Energy rectification in quantum graded spin chains: Analysis of the $\mathit{XXZ}$ model

In this work, with focus on the energy-transport properties in quantum, low-dimensional, graded materials, we address the investigation of the energy (and spin) current in $XXZ$ open chains with graded inner structures and driven out of equilibrium by magnetization pumping applied at the ends. We study several types of graded structures in different situations in order to show a ubiquitous occurrence of energy rectification, even for the system under a homogeneous magnetic field. Due to technical difficulties, we carry out the computation for small chains, but we present arguments that indicate the extension of some results to larger systems. Recalling the generic existence of energy rectification in classical, graded materials, which are described by anharmonic chains of oscillators, and recalling also the anharmonicity of these $XXZ$ models, which involve quartic terms in more transparent representation in terms of fermionic creation and annihilation operators, we may say that our results extend the ubiquity of energy rectification occurrence in classical graded materials to the case of quantum systems.

Figure 1

Energy rectification versus driving strength, for $N=3$ and graded interaction between $S_i^z$ and $S_{i+1}^z$. Here, $B=0.1$, $\alpha =1$, $\delta =1/4$.

Figure 2

Energy rectification versus parameter of asymmetry, for $N=3$ and graded interaction between $S_i^z$ and $S_{i+1}^z$. Here, $\alpha =1$, $\mathrm{\Delta }=0$, $f=0.01$.

Figure 3

Energy rectification versus system size. Here, $\alpha =1$, $B=0.1$, $f=1$; and (a) $\mathrm{\Delta }=1/2$, (b) $\mathrm{\Delta }=3/2$.

Figure 4

Energy rectification ${\mathcal{R}}_E$ versus parameter of asymmetry in several cases with $N=3$. (a) Graded interaction between $S_i^x$ and $S_{i+1}^x$, $S_i^y$ and $S_{i+1}^y:{\alpha }_{1,2}=1-\delta$, ${\alpha }_{2,3}=1+\delta$; and homogeneous interaction between $S_i^z$ and $S_{i+1}^z:{\mathrm{\Delta }}_{i,i+1}=\mathrm{\Delta }=1$. (b) Graded $XXX$ model: ${\alpha }_{1,2}={\mathrm{\Delta }}_{1,2}=1-\delta$, ${\alpha }_{2,3}={\mathrm{\Delta }}_{2,3}=1+\delta$. (c) Completely graded $XXZ$ model: ${\alpha }_{1,2}=1-\delta$, ${\alpha }_{2,3}=1+\delta$, ${\mathrm{\Delta }}_{1,2}=1-\delta$, ${\mathrm{\Delta }}_{2,3}=1+\delta$. In all these cases, (a)–(c), we take $B=0.1$, $f=\pm 0.1$. (d) Energy current versus parameter of asymmetry in the model with graded interaction between $S_i^z$ and $S_{i+1}^z$, but twisted $XY$ boundary gradients: $\langle S_0^y\rangle =\kappa =\langle S_{N+1}^x\rangle$. Here, $\mathrm{\Delta }=1$, $B=0.1$, $\alpha =1$, and $\kappa =1/4$ for the upper (orange) curve; the lower (blue) curve follows for the system with inverted reservoirs, i.e., $\langle S_0^x\rangle =\kappa =\langle S_{N+1}^y\rangle$.

Figure 5

NDR: energy current versus driving strength, for $N=3$ and graded interaction between $S_i^z$ and $S_{i+1}^z$. In (a), $\alpha =1$, $\mathrm{\Delta }=1$, $\delta =0.7$, $B=0.1$. In (b), $\alpha =1$, $\mathrm{\Delta }=2$, $\delta =0.8$, $B=0.1$.