Exact steady states for quantum quenches in integrable Heisenberg spin chains

The study of quantum quenches in integrable systems has significantly advanced with the introduction of the quench action method, a versatile analytical approach to nonequilibrium dynamics. However, its application is limited to those cases where the overlaps between the initial state and the eigenstates of the Hamiltonian governing the time evolution are known exactly. Conversely, in this work we consider physically interesting initial states for which such overlaps are still unknown. In particular, we focus on different classes of product states in spin-$1/2$ and spin-1 integrable chains, such as tilted ferromagnets and antiferromagnets. We get around the missing overlaps by following a recent approach based on the knowledge of complete sets of quasilocal charges. This allows us to provide a closed-form analytical characterization of the effective stationary state reached at long times after the quench, through the Bethe ansatz distributions of particles and holes. We compute the asymptotic value of local correlations and check our predictions against numerical data.

Figure 1

Rapidity distribution functions ${\rho }_n\left(\lambda \right)$ for the tilted ferromagnet state (44) for $n=1,2,3$. The plots correspond to $\mathrm{\Delta }=1.2$ and (a): $\vartheta =\pi /5$, (b): $\vartheta =\pi /3$. The functions are shown for $\lambda >0$ being symmetric with respect to $\lambda =0$.

Figure 2

Normalized densities $D_n$ of the quasiparticles forming $n$ strings [as defined in (31)] for the tilted ferromagnet state (44). Plot (a) corresponds to $\mathrm{\Delta }=1.3$ and shows the dependence on the angle $\vartheta$, while plot (b) corresponds to $\vartheta =\pi /3$ and shows the dependence on the anisotropy parameter $\mathrm{\Delta }$.

Figure 3

Rapidity distribution functions ${\rho }_1\left(\lambda \right), {\rho }_2\left(\lambda \right)$ for the tilted Néel state (53) for different values of $\vartheta$. The plots correspond to $\mathrm{\Delta }=2$. The functions are shown for $\lambda >0$ being symmetric with respect to $\lambda =0$.

Figure 4

Normalized densities $D_n$ of the quasiparticles forming $n$ strings [as defined in (31)] for the tilted Néel state (53). Plot (a) corresponds to $\mathrm{\Delta }=2$ and shows the dependence on the angle $\vartheta$, while plot (b) corresponds to $\vartheta =\pi /4$ and shows the dependence on the anisotropy parameter $\mathrm{\Delta }$.

Figure 5

Rapidity distribution functions ${\rho }_1\left(\lambda \right), {\rho }_2\left(\lambda \right)$ for the spin-1 zero-magnetization product state (61) for different values of $\eta$. The functions are shown for $\lambda >0$ being symmetric with respect to $\lambda =0$.

Figure 6

Normalized densities $D_n$ of the quasiparticles forming $n$ strings [as defined in (31)] for (a): the spin-1 zero magnetization product state (61), (b): the spin-1 Néel state (64).

Figure 7

Rapidity distribution functions ${\rho }_1\left(\lambda \right), {\rho }_2\left(\lambda \right)$ for the spin-1 Néel state (64) for different values of $\eta$. The functions are shown for $\lambda >0$ being symmetric with respect to $\lambda =0$.

Figure 8

Longitudinal correlators for the quench from the tilted ferromagnetic state (44) with (a) $\vartheta =\pi /6$, (b) $\vartheta =\pi /3$. The value of the anisotropy parameter is $\mathrm{\Delta }=4$. The solid lines represent the iTEBD data of Ref. [75]. The exact results for the asymptotic stationary values computed using the method described in the text (dashed lines) are displayed together with the ultralocal GGE predictions computed in Ref. [75] (symbols). Different colors are used to distinguish different correlators.

Figure 9

Transverse correlators for the quench from the tilted ferromagnetic state (44) with $\vartheta =\pi /3$. The value of the anisotropy parameter is $\mathrm{\Delta }=4$. The iTEBD results of Ref. [75] (solid lines) are shown together with the exact results for the asymptotic stationary values (dashed lines) and the ultralocal GGE predictions computed in Ref. [75] (symbols).

Figure 10

Transverse correlators for the quench from the tilted Néel state (53) with $\left(a\right)\vartheta =\pi /18, \left(b\right)\vartheta =\pi /6$. The value of the anisotropy parameter is $\mathrm{\Delta }=8$. The solid lines correspond to the tDMRG data of Ref. [75], which were computed for a chain of $L=64$ sites. They are displayed together with the exact results for the asymptotic stationary values (blue dotted lines) and the ultralocal GGE predictions computed in Ref. [75] (red dashed lines).

Figure 11

Rapidity distribution function ${\rho }_2\left(\lambda \right)$ for the domain-wall state (118) for $p=2$. The exact function computed using the method of Secs. 4a and 4b is compared with the result obtained assuming the $Y$ system (113). The plot corresponds to $\mathrm{\Delta }=1.5$.