Energy transport between two integrable spin chains

We study the energy transport in a system of two half-infinite XXZ chains initially kept separated at different temperatures, and later connected and let free to evolve unitarily. By changing independently the parameters of the two halves, we highlight, through bosonization and time-dependent matrix-product-state simulations, the different contributions of low-lying bosonic modes and of fermionic quasiparticles to the energy transport. In the simulations we also observe that the energy current reaches a finite value which only slowly decays to zero. The general picture that emerges is the following. Since integrability is only locally broken in this model, a preequilibration behavior may appear. In particular, when the sound velocities of the bosonic modes of the two halves match, the low-temperature energy current is almost stationary and described by a formula with a nonuniversal prefactor interpreted as a transmission coefficient. Thermalization, characterized by the absence of any energy flow, occurs only on longer time scales which are not accessible with our numerics.

Figure 1

Sketch of the partitioning protocol: two initially disconnected half chains of semi-infinite length, thermalized at different inverse temperatures ${\beta }_l$ and ${\beta }_r$, are connected at time $t=0$. A net energy current flowing through the junction develops into the system. In this paper, we investigate the properties of such current when the two halves are integrable (but different) spin chains.

Figure 2

The renormalization group flow in presence of the backscattering operator (14). The inhomogeneous fixed point is described by a CFT and is conducting, with energy current given in (16). At the insulating fixed point the renormalized backscattering coupling constant $\lambda$ is formally infinite, and one should expect absence of thermal transport at low energy.

Figure 3

Sketch of the stationary state according to the low-energy Luttinger-liquid description: when the Fermi velocity on the two sides is matched, the discontinuity of the Luttinger parameter $K\left(x\right)$ induces transmission/reflection coefficients: the hot (cold) low-lying bosonic modes coming from left (right) are reflected and transmitted at the junction, determining a nonuniversal value for the stationary energy current.

Figure 4

Transmission coefficient $\mathcal{T}$ as a function of the anisotropy parameter ${\mathrm{\Delta }}_r$ for ${\mathrm{\Delta }}_l=0$. The red dots signal the parameters used in the simulations displayed in Fig. 5. The dashed line indicates the upper limit $\mathcal{T}=1$.

Figure 5

Top panels: the time evolution of energy current $\mathcal{J}\left(t\right)$ for different values of ${\beta }_r$, from bottom to top: $5,7,8.5,10.5J_l^{-1}$ at fixed ${\beta }_l=0.1J_l^{-1}$. Bottom panels: steady-state value of $\mathcal{J}\left(t\right)$ obtained averaging over times $t\ge t_{\min }$ (dashed vertical line in the top panels). The parameters of the right halves are (from left to right) $J_r/J_l=2.092,1.54,0.77,0.709$ and ${\mathrm{\Delta }}_r=-0.7,-0.5,0.5,0.7$ corresponding to $\mathcal{T}=0.89,0.96,0.98,0.96$. In all the cases, ${\mathrm{\Delta }}_l=0$ and $u_l=u_r=J_l$.

Figure 6

Time evolution of the energy current $\mathcal{J}\left(t\right)$ (top panel) and of its derivative (lower panel) for $K=0.768$ (black line), 1 (red line), $1.983$ (blue line) in the $u_l\ne u_r$ case. We set ${\beta }_l=4J_l^{-1},{\beta }_r=5J_l^{-1},{\mathrm{\Delta }}_l=-0.3,J_r/J_l=1$, and ${\mathrm{\Delta }}_r=0.95,0.3,-0.95$ (black, red, and blue lines, respectively).

Figure 7

Top panel: time evolution of the integrated energy current $\mathcal{J}\left(t\right)$ for different values of $\varepsilon$ as indicated in the legend. Bottom panels: local anisotropy parameter ${\mathrm{\Delta }}_i$ and local Fermi velocity $u_i$ as a function of the position. Here, the parameters are ${\beta }_l=4J_l^{-1},{\beta }_r=5J_l^{-1},{\mathrm{\Delta }}_l=-{\mathrm{\Delta }}_r=-0.3$, and $J_r/J_l=1$.

Figure 8

Top panel: snapshots of the local energy current ${\langle {\widehat{j}}_{\lambda ,i}\rangle }_t$ at $t{J}_l/0.4=1,11,21,31,41,51,61$. Lower panel: comparison between the energy current flowing at the center of the chain and the integrated energy current for $a=-5$ and $b=+5$ normalized by $\left(b-a+1\right)$. We set ${\beta }_l=4J_l^{-1},{\beta }_r=5J_l^{-1},{\mathrm{\Delta }}_l=-{\mathrm{\Delta }}_r=-0.3,J_r/J_l=1$.

Figure 9

Top panel: time evolution of the energy current $\mathcal{J}\left(t\right)$ at the center of the chain. Lower panel: time evolution of the derivative of the integrated energy current ${\langle {\partial }_t{\widehat{J}}_{ab}\rangle }_t$. We considered the homogeneous case (black lines), inhomogeneous case with $u_l\ne u_r$ (red lines), and inhomogeneous case with $u_l=u_r$ (blue lines). We set ${\beta }_l=4J_l^{-1},{\beta }_r=5J_l^{-1},{\mathrm{\Delta }}_l=-{\mathrm{\Delta }}_r=-0.3$ corresponding to $u_l=0.798\left(9\right)J_l$ and $u_r=1.183\left(5\right)J_l$. The hopping parameter is $J_r/J_l=1$ apart from the velocity matching case where it is chosen so that $u_l=u_r$ is satisfied.

Figure 10

Top panel: sketch of the energy current profile in the preequilibration behavior as described by the low-energy field theory: two wavefronts propagate in opposite direction with different velocities $v_l,v_r$ determining an almost stationary and homogeneous energy current around the junction. Lower panel: sketch of the asymptotic long-time regime; a thermal region at inverse temperature ${\beta }^{*}$ develops around the junction due to the local integrability breaking. The two transient regions of nonvanishing current keep traveling and increasing their size, as discussed in Sec. 5b.

Figure 11

Distribution of the normalized level spacings for a chain made of $N=16$ spins in the zero magnetization sector. Top panel: homogeneous $\left({\mathrm{\Delta }}_l={\mathrm{\Delta }}_r=0\right)$ and inhomogeneous cases with $u_l\ne u_r$ (choosing ${\mathrm{\Delta }}_r=-{\mathrm{\Delta }}_l>0$). The hopping parameters are fixed as $J_r/J_l=1$. Lower panel: inhomogeneous case with ${\mathrm{\Delta }}_l$ and ${\mathrm{\Delta }}_r$ fixed as in the top panel but tuning $J_r/J_l$ in order to get $u_l=u_r$.

Figure 12

Distribution of the normalized level spacings for a chain made of $N=16$ spins in the zero magnetization sector. Inhomogeneous cases with $u_l\ne u_r$ choosing ${\mathrm{\Delta }}_l=1$ and ${\mathrm{\Delta }}_r=0.95$ (green line), $-0.95$ (blue line). The hopping parameters are fixed as $J_r/J_l=1$. As indicated in the legend, the different values of ${\mathrm{\Delta }}_r$ allow to explore the $K\ne 1$ cases.

Figure 13

The dual defect map $\mathrm{\Omega }$ acting on the incoming and outgoing chiral bosonic waves.

Figure 14

Time evolution of the energy current $\mathcal{J}\left(t\right)$ for different values of ${\chi }_{\max }$, as indicated in the legend. The dashed vertical lines are the times $t_{\mathrm{ex}}\left({\chi }_{\max }\right)$ after which we approximate the time evolution according to the value of ${\chi }_{\max }$. Here, the parameters are ${\beta }_l=4J_l^{-1},{\beta }_r=5J_l^{-1},{\mathrm{\Delta }}_l=-{\mathrm{\Delta }}_r=-0.3$, and $J_r/J_l=1$. In the inset we show a magnification of the data around the times $t_{\mathrm{ex}}\left({\chi }_{\max }\right)$.