Crossover from generalized to conventional hydrodynamics in nearly integrable systems under relaxation time approximation

Abstract

Upon breaking the integrability, the equations of generalized hydrodynamics (GHD) are supplemented by a Boltzmann collision term. Such terms are typically complicated and stem from a perturbative treatment of integrability-breaking terms in the hamiltonian. In our work, we study a simplified version of the collision operator in a form of relaxation time approximation familiar from kinetic theory. We explicitly compute transport coefficients which characterize the Navier-Stokes (NS) hydrodynamic regime emerging at large space-time scales. We also thoroughly study the crossover between GHD and NS hydrodynamic descriptions, identifying relevant characteristic space-time scales for the transition. In particular, we show how the emergence of NS hydrodynamics is visible in dynamics of conserved and non-conserved charge densities, and in hydrodynamic two-point functions.

Figures (4)

• Figure 1: Contributions to the transport coefficients as a function of the interaction strength $c$ for several values of temperature $T$. The contributions $\zeta_{\mathfrak{D}}$ and $\kappa_{\mathfrak{D}}$ are computed using \ref{['eq:transport_coeff_diff']} while the contributions $\zeta_{\mathcal{I}}$ and $\kappa_{\mathcal{I}}$ are obtained from \ref{['eq:transport_coeff_coll']}. Irrespective of the temperature $T$, all coefficients except $\kappa_{\mathcal{I}}$ show non-monotonic behavior as a function of $c$. Starting near zero for small $c$, they attain a maximum value at a finite value of $c$ and eventually vanish at very large $c$. With increasing temperature $T$, the positions of these maxima shift toward larger values of $c$. In contrast, the $\kappa_{\mathcal{I}}$ displays monotonic behavior with respect to $c$, it begins at a finite value at $c \to 0$, increases monotonically with $c$ and asymptotically approaches another finite value at $c \to \infty$. For reference, a dashed line corresponding to classical free particles, given by the formula $\kappa_{\mathcal{I}}=\frac{3}{2} \varrho \tau T$, which is derived from simple adaptation of \ref{['eq:omega1_simplified']} is shown in the last panel for the temperature $T=10$. For better visibility, we refrain from plotting the data for $\kappa_{\mathcal{I}}$ at temperature $T=50$. In all figures, the density $\varrho$ is set to $1$.
• Figure 2: Panel (a): real part of the spectrum of matrix $\mathcal{M}(k)$ as a function of $k$. We consider interaction coupling $c=2$, temperature $T=4.36$ and density $\varrho=1$ with RTA timescale $\tau=1$. For small momenta $k \ll k_c$ we observe gapless modes (in fact there are three such modes, the sound modes have degenerate $\text{Re}\Lambda(k)$) and gapped ones with a gap $\tau^{-1}$. At larger momenta $k \gg k_c$ we find the modes of GHD. Panels (b)-(d): absolute value of rotation matrix $R(k)$ in three different regimes. For $k=0$ we have $\mathcal{M} = \Gamma$ and the operator is diagonal. For $k$ finite, but much smaller than $k_c$ we observe a NS sector for the first three charges. The coefficients are those of standard sound and heat modes. For $k\gg k_c$ the modes of the systems are GHD modes, which are not simply related to the basis $\{ h_n \}$. This is reflected in featureless $R$ matrix.
• Figure 3: Dynamics of the density, velocity and energy fields starting from an initial perturbation in the $\delta \varrho$ and $\delta e$ fields around a homogeneous thermal state $(\varrho^{\rm th}=\varrho, u^{\rm th}=0, T^{\rm th}=T)$. The perturbation is evolved using the linearized NS equations, \ref{['eqs:NS_linearized']}, with the transport coefficients $\zeta$ and $\kappa$ computed from \ref{['eq:transport_coefficients']}. These results are shown in the main plots as solid circles labeled "NS” at different times during the evolution. They are then compared with the corresponding fields obtained by solving the linearized GHD–Boltzmann equation, \ref{['eq:charge_evolution']}, shown as solid lines labeled "GHD-B.”. The agreement between the two at late times confirms that the large-scale space–time behavior of the conserved fields is accurately captured by the NS dynamics with correctly computed transport coefficients. The parameters used here are: $c = 2.0$ and $\tau = 2.0$, $\varrho=1.0$, $T=1.36$. In this setup, the crossover momentum $k_c$ obtained from \ref{['eq:k_critical']} is approximately $0.57$, while the characteristic momentum of initial perturbation is $k_{\rm ini} \approx0.25$ (upper panels) and $k_{\rm ini} \approx 1.9$ (lower panels). In the top row, where $k_{\rm ini}<k_c$, the original dynamics (GHD-Boltzmann) match NS results even at short times $t \sim \tau$. In contrast, in the bottom row, where $k_{\rm ini}>k_c$, it takes longer time $(t \sim 10 \tau )$ for the dynamics to converge to the corresponding NS fields. In the insets of the lower panels, we compare our GHD-Boltzmann data with the pure GHD results [i.e., without the collision term in \ref{['GHD_full']}], shown as void circles labeled "GHD”. At very short times, $t \sim \tau$, the two results match, indicating that the effect of collision integral has not yet become significant. As time increases further, deviations between the two start to appear and at very large time NS dynamics start to appear which is shown in the main plots.
• Figure 4: Dynamics of correlation functions of conserved (left column) and non-conserved (right column) charges in GHD-Boltzmann equation. At late times, the density-density correlator exhibits NS behavior \ref{['eq:S00NS']} with two sound modes propagating with sound velocity $v_s$ and heat mode in the center. On the other hand, correlator of non-conserved charge $S_{q_4,q_0}(x,t)$ decays in time. At early times the dynamics in both cases is well described by the GHD without the collision term.