Generalized virial theorem for massless electrons in graphene and other Dirac materials

The virial theorem for a system of interacting electrons in a crystal, which is described within the framework of the tight-binding model, is derived. We show that, in the particular case of interacting massless electrons in graphene and other Dirac materials, the conventional virial theorem is violated. Starting from the tight-binding model, we derive the generalized virial theorem for Dirac electron systems, which contains an additional term associated with a momentum cutoff at the bottom of the energy band. Additionally, we derive the generalized virial theorem within the Dirac model using the minimization of the variational energy. The obtained theorem is illustrated by many-body calculations of the ground-state energy of an electron gas in graphene carried out in Hartree-Fock and self-consistent random-phase approximations. Experimental verification of the theorem in the case of graphene is discussed.

Figure 1

Schematic tight-binding dispersion of filled single-electron states in electron-doped graphene (solid line) and its approximation in the Dirac model (dotted line). The electron states in the Dirac model are filled in both valleys $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ down to the cutoff momentum $p_{\mathrm{c}}$.

Figure 2

The diagrams presenting the interaction-induced correction $\delta \mathrm{\Omega }$ to the thermodynamic potential and its Brueckner-Goldstone part $\delta {\mathrm{\Omega }}_{\mathrm{BG}}$, which contains no “anomalous” diagrams, in the Hartree-Fock approximation. Thin and thick straight lines are bare and dressed electron Green functions, and the wiggly line is the Coulomb interaction.

Figure 3

The same as Fig. 2, but in the self-consistent random-phase approximation. The thick wiggly line is the screened Coulomb interaction.

Figure 4

(a) Ground-state energy of electron gas in graphene per unit area $E/S$ (thick lines) and its derivative $(p_{\mathrm{c}}/S)(\partial E/\partial p_{\mathrm{c}})$ (thin lines), calculated as functions of carrier density $n$ at different values of $\varepsilon$. (b) The normalized derivative $K=(p_{\mathrm{c}}/E_0)(\partial E/\partial p_{\mathrm{c}})$, as defined by (33). All curves present calculations in the Hartree-Fock (dashed lines) and self-consistent random-phase (solid line) approximations. Inset: $K$ at $n\to 0$ as a function of the dielectric constant $\varepsilon$.