Figure 1

(a) The excitonic condensation due to an electron-hole pairing is studied in the system of two spatially separated graphene monolayers with an excess of electrons on layer 1 and a lack of electrons on layer 2. (b) The schematic phase diagram of the excitonic condensation in the system at different values of a Zeeman splitting and different values of the asymmetry between Fermi energies in layer 1 and 2, ${\epsilon }_Z={\mu }_B\left|\mathbf{h}\right|$ is the Zeeman energy in an in-plane magnetic field $\mathbf{h}.$Reuse & Permissions

Figure 2

A typical transition which is described by ${\widehat{H}}_{12}$ in Eq. (2). Indices $l=1\text{or}2,\zeta ,\mathbf{p},\text{and}s$ denote a layer, a pair of the spin projection and valley, a momentum of an electron, and a conduction $s=+$ or valence $s=-$ band.Reuse & Permissions

Figure 3

The excitonic electron-hole bound state in the two-layer graphene. The left-hand side of the figure shows the electron's spectrum in graphene layers 1 and 2. An electron on the Fermi surface in the layer 1 is shown as a fulfilled circle. Absence of an electron on the Fermi surface in layer 2 is shown as an empty circle. Closed line around both circles represents an excitonic pairing, which is developed due to a Coulomb interaction (shown as a wavy line). The right-hand side of the figure shows the coincided Fermi circles in both layers at Fermi momentum $p_F.$Reuse & Permissions

Figure 4

In the $B$ phase within the excitonic paired state, the electron on the Fermi surface in the layer 1 (grey circle) is characterized by the index $\zeta$ in the spin$\otimes$valley basis $\Phi ,$ and the absent electron (white circle) in the layer 2 is characterized by the same index $\zeta$ but in the spin$\otimes$valley basis $\mathrm{V}\Phi ,$ which is transformed by the matrix of the order parameter $\mathrm{V}$.Reuse & Permissions

Figure 5

Excitonic correlations in the two-layer graphene system with an in-plane magnetic field $h$ in the case of equal charge densities in layers $n_{1e}=n_{2h}.$ Because of a Zeeman splitting $2{\epsilon }_Z$ there are two Fermi circles with radiuses $p_F\pm {\epsilon }_Z/v.$ The absent electron with a particular spin projection is considered as a quasiparticle (hole) with an opposite spin projection.Reuse & Permissions

Figure 6

Excitonic correlations in the $A_1$ phase. Starting from a zero magnetic field the asymmetry between charge carrier densities $n_{1e}>n_{2h}$ is created. In terms of Fermi energies, it means $E_F^{\left(1\right)}>\left|E_F^{\left(2\right)}\right|.$ A magnitude of an in-plane magnetic field is chosen such that a Zeeman energy ${\epsilon }_Z$ satisfies the condition: $E_F^{\left(1\right)}-{\epsilon }_Z=\left|E_F^{\left(2\right)}\right|+{\epsilon }_Z,$ where $(E_F^{\left(1\right)}-{\epsilon }_Z)/v$ is the radius of the Fermi circle for electrons with spin up in layer 1, and $\left(\right|E_F^{\left(2\right)}|+{\epsilon }_Z)/v$ is a radius of the Fermi circle for electrons with spin up in layer 2. Both of these Fermi circles are situated at the same Fermi momentum $p_F$. Therefore two out of four Fermi circles coincide, leading to excitonic correlations between only half of electron's species. Using the expression for the Fermi energies $E_F^{\left(1\right)/\left(2\right)}=\pm v{p}_F+v\delta p_F,$ $p_F>\delta p_F,$ where $\delta p_F=(\sqrt{n_{1e}}-\sqrt{n_{2h}})\sqrt{\pi }/2,$ the condition on the Zeeman energy is the following: ${\epsilon }_Z=v\delta p_F.$Reuse & Permissions