble that incorporates the constraints imposed by the full set of local and quasi-local conserved quantities [9, 31, 32]. This ensemble is known as the generalized Gibbs ensemble (GGE) [33]. The GGE extends the conventional Gibbs description by assigning a Lagrange multiplier to each conserved quantity, thereby encoding the strong memory that integrable systems retain of their initial conditions. It provides the correct stationary description for local observables after quantum quenches in a broad class of integrable models and has become one of the central concepts in the modern understanding of non-equilibrium quantum dynamics.

GGEs have been extensively studied in the literature [9, 31], in particular with regard to their thermodynamic properties [34–38], entropies [39–47], expectation values of local observables [48–56], and correlation functions [57–63].

A very natural question, which has nevertheless received little attention and has been addressed mostly by means of numerical calculations in finite systems [64–66] or rather indirectly by integrability techniques (typically through comparisons of expectation values and correlation functions) [67–71], concerns the precise relation between GGEs and ordinary thermal ensembles. In particular, how different is a GGE from a standard GE? Is there a simple and physically transparent way to quantify this difference? Addressing these questions is important not only for understanding the structure of integrable stationary states, but also for clarifying in which sense integrable systems fail to thermalize in the conventional way.

The goal of the present work is to address these questions. We will show that the concept of athermality [72–75], rooted in quantum resource theory [76, 77], provides a particularly natural and effective framework for this purpose. Here we define the athermality as the relative entropy between a given state (or ensemble) and the closest GE, where closeness is measured by the minimization of the same relative entropy. Such closest thermal ensemble is simply the GE with the same average energy of the chosen state, and as a consequence the athermality is simplified to equal the difference of their von Neumann entropies. In the context of integrable quenches we consider the GGE as the given state (cf. ref. [78] for the use of a minimized relative entropy in a similar context), and hence the closest GE is precisely the thermal ensemble to which the system would have locally relaxed in the absence of integrability. For this reason, the athermality coincides with the difference between the GE and GGE entropies employed in the numerical studies of refs. [64–66, 79]. We stress, however, that by means of integrability techniques we obtain exact thermodynamic-limit results for the athermality, which enable us to shed new light on the fundamental questions above.

We compute the GGE athermality for several different quench protocols in integrable models, including the free XY spin chain in a transverse field, the interacting XXZ spin chain, the interacting Lieb-Liniger gas, and the harmonic chain. The derivations, technical details, and general formalism are presented in the main body of the paper. Here, however, we would like to emphasize the most surprising outcomes of our analysis.

1. The athermality, as a function of the post-quench parameters in the Hamiltonian, is always singular at quantum critical points [80]. This appears at first rather remarkable because the athermality after a quench probes physics at a finite energy density. This sharp signature of quantum criticality in the middle of the many-body spectrum is due to the singular behaviour of the GGE entropy, while the GE entropy remains analytic everywhere as expected at any nonzero temperature [81]. Singularities in the GGE entropy at quantum criticality were observed in ref. [82] (cf. also refs. [83–87]). Here we analytically prove their emergence in the XY chain in a transverse field, and demonstrate that they arise also in the XXZ chain. For the XXZ chain we also uncover the presence of a fractal behaviour of the GGE entropy in the gapless phase, similarly to what found for other physical quantities [88–90]. Singular behaviours at criticality in the context of GGEs have also been uncovered by probing local observables [82, 91–93], while influence of quantum criticality has been observed in the context of integrable transport [32, 94, 95]. More generally, signatures of quantum criticality emerging at intermediate and late times following quantum quenches have been observed in both integrable [96–102] and non-integrable settings [103–105]. Despite the growing body of evidence for such phenomena, a unified picture and theoretical explanation for their microscopic origin and significance is still missing. In the present integrable context, however, sharp singularities can be characterized exactly thanks to the analytical tractability of the models.
2. We find that the athermality is generically an extensive quantity in the subsystem size, with a prefactor of order unity. However, often the athermality becomes surprisingly small at quantum criticality, where it takes the form of a sharp local-minimum cusp. Importantly, the dip at criticality emerges from the interplay between the GE and GGE entropies, and hence it is in general a distinctive property only of the athermality. We uncover that this enhanced proximity between the GGE and a thermal ensemble is generally associated with a pinning mechanism: the GE and GGE root densities (or mode occupation functions in free models) get pinned to a common maximal value at a critical momentum; moreover, the two distributions exhibit very similar behaviour in a broad neighbourhood of this point. As a consequence, the differences between the two ensembles are strongly suppressed over the region of momentum space that contributes most significantly to thermodynamic quantities and athermality.

We conclude this introduction by highlighting a connection with known results in conformal field theory (CFT) that provide us with a heuristic explanation for why the athermality often becomes surprisingly small at quantum criticality. Our starting point is the well-known and rather remarkable observation that, for quantum quenches in CFTs from a special class of initial states of the form $|{\psi }_0⟩=e^{-{\tau }_{0}H}|B⟩,$ the stationary state is exactly thermal, with an effective inverse temperature $\beta =4{\tau }_0$ [106–108]. This result is particularly striking because the underlying dynamics remains fully integrable. The origin of this emergent thermal behavior can be traced back to the structure of the initial state, which excites only the Hamiltonian while leaving all higher conserved charges effectively unactivated [108, 109]. As a consequence, the corresponding GGE reduces to an ordinary GE. We expect that many of the quenches analyzed in this work admit, in the scaling limit and close to a quantum critical point, an effective description that is very similar to the one above. In this regime, the initial states become approximately conformal boundary states and the higher conserved quantities are only weakly activated. The resulting stationary state is therefore expected to be very close to the corresponding thermal ensemble, naturally leading to a small value of the athermality. While this argument provides a qualitative understanding of the suppression of athermality near criticality, the actual values that we find are often extraordinarily small. This remains somewhat surprising.

The remainder of the work is organized as follows. In section 2 we define the athermality and highlight the simplifying features emerging from our minimization prescription. In section 3 we discuss how the athermality can be calculated exactly in the framework of the thermodynamic Bethe ansatz (TBA) [110, 111] and its relevance in the context of quantum quenches. In section 4 we study the athermality for integrable quantum quenches in the XY chain in a transverse field, and analyze its behaviour across both quantum critical lines. In section 5 we turn to the XXZ chain and demonstrate that the main conclusions found for the free XY chain survive in the presence of interactions. In section 6 we consider integrable quenches in the Lieb-Liniger gas, and analyze the athermality for repulsive interactions and the GGE entropy for both repulsive and attractive couplings. The absence of a quantum phase transition in the repulsive gas is reflected in the absence of striking features in the athermality. In section 7 we consider the harmonic chain, and study the differences in athermality due to the bosonic statistics of the elementary modes. Finally, in section 8, we provide some remarks about the time evolution of the athermality after quantum quenches. Several appendices follow which review basic concepts, give further details of the derivations and present additional results.

2 Quantum athermality

Consider a system described by a Hamiltonian $H$, and a generic (reduced) density matrix $\rho$. We are interested in quantifying how far $\rho$ is from a GE at inverse temperature $\beta$

A possible choice would be to employ a proper distance between $\rho$ and $\rho \mathrm{GE}$, such as the trace distance, fidelity or Hilbert-Schmidt distance [112]. The former two possess a clear operational meaning in experimental settings but are usually hard to calculate, while the latter lacks a simple physically relevant operational interpretation. For these reasons we turn our attention to the quantum relative entropy (QRE) [112, 113], defined as

where $\rho$ and $\sigma$ are two arbitrary density matrices (see also, e.g., refs. [114, 115] for the use of the QRE in similar contexts). The QRE is not a metric, as it is not symmetric under $\rho \leftrightarrow \sigma$ and it violates the triangle inequality. However, it respects positivity

and has a clear operational interpretation via the quantum Stein’s lemma [113, 116, 117].

It is well known that the use of the QRE with respect to GEs can be drastically simplified [72]. This is a consequence of the equality

where $S_{\mathit{𝑣𝑁}}$ denotes the von Neumann entanglement entropy and we have used $\ln <!--\; FUNCTION\; APPLICATION\; -->Z_{\mathrm{GE}}=-\beta \mathit{𝑇𝑟}[\rho \mathrm{GE}H]+S_{\mathit{𝑣𝑁}}(\rho \mathrm{GE})$. We define the athermality $𝒜(\rho )$ of a generic density matrix $\rho$ in terms of the QRE as

where as in equation (2.1) the dependence of $\rho \mathrm{GE}$ on $\beta$ is implicit. From the perspective of quantum resource theories [77], the athermality $𝒜(\rho )$ quantifies how resourceful a state $\rho$ that differs from a GE is within an experimental setup in which preparing a GE at any inverse temperature $\beta$ has negligible cost. We stress that, due to the presence of the minimization over $\beta$, the definition (2.5) differs from the one relevant to situations in which the only resource-free states that can be prepared at negligible cost are GEs at a fixed inverse temperature ${\beta }_{\mathit{𝑓𝑖𝑥𝑒𝑑}}$ [72, 74, 75, 118]. To find the global minimum of $S(\rho \Vert \rho \mathrm{GE})$ as a function of $\beta$ we consider its derivatives

and note

Setting equation (2.6) to zero, and given equation (2.8), we find that the minimum of the QRE is obtained at the inverse temperature ${\beta }^{\ast }$ for which the average energy of $\rho$ and $\rho \mathrm{GE}({\beta }^{\ast })$ is the same. From this it follows that the quantum athermality (2.5) is simply given by a difference of two von Neumann entropies

From equations (2.7) and (2.8) we see that the GE energy is a monotonically decreasing function of $\beta$, therefore ${\beta }^{\ast }$ is always unique (it can be negative if the spectrum is bounded from above). The semipositive definiteness of $𝒜(\rho )$, guaranteed by the inequality (2.3), is re-interpreted in equation (2.9) as a consequence of the GE being the highest-entropy ensemble at a fixed average energy.

In many cases of interest the Hamiltonian $H$ might possess a $U(1)$ symmetry generated by an observable $N$ ($[N,H]=0$) that is of operational importance. For example, $N$ might indicate the number of particles within the system subject to full experimental control. In such cases it is also meaningful to consider a GE with the addition of an extra Lagrange multiplier (grand canonical ensemble)

The QRE with respect to such GE retains the same form of equation (2.4) but with the substitution $H\to H-\mathit{𝜇𝑁}$. Following similar logic and simple minimization steps as above we define the athermality $𝒜(\rho )$ as

It is straightforward to generalize these statements in order to include an arbitrary finite number of operationally relevant $U(1)$ symmetries.

3 Athermality of Generalized Gibbs Ensembles

The distinguishing feature of quantum integrable Hamiltonians [12] is that they possess an extensive number of algebraically independent charges $Q^{(n)}$ ($[H,Q^{(n)}]=0,\forall <!--\; FUNCTION\; APPLICATION\; -->n=1,2,\dots$), where each $Q^{(n)}$ can be written as a sum or integral of (quasi)local [32, 119–125] densities $q^{(n)}(x)$ [12, 126]. For this reason, the Generalized Gibbs Ensemble (GGE)

plays a special role in quantum integrability (note that we implicitly included $H$ in the set $Q^{(n)}$). In the framework of quantum quenches in integrable systems, the GGE correctly captures local properties of the system at late times [9, 31]. In particular, given an initial lowly entangled state $|\psi (0)⟩$ (often the ground state of a pre-quench Hamiltonian $H^{\prime }$) that for times $t>0$ evolves unitarily under a post-quench integrable Hamiltonian $H$, in the thermodynamic limit and at late times one expects in any finite (or at most subextensive) subsystem $A$ (with complement $B$) [9]

where $L$ denotes the system size and the order of limits on the left-hand side is important. The Lagrange multipliers ${\beta }_n$ in equation (3.1) enforce that the average densities $⟨Q^{(n)}⟩∕L$ at infinite times are the same as in the initial state $|\psi (0)⟩$. We stress that in the quench context, where the partial trace over $B$ plays a crucial role, the GGE is only one of many possible choices of ensembles capturing the correct local properties at late times [9].

In the following we focus on the GGE density of athermality

This choice is convenient for several reasons: (i) it captures the leading order (in system size) value of the athermality between arbitrary GEs and GGEs, which is an interesting quantity irrespective of the many (isolated, open, weakly nonintegrable or driven) setups where GGEs emerge [9, 14, 26, 31, 127–135]; (ii) the intensive value $\alpha$ describes the leading order (in subsystem size $|A|$) of the athermality in the reduced density matrix ${\rho }_A$ from equation (3.2), relevant in the context of quantum quenches in isolated systems; (iii) $\alpha$ can be computed solely within the framework of the TBA [12, 110, 111], as we explain in the next section. Generalizations of equation (3.3) to accommodate more Lagrange multipliers in the GE, as in equations (2.10) and (2.11), are straightforward.

3.1 Entropy density of GGEs and TBA

Here we just recall the fundamental ingredients we will need in the rest of the discussion, while a brief review of some of the basic concepts can be found in appendix A.

Eigenstates of Bethe-integrable models can be parametrized in terms of sets of rapidities $𝝀=({\lambda }_1,{\lambda }_2,\dots )$. In the limit $L\to \infty$, the essence of TBA is to switch attention from individual eigenstates to classes of eigenstates, or macrostates, with the same local properties. For the simplest models hosting only one type of quasiparticles, in the limit of large $L$ two eigenstates are said to belong to the same macrostate if they possess an identical distribution $\varrho (\lambda )$ of rapidities, where $\varrho (\lambda )$ is known as root density. In models hosting multiple species of quasiparticles, e.g. bound states of different length, a macrostate is specified by a set $𝝔(\lambda )={\{{\varrho }_{\ell }(\lambda )\}}_{\ell =1}^{{\ell }_{\mathit{𝑚𝑎𝑥}}}$ of root densities corresponding to different quasiparticle species labelled by $\ell =1,2,\dots$. The insight of TBA is that many physical quantities depend only on macrostate information. For example, the density of thermal entropy from equation (3.3) is given by

where $s_{\mathrm{YY}}[{𝝔}_{{𝜷}^{\ast }}(\lambda )]$ denotes the Yang-Yang entropy density of the thermal macrostate ${𝝔}_{{\beta }^{\ast }}(\lambda )$ at inverse temperature ${\beta }^{\ast }$ [12] (see appendix A).

Crucially, also the GGE entropy density in equation (3.3) depends solely on macrostate information [35, 136, 137]. This is particularly simple to see in cases where the charges $Q^{(n)}$ with nonzero Lagrange multipliers ${\beta }_n$ in equation (3.1) form an Abelian algebra

In such cases ones speaks of an Abelian GGE. Averages with respect to such GGEs are dominated by eigenstates belonging to a specific non-thermal saddle-point macrostate ${𝝔}_{\mathrm{GGE}}(\lambda )$. Similarly to equation (3.4), the GGE entropy density depends solely on the macrostate information encoded in ${𝝔}_{\mathrm{GGE}}(\lambda )$

When the GGE emerges from quantum quenches as in equation (3.2), the macrostate ${𝝔}_{\mathrm{GGE}}(\lambda )$ can be obtained from the quench action method [138, 139] if expressions for overlaps $⟨\psi (0)|𝝀⟩$ are known. In the absence of the overlaps, it is possible to reconstruct ${𝝔}_{\mathrm{GGE}}(\lambda )$ if the expectation values $⟨\psi (0)|Q^{(n)}|\psi (0)⟩$ of all the charges entering the GGE (3.1) are available [123, 140]. Moreover, for integrable initial states [141–145], one can apply the quantum transfer matrix approach to find analytical expressions for the rapidity distributions [146–149]. In the following sections we will make use of results obtained from all these approaches (see appendix A for further details).

We stress that it has been shown explicitly in many contexts [10, 40, 47, 57, 69, 84, 150, 151] that the entropy density (3.6) does indeed describe the leading order (in subsystem size $A$) saturation value of the entanglement entropy at late times after quantum quenches in integrable models.

In summary, we see from equations (3.3), (3.4) and (3.6) that the density of GGE athermality $\alpha$ is simply given by

As before, these results generalize easily to cases in which we include additional charges in the GE description, cf. equations (2.10) and (2.11).

We remark that integrable models in general possess discrete and continuous non-Abelian symmetries. In some models, even the full extensive algebra of higher (quasi)local conserved charges $Q^{(n)}$ might be non-Abelian. This is for example the case in the XY model (although in the absence of a transverse field) [9, 136, 137] and in the XXZ chain (only at specific roots of unity values of the anisotropy $\mathrm{\Delta }$) [32, 152–155], considered respectively in sections 4 and 5. As pointed out in ref. [136] (see also refs. [9, 137]), however, for any given quench the initial state selects an Abelian subset of charges that can be used to construct an Abelian GGE. Therefore, a description in terms of rapidity distributions should always be possible [38, 156].

For sake of completeness we mention that in certain cases the GGE functional form in equation (3.1) cannot be directly applied for interacting models (at least not without modifications) [38, 156–158]. This is for example the case in the BEC quench [157] in the Lieb-Liniger model analyzed in section 6 and in the Néel quench in the gapped XXZ chain of section 5 [158]. In such cases, however, the correct ensemble ${\rho }_{\mathrm{GGE}}$ behind the reduced density matrix ${\rho }_A$ of equation (3.2), even if not of the form (3.1), is exactly captured within the TBA framework we employ.

4 XY chain in a transverse field

We consider the XY spin chain in a transverse magnetic field [159–161] with periodic boundary conditions (PBC), given by the spin-$1∕2$ Hamiltonian

Here ${\sigma }_j^{\delta }$ denotes a Pauli matrix at site $j$, $\gamma \in [-1,1]$ the anisotropy parameter and we restrict our analysis to the regime $J>0$, $h>0$. $H_{\mathit{𝑋𝑌}}$ features a ${\mathbb{Z}}_2$ symmetry of rotations by $\pi$ around the $ẑ$ axis, i.e. ${\sigma }_j^{\delta }\to -{\sigma }_j^{\delta }$ for $\delta =x,y$. For the special choices $\gamma =0$ and $\gamma =1$ one obtains, respectively, the XX chain in a transverse field (where ${\mathbb{Z}}_2$ is promoted to $U(1)$) and the transverse field Ising chain (TFIC). The $T=0$ phase diagram of the model, represented in figure 1, hosts 2 different classes of quantum critical lines [162]:

1. Each point on the line $(\gamma \ne 0,h=1)$ is a quantum critical point in the Ising universality class, separating the ferromagnetic ordered phase of $h<1$ (order along $\widehat{x}$ for $\gamma >0$ and along $ŷ$ for $\gamma <0$) from the paramagnetic phase of $h>1$.
2. The XX line $(\gamma =0,h<1)$ represents a gapless Luttinger liquid [163, 164] phase with quasi-long-range order. Each point on the line is a quantum critical point in the compact free boson universality class, separating the two ($\gamma >0$ and $\gamma <0$) gapped ferromagnetic phases in (i).

The spin-$1∕2$ operators in equation (4.1) can be mapped to spinless fermions $c_j,c_j^{†}$ by the nonlocal Jordan-Wigner transformation [165], which leaves $H_{\mathit{𝑋𝑌}}$ as a sum of local terms in the fermions [159]. Given the ${\mathbb{Z}}_2$ symmetry $[H_{\mathit{𝑋𝑌}},e^{\mathit{𝑖𝜋𝑁}}]=0$, where $N={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^L{c}_j^{†}{c}_j$, $H_{\mathit{𝑋𝑌}}$ is block diagonal in the subspaces of the fermionic Fock space with even (e) and odd (o) numbers of fermions, i.e. $H_{\mathit{𝑋𝑌}}=H_e\oplus H_o$. The diagonalization of $H_e$ and $H_o$ by Fourier and Bogoliubov transformations is standard and leads to [159, 160]

Here ${\alpha }_k$ and ${\alpha }_k^{†}$ (for which the dependence on $\gamma$ and $h$ is implicit) indicate fermionic Bogoliubov operators associated with the free momenta ($L$ even)

The eigenstates of $H_{\mathit{𝑋𝑌}}$ are generated by repeated actions of ${\alpha }_{k_j}^{†}$ on the Bogoliubov vacua ${|0;\gamma ,h⟩}_S$, defined by ${\alpha }_k{|0;\gamma ,h⟩}_S=0\forall <!--\; FUNCTION\; APPLICATION\; -->k\in S$. For $h>1$ the ground state of $H_{\mathit{𝑋𝑌}}$ coincides with ${|0;\gamma ,h⟩}_e$, while for $h<1$ the two $S=$ e, o vacua ${|0;\gamma ,h⟩}_S$ become degenerate ground states for $L\to \infty$ and for $\gamma \ne 0$ hybridize by spontaneous symmetry breaking to yield ferromagnetic order.

4.1 Quench protocol and GGE root density

We focus on the standard quench protocol in which the initial state is chosen to be the ground state $|\psi (0)⟩=|\mathit{𝐺𝑆};{\gamma }_0,h_0⟩$ of the pre-quench Hamiltonian $H_{\mathit{𝑋𝑌}}({\gamma }_0,h_0)$, and we consider the stationary state locally reached at late times after evolution with $H_{\mathit{𝑋𝑌}}(\gamma ,h)$, where $({\gamma }_0,h_0)\ne (\gamma ,h)$. These initial states $|\psi (0)⟩$ are a paradigmatic example of integrable initial states [141], for which the overlaps with the eigenstates of $H_{\mathit{𝑋𝑌}}(\gamma ,h)$ can be easily obtained [166]. For $h_0>1$, where $|\psi (0)⟩={|0;{\gamma }_0,h_0⟩}_e$, one finds

where ${\alpha }_k$ refer to the post-quench Hamiltonian $H_{\mathit{𝑋𝑌}}(\gamma ,h)$. From equation (4.5) one readily obtains overlaps with the even eigenstates ${|k_1,\dots ,k_{2m};\gamma ,h⟩}_e={\alpha }_{k_1}^{†}\dots {\alpha }_{k_{2m}}^{†}{|0;\gamma ,h⟩}_e$ of $H_{\mathit{𝑋𝑌}}(\gamma ,h)$, from which we see that only parity-invariant eigenstates in the $S=e$ sector constructed by creation of pairs $(k,-k)$ on top of the vacuum ${|0;\gamma ,h⟩}_e$ have nonzero overlap with $|\psi (0)⟩$ (this being the characteristic property of integrable initial states [141] in free models). In free models ${\varrho }_{\mathrm{GGE}}(k)$ is easily determined by exploiting the fact that

For $h_0<1$, the physical ground state at $L\to \infty$ spontaneously breaks the ${\mathbb{Z}}_2$ symmetry and assumes the form

Taking into account the impossibility of thermodynamic phase transitions at finite energy densities in 1D, we deduce that all local operators connecting the even and odd sectors have vanishing expectation value at infinite times after the quench from (4.9). Therefore, the macrostate ${\varrho }_{\mathrm{GGE}}(k)$ associated with the stationary reduced density matrix reached at late times is given again by equations (4.7) and (4.8), because both the even and odd sectors give identical diagonal contributions for $L\to \infty$. Having access to ${\varrho }_{\mathrm{GGE}}(k)$ from equation (4.7), and to the thermal root density

at the inverse temperature ${\beta }^{\ast }$ fixed by equation (2.9), one obtains the entropy densities $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ and $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ needed in equation (3.7) for the density of athermality $\alpha$ as

4.2 Varying the postquench Hamiltonian

Given the large number of available parameters ($\gamma ,h,{\gamma }_0,h_0$), we start by considering the athermality as a function of the postquench $(\gamma ,h)$ when the prequench ones $({\gamma }_0,h_0)$ are fixed.

In figure 2 we show a few density plots for the density of athermality $\alpha$ in the $(\gamma ,h)$ plane. The most important feature, already anticipated in the Introduction, is that from these plots one is able to clearly identify the critical lines of the $T=0$ quantum phase diagram (see figure 1), despite the fact that $\alpha$ probes only physics at a finite energy density above the ground state. They emerge as non-analytic features of $\alpha$ at the $T=0$ critical lines, as evident from figures 3 to 5 where we plot some cuts in the $(\gamma ,h)$ plane. In addition to these singular points, all subplots in figure 2 feature trivial zeros due to the absence of the quench for $(\gamma ,h)=({\gamma }_0,h_0)$.

The singular behaviour of $\alpha$ comes from the GGE entropy density $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$, see section 4.2.3 below for further details and appendix B for a proof. The thermal entropy density $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ is instead a smooth function of $(\gamma ,h)$ as a consequence of the absence of phase transitions in 1D system with short-range interactions at finite temperatures (see, e.g., ref. [81]). In the context of the XY chain, the smoothness of $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ with respect to $\gamma$ and $h$ is easily proved by means of Leibniz’s integral rule by noticing that the integrand in equation (4.11) is an even function of ${𝜀}_{\gamma ,h}(k)$ (hence its Taylor expansion involves only the analytic function ${𝜀}_{\gamma ,h}^2(k)$) and that ${\beta }^{\ast }={\beta }^{\ast }(\gamma ,h)$ is also a smooth function (this comes from the fact that the energy density $e$ injected by the quench is trivially smooth). Apart from the singular points, a natural question is whether the qualitative behaviour of $\alpha$ in the $(\gamma ,h)$ plane merely reflects analogous behaviour of the energy density $e$ injected by the quench, which we show in figure 16 of appendix C. We find that $\alpha$ possesses much more structure than $e$, and that there is no strong correlation between the behaviour of $e$ and $\alpha$.

4.2.1 $h$ dependence

Figure 3 shows cuts where $\gamma ={\gamma }_0$ and hence the quench is entirely driven by $h_0\to h$. Figure 3(a) and (b) correspond to $h$-quenches in the TFIC ($\gamma ={\gamma }_0=1$). We see that, in addition to the trivial zeros at $h_0=h$, the plots for $\alpha$ present a very pronounced local minimum at criticality $h=1$, where a cusp marks non-analytic behaviour and $\alpha$ is anomalously close (but not equal) to zero. Figure 3(c) shows that analogous behaviour emerges in $h$-quenches even away from the TFIC limit (${\gamma }_0=\gamma <1$). In the density plots of figure 2(a) and (b) these dips at criticality coincide with the narrow purple regions around the critical line $(\gamma >0,h=1)$.

Figure 4 considers again the dependence of $\alpha$ on $h$, but now in quenches where ${\gamma }_0\ne \gamma$ in addition to $h_0\ne h$ (2-parameter quenches). While for ${\gamma }_0,\gamma >0$ one again generally finds a dip at criticality $h=1$ (cf. figure 2(a) and (b)), the behaviour is richer when one allows large jumps $0<{\gamma }_0\to 0>\gamma$. In such cases one can find again a dip at criticality $h=1$, as shown in figure 4(a) (cf. figure 2(b)), but it is also possible to find situations where $h=1$ features a non-analytic point that does not represent a pronounced local minimum, as shown in figure 4(b) (cf. figure 2(a)).

In figure 2(c) and figure 4(c) we show, respectively, a density plot and a cut for quenches from the critical point $h_0=1$. In this case the situation is yet again different, with the critical point $h=1$ exhibiting a local maximum (instead of a minimum) emerging as a non-analytic cusp.

We stress that while the singular nature of $\alpha$ at the critical points is due solely to $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$, the behaviour as a pronounced local minimum or maximum depends in general on the interplay between $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ and $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ (cf. density plots in appendix C), and it is therefore in general only a property of the athermality.

4.2.2 $\gamma$ dependence

Figure 5 shows cuts for the dependence of $\alpha$ on $\gamma$, with $h_0$, $h$ and ${\gamma }_0$ fixed. Based on the features of figure 2(a) around the XX critical line we expect situations where the critical point $\gamma =0$ corresponds to a local-minimum cusp. This is shown in the cut of figure 5(a), for choices of $({\gamma }_0,h_0)$ similar to those of figure 2(a). On the other hand, figure 5(b) shows a cut for parameters choices similar to those of figure 2(b), and indeed we find a local-maximum cusp, as expected from the density plot. The density plot of figure 2(c), corresponding to a critical initial state $({\gamma }_0,h_0=1)$ features both regions where the critical point $\gamma =0$ corresponds to a minimum cusp ($h$ close to 1), and regions where $\gamma =0$ corresponds to a maximum cusp ($h$ close to $0$), which we have also verified by appropriate cuts. As for the case of the Ising critical line, also here the non-analytic behaviours are entirely due to $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$, while the local-minimum (local-maximum) nature of the critical points arises in general from the nontrivial interplay between $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ and $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ (cf. density plots in appendix C).

4.2.3 Non-analytic behaviour from root densities and pinning mechanism

The non-analytic behaviour of $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ at the critical lines (cf. density plots in appendix C) is inherited from the singular points in the $\gamma$ and $h$ dependence of ${\varrho }_{\mathrm{GGE}}(k)$ (equations (4.7) and (4.8)), which in turn can be traced back to the dispersion ${𝜀}_{\gamma ,h}(k)$ becoming gapless at criticality. For example, focusing directly on the density of occupation numbers $n_{\mathrm{GGE}}(k)=2\pi {\varrho }_{\mathrm{GGE}}(k)$, it is easy to see that

where $\mathit{𝑠𝑔𝑛}(x)$ is the sign function. Therefore $n_{\mathrm{GGE}}(k)$ is subject to a discontinuous jump from $0$ to $1$ whenever $h$ or $h_0$ cross the critical points $h=1$ or $h_0=1$. Exactly on the critical Ising line we find

and analogously for $h_0=1$, $h\ne 1,{\gamma }_0\ne 0$ by the symmetry $(\gamma ,h)\leftrightarrow ({\gamma }_0,h_0)$ of $n_{\mathrm{GGE}}(k)$. Similar discontinuities are found along the XX line when $\gamma$ or ${\gamma }_0$ cross zero, at the values of $k^{\ast }$ for which ${𝜀}_{0,h}(k^{\ast })=0$. Examples of the discontinuous jumps across the Ising line are given in figure 6 (see figure 17 for additional plots). In appendix B we prove analytically that the existence of singular points for $n_{\mathrm{GGE}}(k)$ translates into non-analytic features at criticality of $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$.

Some of the most striking features arising from the previous athermality plots are the dips at criticality for which $\alpha \approx 0$, see figure 3 and figure 4(a). In the case of the Ising line, they emerge due to a pinning mechanism for which both $n_{\mathrm{GGE}}(k)$ (see equation (4.13)) and $n_{{\beta }^{\ast }}(k)=2\pi {\varrho }_{{\beta }^{\ast }}(k)$ (see equation (4.10) and remember ${𝜀}_{\gamma ,1}(0)=0$) are pinned to the value of $1∕2$ for $h\to 1$ and $k\to 0$, while also featuring very similar decays away from $k=0$. This forces the curves $n_{\mathrm{GGE}}(k)$ and $n_{{\beta }^{\ast }}(k)$ to be anomalously close to each other, drastically suppressing $\alpha$. An example of the Ising-line pinning is shown in log-scale in figure 6(c). Similar (quasi)pinning mechanisms are also responsible for the dips along the XX line, like the one in figure 5(a).

4.3 Varying the prequench Hamiltonian

We now investigate the structure of $\alpha$ as a function of the prequench parameters $({\gamma }_0,h_0)$, for fixed postquench $(\gamma ,h)$. In this setup it is perhaps more natural to expect signatures of quantum criticality in the plots for $\alpha$, given that we vary the parameters ${\gamma }_0$ and $h_0$ that define the $T=0$ ground state.

Due to the invariance of ${\varrho }_{\mathrm{GGE}}(k)$ under the swap $({\gamma }_0,h_0)\leftrightarrow (\gamma ,h)$, for any density plot or cut of $\alpha$ presented in section 4.2 as a function of $(\gamma ,h)$ (for fixed ${\gamma }_0$ and $h_0$), the contribution $-s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ to $\alpha$ remains identical if we consider now the athermality as a function of $({\gamma }_0,h_0)$ (for fixed $(\gamma ,h)$ chosen to be identical to the $({\gamma }_0,h_0)$ used in the figures of section 4.2). However, the overall $\alpha$ changes with respect to the plots of section 4.2 because of the now different contribution $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$, where unlike before ${\beta }^{\ast }$ is determined by the average energy of the fixed Hamiltonian $H_{\mathit{𝑋𝑌}}(\gamma ,h)$ in the varying initial state $|\mathit{𝐺𝑆};{\gamma }_0,h_0⟩$. Indeed, while in section 4.2 the postquench energy density $e$, and by consequence $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$, did not feature singular points as we varied $\gamma$ and $h$, here $e$ is singular at the critical lines as we vary ${\gamma }_0$ and $h_0$. For fixed postquench $\gamma$ and $h$ we have

and by application of the Hellmann-Feynman theorem we obtain

where $e_{\mathrm{GS}}({\gamma }_0,h_0)$ is the ground state energy of $H_{\mathit{𝑋𝑌}}({\gamma }_0,h_0)$. By definition of quantum phase transition $e_{\mathrm{GS}}({\gamma }_0,h_0)$ is singular at the critical points (in the XY chain logarithmic divergences appear in the second-order derivatives of $e_{\mathrm{GS}}$), therefore equation (4.16) proves that the postquench energy density $e({\gamma }_0,h_0)$, and by consequence $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ via ${\beta }^{\ast }$, is singular too (see appendix C for some plots).

In figure 7 we show density plots analogous to those of figure 2 but swapping $({\gamma }_0,h_0)\leftrightarrow (\gamma ,h)$, i.e. we now vary $({\gamma }_0,h_0)$ and fix $(\gamma ,h)$ equal to the values $({\gamma }_0,h_0)$ from figure 2. By visual comparison we see that the structure of $\alpha$ is very similar to that of figure 2, in particular with the critical lines emerging as non-analytic features. However, as discussed, the results are not expected to be equal and indeed one can easily identify differences, e.g. the absence of a local-minimum cusp (which we verified also by means of cuts) on the XX line of figure 7(a) as compared to figure 2(a) (cf. also figure 5(a)). Density plots and cuts for $e$, $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ and $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ corresponding to the parameter choice of figure 7(a) are shown in figure 18 of appendix C. Similarly to before, we find that in $\alpha$ some interesting structure (see e.g. figure 7(a) for ${\gamma }_0>0$ and $h_0<1$) emerges from the interplay (i.e. the difference) between $s_{\mathrm{YY}}[{\varrho }_{{\beta }^{\ast }}]$ and $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ (cf. same parameter region in figure 18, where the behaviour appears qualitatively different).

5 XXZ chain

The spin-1/2 XXZ chain is a paradigmatic interacting integrable model solvable by Bethe ansatz [12, 111, 167–171]. In the absence of external magnetic fields its Hamiltonian reads

where we assume PBC and focus on antiferromagnetic couplings $J=1$, $\mathrm{\Delta }>0$. The quantum phase diagram for anisotropies $\mathrm{\Delta }>0$ features two phases separated by a quantum critical point at $\mathrm{\Delta }=1$:

1. $\mathrm{\Delta }>1$ is a gapped phase where the GS spontaneously breaks the ${\mathbb{Z}}_2$ symmetry ${\sigma }^z\to -{\sigma }^z$, giving rise to antiferromagnetic order along the $ẑ$ axis.
2. $0<\mathrm{\Delta }<1$ is a gapless Luttinger liquid with quasi long-range order.

The Hamiltonian (5.1) possesses a $U(1)$ symmetry associated with conservation of the magnetization ${\sigma }_{\mathit{𝑡𝑜𝑡}}^z={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^L{\sigma }_j^z$. The eigenvalues of ${\sigma }_{\mathit{𝑡𝑜𝑡}}^z$ are $L-2N$, where $N$ denotes the number of spins flips relative to the fully polarized state with all spins up. Below we focus only on quenches from initial states $|\psi (0)⟩$ that have zero magnetization $(N=L∕2)$ and therefore no additional chemical potential $\mu$ (i.e. external magnetic field) must be included in the GE ($\mu =0$ automatically in equation (2.10)).

The spectrum of the XXZ chain features bound states [111], i.e. solutions $𝝀$ of the Bethe equations forming regular string patterns in the complex plane (see appendix D). Therefore each macrostate is identified by a set of root densities $𝝔(\lambda )={\{{\varrho }_{\ell }(\lambda )\}}_{\ell =1}^{{\ell }_{\mathit{𝑚𝑎𝑥}}}$, which specify the distribution of the real string centers for bound states composed of $n_{\ell }$ particles (i.e. $n_{\ell }$ spin flips relative to the fully polarized state). In appendix D we report the TBA equations determining the thermal macrostates ${𝝔}_{{\beta }^{\ast }}(\lambda )$ at inverse temperature ${\beta }^{\ast }$ [111]. For our purposes, the most important facts about the TBA description of the XXZ chain are the following:

• In the regime $\mathrm{\Delta }>1$ there are an infinite number (${\ell }_{\mathit{𝑚𝑎𝑥}}=\infty$) of distinct string types and $n_{\ell }=\ell$. The string centers are restricted to the real interval $\lambda \in (-\pi ∕2,\pi ∕2]$.

• In the regime $0<\mathrm{\Delta }<1$ we restrict our attention to values of the anisotropy parametrized by

  $$\mathrm{\Delta }=\cos <!--\; FUNCTION\; APPLICATION\; -->\left(\frac{\pi }{p+1}\right)\text{with}p\in {\mathbb{Z}}^{+}.$$

  (5.2)

  This subclass of the roots of unity points is characterized by a relatively simple TBA description [111] (due to a significantly enriched symmetry algebra [152–154]), in which only ${\ell }_{\mathit{𝑚𝑎𝑥}}=p+1$ distinct string types are allowed. The number of particles in each string fulfils $n_{\ell }=\ell (1-{\delta }_{\ell ,p+1})+{\delta }_{\ell ,p+1}$. The cases of $\ell =1,p+1$ (simple unbounded magnons) are distinguished by their parities $v_1=1$ and $v_{p+1}=-1$ (see appendix D), which are associated respectively with a real and complex rapidity. For for $\ell \ne 1,p+1$ we have instead $v_{\ell }=1$. The string centers live on the full real line $\lambda \in (-\infty ,\infty )$.

Given a macrostate specified by the root densities ${\varrho }_{\ell }(\lambda )$ for $\ell =1,2,\dots ,{\ell }_{\mathit{𝑚𝑎𝑥}}$, the Yang-Yang entropy density is given by

Here ${\varrho }_{\ell }^{(h)}(\lambda )$ denotes the density of hole centers associated with ${\varrho }_{\ell }(\lambda )$ (see appendix D), and $A=\pi ∕2$ ($A=\infty$) for $\mathrm{\Delta }>1$ ($0<\mathrm{\Delta }<1$ fulfilling equation (5.2)).

5.1 Initial states and GGE root densities

We focus on quenches from the class of tilted-Néel product states

These have zero magnetization ${\sigma }_{\mathit{𝑡𝑜𝑡}}^z$ irrespective of the tilting angle $𝜃$, and their average energy density $e$ is given for arbitrary anisotropies $\mathrm{\Delta }$ and tilting angles $𝜃$ by

Exact formulas for the overlaps between the Néel state ($𝜃=0$) and the XXZ eigenstates have been derived in refs. [172, 173], while a conjecture for the overlaps in the full class (5.4) has been proposed in ref. [174].

Actually, in the regime $\mathrm{\Delta }>1$ ref. [175] (see also ref. [146]) obtained explicit results for the GGE root densities ${𝝔}_{\mathrm{GGE}}(\lambda )$ for the full class (5.4), by reconstructing the rapidity distributions from the initial state [123, 140] (see also refs. [70, 176]). Simple and compact recursive formulas to obtain the GGE ${\varrho }_{\ell }(\lambda )$ and ${\varrho }_{\ell }^{(h)}(\lambda )$ up to arbitrary $\ell$ can be found directly in ref. [175]. For arbitrary tilting angles $𝜃$ the majority of particles are unbounded magnons ($\ell =1$), while bound states ($\ell \ge 2$) contribute a finite but lower amount to the total density $d={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{\ell }{d}_{\ell }=\underset{L\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}N∕L=1∕2$.

In the regime $0<\mathrm{\Delta }<1$ of equation (5.2) we restrict our attention to the Néel state ($𝜃=0$) for which the quench action TBA equations have been obtained for the anisotropies of equation (5.2) in ref. [147] (see also ref. [177]). The equations are reported in appendix D. They can be numerically solved to determine the GGE root and hole densities ${\varrho }_{\ell }(\lambda )$ and ${\varrho }_{\ell }^{(h)}(\lambda )$. Also in this regime unbounded magnons are generally the dominant contribution.

Having access to ${\varrho }_{\ell }(\lambda )$ and ${\varrho }_{\ell }^{(h)}(\lambda )$ both for the GGE and the thermal macrostate, we compute the athermality $\alpha$ by applying equations (3.7) and (5.3).

5.2 Results

In figure 8(a) we report the density of athermality $\alpha$ as a function of $\mathrm{\Delta }$ in a quench from the Néel state ($𝜃=0$). Remarkably, we observe the same phenomenology frequently found in the XY chain, namely the presence of a dip at criticality ($\mathrm{\Delta }=1$) in the form of a pronounced local-minimum cusp. The singular nature at $\mathrm{\Delta }=1$ is again due to the GGE entropy density $s_{\mathrm{YY}}[{𝝔}_{\mathrm{GGE}}]$, as visible in figure 8(b), while the GE entropy in figure 8(c) appears smooth, as expected. These results show that a reduction in the degree of athermality at quantum critical points is not an artifact of free models like XY, but can occur also in the presence of interactions. Figure 8(a) and (b) also uncover in the gapless regime $0<\mathrm{\Delta }<1$ what appears as a strongly non-smooth behaviour of $s_{\mathrm{YY}}[{𝝔}_{\mathrm{GGE}}]$ and $\alpha$ as functions of $\mathrm{\Delta }$. This is analogous to the fractal behaviour found in other physical quantities like Drude weights [88, 90], which emerges due to the discontinuous quasiparticle content at different roots of unity points for $\mathrm{\Delta }$ (equation (5.2) being the simplest class of such points). On the other hand, $s_{\mathrm{YY}}[{𝝔}_{{\beta }^{\ast }}]$ shown in figure 8(c) appears smooth also for $0<\mathrm{\Delta }<1$, again due to the impossibility of finite-temperature phase transitions in 1D. Indeed, it is analytically well-understood that discontinuities in the quasiparticle content at the roots of unity points play no role in GE thermodynamics [178].

Given the simple pinning mechanism behind the dips at criticality discussed for XY, it is natural to wonder if something similar happens for the XXZ chain. In figure 9 we plot in log-scale the GE and GGE root densities ${\varrho }_{\ell }(\lambda )$ for a few values of $\ell \ge 1$ and $\mathrm{\Delta }>1$ in the Néel quench ($𝜃=0$). We observe a behaviour that is qualitatively similar to the pinning mechanism of figure 6 for the XY chain. In particular, as we approach the quantum critical point $\mathrm{\Delta }=1$, both the GE and GGE root densities for each $\ell$ shrink around $\lambda =0$ and develop similar pronounced peaks. An important difference compared to XY is that the pinning value is unbounded as the limit $\mathrm{\Delta }\to 1$ is approached. This is a consequence of the $\lambda$-dependent density of total vacancies ${\varrho }_{\ell }^{(\mathit{𝑡𝑜𝑡})}(\lambda )={\varrho }_{\ell }(\lambda )+{\varrho }_{\ell }^{(h)}(\lambda )$, which is instead a constant in free theories like XY. In particular, ${\varrho }_{\ell }^{(\mathit{𝑡𝑜𝑡})}(\lambda )$ develops a divergence at $\lambda =0$ in the limit $\mathrm{\Delta }\to 1$. This is due to the fact that, in the limit considered, the driving term $a_{\ell }(\lambda )$ in TBA (see equation (D.3) for $\mathrm{\Delta }>1$ and equation (D.13) for $\mathrm{\Delta }<1$ in appendix D) approaches a Dirac delta centred at zero. Given that the entropies (5.3) depend also on the hole densities, in appendix D we report plots identical to those in figure 9 but for ${\varrho }_{\ell }^{(h)}(\lambda )$, and find that a similar pinning mechanism emerges. It is natural to wonder if a pinning mechanism to finite values, hence more similar to the one seen in XY, appears if one focuses on the bounded filling functions ${𝜗}_{\ell }(\lambda )={\varrho }_{\ell }(\lambda )∕{\varrho }_{\ell }^{(\mathit{𝑡𝑜𝑡})}(\lambda )$. We find this to not be the case, given that the similarity between the GE and GGE filling functions ${𝜗}_{\ell }(\lambda )$ emerges on a large range of $\lambda$ values as an almost constant plateau, see plots in appendix D. In appendix D we show that a pinning mechanism for ${\varrho }_{\ell }(\lambda )$ arises also in the gapless regime $0<\mathrm{\Delta }<1$ as $\mathrm{\Delta }\to 1$ is approached (i.e. $p\gg 1$ in the parametrization (5.2)), giving rise to the dip at criticality for $\alpha$ on the left side of figure 8(a).

Thanks to the results of ref. [175], in the gapped regime $\mathrm{\Delta }>1$ we can also consider nonzero tilting angles $𝜃\ne 0$ for the initial states (5.4). From figure 10(a) we observe that a decrease in the athermality $\alpha$ close to criticality is present also for $𝜃=\pi ∕5$. In figure 10(b) we plot the GE and GGE entropy densities giving rise to the dip. Figure 10(c) compares the temperatures $1∕{\beta }^{\ast }$ as a function of $\mathrm{\Delta }$ in both the $𝜃=0$ and $𝜃=\pi ∕5$ quenches, demonstrating that in general there is no simple direct correlation between the behaviour of the temperature and that of $\alpha$ (cf. the minimum of $1∕{\beta }^{\ast }$ at intermediate values of $\mathrm{\Delta }>1$ in the $𝜃=\pi ∕5$ case).

Overall, the results of this section demonstrate that sharp signatures of quantum criticality in the athermality and GGE entropy survive also in the presence of interactions.

6 Lieb-Liniger model

The Lieb-Liniger (LL) model [179, 180] describes a one-dimensional gas of bosons interacting with Dirac delta potentials (we set $\hslash =2m=1$)

Here $\varphi (x)$ denotes the annihilation operator for a canonical Bose field, $c$ is the interaction strength and we assume PBC. The model cannot be mapped to a free theory aside for the limits $c\to 0$ (free bosons) and $c\to \infty$ (free fermions via a continuum Jordan-Wigner transformation [181, 182]). In the following we consider both the repulsive ($c>0$) and attractive ($c<0$) regimes [111, 183].

For $c>0$ the model admits no bound states in the spectrum, and the eigenstates are parametrized by real rapidities $𝝀$ only. Therefore, in the TBA description the macrostates are specified by a single root density $\varrho (\lambda )$ (see appendix E for details). The ground state sector is gapless irrespective of the interaction strength $c>0$, and the low-energy physics is described by a Luttinger liquid [184, 185].

Given the absence of a quantum critical point, based on the results from the previous sections we can already anticipate that the athermality will feature less structure than in the XY and XXZ cases. We stress that in repulsive LL the thermal macrostate ${\varrho }_{{\beta }^{\ast },{\mu }^{\ast }}(\lambda )$ features an additional experimentally relevant Lagrange multiplier $\mu$, associated with the conservation of the number of particles (see equation (2.11)). The TBA nonlinear integral equations defining ${\varrho }_{{\beta }^{\ast },{\mu }^{\ast }}(\lambda )$ are reported in appendix E.

In the attractive regime $c<0$ bound states of arbitrary number of particles exist (corresponding to strings of arbitrary length) [111, 183, 186, 187] and macrostates are specified by an infinite sets of root densities ${\varrho }_{\ell }(\lambda )$, $\ell =1,2,\dots ,\infty$ corresponding to the densities of $n_{\ell }=\ell$ bound particles with string center $\lambda$. Crucially, the attractive LL does not admit a stable thermodynamic limit in equilibrium. For $c<0$, the system undergoes a collapse into a macroscopic bound state whose ground-state energy is superextensive ($\propto -L^3$ at fixed nonzero density) [111]. Because the entropy scales only extensively ($\propto L$), temperature cannot stabilize the free energy, precluding the existence of a well-defined thermal ensemble. Therefore at $c<0$ the athermality cannot be defined. However, quantum quenches to the attractive LL are perfectly meaningful [186, 187] and can be experimentally realized [188–192]. Indeed, in the quench setup the extensive charges (including the energy) associated with the initial state $|\psi (0)⟩$ are conserved and therefore at late times the system is locally described by a well-defined macrostate of $H_{\mathit{𝐿𝐿}}(c<0)$. This enables us to consider the behaviour of both $s_{\mathrm{YY}}[{\varrho }_{\mathrm{GGE}}]$ ($c>0$) and $s_{\mathrm{YY}}[{𝝔}_{\mathrm{GGE}}]$ ($c<0$). These, together with the thermal Yang-Yang density $(c>0)$, are obtained from equation (5.3) by setting $A=\infty$ and ${\ell }_{\mathit{𝑚𝑎𝑥}}=1$ (${\ell }_{\mathit{𝑚𝑎𝑥}}=\infty$) for $c>0$ ($c<0$).

6.1 Initial state and GGE root density

We focus on a quench with Hamiltonian $H_{\mathit{𝐿𝐿}}(c)$ starting from the ground state of the $c=0$ free bosonic theory