Symmetry breaking in zero-field two-dimensional electron bilayers

We theoretically consider bilayers of two-dimensional (2D) electron gases as in semiconductor quantum wells, and investigate possible spontaneous symmetry breaking transitions at low carrier densities driven by interlayer Coulomb interactions. We use a self-consistent technique implementing mean-field truncations of the interacting four-fermion terms, and find a U(1) layer symmetry breaking transition at low carrier densities where the individual layer identities are lost leading to an effective pseudospin XY ferromagnet in the 2D plane. Our results validate earlier theoretical studies using simpler restricted Hartree-Fock techniques, and establish the pseudospin XY ferromagnet as a possible low-density symmetry broken phase of 2D bilayers.

Figure 1

We include cartoon depictions of the trial states $S_0, S_1, S_2$, and $S_{\chi }$. (a) The $S_0$ phase is defined as having no spin polarizations in each layer. (b) The $S_1$ phase occurs when both layers are spin polarized. (c) The $S_2$ phase occurs when one layer is spin polarized and the other layer is not. In $S_0, S_1$, and $S_2$, there is no interlayer coherence and the only variational degree of freedom is how many electrons are in each layer. When the two layers have equal density of positive charges, $R={\rho }_B/{\rho }_T=1$, the number of electrons in each layer $S_0$ and $S_1$ are the same, but when $R<1$, the two layers can have different numbers of electrons and hence different Fermi surface sizes. In (d) we depict $S_{\xi }$ where the red lines indicate the interlayer coherence. Here, both layers have the same Fermi surface and the variational degree of freedom is the strength of the coherence, which is directly related to how many electrons are in each layer. When the coherence is zero, all the electrons are in one layer or the other, which we still define as $S_{\xi }$.

Figure 2

In (a)–(c), we plot the energy $\langle \tilde{H}\rangle$ as a function of $r_T$, the dimensionless density parameter, for the trial states $S_0, S_1, S_2$, and $S_{\chi }$ as well as our self-consistent solution to the Hartree-Fock equations (SCHF) with the dimensionless distance between layers $\tilde{d}=1$ ($R={\rho }_B/{\rho }_T$ is the density ratio between the two layers). We accurately reproduce the ground-state energy predicted from the four trial states. In (d)–(f), we plot the interlayer coherence parameter ${\mathrm{\Theta }}_{(\uparrow T),(\uparrow ,B)}^0=\langle c_{0,(\uparrow ,T)}^{†}{c}_{0,(\uparrow ,B)}\rangle$ obtained from our SCHF solution as a function of $r_T$. In the balanced case, $R=1.0$, the interlayer coherence is always present above $r_T\approx 5$, but when $R<1.0$, there is a maximum $r_T$ for which there is interlayer coherence, beyond which all the electrons have been transferred to the top layer. The value of $R\in \{1.0,0.9,0.8\}$ is the same for the figures in the same column. The numerical parameters ${\tilde{k}}_{\text{max}}=1.5$ and $N_k={\tilde{k}}_{\text{max}}/\delta \tilde{k}=4000$.

Figure 3

We plot ${\mathrm{\Theta }}_{(\uparrow T,\uparrow T)}^{\tilde{k}}/{\mathrm{\Theta }}_{(\uparrow T,\uparrow T)}^0$ as a function of $\tilde{k}$ for $\tilde{d}=1, r_T=5.2$, and $R=0.9$. Although there is minimal $k$ dependence, it does not decrease as $N_k={\tilde{k}}_{\text{max}}/\delta \tilde{k}$ increases implying that the ground state is slightly different than expected from the state $S_{\xi }$. However, the energy is only slightly lower: $E_{\text{SCHF}}(N_k=8000)-E_{S_{\xi }}=-6\times {10}^{-4}$.