Rényi entropies of generic thermodynamic macrostates in integrable systems

The XXZ chain can be solved by the Bethe ansatz [103], which allows one to construct all the eigenstates of (2.1) analytically. It is convenient to work in the sector with fixed magnetisation $S_T^z$, or, equivalently with fixed number N of down spins. Here we follow the standard Bethe ansatz framework referring to down spins as particles. The eigenstates of the XXZ chain are easily constructed starting from the ferromagnetic state, i.e. the state with all the spins up $|\uparrow\uparrow\cdots\rangle$. The Bethe ansatz expression for the generic eigenstate in the sector with N down spins reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle | \boldsymbol \lambda \rangle = \sum_{n_{1}< n_{2} < \dots < n_{N}} \psi(n_{1},n_{2},\dots,n_{N}) \sigma^{-}_{n_{1}} \sigma^{-}_{n_{2}} \dots \sigma^{-}_{n_{N}} |\! \uparrow \uparrow\dots \uparrow \rangle, \label{eq:pBetheAnsatzEigenstate1} \nonumber \end{align} \tag{ 2.6 }$$

where $\sigma_i^-$ is the spin-1/2 lowering operator. Here the sum is over the positions n_i of the particles. The amplitudes $\psi(n_1, n_2, \dots, n_N)$ read

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:pBetheAnsatzEigenstate2} \psi(n_{1},n_{2},\dots,n_{n}) = \sum_{P} A(P) \prod_{j=1}^{N} \left(\frac{\sin(\lambda_{P_j}+{\rm i} \eta/2)}{\sin(\lambda_{P_j} - {\rm i}\eta/2)}\right)^{n_{j}}, \nonumber \end{align} \tag{ 2.7 }$$

where $\newcommand{\e}{{\rm e}} \eta\equiv {\rm arcosh}(\Delta)$, and the sum is now over the permutations P of the indices $j\in[1, N]$. The overall amplitude $A(P)$ is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A(P) = (-1)^{{\rm sign}(P)} \prod_{j=1}^{N} \prod_{k=j+1}^{N} \sin(\lambda_{P_{j}}-\lambda_{P_{k}} + {\rm i}\eta). \label{eq:pBetheAnsatzEigenstate3} \nonumber \end{align} \tag{ 2.8 }$$

The state is then specified by a set of N rapidities $\lambda_j$ that play the same role of the quasimomenta in free models. The total energy of the eigenstate is obtained by summing independently the contributions of each particle, to obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E = \sum_{j=1}^{N} e(\lambda_{j}),\qquad e(\lambda) \equiv - \frac{4 \sinh(\eta)^{2}}{\cosh(\eta)-\cos(2\lambda_{j})}. \label{eq:eEigenEnergy} \nonumber \end{align} \tag{ 2.9 }$$

In order for (2.6) to be an eigenstate of the XXZ chain, the rapidities have to be solutions of the Bethe equations [103]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(\frac{\lambda_{j} + {\rm i}\eta/2}{\lambda_{j} - {\rm i}\eta/2} \right)^{L} = - \prod_{k=1}^{N} \frac{\lambda_{j} - \lambda_{k} + {\rm i} \eta}{\lambda_{j} - \lambda_{k} - {\rm i}\eta} \qquad (\,j=1,\dots,N). \label{eq:lBetheEquationsFiniteSize} \nonumber \end{align} \tag{ 2.10 }$$

Each independent set of solutions $\{\lambda_j\}_{j=1}^N$ identifies a different eigenstate of the XXZ chain.

A distinctive property of the XXZ model, which underlies its integrability, is the existence of an infinite number of pairwise commuting local and quasilocal conserved quantities (charges) [104]

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle Q_{s}^{(\,j)} = \frac{{\rm d}^{\,j-1}}{{\rm d} \lambda^{\,j-1}} T_{\rm s}(\lambda) \bigg|_{\lambda=0}, \qquad [Q_{r}^{(\,j)},Q_{s}^{(k)}] = 0, \qquad Q_{1/2}^{(2)} = \frac{1}{2 \sinh \eta} H_{\mathrm{XXZ}}. \label{eq:commutingConservedCharges} \nonumber \end{align} \tag{ 2.11 }$$

Here $T_{\rm s}(\lambda)$ is the transfer matrix of the XXZ model with an s-dimensional auxiliary space. The dimension s of the auxiliary space can be any positive integer or half-integer. In fact, choosing s  =  1/2 yields the usual transfer matrix of the six-vertex model [105]. The corresponding charges $Q_{1/2}^{(\,j)}$ are local, which means they are of the sum of operators that act non-trivially at most at m^(j) sites, where m^(j) is an integer depending on j. For example, the charge $Q_{1/2}^{(2)}$ has m⁽²⁾  =  2 and it is proportional to the system Hamiltonian (2.1). On the other hand, there is no such limit on the size of the terms in the charges $Q_{s}^{(\,j)}$ when s  >  1/2. These charges, called quasilocal charges, are crucial to correctly describe the steady state arising after a quantum quench in integrable models [50].

Since the charges (2.11) commute with the Hamiltonian (2.1), all of them are diagonal in the basis of Bethe ansatz eigenstates (2.6)–(2.10). The corresponding eigenvalues are given as [104]

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle Q^{(\,j)}_{s} | \boldsymbol \lambda \rangle = \sum_{k=1}^{N} q_{s}^{(\,j)} (\lambda_{k})| \boldsymbol \lambda \rangle + o(L), \quad q_{s}^{(\,j)}(\lambda) = \left(-{\rm i} \frac{\rm d}{{\rm d} \mu} \right)^{\,j-1} \ln \left(\frac{\sin(\mu - \lambda +{\rm i} s \eta)}{\sin(\mu - \lambda - {\rm i} s \eta)}\right)\bigg|_{\mu=0}, \label{eq:conservedChargeEigenvalues} \nonumber \end{align} \tag{ 2.12 }$$

where $o(L)$ indicates terms that are either zero (in the case of local charges with s  =  1/2) or vanish in the thermodynamic limit (in the case of quasilocal charges with s  >  1/2).

In this paper we are interested in the thermodynamic limit $L, N\to\infty$, with the ratio N/L (i.e. the particle density) fixed. The solutions of the Bethe equations (2.10) are in general complex. However, a remarkable feature of the Bethe equations is that in the thermodynamic limit their solutions are organised into strings, i.e. multiplets of solutions having the same real part, but different imaginary components. This is the famous string hypothesis [103]. Precisely, the generic structure of a string multiplet reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lambda_{n,j}^{\alpha} = \lambda_{n}^{\alpha} + \frac{{\rm i} \eta}{2} (n + 1 - 2j) + \delta_{n,j}^{\alpha}. \label{eq:lStringRapiditiesTDL} \nonumber \end{align} \tag{ 2.13 }$$

Here n is the string length, i.e. the number of solutions with the same real part, α labels the different strings of the same size n, and j labels the different components within the same string multiplet. The real number $\lambda_{n}^{\alpha}$ is called string center. The string hypothesis holds only in the thermodynamic limit: for finite chains, string deviations (denoted as $\delta_{n, j}^{\alpha}$ in (2.13)) are present, but for thermodynamically relevant states, they vanish exponentially in L.

In the thermodynamic limit the string centers become dense on the real axis. Thus, instead of working with individual eigenstates, it is convenient to describe the thermodynamic quantities by introducing the densities $\rho_{n}(\lambda)$, one for each string type n. $\rho_n(\lambda)$ are the densities of string centers on the real line. Similarly, one can introduce the densities of holes $\rho_{\mathrm{h}, n}(\lambda)$, which is the density of unoccupied string centers, and the total density (density of states) $\newcommand{\e}{{\rm e}} \rho_{\mathrm{t}, n}(\lambda)\equiv \rho_{n}(\lambda)+ \rho_{\mathrm{h}, n}(\lambda)$. Each set of particle and hole densities identify a thermodynamic macrostate of the XXZ chain. Moreover, a given set of densities $\rho_{n}(\lambda)$ corresponds to an exponentially large (with L and N) number of microscopic eigenstates. The rapidities identifying all these eigenstates converge in the thermodynamic limit to the same set of densities. The logarithm of the number of thermodynamically equivalent eigenstates is given by the Yang–Yang entropy [106]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S_{\mathrm{YY}}= s_{\rm YY}L=L\sum_{n=1}^{\infty} \int\! \mathrm{d} \lambda\, \left[\rho_{\mathrm{t}, n}(\lambda) \ln \rho_{\mathrm{t}, n}(\lambda) - \rho_{n}(\lambda) \ln \rho_{n}(\lambda) - \rho_{\mathrm{h}, n}(\lambda) \ln \rho_{\mathrm{h}, n}(\lambda) \right]. \label{eq:sYangYangEntropy} \nonumber \end{align} \tag{ 2.14 }$$

The Yang–Yang entropy is extensive, and its density is obtained by summing independently the contributions of the different rapidities and string types n. The only constraint that a legitimate set of densities has to satisfy is given by the continuum limit of the Bethe equations, which are called Bethe–Gaudin–Takahashi (BGT) equations [103]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{\mathrm{t}, n} = a_{n} - \sum_{m=1}^{\infty} A_{nm} \star \rho_{m}, \label{eq:rBetheGaudinTakahashiCoupled} \nonumber \end{align} \tag{ 2.15 }$$

where we defined

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a_{n}(\lambda) \equiv \frac{1}{\pi} \frac{\sinh(n \eta)}{\cosh(n \eta) - \cos (2 \lambda)}. \label{eq:aBGTCoupledSourceAN} \nonumber \end{align} \tag{ 2.16 }$$

In (2.15), A_nm are the scattering phases between the bound states, defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A_{nm}(\lambda) \equiv (1-\delta_{n,m})a_{|n-m|}(\lambda) + a_{|n-m|+2}(\lambda) + \dots + a_{n+m-2}(\lambda) + a_{n+m}(\lambda), \label{eq:aBGTCoupledKernelANM} \nonumber \end{align} \tag{ 2.17 }$$

and the star symbol $\star$ denotes the convolution

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle [\,f \star g](\lambda) \equiv \int_{-\pi/2}^{\pi/2} \! \mathrm{d} \mu\, f(\lambda-\mu) g(\mu). \label{eq:fConvolutionDefinition} \nonumber \end{align} \tag{ 2.18 }$$

Importantly, a standard trick in Bethe ansatz allows one to simplify (2.15) obtaining a system of partially decoupled integral equations as [103]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{\mathrm{t}, n} = s \star (\rho_{\mathrm{h}, n-1} + \rho_{\mathrm{h}, n+1}) \quad (n=1,2,\dots), \label{eq:rBetheGaudinTakahashiDecoupled} \nonumber \end{align} \tag{ 2.19 }$$

where, for the sake of simplicity, we defined $\newcommand{\e}{{\rm e}} \rho_{\mathrm{h}, 0}(\lambda) \equiv \delta(\lambda)$, and we introduced $s(\lambda)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle s(\lambda) \equiv \frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \frac{{\rm e}^{-2 {\rm i} k \lambda}}{\cosh{k \eta}} = \frac{1}{2 \pi} + \frac{1}{2 \pi} \sum_{k=1}^{\infty} \frac{\cos 2 k \lambda}{\cosh k \eta}. \label{eq:sFunction} \nonumber \end{align} \tag{ 2.20 }$$

The partially decoupled system is typically easier to solve numerically than (2.15).

In the thermodynamic limit, the eigenvalues of the conserved quantities $Q_s^{(\,j)}$ (2.12) can be written in terms of the densities $\rho_n$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \langle \boldsymbol \rho | Q^{(\,j)}_{s} | \boldsymbol \rho \rangle = \sum_{n=1}^{\infty} \int_{-\pi/2}^{\pi/2} {\rm d} \lambda \rho_{n}(\lambda) q_{s,n}^{(\,j)}(\lambda), \qquad q_{s,n}^{(\,j)} = \sum_{k=1}^{n} q_{s}^{(\,j)}\left(\lambda + \frac{{\rm i} \eta}{2} (n + 1 - 2k)\right).\qquad \label{eq:conservedChargeEigenvaluesTBA} \nonumber \end{align} \tag{ 2.21 }$$

Here $q_s^{(\,j)}$ is the same as in (2.12).

The GGE Rényi entropy as expressed in (3.13) are functionals of an infinite set of densities $\rho_{n}(\lambda)$. In this section we show that it is possible to simplify (3.13) writing the Rényi entropies only in terms of $\newcommand{\e}{{\rm e}} \eta_1^{(\alpha)}$. A formula similar to the one we are going to derive is known for the Gibbs (thermal) free energy since many years [103].

The first step in this derivation is to rewrite (3.13) as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle S^{(\alpha)}_{\mathrm{GGE}} = &\frac{L}{\alpha-1}\sum_{n=1}^{\infty} \int_{-\pi/2}^{\pi/2}{\mathrm d} \lambda\bigg[\alpha \rho^{(\alpha)}_{n}(\lambda) g_{n}(\lambda) - \rho_{n}^{(\alpha)}(\lambda) \ln (1 + \eta^{(\alpha)}_{n}(\lambda)) - \rho_{\mathrm{h},n}^{(\alpha)}(\lambda) \ln (1 + 1/\eta_{n}^{(\alpha)}(\lambda))\bigg] \nonumber \\ & + \frac{\alpha L}{\alpha-1} f_{\mathrm{GGE}}. \label{eq:sRenyiEntropyGGEIntegralExpressionFull} \nonumber \end{align} \tag{ 3.14 }$$

The function $\alpha g_{n}(\lambda)$ can be obtained from equation (3.9), whereas $\rho_{\mathrm{h,}n}^{(\alpha)}(\lambda)$ can be obtained from the BGT equations in (2.15). After some algebra this yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(\alpha)}_{\mathrm{GGE}} = \frac{L}{1-\alpha} \left[\sum_{n=1}^{\infty} \int_{-\pi/2}^{\pi/2}\! \mathrm{d} \lambda\, a_{n}(\lambda) \ln (1 + 1/\eta_{n}^{(\alpha)}(\lambda)) - \alpha f_{\mathrm{GGE}} \right]. \label{eq:sRenyiEntropyStrangeSum} \nonumber \end{align} \tag{ 3.15 }$$

The infinite sum in (3.15) can be further simplified by considering the first of the saddle point equations in (3.9)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \ln (1 + \eta^{(\alpha)}_{1}(\lambda)) = g_{1}(\lambda) + \sum_{m=1}^{\infty} \int_{-\pi/2}^{\pi/2} \! {\mathrm d} \mu \, \left[a_{m-1}(\lambda-\mu) + a_{m+1}(\lambda-\mu)\right]\, \ln (1 + 1/\eta^{(\alpha)}_{m}(\mu)). \label{eq:eFirstSaddlePointEquation} \nonumber \end{align} \tag{ 3.16 }$$

One then has to multiply (3.16) by $s(\lambda)$ (see (2.20)) and integrate over λ. Finally, after some manipulations (identical to those appearing in [107]), one obtains

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int_{-\pi/2}^{\pi/2}\! \mathrm{d} \lambda \, s(\lambda) \left[\ln (1 + \eta^{(\alpha)}_{1}(\lambda)) - g_{1}(\lambda) \right] = \sum_{n=1}^{\infty} \int_{-\pi/2}^{\pi/2}\! {\mathrm d} \lambda \, a_{n}(\lambda) \ln (1 + 1/\eta_{n}^{(\alpha)}(\lambda)). \label{eq:eFirstSaddlePointEquationConvolved} \nonumber \end{align} \tag{ 3.17 }$$

The right-hand side of (3.17) is precisely the first term in the square brackets in (3.15). Plugging (3.17) into (3.15), one obtains the simplified formula for the Rényi entropy as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(\alpha)}_{\mathrm{GGE}} = \frac{L}{\alpha-1} \left\{\int_{-\pi/2}^{\pi/2} \mathrm{d}\lambda\, s(\lambda) \left[\alpha g_{1}(\lambda) - \ln (1+\eta^{(\alpha)}_{1}(\lambda)) \right] + \alpha f_{\mathrm{GGE}} \right\}, \label{eq:sRenyiEntropyTheta1Expression} \nonumber \end{align} \tag{ 3.18 }$$

which is our final result depending only on $\newcommand{\e}{{\rm e}} \eta^{(\alpha)}_{1}(\lambda)$.

An important remark is that while (3.18) depends only on one rapidity density, it is still necessary to solve the full set of TBA equation (3.9) in order to determine $\newcommand{\e}{{\rm e}} \eta^{(\alpha)}_{1}$, because all the $\newcommand{\e}{{\rm e}} \eta_n^{(\alpha)}$ are coupled. However, equation (3.18) has at least two advantages. First, it is more convenient than (3.13) from the numerical point of view, because it contains less integrals to be evaluated. Second, equation (3.18) is more convenient for analytical manipulations.

In this section we employ the TBA approach described above to calculate the Rényi entropies after the quench from the dimer state, generalising the results of [100] for the quench from the Néel state. For the dimer state, the overlaps are analytically known [108, 112] and hence the quench action provides the driving functions g_n as [108]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_{1}(\lambda) &= -\ln \left(\frac{\sinh^{4}(\lambda) \cot^{2}(\lambda)}{\sin(2 \lambda + {\rm i}\eta)\, \sin(2 \lambda- {\rm i}\eta)}\right), \nonumber \\ g_{n}(\lambda) &= \sum_{k=1}^{n} g_{1}(\lambda+{\rm i} \eta (n+1-2k)/2), \qquad (n \geqslant 2),\nonumber \label{eq:gSourceFunction} \nonumber \end{align} \tag{ 3.19 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_{n}(\lambda) = -\ln \! \left(\frac{\vartheta_{4}(\lambda)}{\vartheta_{1}(\lambda)} \right)^2 + (-1)^{n}\ln \! \left(\frac{\vartheta_{2}(\lambda)}{\vartheta_{3}(\lambda)} \right)^{2}, \label{eq:gDimerSourceFunctionD} \nonumber \end{align} \tag{ 3.20 }$$

where $\newcommand{\e}{{\rm e}} \vartheta_{\ell}(x)$ are the Jacobi ϑ-functions with nome $\newcommand{\e}{{\rm e}} {\rm e}^{-2 \eta}$.

The strategy to calculate the Rényi entropies is to use the driving function g_n in the TBA equations for $\newcommand{\e}{{\rm e}} \eta_n^{(\alpha)}$ (see (3.9)). After solving for $\newcommand{\e}{{\rm e}} \eta^{(\alpha)}_n$, the GGE Rényi entropies are obtained from (3.18). Obviously, the term $f_{{\rm GGE}}$ has to be evaluated using the density $\rho_n^{(1)}$ (see (3.13)), but normalisation provides $f_{{\rm GGE}}=0$.

The numerical results for the Rényi entropies obtained with this procedure are shown in figure 1. The figure shows the entropy densities $S^{(\alpha)}/L$ plotted versus the chain anisotropy Δ. Different lines correspond to different values of α. As expected, one has that for any Δ, $S^{(\alpha)}<S^{(\alpha')}$ if $\alpha>\alpha'$. For completeness we report the result for $\alpha\rightarrow1$. An interesting observation is that the Rényi entropies do not vanish in the Ising limit for $\Delta\rightarrow\infty$. This is in contrast with what happens for the quench from the Néel state, for which the steady-state entropies at $\Delta\to\infty$ vanish. The reason is that the Néel state becomes the ground state of the XXZ chain in that limit, and there is no dynamics after the quench. In some limiting cases it is possible to derive closed analytic formulas for the post-quench stationary Rényi entropy: in the following, we will provide analytical results in the Ising limit $\Delta\rightarrow\infty$ and for the min entropy, i.e. the limit $\alpha\rightarrow\infty$.

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In this section we perform an expansion of the steady-state Rényi entropies in the large Δ limit, by closely following the procedure introduced in [107]. A similar expansion for the Rényi entropies after the quench from the Néel state has been carried out in [97]. In that case, the $\Delta\rightarrow\infty$ limit is very special, since the Néel state becomes the ground state of the model, and there is no dynamics. The quench from the dimer state is more generic, because the dimer state is never an eigenstate of the chain, and consequently the post-quench dynamics is nontrivial, implying that the stationary value of the Rényi entropy is nonzero also for $\Delta\to\infty$. We anticipate that in the Ising limit the Rényi entropies have the same form as for free-fermion models [97], but deviations from the free-fermion result appear already at the first non trivial order in $1/\Delta$.

In the following we obtain the steady-state Rényi entropies as a power series in the variable $\newcommand{\e}{{\rm e}} z\equiv {\rm e}^{-\eta}$ with $\newcommand{\e}{{\rm e}} \eta={\rm arccosh}(\Delta)$. Following [100], we use the ansatz for $\newcommand{\e}{{\rm e}} \eta^{(\alpha)}_n$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta_{n}^{(\alpha)}(\lambda) = z^{\beta_{n}(\alpha)} \eta^{(\alpha)}_{n,0}(\lambda) \exp \left(\Phi^{(\alpha)}_{n}(z,\lambda) \right). \label{eq:eLargeDeltaAnsatz} \nonumber \end{align} \tag{ 3.21 }$$

Here the exponents $\beta_n(\alpha)$, the functions $\Phi^{(\alpha)}_n(\lambda)$, and $\newcommand{\e}{{\rm e}} \eta^{(\alpha)}_{n, 0}$ have to be determined by plugging the ansatz (3.21) into the TBA equation (3.11). We also need the expansion of the driving functions d_n around z  =  0:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_{n}=\left\{\begin{array}{@{}ll@{}} \ln (4\sin^{2}(2\lambda))\, z^{2} + 4 \cos(4\lambda)\, z^{4} + O(z^{6})& n {{\rm ~even,}} \nonumber \\ \ln (\tan^{2}(\lambda))\, + 8\cos(2\lambda)\, z^{2} - 8\cos(2\lambda)\, z^{4} + O(z^{6}) & n {{\rm ~odd}}. \end{array}\right. \label{eq:dExpansion} \nonumber \end{align} \tag{ 3.22 }$$

The expansion of the kernel $s(\lambda)$ appearing in (3.11) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle s(\lambda) = \frac{1}{2\pi} + \frac{2}{\pi} \cos(2\lambda) z + \frac{2}{\pi} \cos (4\lambda) z^{2} + O(z^3). \label{eq:sExpansion} \nonumber \end{align} \tag{ 3.23 }$$

After plugging the ansatz (3.21) into (3.11), and considering the leading order in powers of z, one can fix the exponents $\beta_n$. By treating separately the cases of even and odd n in (3.22), one finds

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta_{n} =\left\{\begin{array}{@{}cc@{}} 2\alpha & n {{\rm ~even,}} \nonumber \\ 0 & n {{\rm ~odd}}. \end{array}\right. \label{eq:betaExponents} \nonumber \end{align} \tag{ 3.24 }$$

This choice is not unique, but it is the only one that is consistent with the BGT equations (2.19), see [107]. The leading order in z of (3.11) fixes the functions $\newcommand{\e}{{\rm e}} \eta_{n, 0}(\lambda)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta_{n,0}^{(\alpha)}(\lambda) =\left\{\begin{array}{@{}ll@{}} 4^{\alpha}\,{\rm e}^{c(\alpha)}\, |\sin(\lambda)|^{2\alpha} & n {{\rm ~even,}} \nonumber \\ |\tan(\lambda)|^{2\alpha} & n {{\rm ~odd,}} \end{array}\right. \label{eq:etaExpansionLeadingCoefficients} \nonumber \end{align} \tag{ 3.25 }$$

where the constant $c(\alpha)$ is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle c(\alpha) = \frac{1}{\pi} \int_{-\pi/2}^{\pi/2}\! {\mathrm d} \mu \ln (1 + |\tan(\mu)|^{2\alpha}). \label{eq:cExpansionLeadingCoefficientConstant} \nonumber \end{align} \tag{ 3.26 }$$

In the limit $\alpha \rightarrow 1$, one has $c(1)=\ln 4$. Interestingly, equation (3.25) shows that for n odd, $\newcommand{\e}{{\rm e}} \eta_n^{(\alpha)}$ is a regular function for any value of λ, whereas for even n it diverges for $\lambda=\pm\pi/2$. This is a striking difference compared to the quench from the Néel state, for which $\newcommand{\e}{{\rm e}} \eta_n^{(\alpha)}$ diverges in the limit $\lambda\to 0$ for even n (see [100]), whereas it is regular for odd n.

Also, at the leading order in z, one has that $\Phi_n^{(\alpha)}=0$. By combining the results in (3.25) and (3.24) with the TBA equation (2.19), the leading order of the rapidity densities $\rho^{(\alpha)}_{{\rm t}, n}$ are

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:r1} \rho_{{\rm t},1}^{(\alpha)} = \frac{1}{2\pi}(1 + 4\cos(2\lambda)\, z) + O(z^{\min (2,2\alpha)}), \nonumber \end{align} \tag{ 3.27 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:r2} \rho_{{\rm t},2}^{(\alpha)} = \frac{1}{8\pi} + O(z^{\min (1,2\alpha)}), \nonumber \end{align} \tag{ 3.28 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{{\rm t},n}^{(\alpha)} = O(z^{2\alpha}) \qquad (n\geqslant 2). \label{eq:r3} \nonumber \end{align} \tag{ 3.29 }$$

Notice that for any α, only $\rho_n^{(\alpha)}$ and $\rho_{{\rm t}, n}^{(\alpha)}$ with $n\leqslant2$ remain finite in the limit $z\to0$, whereas all densities with n  >  2 vanish. This is different from the Néel state, for which only the densities with n  =  1 are finite [100]. Physically, this is expected because in the dimer state only components with at most two aligned spins can be present. Furthermore, the leading order of $\rho_{{\rm t}, n}^{(\alpha)}$ in (3.27)–(3.29) does not depend on the Rényi index α. Finally, in the limit $z\to0$, the densities become constant, similar to free-fermion models. This suggests that in the limit $z\to0$ the form of the Rényi entropies may be similar to that of free models, as we are going to show in the following.

It is now straightforward to derive the leading behaviour of the Rényi entropies for any α in the limit $z\to0$. First, we obtain the expansion of the driving functions g_n (3.19) as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_{1}(\lambda) = \ln (4 \tan^{2}(\lambda)) + 4z + 4 \sin^{2}(2\lambda) z^2 + O(z^{3}), \label{eq:1} \nonumber \end{align} \tag{ 3.30 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_{2}(\lambda) = \ln (64 \sin^{2}(2 \lambda)) + 16\sin^{2}(\lambda)\,z + 4 z^{2} + O(z^{3}). \label{eq:2} \nonumber \end{align} \tag{ 3.31 }$$

The contributions of g_n for n  >  2 are subleading for $z\to0$ and may be neglected. Using the expansions (3.30) and (3.31), the leading order of the densities in (3.27)–(3.29), and (3.25), one obtains the $z \rightarrow 0$ limit of ${{\mathcal E}}$ (see (3.6)) as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{E} = \frac{L}{4\pi} \Big[\int_{0}^{\pi/2}\! {\mathrm d}\lambda\, \frac{\ln(4|\tan(\lambda)|^{2})} {1 + |\tan(\lambda)|^{2\alpha}} + \frac{1}{4} \int_{0}^{\pi/2}\!{\mathrm d}\lambda\, \ln(64|\sin(2 \lambda)|^{2}) \Big]= \frac{L}{8}\,\ln(2) +\frac{L}{4\pi} \int_{0}^{\pi/2}\! {\mathrm d}\lambda\, \frac{\ln(4|\tan(\lambda)|^{2})}{1 + |\tan(\lambda)|^{2\alpha}}. \label{eq:ePseudoEnergyZ0Limit} \nonumber \end{align} \tag{ 3.32 }$$

The two terms in the right-hand-side of (3.32) are the contributions of the densities with n  =  1 and n  =  2. Similarly, the $z\rightarrow 0$ limit of the Yang–Yang entropy (2.14) is obtained as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S_{\mathrm{YY}} = \frac{L}{2 \pi}\, \int_{-\pi/2}^{\pi/2}\!{\mathrm d}\lambda \left(\frac{1}{1+|\tan(\lambda)|^{2 \alpha}}\ln (1 + |\tan(\lambda)|^{2 \alpha}) + \frac{1}{1+|\cot(\lambda)|^{2 \alpha}} \ln (1 + |\cot(\lambda)|^{2 \alpha}) \right). \label{eq:sYangYangEntropyZ0Limit} \nonumber \end{align} \tag{ 3.33 }$$

Plugging equation (3.32)–(3.33) into the definition of the Rényi entropies (3.2), one obtains at the leading order in z

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(\alpha)}_{{\rm GGE}} = \frac{L}{1-\alpha} \int_{-\pi/2}^{\pi/2}\!\frac{{\mathrm d} \lambda}{2 \pi} \ln \left[\left(\frac{1}{1+\tan^{2}(\lambda)} \right)^{\alpha}+\left(1 - \frac{1}{1+\tan^{2}(\lambda)} \right)^{\alpha}\right]. \label{eq:sRenyiEntropyZ0Limit} \nonumber \end{align} \tag{ 3.34 }$$

Equation (3.34) shows that for any α the Rényi entropies are not vanishing in the limit $z\to0$.

The Rényi entropy obtained from (3.34) are plotted in figure 2 as a function of the Rényi index α. Like for finite Δ, the Rényi entropies are monotonically decreasing functions of α. For some values of α the integrals in (3.34) can be computed analytically. For instance, for the max entropy, i.e. in the limit $\alpha\to0$, one obtains $S^{(0)}/L = \ln(2)/2$: since the max entropy is twice the logarithm of the total number of eigenstates that have nonzero overlap with the initial state [97], we have that that this number is $\propto {\rm e}^{L/2}$ (which is the same result for the quench from the Néel state [100]). For the min entropy we have $S^{(\infty)}/L = \ln(2) - 2G/\pi$, where G is the Catalan constant. Some other analytical results are reported in the figure.

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It is relatively easy to obtain the higher-order corrections in powers of z for any fixed α. Instead, it is rather cumbersome to carry out the expansion for general real α. Therefore, here we only show the explicit calculation for the case $\alpha = 2$. Specifically, we determine S⁽²⁾ up to ${{\mathcal O}}(z^2)$. For convenience, instead of $\newcommand{\e}{{\rm e}} \eta_n^{(\alpha)}$ we consider the filling functions $\newcommand{\e}{{\rm e}} \vartheta_n\equiv 1/(1+\eta_n)$, where we suppressed the dependence on α, because we consider $\alpha=2$. The derivation of the higher-order expansion for the filling functions is the same as in [100] and we will omit it. The idea is that one has to plug the ansatz (3.21) into the equations for $\newcommand{\e}{{\rm e}} \eta_n$ (see (3.11)) solving the system order by order in powers of z. Similar to the Néel quench, we observe that the system (3.11) contains an infinite set of equations (one for each string type). However, to obtain the filling functions up to terms ${{\mathcal O}}(z^\omega)$, only the first $m(\omega)\sim \omega$ equations matter because the leading order of higher strings is given by higher orders in powers of z. The expansion of the filling functions $\vartheta_n$ around z  =  0 reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vartheta_{1}(\lambda) = \frac{1}{1 + \tan(\lambda)^{4}(\lambda)} - \frac{16 \cos(2\lambda)\tan^{4}(\lambda)}{(1 + \tan^{4}(\lambda))}z^{2} + O(z^{3}), \label{eq:theta1SmallZExpansion} \nonumber \end{align} \tag{ 3.35 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vartheta_{2}(\lambda) = 1 + O(z^{4}), \label{eq:theta2SmallZExpansion} \nonumber \end{align} \tag{ 3.36 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vartheta_{n}(\lambda) = O(z^{4}), \qquad (n \geqslant 3). \label{eq:thetaLargerNSmallZExpansion} \nonumber \end{align} \tag{ 3.37 }$$

A similar procedure for the TBA equations for the particle densities (see (2.19)) gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{\mathrm t,1} = \frac{1}{2 \pi} + \frac{2}{\pi}\cos(2\lambda) z + \frac{2}{\pi} \cos(4 \lambda) z^{2} + O(z^{3}), \label{eq:rho1tSmallZExpansion} \nonumber \end{align} \tag{ 3.38 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{\mathrm t,2} = \frac{1}{8 \pi} + \frac{2 - \sqrt{2}}{\pi} \cos^{2}(\lambda) z + \frac{1}{\pi} \cos (2 \lambda) z ^{2}, \label{eq:rho2tSmallZExpansion} \nonumber \end{align} \tag{ 3.39 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{{\rm t},n} = O(z^{4}), \qquad (n \geqslant 3). \label{eq:rhoLargerNtSmallZExpansion} \nonumber \end{align} \tag{ 3.40 }$$

The next-to-leading order in powers of z of the Rényi entropies can be computed by plugging (3.23), (3.30), (3.31) and (3.35)–(3.40) into (3.6) and (2.14). Given that the first-order in z cancels in both ${{\mathcal E}}$ and the Yang–Yang entropy and also that the ${{\mathcal O}}(z^2)$ contribution to S_YY vanishes, the first nonzero contribution is ${{\mathcal O}}(z^2)$ in ${{\mathcal E}}$, i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{E}= \frac{1}{4}\ln2 - \frac{\sqrt{2} \pi}{32} + \frac{z^{2}}{2} + O(z^{3}). \label{eq:ePseudoEnergySmallZCorrection} \nonumber \end{align} \tag{ 3.41 }$$

Thus, putting the various pieces together, the second Rényi entropy S⁽²⁾ is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(2)} = 2\ln2 - \ln(2+\sqrt{2}) + 2z^{2} + O(z^{3}). \label{eq:sRenyiEntropySmallZCorrection} \nonumber \end{align} \tag{ 3.42 }$$

Equation (3.42) implies that the asymptotic value of S⁽²⁾ for $\Delta\to\infty$ is approached as $1/\Delta$.

It is tempting to investigate the structure of equation (3.34) which has the same structure as the Rényi entropy of free-fermion models, written usually as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(\alpha)}_{{\rm GGE}} = \frac{L}{1-\alpha} \int_{-\pi/2}^{\pi/2}\! \frac{{\mathrm d} k}{2 \pi} \ln \left[\vartheta({k})^{\alpha}+ \left(1 - \vartheta({k}) \right)^{\alpha}\right], \label{eq:RenyiEntropyLimit} \nonumber \end{align} \tag{ 3.43 }$$

where $\vartheta(k)$ are now the free-fermion occupation numbers identifying the macrostate. Equation (3.43) is the same as (3.34) after defining $\vartheta(k)=1/(1+\tan^2 (k))$. Also, the factor $1/(2\pi)$ in (3.34) is the fermionic total density of states $\rho_{t}=1/(2\pi)$. A natural question is whether the free-fermion formula (3.43) holds true beyond the leading order in z. For instance, it is interesting to check whether (3.42) can be written in the free-fermion form (3.43). However, as the XXZ chain is interacting, equation (3.43) requires some generalisation. First, as there are different families of strings it is natural to sum over the string content of the macrostate. Moreover, in contrast with free fermions, for Bethe ansatz solvable models the total density of states $\rho_t$ is not constant, but it depends on the string type. The most natural generalisation of the free-fermion formula (3.43) would be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ff-last} S^{(\alpha)}_{{\rm GGE}}\stackrel{?}{=}\frac{L}{1-\alpha}\sum_n\int_{-\pi/2}^{\pi/2} {\rm d}\lambda\rho_{n,t}\ln[\vartheta_{n}^\alpha+(1-\vartheta_n)^\alpha]. \nonumber \end{align} \tag{ 3.44 }$$

Equation (3.44) is the same as the free-fermion formula (3.43) except for the overall term $\rho_{{\rm t}, n}$ in the integrand, which takes into account that for interacting models the density of states is not constant. In equation (3.44), the filling functions are given in (3.35)–(3.37).

Unfortunately, equation (3.44) does not give the correct value for the steady-state Rényi entropies. A very simple counterexample is provided by the standard Gibbs ensemble at infinite temperature. For the XXZ chain the macrostate describing this ensemble can be derived using the standard TBA approach (see [103]). In particular, for the XXX chain the exact infinite temperature Rényi entropies can be worked out analytically: they become equal and their density is $S^{(\alpha)}=L \ln2$ for any α. However, by using the analytical expression [103] for the infinite-temperature filling functions $\vartheta_n$ for the XXX chain in equation (3.44), one can verify that $S^{(2)}/L\ne\ln2$.

Still, since the free-fermion formula (3.44) holds true exactly at $\Delta\to\infty$ (see (3.42)), it is natural to wonder at which order in $1/\Delta$ (equivalently in z) it breaks down. By using equation (3.38)–(3.40) and (3.35)–(3.37) in (3.44), one can check that $\rho_{{\rm t}, 2}$ (see equation (3.39)) gives rise to an ${{\mathcal O}}(z)$ term in the entropy, which is absent in (3.42). This shows that (3.44) breaks down already at the first nontrivial order beyond the Ising limit.

In this section we discuss the Rényi entropy in the limit $\alpha\rightarrow\infty$ also known as min entropy, for which analytical results are obtainable. The analysis of the min entropy after the dimer quench is similar to that for the Néel quench [100]. In the following we remove the dependence on α in the saddle point densities to simplify the notation. After defining the functions $\newcommand{\e}{{\rm e}} \gamma_{n}=\ln (\eta_{n})/\alpha$, the $\alpha\rightarrow\infty$ limit of the saddle-point equations (3.11) yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma_{n} = d_{n} + s \star (\gamma_{n-1}^{+}+\gamma_{n+1}^{+}), \label{eq:gSaddlePointEquationsAlphaInfinity} \nonumber \end{align} \tag{ 3.45 }$$

where $\gamma_{n}^{+}=(\gamma_{n}+|\gamma_{n}|)/2$. Some insights on the structure of the solutions of (3.45) can be obtained by looking at the limit $\Delta\to\infty$. Precisely, from equation (3.25) one has that $\newcommand{\e}{{\rm e}} \eta_n\to 0$ for even n in the limit $\alpha \to\infty$. On the other hand, one has that $\newcommand{\e}{{\rm e}} \eta_n$ diverges for odd n for $\lambda\in[-\pi/4, \pi/4]$, which implies $\newcommand{\e}{{\rm e}} \ln\eta_n=\alpha d_n$ for n odd in that interval. Thus we have that at large Δ, $\gamma_{2n} (\lambda)<0$ and $\gamma_{2n+1}(\lambda)=d_{2n+1}(\lambda)$. As a consequence, the filling functions $\vartheta_n$ become

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{theta1} \vartheta_{2n}(\lambda)= \lim_{\alpha\rightarrow\infty} \frac{1}{1+{\rm e}^{\alpha\gamma_{2n}(\lambda)}} = 1, \nonumber \end{align} \tag{ 3.46 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{theta2} \vartheta_{2n+1}(\lambda)= \lim_{\alpha\rightarrow\infty} \frac{1}{1+{\rm e}^{\alpha d_{2n+1}(\lambda)}} = \Theta_{\mathrm H}(|\lambda|-\pi/4), \nonumber \end{align} \tag{ 3.47 }$$

where $\Theta_{\mathrm H}(x)$ is the Heaviside step function. The associated total densities are obtained using the BGT equations as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{\mathrm t, 1}(\lambda)=s(\lambda), \nonumber \end{align} \tag{ 3.48 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ccc} \rho_{\mathrm t, 2}(\lambda)=&~[s \star (s \cdot \Theta_{\mathrm H}(|x|-\pi/4))](\lambda) \nonumber \\ =&~ \frac{1}{4 \pi^{2}}\, \sum_{k \in {\mathbb Z}} \frac{{\rm e}^{-2 {\rm i} k \lambda}}{\cosh(k\eta)} \left(\sum_{\ell \ne k} \frac{\sin((k-\ell) \pi) - \sin ((k - \ell)\pi/2)}{(k-\ell) \cosh(\ell \eta)} + \frac{\pi}{2 \cosh(k \eta)} \right),\nonumber \end{align} \tag{ 3.49 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ccc1} \rho_{\mathrm t, n}(\lambda)=0, \qquad (n>2). \nonumber \end{align} \tag{ 3.50 }$$

These results imply that the min entropy is completely determined by the first two densities with n  =  1 and n  =  2, in contrast with the quench from the Néel state [100], where only the first density enters in the expression for $S^{(\infty)}$.

To derive the general expression for the min entropy, a crucial preliminary observation is that the macrostate describing $S^{(\infty)}$ has zero Yang–Yang entropy. This is a general result that holds for quenches from arbitrary states. Indeed, first we notice that the ansatz $\newcommand{\e}{{\rm e}} \eta_n={\rm e}^{\alpha\gamma_n}$ implies that, in the limit $\alpha\to\infty$, $\vartheta_n$ can be only zero or one. Then, assuming that $\rho_{{\rm t}, n}$ is finite, the Yang–Yang entropy

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S_{\rm YY}=-L\sum_n\int {\rm d}\lambda \rho_{{\rm t},n}[\vartheta_{n}\ln\vartheta_n+ (1-\vartheta_n)\ln(1-\vartheta_{n})], \nonumber \end{align} \tag{ 3.51 }$$

must vanish in the limit $\alpha\to\infty$. Consequently the $S^{(\infty)}_{{\rm GGE}}$ is determined only by the driving functions as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S^{(\infty)}_{{\rm GGE}}=L\sum_n\int_{-\pi/2}^{\pi/2} {\rm d}\lambda g_n\rho_n+L f_{\rm GGE}. \label{scopy} \nonumber \end{align} \tag{ 3.52 }$$

Interestingly, in the large Δ limit equation (3.52) simplifies. Specifically, only the first two strings with $n=1, 2$ contribute in (3.52), as is clear from (3.50).

Upon lowering Δ, we observe a sharp transition in the behaviour of $S^{(\infty)}$. Indeed, there is a 'critical' value $\Delta^*$, such that for $\Delta<\Delta^*$, higher-order strings become important. The condition that determines $\Delta^*$ is that $\gamma_2$ becomes positive, i.e. that for some λ

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_2+2s\star d_1\geqslant0. \label{cond} \nonumber \end{align} \tag{ 3.53 }$$

The value of $\Delta^{*}$ can be found by numerically by imposing equality in (3.53) and the final result is $\Delta^{*}\approx 1.7669$. This is the same value of $\Delta^*$ found for the quench from the Néel state [100], although the condition for higher strings to contribute for the Néel state (i.e. $d_1+2s\star d_2\geqslant0$) may appear different. This, however, is equivalent to (3.53) after noticing that $d_{2n}^{{\rm Dimer}}=d_{2n+1}^{{\rm Neel}}$.

Finally, in contrast with the large Δ limit, for $\Delta<\Delta^{*}$, an analytical solution of (3.45) is not possible. However, the system (3.45) can be effectively solved numerically. The result for $S^{(\infty)}$ is reported in figure 1 (bottom line).

Here we numerically extract the driving functions $g_n(\lambda)$ for the quench from the tilted Néel state. The key ingredients are the rapidity densities describing the post-quench steady state that have been determined in [101]. These are used in the system (4.1) to extract g_n. To make the paper self contained, we report the results for the saddle point densities. The starting point is $\newcommand{\e}{{\rm e}} \eta_1(\lambda)$, which is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eta1} \eta_{1}(\lambda) = -1 + \frac{T_{1}\!\left(\lambda + {\rm i} \frac{\eta}{2}\right) T_{1}\!\left(\lambda - {\rm i} \frac{\eta}{2}\right)}{\phi\!\left(\lambda + {\rm i} \frac{\eta}{2} \right) \bar \phi\!\left(\lambda - {\rm i} \frac{\eta}{2} \right)}, \nonumber \end{align} \tag{ 4.2 }$$

where the auxiliary functions $\phi, \bar\phi$ and T₁ are defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{1}(\lambda) =& - \frac{1}{8} \cot(\lambda) \{8 \cosh(\eta) \sin^{2}(\theta) \sin^{2}(\lambda) - 4 \cosh(2\eta) \nonumber \\ & + [\cos(2\theta) + 3][2 \cos(2\lambda) - 1] + 2 \sin^{2}(\theta) \cos(4\lambda) \} , \label{eq:eSaddlePointTiltedNeelFunctionT1} \nonumber \end{align} \tag{ 4.3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi(\lambda) = \frac{1}{8} \sin(2\lambda + {\rm i} \eta) [2 \sin^{2}(\theta) \cos(2\lambda - {\rm i} \eta) + \cos(2\theta) + 3], \label{eq:eSaddlePointTiltedNeelFunctionPhi} \nonumber \end{align} \tag{ 4.4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \bar \phi(\lambda) = \frac{1}{8} \sin(2\lambda - {\rm i} \eta) [2 \sin^{2}(\theta) \cos(2\lambda + {\rm i} \eta) + \cos(2\theta) + 3]. \label{eq:eSaddlePointTiltedNeelFunctionPhiBar} \nonumber \end{align} \tag{ 4.5 }$$

Here θ denotes the tilting angle. For n  >  1, $\newcommand{\e}{{\rm e}} \eta_{n}(\lambda)$ is determined recursively from the Y-system

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta_{n+1}(\lambda) = \frac{\eta_{n}(\lambda + {\rm i} \eta /2) \eta_{n}(\lambda - {\rm i} \eta/2)}{1 + \eta_{n-1}(\lambda)}-1, \label{eq:YRelations} \nonumber \end{align} \tag{ 4.6 }$$

with the convention $\newcommand{\e}{{\rm e}} \eta_{0}\equiv 0$. From the densities $\newcommand{\e}{{\rm e}} \eta_n$, the particle densities $\rho_n$ are obtained, as usual, by solving the thermodynamic version of the TBA equation (2.15).

The driving function g_n can be easily extracted numerically by plugging the above root densities in equation (4.1). The results for quenches for the XXZ chain with $\Delta=2$ and the quench from the tilted Néel state with tilting angle $\theta=\pi/3$ are reported in figure 3. These results may be compared with the recently conjectured form of the overlaps [102]. So far, this conjecture has been tested numerically for Bethe states containing few particles, and it has been shown to give the correct thermodynamic macrostate after the quench. The thermodynamic limit of the overlaps in [102] can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \ln \langle \Psi_{0} |\rho_n\rangle = L\sum_{n=1}^{\infty} \int_{-\pi/2}^{\pi/2} \mathrm{d} \lambda \rho_{n}(\lambda) g_{n}(\lambda), \label{eq:pTiltedNeelOverlaps} \nonumber \end{align} \tag{ 4.7 }$$

where $\rho_n$ are the particle densities describing the thermodynamic macrostate and the explicit forms of g_n read [102]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_{1}(\lambda) = \frac{\tan(\lambda + {\rm i} \eta / 2) \tan(\lambda - {\rm i} \eta / 2)}{4 \sin^{2}(2\lambda)}\cdot \frac{\cos^{2}(\lambda+{\rm i} \xi)\cos^{2}(\lambda-{\rm i} \xi)}{\cosh^{4}(\xi)}, \label{eq:pTiltedNeelOverlapKernel1} \nonumber \end{align} \tag{ 4.8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_{n}(\lambda) = \sum_{j=1}^{n} g_{1}(\lambda + {\rm i} \eta / 2 (n+1-2j)), \label{eq:pTiltedNeelOverlapKernelN} \nonumber \end{align} \tag{ 4.9 }$$

where ξ is related to the tilting angle θ as $\newcommand{\e}{{\rm e}} \xi\equiv -\ln(\tan(\theta / 2))$.

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We now compare the driving functions g_n as extracted from the TBA equation (4.1) with the conjectured result in equations (4.8) and (4.9). The comparison is presented in figure 3 for the XXZ chain with $\Delta=2$ and the quench from the tilted Néel state with tilting angle $\theta=\pi/3$ (we tested them also for other tilting angles, finding equivalent results that we do not report here). The continuous lines are the numerical results for g_n for $n\leqslant 4$ (higher strings are not reported). The different symbols (crosses) are the numerical results using the conjecture (4.8) and (4.9). As it is clear from the figure, the agreement between the two results is perfect for all values of λ.

Now we are in the position to obtain results for the steady-state Rényi entropies after the quench from the tilted Néel state in the XXZ chain. The theoretical predictions for the entropies are obtained by combining the results of section 4.1 to extract the driving functions g_n with the procedure of [97] (see section 3). Our results are reported in figure 4 plotting $S^{(\alpha)}/L$ versus the chain anisotropy Δ. The data shown in the figure are for the tilted Néel with tilting angle $\theta=\pi/3$ and $\theta=\pi/6$ (right and left panel, respectively). As expected, one has that $S^{(\alpha)}<S^{(\alpha')}$ if $\alpha>\alpha'$. For all values of α and $\theta\neq0$ the entropy densities are finite in the limit $\Delta\to\infty$, in contrast with the quench from the Néel state [100], where all the entropies vanish for $\Delta\to\infty$. Finally, an intriguing feature is that for $\theta=\pi/6$ the behaviour of the entropies is not monotonic as a function of Δ, but $S^{(\alpha)}$ exhibits a minimum around $\Delta\approx 5$, although the minimum is not very pronounced. This remains true for a window of tilting angle θ close to $\pi/6$.

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We now focus on the steady-state value of the min entropy. Similar to the Néel and dimer states, in the limit $\alpha\to\infty$ one can use the ansatz $\newcommand{\e}{{\rm e}} (\ln\eta_n)/\alpha=\gamma_n$. The equations for $\gamma_n$ are the same as for the dimer (see (3.45)), i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma_n=d_n+s\star(\gamma_{n-1}^++\gamma_{n+1}^+), \nonumber \end{align} \tag{ 4.10 }$$

where now the driving d_n is obtained from the driving functions g_n for the quench from the tilted Néel as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_n=g_n-s\star(g_{n+1}+g_{n-1}). \nonumber \end{align} \tag{ 4.11 }$$

For the Néel quench, i.e. for $\theta=0$, there is a 'critical' value $\Delta^*$ such that for $\Delta>\Delta^*$ the thermodynamic macrostate that the describes the min entropy is the ground state of the XXZ chain [100]. For $\Delta<\Delta^*$ this is not the case, and the macrostate is an excited state with zero Yang–Yang entropy. The 'critical' $\Delta^*$ at which the behaviour of the thermodynamic macrostate changes is determined for the Néel state (and for the dimer state as well) by the condition that $\gamma_2$ becomes positive. It is natural to investigate how this scenario is modified upon tilting the initial state. Here we show that the macrostate describing the min entropy is the ground state of the XXZ chain provided that the tilting angle θ is not too large.

To clarify this issue, we numerically observed that for large Δ

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{cond1} & \gamma_1(\lambda)<0,\nonumber \\ & \gamma_3(\lambda)<0.\nonumber \end{align} \tag{ 4.12 }$$

The conditions in equation (4.12) have important consequences for the particle densities $\rho_n$. In particular, it implies that the macrostate describing the min entropy is the ground state of the XXZ chain. To show that, let us consider the TBA equations for $\rho_{{\rm t}, n}$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{n,{\rm t}}=s\star[(1-\vartheta_{n-1})\rho_{n-1,{\rm t}}+ (1-\vartheta_{n+1})\rho_{n+1,{\rm t}}]. \label{sys} \nonumber \end{align} \tag{ 4.13 }$$

First, since $\vartheta_0=0$ and $\rho_0=\delta(\lambda)$, one has that $\rho_1=s$, which is the density of the ground state of the XXZ chain. Clearly, the conditions in (4.12) together with the system (4.13) imply that $\rho_{2, {\rm t}}=0$. Another important consequence is that the first two equations in (4.13) are decoupled from the rest, which form a linear homogeneous system of integral equations. Moreover, for $n\to\infty$ one expects that $\rho_{{\rm t}, n} \to 0$. Thus, it is natural to conjecture that $\rho_{{\rm t}, n}=0$ for any n  >  2. Finally, we observe that a similar decoupling occurs for the quench from the Néel state [100], although via a different mechanism. Precisely, for the Néel state one has that $\gamma_{2n+1}<0$ for all n.

We now use the conditions (4.12) to characterise the behaviour of the min entropy. Our results are summarised in the 'phase diagram' in figure 5. The blue region in the figure corresponds to the region in the parameter space $(\theta, \Delta)$ where the thermodynamic macrostate describing the min entropy is the ground state of the XXZ chain (at that value of Δ). For $\theta=0$ we recover the result of [100], i.e. that the ground state describes the min entropy for $\Delta>\Delta^*\approx 1.766$. The ground state remains the correct macrostate for the min entropy in a region of not too large θ. Conversely, for $\theta\gtrsim 0.1$ the macrostate is an excited state of the XXZ at any Δ. However, at smaller θ there is always an extended region where the min entropy is described by the ground state of the XXZ chain. The extension of this region shrinks upon increasing Δ, and it is likely to vanish in the limit $\Delta\to\infty$. The dashed line in figure 5 marks the 'transition' between the two regimes. The line is obtained by numerically finding the values of $(\theta, \Delta)$ for which either $\gamma_1$ or $\gamma_3$ vanish, violating the conditions in equation (4.12).

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