Finite-volume effects on the chiral phase transition of thermal ${\mathrm{QED}}_3$

The influence of finite-volume effects on dynamical chiral symmetry breaking of thermal ${\mathrm{QED}}_3$ is investigated. We show that the chiral symmetry gets restored at all temperatures when the size of system is less than some certain value $L_c^{\prime }=189$; the critical temperature of chiral phase transition generally increases with the size of the system and the critical temperature when the system volume tends to infinity is slightly less than that in the case of infinite volume.

Figure 1

The dependence of boson polarization value on the summation upper limit at $L=500$ (top) and $L=2000$ (bottom) at $T=0.01$. As a general discussion, we only give the points when $m_1=m_2$ of $\mathrm{\Pi }(Q_v^2)$, and the solid line $L_{\inf }$ illustrates the behavior of Eq. (7).

Figure 2

The boson polarization value with a range of volume at the summation upper limit $M=250$ at $T=0.01$.

Figure 3

The dependence of fermion self-energy on the summation upper limit $L=300$ (top) and $L=1000$ (bottom) at $T=0.01$ (where ${\omega }_1={\omega }_2$ is adopted).

Figure 4

The typical dependence of fermion self-energy on the volume size at $T=0.01$ (top) and on temperature at $L=300$ (bottom), where the solid line $L_{\inf }$ gives the behavior of B at infinite volume.

Figure 5

The relation between order parameter and temperature with several volume size (left) and with different temperature (right).

Figure 6

The relation of inverse volume size $1/L$ and the corresponding critical temperature $T$.