Mixed-state fidelity and quantum criticality at finite temperature

We extend to finite temperature the fidelity approach to quantum phase transitions (QPTs). This is done by resorting to the notion of mixed-state fidelity that allows one to compare two density matrices corresponding to two different thermal states. By exploiting the same concept we also propose a finite-temperature generalization of the Loschmidt echo. Explicit analytical expressions of these quantities are given for a class of quasifree fermionic Hamiltonians. A numerical analysis is performed as well showing that the associated QPTs show their signatures in a finite range of temperatures.

Figure 1

(Color online) Mixed-state fidelity of finite-temperature thermal state of the $XY$ model. The fidelity is a function of $\lambda$ and $\gamma$. Here, we choose the spin number $N=200$ and the parameter perturbation $\delta \lambda =\delta \gamma ={10}^{-2}$. The temperatures of system in the above four figures are chosen to be $\beta =1{\mathrm{J}}^{-1}$, $\beta =10{\mathrm{J}}^{-1}$, $\beta =20{\mathrm{J}}^{-1}$, and $\beta =100{\mathrm{J}}^{-1}$, respectively. At low temperatures, e.g., $\beta =100{\mathrm{J}}^{-1}$, the mixed-state fidelity clearly shows the signature of QPT which occurs at absolute zero temperature. When the temperature increases, the decay of fidelity becomes less sharp and finally the signature of QPT disappears due to the thermal excitation.

Figure 2

(Color online) A cross section of Fig. 1 with $\gamma =1$, i.e., mixed state fidelity of finite-temperature thermal state of transverse Ising model. Here, the same as that in Fig. 1, the spin number is chosen to be $N=200$ and the parameter perturbation $\delta \lambda =\delta \gamma ={10}^{-2}$. Four curves represent four different temperatures: $\beta =1{\mathrm{J}}^{-1}$, $\beta =10{\mathrm{J}}^{-1}$, $\beta =20{\mathrm{J}}^{-1}$, and $\beta =100{\mathrm{J}}^{-1}$, respectively. The decay of fidelity, though becomes less sharp with the increase of the temperature, well indicates the critical point of the transverse Ising model.

Figure 3

(Color online) Mixed-state Loschmidt echo finite-temperature thermal state of the $XY$ model at an instant $t=10\mathrm{s}$. Here, we choose the spin number $N=200$ and the parameter perturbation $\delta \lambda =\delta \gamma ={10}^{-2}$. The temperatures of system in the above four figures are chosen to be $\beta =1{\mathrm{J}}^{-1}$, $\beta =10{\mathrm{J}}^{-1}$, $\beta =20{\mathrm{J}}^{-1}$, and $\beta =100{\mathrm{J}}^{-1}$, respectively. Similar to the mixed-state fidelity approach, the critical region is well indicated by Loschmidt echo at an instant when the system is at a low temperature. However, the signature of QPT disappears when the temperature increases to very high value.

Figure 4

(Color online) A cross section of Fig. 3 with $\gamma =1$, i.e., mixed-state Loschmidt echo (at an instant) of finite-temperature thermal state of transverse Ising model. Here, the same as that in Fig. 3, the spin number is chosen to be $N=200$ and the parameter perturbation $\delta \lambda =\delta \gamma ={10}^{-2}$. Four curves represent four different temperatures: $\beta =1{\mathrm{J}}^{-1}$, $\beta =10{\mathrm{J}}^{-1}$, $\beta =20{\mathrm{J}}^{-1}$, and $\beta =100{\mathrm{J}}^{-1}$, respectively. The critical point is well indicated by Loschmidt echo when the temperature is not too high.