Heat current magnification in classical and quantum spin networks

We investigate heat current magnification (CM) due to asymmetry in the number of spins in two-branched classical as well as quantum spin systems that are kept between two heat baths at different temperatures. We study the classical Ising-like spin models using Q2R and Creutz cellular automaton dynamics. We show that just the difference in the number of spins is not enough and some other source of asymmetry like unequal spin-spin interaction strengths in the upper and lower branches is required for heat CM. We also provide a suitable physical motivation for CM along with ways to control and manipulate it. We then extend this study to a quantum system with modified Heisenberg XXZ interaction and preserved magnetization. Interestingly, in this case, just the asymmetry in the number of spins in the branches is enough to achieve heat CM. We observe that the onset of CM is accompanied by a dip in the total heat current flowing through the system. We then discuss how the observed CM characteristics can be attributed to the intersection of nondegenerate energy levels, population inversion, and atypical magnetization trends as a function of the asymmetry parameter in the Heisenberg XXZ Hamiltonian. Finally we use the concept of ergotropy to support our findings.

Figure 1

Possible ways for current to flow in a two-branch system (motivated from Ref. [31]).

Figure 2

Schematic representation of the model system.

Figure 3

Illustration of the model system consisting of three spins with $N_U=1$ and $N_D=0$.

Figure 4

(a) Comparison of total current ($I_L$) between analytical, numerical, and simulation results for the three-spin system with $N_U=1, N_D=0$, (b) variation of heat currents ($I$) with $T_L$ for $N_U=1, N_D=0$, (c) variation of $I_L$ with $T_L$ for an eight-spin system but with different branch spin distributions, (d) variation of heat currents ($I$) with $N_U$ for $N_D=4,T_L=2,T_R=0.1$. For all cases unless otherwise specified, we use $T_R=1, J=1$.

Figure 5

Variation of heat currents ($I$) with (a) $T_L$, with (b) $T_L$ for $N_D=2,N_U=3$, with (c) $T_R$, with (d) $J_2$, with (e) $N_U$ for $N_D=4$, and (f) variation of heat currents ($I$) and $\langle {\sigma }_2{\sigma }_3\rangle$ with $T_L$. For all cases, unless otherwise specified we use $J_1=2,T_L=2,T_R=0.1,J_2=1.9,N_D=3,N_U=2$. In all figures where multiple regions of current circulation directions are present, we use green, gray, and yellow background colors to indicate clockwise, parallel, and anticlockwise circulating currents, respectively. For all other cases, we use a white background.

Figure 6

Example of a process which results in the transfer of energy $2(J_1-J_2)$ from left bath to right bath.

Figure 7

Variation of Heat Currents ($I$) with (a) $T_L$, with (b) $J_2$ for the system shown in Fig. 6 with $N_D=1,N_U=1$, For all the cases, unless otherwise specified we use $J_1=2,T_L=2,T_R=0.1,J_2=1.9.$

Figure 8

Variation of Heat Currents ($I$) for the CCA model with (a) $T_L$ for $J_1=J_2=1$, with (b) $T_L$, with (c) $T_R$ and (d) with $\mu$. For all the cases unless otherwise specified $J_1=2,T_L=2,T_R=0.1,J_2=1,N_D=3,N_U=2$, and $\mu =4$.

Figure 9

Illustration of the quantum spin system consisting of five spins with $N_U=2$ and $N_D=1$.

Figure 10

Variation of heat currents ($I$) with the asymmetry parameter $\mathrm{\Delta }$ for (a) setup 5($3\uparrow 2\downarrow$), (b) setup 5($2\uparrow 3\downarrow$), (c) a six-spin system with setup 6($4\uparrow 2\downarrow$) with $N_U=3,N_D=1$, and (d) variation of heat currents ($I$), correlation, and concurrence between spins 2 and 3 with $T_L$ for $\mathrm{\Delta }=2.1$. For all cases unless otherwise specified, we work with the setup $5(3\uparrow 2\downarrow )$ and $T_L=2, T_R=1, J=1$. In (d), we scale the correlation and concurrence by a factor of 0.2 for the visual convenience.

Figure 11

(a) Variation of energy levels with $\mathrm{\Delta }$, (b) variation of occupation probability ($P$) with $\mathrm{\Delta }$, (c) variation of individual spin magnetization ($M_k$) with $\mathrm{\Delta }$, (d) variation of ergotropy ($\epsilon$) with $\mathrm{\Delta }$, variation of (e) energy levels and (f) currents for a larger range of asymmetry parameter $\mathrm{\Delta }$. Note that only the magnitude of currents are considered for the log plot in (f) but the background colors for parallel, clockwise, or anticlockwise circulating currents are kept. For all cases unless otherwise specified, we work with the setup $5(3\uparrow 2\downarrow )$ and $T_L=2, T_R=1, J=1$.