Coexistence of energy diffusion and local thermalization in nonequilibrium $XXZ$ spin chains with integrability breaking

In this work we analyze the simultaneous emergence of diffusive energy transport and local thermalization in a nonequilibrium one-dimensional quantum system, as a result of integrability breaking. Specifically, we discuss the local properties of the steady state induced by thermal boundary driving in a $XXZ$ spin chain with staggered magnetic field. By means of efficient large-scale matrix product simulations of the equation of motion of the system, we calculate its steady state in the long-time limit. We start by discussing the energy transport supported by the system, finding it to be ballistic in the integrable limit and diffusive when the staggered field is finite. Subsequently, we examine the reduced density operators of neighboring sites and find that for large systems they are well approximated by local thermal states of the underlying Hamiltonian in the nonintegrable regime, even for weak staggered fields. In the integrable limit, on the other hand, this behavior is lost, and the identification of local temperatures is no longer possible. Our results agree with the intuitive connection between energy diffusion and thermalization.

Figure 1

Scheme of the nonequilibrium system studied. At the left (L) and right (R) boundaries of a spin chain, thermal reservoirs of temperatures $T_{\text{targ}}^{L,R}$ and chemical potentials ${\mu }_{\text{targ}}^{L,R}$ induce local grand-canonical states on two neighboring spins. This leads to energy currents $J^{XXZ}$ [Eq. (5)] and/or spin currents $J^{\text{S}}$ [Eq. (6)] through the chain. In the scheme, they flow from the left (red) to the right (blue) reservoir, assuming $T_{\text{targ}}^L>T_{\text{targ}}^R$ and/or ${\mu }_{\text{targ}}^L>{\mu }_{\text{targ}}^R$.

Figure 2

Energy transport properties of integrable $XXZ$ spin chains. The results correspond to $T_{\text{targ}}^L=\infty ,T_{\text{targ}}^R=20$ and different interactions $\Delta$. (a) Energy current as a function of $N$. (b) Energy profiles for $N=80$. Note the strong boundary effects of the thermal driving.

Figure 3

Energy transport properties of the $XXZ$ model with staggered magnetic field. The simulations correspond to $\Delta =1.5,T_{\text{targ}}^L=\infty$, and $T_{\text{targ}}^R=20$. (a) Examples of energy profiles for two staggered magnetic fields. (b) Corresponding energy currents as a function of $N$. The symbols represent the TNT results, and the lines are guides to the eye. (c) Scaling of the ratio $\langle J^{XXZ}\rangle /\Delta E$ with the size of the system. The symbols are the numerical data, and the lines represent the fits to Eq. (7). To perform the fit, we have discarded $n=15$ sites at each boundary of the chain for all values of $N$ considered. Larger values of $n$ do not modify the results, since the energy gradient is homogeneous in the region of the chain retained. For $B=0.15$, the fit gives ${\kappa }_{XXZ}=145\left(5\right)$ and $\alpha =0.98\left(1\right)$, and for $B=0.30$ it gives ${\kappa }_{XXZ}=40.6\left(5\right)$ and $\alpha =0.97\left(1\right)$. (d) Conductivity of the system as a function of $B$. The solid line corresponds to the fit ${\kappa }_{XXZ}=4.0\left(1.3\right)B^{-1.9\left(1\right)}$.

Figure 4

Spin-spin correlations of the $XXZ$ model with staggered magnetic field, for $\Delta =1.5,N=100,T_{\text{targ}}^L=\infty$, and $T_{\text{targ}}^R=20$. From top to bottom, the lines correspond to $B=0,0.15,0.20,0.30,0.35$. The field amplitude $B$ thus increases as indicated by the arrow.

Figure 5

Trace distance between the reduced density operator ${\tilde{\rho }}_2$ of the two central sites and two-site states (9) with temperature $T_{\frac{N}{2},\frac{N}{2}+1}$ for various staggered fields $B$. The calculations correspond to $N=100,T_{\text{targ}}^{\text{L}}\to \infty ,T_{\text{targ}}^{\text{R}}=20,\Delta =1.5,\delta T={10}^{-2}$, and the final iteration of the self-consistent procedure.

Figure 6

Temperature profiles across the system for the parameters of Fig. 5. The (flat) profile of $B=0$ is not shown since it corresponds to temperatures $\approx 95$.

Figure 7

Comparison between expectation values $\langle {\sigma }_i^z{\sigma }_{i+1}^z\rangle$ directly obtained from the numerical simulations of driven systems with temperature imbalance (dashed lines) and those of the chosen two-site thermal states (solid lines). Each indicated value of $B$ refers to the closest solid and dashed line. For clarity, the dashed lines correspond to $B=0$ (dash-dot), $B=0.1$ (long-dashed), $B=0.2$ (medium-dashed), and $B=0.4$ (short-dashed). Also, the results for the 10 leftmost and rightmost sites have not been plotted. The calculations correspond to the parameters of Fig. 5.

Figure 8

(a) Comparison between expectation values $\langle {\sigma }_j^z\rangle$ directly obtained from the numerical simulations of driven systems with temperature imbalance $(\circ )$ and those of various two-site thermal states in Eq. (9). The red dashed line refers to states with no local chemical potential. The black solid line corresponds to states with the mean-field chemical potentials in Eq. (16). The solid blue (oscillating) line represents the results when using ${\mu }_j$ and ${\mu }_{j+1}$ as fitting parameters. (b) Values of ${\mu }_j$ minimizing the trace distance between the two-site reduced and thermal states of sites $(j,j+1)$. The calculations correspond to $B=0.4$ and the other parameters of Fig. 5.

Figure 9

Effective site-dependent Hamiltonian couplings of local two-site states for different staggered magnetic fields $B$ and interaction $\Delta =1.5$. (a) Effective hopping ${\tilde{\tau }}_j$. (b) Effective coupling in the $z$ direction ${\tilde{\Delta }}_j$. The calculations correspond to the parameters of Fig. 5. The results of $B=0$ are not shown since they are located in significantly different ranges of the $y$ axis, i.e., ${\tilde{\tau }}_j\approx 0.93$ and ${\tilde{\Delta }}_j\approx 1.77$ in the bulk.

Figure 10

Scaling of the difference of effective and real exchange coupling $\left(\square \right)$ and anisotropy $(\circ )$ with the system size, for the central sites and $\Delta =1.5$. (a) For integrable regime, $B=0$. The solid lines are guides to the eye. (b) For nonintegrable regime with $B=0.15$. The solid lines are the corresponding linear fits: $(\tilde{\Delta }-\Delta )/\Delta =1.16\left(8\right)N^{-1}-6\left(13\right)\times {10}^{-4}$, and $(\tau -\tilde{\tau })/\tau =0.49\left(4\right)N^{-1}-3\left(6\right)\times {10}^{-4}$. The calculations correspond to the parameters of Fig. 5.

Figure 11

Scheme of the magnetothermal effects induced in boundary-driven systems with a finite average magnetization. (a) Peltier effect, in which an energy current is induced by a magnetic imbalance $\left({\mu }_{\text{targ}}^R>{\mu }_{\text{targ}}^L\right)$ with no temperature imbalance $\left(T_{\text{targ}}^L=T_{\text{targ}}^R=T\right)$. (b) Seebeck effect, in which a spin current is induced by a temperature imbalance $\left(T_{\text{targ}}^R>T_{\text{targ}}^L\right)$ with no magnetization imbalance $\left({\mu }_{\text{targ}}^L={\mu }_{\text{targ}}^R=\mu \right)$. The arrows indicate spin and energy currents. The solid lines represent magnetization and energy profiles.

Figure 12

Transport properties induced by a magnetization imbalance across an $XXZ$ chain, with finite magnetization. The results correspond to ${\mu }_{\text{targ}}^L/T_{\text{targ}}^L=0.5,{\mu }_{\text{targ}}^R/T_{\text{targ}}^R=0.7$, and $T_{\text{targ}}^{L,R}=100$. The left panels show the spin (a) and energy currents (b) as a function of $N$. The right panels correspond to the magnetization (c) and energy profiles (d) for $\Delta =1.5$ and $N=100$.

Figure 13

Trace distance between the two-site reduced density operator ${\tilde{\rho }}_2(\frac{N}{2},\frac{N}{2}+1)$ and two-site thermal states ${\rho }_2$ with trial temperatures $T_{\frac{N}{2},\frac{N}{2}+1}$ and chemical potentials ${\mu }_{\frac{N}{2}}$ and ${\mu }_{\frac{N}{2}+1}$. The results correspond to a system with $B=0.4$ and the parameters of Fig. 5. (a) Sweep over temperature for three iterations $k=1,2,3$ of the self-consistent process. Iteration $k=1$ corresponds to the mean-field chemical potentials of Eq. (16). (b) Sweep over chemical potential of site $\frac{N}{2}+1$ for various fixed potentials of site $\frac{N}{2}$, for iteration $k=1$. No more iterations are depicted here, since they give the same trace distances than those of $k=1$.