1 Introduction

Entanglement is a fundamental ingredient of quantum theory and has a central role in quantum information theory [1,2,3]. For pure states, this is directly related to correlations, while for mixed states, entanglement has a more complex meaning. A quantum state is entangled, if its density matrix cannot be written as a mixture of product states. Deciding whether a state is entangled or not is a difficult problem. However, there are conditions that are necessary and sufficient for small systems., e.g., for \(2\times 2\) (two-qubit) and \(2\times 3\) bipartite systems [4, 5] and for multi-mode Gaussian states [6]. There are also conditions that are sufficient conditions for entanglement for larger systems, but does not detect all entangled states.

Considering experiments usually only limited information about the quantum state is available and this is true for theoretical calculations for very large systems. Only those approaches for entanglement detection can be applied that require the measurement of a few observables. There are entanglement conditions that are linear in operator expectation values; these are the entanglement witnesses. They are operators that have a positive expectation value for all separable states. Thus, a negative expectation value signals the presence of entanglement. The theory of entanglement witnesses has recently been rapidly developing [7,8,9,10]. It is also known how to optimize a witness operator to detect the most entangled states [9].

Apart from determining optimal entanglement witnesses, it is also important to find witnesses that are easy to measure in an experiment or possible to evaluate in a theoretical calculation. From both point of views, witnesses based on spin-chain Hamiltonians attracted considerable attention [11,12,13,14,15,16,17]. The energy-based witnesses have been used in various physical systems [18,19,20,21,22,23]. However, it has been shown that the optimal witness for the thermal state of the chain is not necessarily the Hamiltonian [17]. Therefore, it is recommended to consider another approach, based on a family of witnesses that detect entanglement whenever the entanglement negativity of the nearest-neighbor two-spin density matrix is nonzero [24], i.e., when the state violates the entanglement criterion based on the positivity of the partial transpose (PPT) [4, 5].

For mixed states, one generally considers thermal states and calculates a temperature bound, below which the state is entangled [11,12,13,14,15,16,17]. Recently, however, the entanglement of other type of mixed states has also been considered, which are postquench nonequilibrium states. These states are obtained through such a protocol, when the quantum system is first placed to its ground state, and then a quench is performed, when the parameters of the Hamiltonian change suddenly [25,26,27,28,29]. Since the state is not an eigenstate of the new Hamiltonian, dynamics start. In the infinite time limit, the system approaches a stationary state, which is some mixture of the states appearing during the dynamics. If the Hamiltonian is nonintegrable, the system is expected to be thermalized and the stationary state is described by a Gibbs ensemble with an effective temperature [30,31,32,33,34,35,36,37,38,39,40]; see, however, Refs. [41,42,43,44]. For integrable systems, such as the transverse Ising chain, XY and XXY chains, the stationary state is assumed to be described by a so-called Generalized Gibbs Ensemble (GGE) [45,46,47,48,49,50,51,52,53], for which different effective temperatures are assigned to each conserved quantities. This type of description has been exactly calculated for the quantum Ising chain [54], and a similar formalism is conjectured for the XY chain [55].

In experiments, one cannot realize such systems, which are purely integrable, since weak integrable breaking perturbations are always unavoidable. In the presence of a perturbation that breaks integrability, usual thermalization is again expected to take place. However, if the perturbation is small, the process may require a long time. On a finite time scale, the dynamics is approximately described by the evolution under the integrable unperturbed Hamiltonian. The system initially relaxes to a stationary state of the unperturbed Hamiltonian, which is called prethermalization, while genuine thermalization only occurs at later times [56,57,58]. This later thermalization is typically involves a thermalization time \(\tau \sim \lambda ^{-2}\), where \(\lambda \) is the perturbation strength [59, 60]. However, for specific Hamiltonians, this timescale can be much longer [61, 62].

In this paper, we aim to compare the entanglement properties of a prethermalized and a genuine thermalized state. For this purpose, we use the XY model, which is exactly solvable, and consider some type of integrable breaking perturbation. For the unperturbed model, several entanglement-based properties have been studied recently [63,64,65,66,67], and also, the postquench nonequilibrium stationary state is analyzed by energy-based and negativity-based entanglement witnesses [68]. Here, we let switch on an integrable breaking perturbation and repeat the calculation.

Our paper is organized as follows. In Sect. 2, we introduce the XY model, present its free-fermion representation, calculate thermal averages, and present its conjectured GGE after a global quench. In Sect. 4, the negativity-based entanglement witness is described. In Sect. 5, the bounds for postquench states are calculated and the entangled areas are compared for prethermalized and genuine thermalized states. In Sect. 6, we close our paper with a discussion.

2 Model and methods

Here, we consider the XY spin-chain defined by the Hamiltonian

$$\begin{aligned} \displaystyle \mathcal{H}_{XY}=-\sum _{l=1}^L \left[ \dfrac{1+\gamma }{2} \sigma ^x_l \sigma ^x_{l+1}+\dfrac{1-\gamma }{2} \sigma ^y_l \sigma ^y_{l+1}\right] -h \sum _{l=1}^L \sigma ^z_l\;, \end{aligned}$$
(1)

in terms of the \(\sigma _l^{x},\) \(\sigma _l^{y},\) and \(\sigma _l^{z}\) Pauli spin operators acting on the spin at site l,  and \(\sigma _{L+1}^{\alpha }\equiv \sigma _{1}^{\alpha }\) for \(\alpha =x,y,z.\) We mention that the special case \(\gamma =1\) represents the transverse Ising model, and for \(h=0\), \(\gamma =0\) the Hamiltonian reduces to the XX chain.

Later, we extend the Hamiltonian with a general integrability breaking term

$$\begin{aligned} \displaystyle \mathcal{H}=\displaystyle \mathcal{H}_{XY}+\lambda \displaystyle \mathcal{V}(\{\pmb {\sigma }\}). \end{aligned}$$
(2)

This term \(\displaystyle \mathcal{V}(\{\pmb {\sigma }\})\) can c.f. contain interaction between more distant neighbors, but we do not specify its form, the rǒle of this perturbation for \(\lambda \ll 1\) is to ensure thermalization of the model after a quench for sufficiently long time, \(\sim \lambda ^{-2}\).

2.1 Integrable model: \(\lambda =0\)

In detail, we consider the integrable model with \(\lambda =0\), and using standard techniques [69, 70], it is expressed in terms of fermion creation and annihilation operators \(\eta ^{\dag }_p\) and \(\eta _p\) as

$$\begin{aligned} \mathcal{H}_{XY}=\sum _p \varepsilon \left( p\right) \left( \eta ^{\dag }_p \eta _p-\frac{1}{2}\right) , \end{aligned}$$
(3)

where the sum runs over L quasi-momenta, which are equidistant in \([-\pi ,\pi ]\) for periodic boundary conditions. The energy of the modes is given by [25, 26, 55]

$$\begin{aligned} \varepsilon \left( p\right) =2\sqrt{\gamma ^2 \sin ^2 p+\left( h-\cos p\right) ^2} \end{aligned}$$
(4)

and the Bogoliubov angle \(\varTheta _p\) diagonalizing the Hamiltonian is given by

$$\begin{aligned} \tan \varTheta _p=-\gamma \sin p/\left( h-\cos p\right) .\end{aligned}$$
(5)

The ground state is the fermionic vacuum, its energy being

$$\begin{aligned} E_0=-\sum _p \frac{\varepsilon \left( p\right) }{2}. \end{aligned}$$
(6)

The model has a so-called disorder line at

$$\begin{aligned} h_d(\gamma )=\sqrt{1-\gamma ^2} \end{aligned}$$
(7)

along which the ground state of the model is a product state, i.e., not entangled. For \(h<h_d(\gamma )\) (\(h>h_d(\gamma )\)), the long-range two-point correlation functions have an (do not have) oscillatory behavior, while at \(h=h_d(\gamma )\), they are constant [25, 26].

At finite temperature, \(T>0\), the average value of the energy is given by

$$\begin{aligned} \langle \mathcal{H} \rangle _T=-\sum _p t(p,T)\frac{\varepsilon \left( p\right) }{2}, \end{aligned}$$
(8)

with

$$\begin{aligned} t(p,T)=\tanh \left( \frac{\varepsilon \left( p\right) }{2T}\right) . \end{aligned}$$
(9)

(Here and in the following, we set \(k_B=1\).)

In the thermodynamic limit, \(L \rightarrow \infty \), the two-point correlation functions are calculated in Refs. [25, 26] and the nearest-neighbor correlations are given by

$$\begin{aligned} \langle \sigma _l^x \sigma _{l+1}^x \rangle _T&=g_c-g_s,\nonumber \\ \langle \sigma _l^y \sigma _{l+1}^y \rangle _T&=g_c+g_s,\nonumber \\ \langle \sigma _l^z \sigma _{l+1}^z \rangle _T&=g_0^2-g_c^2+g_s^2, \end{aligned}$$
(10)

with

$$\begin{aligned} g_c&=\frac{1}{\pi }\int \limits _{-\pi }^{\pi } \textrm{d} p \cos p (\cos p -h)~t(p,T)\varepsilon ^{-1}\left( p\right) ,\nonumber \\ g_s&=-\gamma \frac{1}{\pi }\int \limits _{-\pi }^{\pi } \textrm{d} p \sin ^2 p ~t(p,T)\varepsilon ^{-1}\left( p\right) ,\nonumber \\ g_0&=\frac{1}{\pi }\int \limits _{-\pi }^{\pi } \textrm{d} p (h-\cos p) ~t(p,T)\varepsilon ^{-1}\left( p\right) . \end{aligned}$$
(11)

3 Nonequilibrium stationary states after a quench

We consider global quenches at zero temperature, which suddenly change the parameters of the Hamiltonian from \(\gamma _0\), \(h_0\) for \(t<0\) to \(\gamma \), h for \(t>0\), keeping, however, the value of \(\gamma _0=\gamma \). For \(t<0\), the system is assumed to be in equilibrium, i.e., in the ground state \(\left| \Phi _0\right\rangle \) of the Hamiltonian \(\mathcal{H}_0\) with parameters \(\gamma _0\) and \(h_0.\) After the quench, for \(t>0\), the state evolves coherently according to the new Hamiltonian \(\mathcal{H}\) as

$$\begin{aligned} \left| \Phi _0(t)\right\rangle =\exp (-i\mathcal{H}t)\left| \Phi _0\right\rangle . \end{aligned}$$
(12)

Correspondingly, the time evolution of an operator in the Heisenberg picture is

$$\begin{aligned} \sigma _l\left( t\right) =e^{i\mathcal{H}t} \sigma _l e^{-i\mathcal{H}t}. \end{aligned}$$
(13)

After large enough time and in the thermodynamic limit, the system is expected to reach a stationary state

$$\begin{aligned} {\varvec{\rho }}_{q}=\lim _{\tau \rightarrow \infty }\frac{1}{\tau }\int \limits _{0}^{\tau } e^{-i \mathcal{H}t} |\Phi _0\rangle \langle \Phi _0| e^{+i\mathcal{H}t} \textrm{d}t, \end{aligned}$$
(14)

so that, for an observable \(\mathcal{O}\), the stationary value is given by

$$\begin{aligned} \langle \mathcal{O} \rangle _\textrm{st}=\textrm{Tr}({\varvec{\rho }}_{q} \mathcal{O}). \end{aligned}$$
(15)

3.1 Stationary values in the integrable model

In the integrable model with \(\lambda =0\), the energy of the system after the quench is given as

$$\begin{aligned} \langle \Phi _0| \displaystyle \mathcal{H} |\Phi _0 \rangle =\sum _p \varepsilon \left( p\right) \left( \langle \Phi _0|\eta ^{\dag }_p \eta _p|\Phi _0 \rangle -\frac{1}{2}\right) , \nonumber \\ \end{aligned}$$
(16)

where the occupation probability of mode p in the initial state \(\left| \Phi _0\right\rangle \) is given as

$$\begin{aligned} f_p=\left\langle \Phi _0\right| \eta ^{\dag }_p \eta _p \left| \Phi _0\right\rangle . \end{aligned}$$
(17)

For the XY model, it is expressed through the difference \(\Delta _p=\varTheta ^0_p-\varTheta _p\) of the Bogoliubov angles as

$$\begin{aligned} f_p=\frac{1}{2}\left( 1-\cos \Delta _p\right) \end{aligned}$$
(18)

with the cosine of the difference \(\Delta _p\) given as

$$\begin{aligned} \cos \Delta _p=4\dfrac{\left( \cos p -h_0\right) \left( \cos p -h\right) +\gamma \gamma _0\sin ^2 p}{\varepsilon \left( p\right) \varepsilon _0\left( p\right) }, \nonumber \\ \end{aligned}$$
(19)

where the index 0 refers to quantities before the quench [55]. In the thermodynamic limit, Eq. (16) can be rewritten as

$$\begin{aligned} \frac{\langle \Phi _0| \displaystyle \mathcal{H} |\Phi _0 \rangle }{L}=-\frac{1}{4\pi }\int \limits _{-\pi }^\pi \varepsilon (p)\cos \Delta _p \textrm{d}p. \end{aligned}$$
(20)

The fermions with occupation probability \(f_p\) are quasiparticles, which are created homogeneously in space and the corresponding wave-packets move ballistically with constant velocity. Such a wave packet is well described by a sharp kink excitation, if the quench is performed deep into the ordered phase. Such a kink is used in the semi-classical description of the correlation functions after the quench [71]. For quenches close to the critical point, the kinks are not sharply localized and the domain walls have a finite extent of the order of the equilibrium correlation length. In this case, the semi-classical treatment should be modified. As shown in Refs. [54, 55] in the thermodynamic limit, this effect can be taken into account using an effective occupation probability

$$\begin{aligned} f_p \rightarrow {\tilde{f}}_p=-\frac{1}{2}\ln |\cos \Delta _p|, \end{aligned}$$
(21)

so that in leading order \({\tilde{f}}_p=f_p + \mathcal{O}(f_p^2)\).

In the stationary state, due to conserved symmetries, averages of correlations are described by a Generalized Gibbs Ensemble (GGE) [45,46,47,48, 50,51,52,53]. In this case to each fermionic mode, an effective temperature \(T_\textrm{eff}(p)\) is attributed through the relation [55]

$$\begin{aligned} \tanh \left( \frac{\varepsilon \left( p\right) }{2T_\textrm{eff}(p)}\right) =e^{-2{\tilde{f}}_p}=|2f_p-1|=|\cos \Delta _p|. \nonumber \\ \end{aligned}$$
(22)

In this way, the nearest-neighbor correlations in the stationary state can be obtained as in Sect. 2.1, just replacing t(p, T) defined in Eq. (9) by \(|\cos \Delta _p|\)

$$\begin{aligned} t(p,T) \rightarrow |\cos \Delta _p|. \end{aligned}$$
(23)

In particular, we have to apply Eq. (23) for the correlation functions in Eqs. (10) and (11).

3.2 Stationary values in the nonintegrable model

In the nonintegrable model with \(0<\lambda \ll 1\) on finite time scale prethermalization takes place and the quasi-stationary state is described by a GGE, as explained in Sect. 3.1. After sufficiently long time, however, the system is expected to be thermalized and the genuine stationary state is expected to be described by a Gibbs ensemble. Possible ways to define a characteristic thermalization temperature, \(T_{th}\), have been discussed in several papers [32, 72,73,74,75,76]. After the quench at the prethermalization state, the different fermionic modes in the system are characterized by a set of effective temperatures in Eq. (22), and at later times in the thermalised stationary state, some average of these effective temperatures is expected. For small \(\lambda \), the average value of the energy remains the same, from which the following condition for the thermalization temperature, \(T_{th}\) follows:

$$\begin{aligned} \int \limits _{-\pi }^\pi \varepsilon (p)\tanh \left( \frac{\varepsilon \left( p\right) }{2T_\textrm{eff}(p)}\right) \textrm{d}p=\int \limits _{-\pi }^\pi \varepsilon (p) \tanh \left( \frac{\varepsilon \left( p\right) }{2T_{th}}\right) \textrm{d}p. \nonumber \\ \end{aligned}$$
(24)

Thus, the thermal stationary state is independent of the exact nature of the integrability breaking perturbations and completely characterized by the temperature \(T_{th}\).

4 Negativity-based entanglement witness

Generally, an operator \(\mathcal{W}\) is called an entanglement witness, if its expectation value, \(\langle \mathcal{W} \rangle \) satisfies the following requirements [77, 78]:

  1. (i)

    \(\langle \mathcal{W} \rangle \ge 0\) for all separable states,

  2. (ii)

    \(\langle \mathcal{W} \rangle < 0\) for some entangled state.

Such a state is detected by the witness as entangled. Entanglement witnesses have been used in various physical systems to verify the presence of entanglement [79,80,81,82,83,84,85,86,87,88,89]. We mention that a single entanglement witness cannot detect all entangled states.

Here, we consider one of the most important entanglement witness, which is connected to the partial transpose of the density matrix [4, 5] and to the entanglement negativity [24]. For a bipartite density matrix given as

$$\begin{aligned} \varvec{\rho }=\sum _{kl,mn}\varvec{\rho }_{kl,mn} {|{k}\rangle }{\langle {l}|}\otimes {|{m}\rangle }{\langle {n}|}, \end{aligned}$$
(25)

the partial transpose according to first subsystem is defined by exchanging subscripts k and l as

$$\begin{aligned} \varvec{\rho }^{T_A}=\sum _{kl,mn}\varvec{\rho }_{lk,mn} {|{k}\rangle }{\langle {l}|}\otimes {|{m}\rangle }{\langle {n}|}. \end{aligned}$$
(26)

It has been shown that for separable quantum states [4, 7]

$$\begin{aligned} \varvec{\rho }^{T_A}\ge 0 \end{aligned}$$
(27)

holds. Thus, if \(\varvec{\rho }^{T_A}\) has a negative eigenvalue, then the quantum state is entangled. For \(2\times 2\) and \(2\times 3\) systems, the PPT condition detects all entangled states [7]. For systems of size \(3\times 3\) and larger, there are PPT entangled states [5, 90]. The entanglement negativity [24] is defined as

$$\begin{aligned} \mathcal{N}(\varvec{\rho })=2\textrm{max}(0,-\textrm{min}(\mu _{\nu })), \end{aligned}$$
(28)

where \(\mu _{\nu }\) are the eigenvalues of the partial transpose \(\varvec{\rho }^{T_A}.\)

Let us turn to XY chains and consider the nearest-neighbor reduced density matrix, \(\varvec{\rho }\), which is defined in the \(\sigma ^z\) basis. As described in details in Refs. [17, 68] due to symmetries of the problem, \(\varvec{\rho }\) is a direct sum of two \(2 \times 2\) matrices and the same property holds also for the partial transpose \(\varvec{\rho }^{T_A}\). The minimal eigenvalues of the \(2 \times 2\) submatrices can be calculated by second-order quadrature in terms of the matrix-elements of \(\varvec{\rho }\). The later for the XY and Heisenberg spin chains can be expressed through nearest-neighbor correlations [64]. The final results for the minimal eigenvalues are given by [17, 68]

$$\begin{aligned}&\mu _\textrm{min}^{(1)}=\frac{\langle \sigma _l^z \sigma _{l+1}^z \rangle +1}{4}\nonumber \\&-\frac{1}{4}\sqrt{(\langle \sigma _l^z\rangle +\langle \sigma _{l+1}^z \rangle )^2+(\langle \sigma _l^x \sigma _{l+1}^x \rangle +\langle \sigma _l^y \sigma _{l+1}^y \rangle )^2}\;, \end{aligned}$$
(29)

and

$$\begin{aligned} \mu _\textrm{min}^{(2)}=-\frac{1}{4}(\langle \sigma _l^x \sigma _{l+1}^x \rangle -\langle \sigma _l^y \sigma _{l+1}^y \rangle +\langle \sigma _l^z \sigma _{l+1}^z \rangle -1). \nonumber \\ \end{aligned}$$
(30)

The entanglement witness related to \(\mu _\textrm{min}^{(2)}\) is given by

$$\begin{aligned} \mathcal{W}_N=-\frac{1}{4}( \sigma _l^x \sigma _{l+1}^x -\sigma _l^y \sigma _{l+1}^y + \sigma _l^z\sigma _{l+1}^z -\mathbb {1}), \end{aligned}$$
(31)

whereas the same for \(\mu _\textrm{min}^{(1)}\) is more complicated and derived in [68].

To proceed, we need to know that the partial transposition of a two-qubit state has at most one negative eigenvalue and all the eigenvalues lie in \([-1/2,1]\) [91, 92]. Thus, only one of the \(\mu _\textrm{min}^{(1)}\) and \(\mu _\textrm{min}^{(2)}\) can be negative, and when they are equal to each other, they must be nonnegative and the state must be separable.

5 Entanglement in nonequilibrium postquench states

In this section, we consider global quenches in the system, as described in Sect. 3, and study the entanglement properties of nonequilibrium stationary states, which are obtained in the large-time limit after the quench. First, we consider the prethermalized state, which is obtained by setting formally \(\lambda =0\) in Eq. (2) and having the integrable model. In this part to calculate averages, we use the GGE protocol and assign different effective temperatures to each fermionic modes, as described in Eqs. (22) and (23). Afterward, we let to switch on a small perturbation, \(\lambda \ll 1\), and consider the genuine thermalized state in which there is a unique thermalization temperature as defined in Eq. (24).

We have calculated postquench states detected as entangled by the entanglement negativity-based witness using the two minimal eigenvalues, \(\mu _\textrm{min}^{(1)}\) and \(\mu _\textrm{min}^{(2)}\) in Eqs. (29) and (30), respectively. These are presented in Fig. 1 having the entangled regions where \(\mu _\textrm{min}^{(2)}<0\), and in Fig. 2 where in the entangled regions, \(\mu _\textrm{min}^{(1)}<0\). Entanglement detected areas in the prethermalized states are colored by violet and in the genuine thermalized states by green. Overlapping areas are colored orange. If the two parameters are equal, \(h_0=h\), which corresponds to the diagonal of the figures, the postquench state is evidently entangled. This behavior remains valid, if \(h_0\) and h do not differ by a large extent. If, however, they are largely different, then the postquench state becomes separable. This happens in particular if the quench is performed from the ordered state (\(h_0 <1\)) to the paramagnetic state (\(h>1\)).

Fig. 1
figure 1

Postquench states after a sudden quench protocol \((h_0,\gamma ) \rightarrow (h,\gamma )\) in the XY chain. Entanglement is detected in the postquench state by the negativity-based method using \(\mu ^{(2)}_\textrm{min}\) in Eq. (30): in the prethermalized state (violet) and in the genuine thermalized state (green). Overlapping areas are colored orange

Full size image

In Fig. 1a, we consider the case with \(\gamma =1\), and hence, we have a quantum Ising chain. In this case, the negativity-based witness with \(\mu ^{(2)}_\textrm{min}\) is applicable in the whole phase diagram; consequently, in Fig. 2a, there is no entangled region detected. In Fig. 1b–f, we have \(\gamma <1,\) and the condition \(\mu _\textrm{min}^{(1)}<\mu _\textrm{min}^{(2)}\) is fulfilled in a part of the phase diagram. Thus, entangled postquench states are also detected based on \(\mu ^{(1)}_\textrm{min}\) and there are entangled regions in Fig. 2b–f.

Let us now consider first Fig. 1 and compare the detected entangled regions of the prethermalised and the genuine thermalised states. Here, most of the entangled regions are overlapping, but at the surfaces, there are extra regions of the genuine thermalized states. These regions are quite considerable for \(\gamma \) close to 1; see the results in Fig. 1a close to \(h_0=0\). But even in this case, there is an extra prethermalized region close to the critical point, \(h_0=1\). We note that the detected entangled region shows non analytical behavior, both for the prethermalized and the genuine thermalized states. By reducing the value of \(\gamma \), the extra thermalized region shrinks, and in the case \(\gamma \ll 1\), there is even an extra entangled prethermalized region detected by the negativity-based witness with \(\mu ^{(2)}_\textrm{min}\), see in Fig. 1f. If we look at Fig. 2, the trend is rather the opposite. For larger values of \(\gamma \) there are extra regions belonging to the prethermalized state. These extra regions start to shrink by decreasing \(\gamma \), and for very small values of \(\gamma \), the thermalized states have larger entangled regions. It is interesting to compare the entangled regions in the limit \(\gamma \rightarrow 0^+\). In this limit in the prethermalized state, just a part of the area is detected entangled, while in the thermalized state, the complete area is entangled. The latter behavior can be understood, while in the genuine thermalised state, the limit \(\gamma \rightarrow 0^+\) is equivalent to \(\gamma =0\), when the Hamiltonians \(\mathcal{H}_0\) and \(\mathcal{H}\) commute and have the same ground state, which is then evidently entangled. In the prethermalised state, however different fermionic modes are occupied, than in the ground state of \(\mathcal{H}_0\), which results in separable regions in Figs. 1 and  2.

Fig. 2
figure 2

The same as in Fig. 1 but using the negativity-based method with \(\mu ^{(1)}_\textrm{min}\) in Eq. (29)

Full size image

6 Discussion

Entanglement in mixed quantum states is a difficult problem, in particular when the degrees of freedom are large and we approach the thermodynamic limit. The possible systems of investigation are usually quantum spin chains, which could have experimental realizations in condensed matter systems [93] or they could be engineered artificially through ultracold atomic gases in an optical lattice. Recently, this latter type of technique is very well developed and different intriguing questions could be studied experimentally [94,95,96,97,98,99,100,101,102].

In this paper, we consider the XY chain, which is integrable through free-fermionic techniques and several exact results are available, mainly in the ground state, but there are some known results even at finite temperature [25, 26]. We consider the entanglement properties of mixed states of the XY chain, which are nonequilibrium stationary states after a quantum quench protocol. To detect entanglement, we use a family of entanglement witnesses that detect states with a nonzero bipartite entanglement negativity. In practice, this witness can detect all states that have nearest-neighbor entanglement.

The mixed states we consider are due to a quench, when parameters of the Hamiltonian of the system are changed abruptly and the time evolution of the system is governed by the new Hamiltonian. After a sufficiently long time, the system will approach a nonequilibrium stationary state, which is a mixed quantum state. For integrable systems, such as the XY chain, the postquench state is described by a so-called Generalized Gibbs Ensemble, while for general, nonintegrable systems, it is expected to be a thermal state, which is described by an appropriate Gibbs ensemble. In experiments, one can not realize such systems, which are purely integrable, since weak integrable breaking perturbations are always present. If this perturbation is small, the relaxation takes part in two steps. The system initially relaxes to a stationary state of the unperturbed Hamiltonian, which is the prethermalized state, while for later times genuine thermalization takes part. In the present paper, we studied the entanglement properties of the two nonequilibrium stationary states, in particular, we want to clarify the difference between the areas detected to be entangled by the entanglement negativity witness.

The entanglement negativity in Eq. (28) for the XY-chain can be nonzero due to one of the two eigenvalues of the partial transpose, which are defined in Eqs.(29) and (30). Therefore, the areas which are detected entangled are indicated separately in Figs. 1 and  2 for the two cases. We observed that the areas corresponding to the prethermalized and the the genuine thermalized states are mainly overlapping; however, there are extra regions at the boundaries of the overlapping areas. For the negativity witness in Eq. (30), corresponding to Fig. 1 generally, the areas to be detected entangled increase during the thermalization process. Even in this case, there are opposite tendencies close to the critical point of the initial state. Considering the other witness in Eq. (29) and the corresponding Fig. 2 here in the prethermalized state are larger entangled areas for larger values of \(\gamma \), which trend however reverse for small values of \(\gamma \). We mention that it would be interesting to check the entanglement properties of postquench states of other (Bethe–Ansatz) integrable models.

While we studied the nearest-neighbor entanglement of the postquench state, other properties uncovering hidden criticality of the initial system not detectable by local quantities have recently been considered [103]. The method has been based on efficient upper bounds on the negativity in XY chains [104, 105]. We have shown that the criticality of the initial state can still be seen in the boundaries of the regions with nearest-neighbor entanglement.