Quantum thermodynamic cycle with quantum phase transition

With the Lipkin-Meshkov-Glick (LMG) model as an illustration, we construct a thermodynamic cycle composed of two isothermal processes and two isomagnetic field processes, and we study the thermodynamic performance of this cycle accompanied by the quantum phase transition (QPT). We find that for a finite particle system working below the critical temperature, the efficiency of the cycle is capable of approaching the Carnot limit when the external magnetic field ${\lambda }_1$ corresponding to one of the isomagnetic processes reaches the cross point of the ground states' energy level, which can become the critical point of the QPT in the large-$N$ limit. Our analysis proves that the system's energy level crossings at low-temperature limits can lead to a significant improvement in the efficiency of the quantum heat engine. In the case of the thermodynamics limit $(N\to \infty )$, the analytical partition function is obtained to study the efficiency of the cycle at high- and low-temperature limits. At low temperatures, when the magnetic fields of the isothermal processes are located on both sides of the critical point of the QPT, the cycle reaches maximum efficiency, and the Carnot efficiency can be achieved. This observation demonstrates that the QPT of the LMG model below critical temperature is beneficial to the thermodynamic cycle's operation.

Figure 1

Entropy-magnetic diagram ($S-\lambda$) of the thermodynamic cycle based on the LMG model. Here, ${\lambda }_1$ and ${\lambda }_2$ are the magnetic fields in the two isomagnetic field processes from $B$ to $C$ and from $D$ to $A$. $T_H$ and $T_C$ are the temperatures of the two heat baths.

Figure 2

Diagrams of the efficiency $\eta$ as a function of ${\lambda }_1$ for the system with $N=20$ in (a) and $N=2$ in (b). The blue solid line and the black dotted line correspond to the high-temperature heat bath at $T_H=0.8$ and 80, respectively. Here we guarantee that the Carnot efficiency of each curve equals 0.5, as indicated by the red solid line.

Figure 3

(a) Diagram of the efficiency $\eta$ as a function of ${\lambda }_1$ for the system with $N=2$. $T_H=0.6$ is the temperature of the hot heat bath and $T_C=0.3$ is the temperature of the cold heat bath. The black solid line represents the efficiency of the heat engine, while the red solid line represents the corresponding Carnot efficiency. (b) Diagram of the entropy $S$ as a function of ${\lambda }_1$, where the red solid line, the blue solid line, and the black circle line represent the change in the entropy when the temperatures ($T$) of the system are at 0.6, 0.3, and 0, respectively.

Figure 4

Diagrams of the efficiency $\eta$ as a function of ${\lambda }_1$, where the black solid lines represent the efficiency of the heat engine while the red solid lines represent the corresponding Carnot efficiency. (a) For the system with $N=4$, the parameters $T_H=0.3,T_C=0.15,{\lambda }_1\in [0,4]$, and ${\lambda }_2=4$. (b) For the system with $N=6$, the parameters $T_H=0.2,T_C=0.1,{\lambda }_1\in [0,4]$, and ${\lambda }_2=4$. (c) For the system with $N=8$, the parameters $T_H=0.1,T_C=0.06$, ${\lambda }_1\in [0,2]$, and ${\lambda }_2=2$. (d) For the system with $N=10$, the parameters $T_H=0.1,T_C=0.06,{\lambda }_1\in [0,2]$, and ${\lambda }_2=2$.

Figure 5

Diagrams of ${\eta }^{\prime }=\partial \eta /\left(\partial {\lambda }_1\right)$ as a function of ${\lambda }_1$, where ${\lambda }_1\in [0,2]$ and ${\lambda }_2=2$. (a) For the system with $N=6$, the parameters $T_H=0.3$ and $T_C=0.2$. (b) For the system with $N=10$, the parameters $T_H=0.2$ and $T_C=0.1$. (c) For the system with $N=20$, the parameters $T_H=0.12$ and $T_C=0.06$. (d) For the system with $N=30$, the parameters $T_H=0.2$ and $T_C=0.1$.

Figure 6

Diagrams of $\eta$ as a function of ${\lambda }_1$. Here, the black circle line denotes the numerical results, the blue solid line denotes the analytical results of Eq. (30), and the red solid line denotes ${\eta }_0={\eta }_C/\left(1+{\eta }_C\right)$, where ${\eta }_C$ is the corresponding Carnot efficiency. The temperatures of the two heat baths are $T_H=800$ and $T_C=500$, respectively.

Figure 7

Diagrams of the efficiency $\eta$ as a function of ${\lambda }_1$, where the black crosses are the numerical results and the red solid lines represent the corresponding Carnot efficiency. (a) For the system with $N=30$, the parameters $T_H=0.5,T_C=0.3,{\lambda }_1\in [0,0.8]$, and ${\lambda }_2=0.8$. (b) For the system with $N=50$, the parameters $T_H=0.2,T_C=0.1$, ${\lambda }_1\in [0,0.2]$, and ${\lambda }_2=0.2$.

Figure 8

Diagram of $\eta$ as a function of ${\lambda }_1$ for the system with $N=20$, where ${\lambda }_1\in [0,4]$ and ${\lambda }_2=4$. $T_H=0.3$ is the temperature of the hot heat bath. $T_C=0.2$ is the temperature of the cold heat bath. Here, the blue square points are the numerical results and the red solid line is the corresponding Carnot efficiency.