Periodic ground state for the charged massive Schwinger model

It is shown that the charged massive Schwinger model supports a periodic vacuum structure for arbitrary charge density, similar to the common crystalline layout known in solid state physics. The dynamical origin of the inhomogeneity is identified in the framework of the bosonized model and in terms of the original fermionic variables.

Figure 1

The scalar field ${\varphi }_z$ for $b=0.025$, $m=e=1$. The slope $b_s=-0.001078$ fitted in the central region should be compared to $b_s=b{e}^2/({\kappa }^2\pi )=-0.001074$ from Eq. (19).

Figure 2

Dependence of the point $b_L$ on the size $L$ of the system for $m=e=1$.

Figure 3

The periodic part ${\tilde{\varphi }}_z$ of the scalar ground-state field configuration for $b=0.5$, $m=e=1$.

Figure 4

Amplitude $A$ of the periodic part ${\tilde{\varphi }}_z$ vs the parameter $b$ for various values of the electron mass $m$ and $e=1$. The solid line represents our perturbative estimate for the amplitude given by Eq. (24). The numerical values justify the perturbative estimate for small values of $A$.

Figure 5

Wavelength $\ell$ of the periodic part ${\tilde{\varphi }}_z$ vs the parameter $b$ for various values of the electron mass $m$ and $e=1$. The solid line refers to the curve $\ell =\sqrt{\pi }/b\equiv 1/{\rho }_{\mathrm{e}\mathrm{x}\mathrm{t}}$.

Figure 6

(Top) Energy densities vs $e\mu /m$ and (middle) the one-loop and (bottom) two-loop contributions to those. The solid and dashed lines refer to the periodic and the homogeneous phases, respectively.

Figure 7

Density dependences of (a) the energy density, (b) the amplitude $a$ of the photon condensate, and (c) the product of the filling factor $f$ and the wave number $Q$ of the photon condensate for $m=2$. The dotted lines indicate the “critical charge density” ${\rho }_c$; solid and dashed lines correspond to the periodic and normal phases, respectively.

Figure 8

Dispersion relations in the first Brillouin zone with (left) one occupied band and (right) two occupied bands sunk into the Dirac sea for $ea=0.5$ and $ea=0.2$, respectively, $m=2$.

Figure 9

Allowed bands (shaded regions) for (a) undercritical $ea=0.9$ and (b) overcritical $ea=2$ static periodic external fields with wave number $Q$ for $m=e=1$.

Figure 10

Momentum dependence of the difference of the single-particle energies in the expression (c16) of the one-loop energy for $m=2$, $Q=1.1$, and $a=2$.