Entanglement and thermodynamics after a quantum quench in integrable systems

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the low-lying excitations around it. We provide a thorough check of our result focusing on the spin-1/2 Heisenberg XXZ chain, and considering quenches from several initial states. We compare our results with numerical simulations using both tDMRG and iTEBD, finding always perfect agreement.

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics. Recently, the study of quantum quenches revealed that these concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure state maintains the system at zero entropy, local properties at long times are captured by a statistical ensemble with nonzero thermodynamic entropy, which is the entanglement accumulated during the dynamics. Therefore, understanding the entanglement evolution unveils how thermodynamics emerges in isolated systems. Alas, an exact computation of the entanglement dynamics was available so far only for noninteracting systems, whereas it was deemed unfeasible for interacting ones. Here, we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the steady state and its excitations, leads to a complete understanding of the entanglement dynamics in the space-time scaling limit. We thoroughly check our result for the paradigmatic Heisenberg chain. entanglement | quantum quench | integrability | thermodynamics S ince the early days of quantum mechanics, understanding how statistical ensembles arise from the unitary time evolution of an isolated quantum system has been a fascinating question (1)(2)(3)(4)(5)(6)(7). A widely accepted mechanism is that, although the entire system remains in a pure state, the reduced density matrix of an arbitrary finite compact subsystem attains a long time limit that can be described by a statistical ensemble (8). In the last decade, groundbreaking experiments with cold atoms (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) simulated with incredible precision the unitary time evolution of many-body quantum systems, reviving the interest in this topic. The simplest out-of-equilibrium protocol in which these ideas can be theoretically and experimentally tested is the quantum quench (20,21). A system is prepared in an initial state |Ψ0 , typically the ground state of a local Hamiltonian H0, and it evolves with a many-body Hamiltonian H . At asymptotically long times, physical observables relax to stationary values, which for generic systems, are described by the Gibbs (thermal) ensemble (3-7), whereas for integrable systems, a generalized Gibbs ensemble has to be used (8,. Although these results suggest a spectacular compression of the amount of information needed to describe steady states, state of the art numerical methods, such as the time-dependent density matrix renormalization group (46-49) (tDMRG), can only access the short-time dynamics. Physically, the origin of this conundrum is the fast growth of the entanglement entropy S ≡ −TrAρA ln ρA, with ρA being the reduced density matrix of an interval A of length embedded in an infinite system. It is well-understood that S grows linearly with the time after the quench (50). This linear behavior implies an exponentially increasing amount of information manipulated during typical tDMRG simulations. Remarkably, the entanglement dynamics has been successfully observed in a very recent cold-atom experiment (19).
In this work, using a standard quasiparticle picture (50), we show that, in integrable models, the steady state and its lowlying excitations encode sufficient information to reconstruct the entanglement dynamics up to short times. According to the quasiparticle picture (50), the prequench initial state acts as a source of pairs of quasiparticle excitations. Let us first assume that there is only one type of quasiparticles identified by their quasimomentum λ and moving with velocity v (λ). Although quasiparticles created far apart from each other are incoherent, those emitted at the same point in space are entangled. Because these propagate ballistically throughout the system, larger regions get entangled. At time t, S (t) is proportional to the total number of quasiparticle pairs that, emitted at the same point in space, are shared between A and its complement (Fig.  1A). Specifically, one obtains where f (λ) depends on the production rate of quasiparticles. Eq. 1 holds in the space-time scaling limit t, → ∞ at t/ fixed. When a maximum quasiparticle velocity vM exists [e.g., because of the Lieb-Robinson bound (51)], for t ≤ /(2vM ), S grows linearly in time, because the second term in Eq. 1 vanishes. In contrast, for t /(2vM ), the entanglement is extensive (i.e., S ∝ ). This light-cone spreading of entanglement has been confirmed analytically only in free models (52)(53)(54)(55)(56)(57), numerically in several studies (58)(59)(60), in the holographic framework (61)(62)(63)(64)(65)(66)(67)(68), and in a recent experiment (19). The validity of the quasiparticle picture Eq. 1 for the entanglement dynamics has been proven for free models in ref. 52. In the presence of interactions, few results are known. For instance, the validity of Eq. 1 has been proven for rational CFTs (Conformal Field Theories) in ref. 69. In interacting integrable modelsà la Yang-Baxter, the quasiparticle picture has been used to describe the out-of-equilibrium dynamics after Significance Understanding how statistical ensembles arise from the outof-equilibrium dynamics of isolated pure systems has been a fascinating question since the early days of quantum mechanics. Recently, it has been proposed that the thermodynamic entropy of the long-time statistical ensemble is the stationary entanglement of a large subsystem in an infinite system. Here, we combine this concept with the quasiparticle picture of the entanglement evolution and integrability-based knowledge of the steady state to obtain exact analytical predictions for the time evolution of the entanglement in arbitrary 1D integrable models. These results explicitly show the transformation between the entanglement and thermodynamic entropy during the time evolution. Thus, entanglement is the natural witness for the generalized microcanonical principle underlying relaxation in integrable models. an inhomogeneous quench, and it is at the foundation of the integrable hydrodynamics approach for transport in integrable models (70,71).

Results
In a generic interacting integrable model, there are different species of stable quasiparticles corresponding to bound states of an arbitrary number of elementary excitations. Integrability implies that different types of quasiparticles must be treated independently. It is then natural to conjecture that where the sum is over the types of particles n, vn (λ) is their velocity, and sn (λ) is their entropy. To give predictive power to Eq. 2, in the following, we show how to determine vn (λ) and sn (λ) in the Bethe ansatz framework for integrable models. Eq. 2 is straighforwardly generalized to the mutual information between two intervals (see SI Materials and Methods and Fig. S1).
The eigenstates of Bethe ansatz solvable models are in correspondence with a set of pseudomomenta (rapidities) λ. In the thermodynamic limit, these rapidities form a continuum. One then introduces the particle densities ρn,p(λ), the hole (i.e., unoccupied rapidities) densities ρ n,h (λ), and the total densities ρn,t (λ) = ρn,p(λ) + ρ n,h (λ). Every set of densities identifies a thermodynamic "macrostate." This macrostate corresponds to an exponentially large number of microscopic eigenstates, any of which can be used as a "representative" for the macrostate. Physically, SYY corresponds to the total number of ways of assigning the quasimomentum label to the particles, similar to free fermion models.
We now present our predictions for the entanglement dynamics ( Fig. 1B gives a survey of our theoretical scheme). First, in the stationary state, the density of thermodynamic entropy coincides with that of the entanglement entropy in Eq. 2, as it has been shown analytically for free models (52,88,89). This identification implies that sn (λ) = sYY [ρ * n,p , ρ * n,h ](λ). Moreover, it is natural to identify the entangling quasiparticles in Eq. 2 with the low-lying excitations around the stationary state ρ * . Their group To substantiate our idea, we focus on the spin-1/2 anisotropic Heisenberg (XXZ ) chain, considering quenches from several low-entangled initial states, namely the tilted Néel state, the Majumdar-Ghosh (dimer) state, and the tilted ferromagnetic state (Materials and Methods). For these initial states, the densities ρ * n(h),p are known analytically (77)(78)(79). Fig. 2 summarizes the expected entanglement dynamics in the space-time scaling limit, plotting S / vs. vM t/ . Interestingly, S / is always smaller than ln 2 (i.e., the entropy of the maximally entangled state). For the Néel quench, because the Néel state becomes the ground state of Eq. 4 in the limit ∆ → ∞, S / ≈ ln(∆)/∆ 2 vanishes, whereas it saturates for all of the other quenches. For the Majumdar-Ghosh state, one obtains S / = −1/2 + ln 2 at ∆ → ∞. For the tilted ferromagnet with ϑ → 0 (Fig. 2E), S / is small at any ∆, reflecting that the ferromagnet is an eigenstate of the XXZ chain. Surprisingly, the linear growth seems to extend for vM t/ > 1. However, dS /dt (Fig. 2F) is flat only for vM t/ ≤ 1, which signals true linear regime only for vM t/ ≤ 1. This peculiar behavior is caused by the large entanglement contribution of the slow quasiparticles. In Fig. 3, we report the bound-state resolved contributions to the entanglement dynamics. Fig. 3 A and C focuses on the steady-state entropy (second term in Eq. 2), whereas Fig. 3 B and D shows the bound-state contributions to the slope of the linear growth (first term in Eq. 2). The contribution of the bound states, although never dominant, is crucial to ensure accurate predictions. Fig. 4A shows tDMRG results for S (t) for the quench from the symmetrized Néel state (|↑↓↑ . . . +|↓↑↓ . . . )/ √ 2. The data are for the open XXZ chain and subsystems starting from the chain boundary. The qualitative agreement with Eq. 2 is apparent. Fig. 4B reports the steady-state entanglement entropy as a function (data at t ≈ 8 in Fig. 4A). The volume law S ∝ is visible. The dashed-dotted lines in Fig. 4B are fits to S ∝ s * YY , supporting the equivalence between entanglement and thermodynamic entropy. Fig. 4C focuses on the full-time dependence, plotting S / vs. vM t/ . The dashed-dotted line in Fig. 4C is Eq. 2 with t → t/2 because of the open boundary conditions (58). Deviations from Eq. 2 because of the finite are visible. The diamonds in Fig. 4C are numerical extrapolations to the thermodynamic limit. The agreement with Eq. 2 is perfect. Finally, we provide a more stringent check of Eq. 2, focusing on the linear entanglement growth. Fig. 5 shows infinite time-evolving block decimation (iTEBD) results in the thermodynamic limit for S (vM t), with S (x ) ≡ dS (x )/dx taken from ref. 91. For all of the quenches, the agreement with Eq. 2 (horizontal lines in Fig. 5) is spectacular.

Conclusions
The main result of this work is the analytical prediction in Eq. 2 for the time-dependent entanglement entropy after a generic quantum quench in an integrable model. We tested our conjecture for several quantum quenches in the XXZ spin chain, although we expect Eq. 2 to be more general. Additional checks of Eq. 2 (e.g., for the Lieb-Liniger gas) are desirable. It would be also interesting to generalize Eq. 2 to quenches from inhomogeneous initial states, exploiting the recent analytical results (70,71,92). Although we are not able yet to provide an ab initio derivation of Eq. 2, we find it remarkable that it is possible to characterize analytically the dynamics of the entanglement entropy, whereas its equilibrium behavior is still an open challenge. Finally, we believe that Eq. 2 represents a deep conceptual breakthrough, because it shows in a single compact formula the relation between entanglement and thermodynamic entropy for integrable models. An analogous description for nonintegrable systems, where quasiparticles have finite lifetime or do not exist at all, could lead to a deeper understanding of thermalization (19).

Materials and Methods
The anisotropic spin-1/2 Heisenberg chain is defined by the Hamiltonian where S α i are spin-1/2 operators, and ∆ is the anisotropy parameter. Here, we considered as prequench initial states the tilted Néel state |ϑ, · · · ≡ e iϑ j S y j |↑↓ · · · , the Majumdar-Ghosh (dimer) state |MG ≡ ((|↑↓ − |↓↑ )/2) ⊗L/2 , and the tilted ferromagnetic state |ϑ, ≡ e iϑ j S y j |↑↑ · · · . The Heisenberg spin chain is the prototype of all integrable models. Moreover, for all of the initial states considered in this work, the postquench steady state can be characterized analytically via the macrostate densities ρ * p(h) . Specifically, a set of recursive relations for these densities can be obtained (SI Materials and Methods). The group velocities of the low-lying excitations around the steady state (i.e., the entangling quasiparticles) are obtained by solving numerically an infinite set of secondtype Fredholm integral equations (details are in SI Materials and Methods).
The numerical data for the postquench dynamics of the entanglement entropy presented in Fig. 4 were obtained using the standard tDMRG (46)(47)(48)(49) in the framework of matrix product states. For the implementation, we used the ITENSOR library (itensor.org/). The data presented in Fig. 5 are obtained using the iTEBD method (93) and they are a courtesy of Mario Collura.

ACKNOWLEDGMENTS.
We thank Lorenzo Piroli and Eric Vernier for sharing their results before publication. The iTEBD data presented in Fig. 5 are a courtesy of Mario Collura. We acknowledge support from European Research Council Starting Grant 279391 EDEQS. This project has received funding from the European Union's Horizon 2020 research and innovation program under Marie Sklodowoska-Curie Grant 702612 OEMBS.