Zigzag Phase Transition in Quantum Wires

We study the quantum phase transition of interacting electrons in quantum wires from a one-dimensional (1D) linear configuration to a quasi-1D zigzag arrangement using quantum Monte Carlo methods. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal, then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase, and, finally, the electrons form a disordered liquidlike phase. We show that the linear to zigzag phase transition is not destroyed by the strong quantum fluctuations present in narrow wires; it has characteristics which are qualitatively different from the classical transition.

Figure 1

(a) Electrons (red spheres) confined by a harmonic potential form a linear Wigner crystal. (b) As electron density increases, symmetry about the wire axis breaks and the electrons form a zigzag structure.

Figure 2

Pair density, $⟨\widehat{\rho }(r,\theta )\widehat{\rho }(r_{\star },0)⟩/⟨\widehat{\rho }(r_{\star },0)⟩$, for electrons at progressively higher densities in a wire with $\omega =0.1$. The red stars mark $r_{\star }$. ($r$ is plotted relative to $\overline{r}$ in units of the effective Bohr radius $a_0^{*}$, and $\theta$ is plotted in units of the interparticle spacing $2\pi /N$.) (a) $r_s=4.0$, $N=30$. At low densities, the electrons form a linear Wigner crystal. (b) $r_s=3.6$, $N=30$. As the density increases, a zigzag structure forms. (c) $r_s=3.0$, $N=30$. The amplitude of the zigzag increases. (d) $r_s=2.0$, $N=60$. At higher densities, the zigzag structure is destroyed. The color scale shows the pair density relative to the maximum value for each plot. Only a portion of the full periodic system is shown.

Figure 3

Pair densities for electrons in a wire with $\omega =0.6$. The red stars mark $r_{\star }$. ($r$ is plotted relative to $\overline{r}$, and $\theta$ is plotted in units of the interparticle spacing $2\pi /N$.) (a) $r_s=1.5$, $N=30$. At low densities, electrons in a quantum wire form a linear Wigner crystal. (b) $r_s=1.3$, $N=30$. As the density increases, a zigzag structure forms, though quantum fluctuations smear our correlations in the pair density. (c) $r_s=0.5$, $N=60$. At higher densities, the zigzag structure vanishes. The color scale shows the pair density relative to the maximum value for each plot. Only a portion of the full periodic system is shown.

Figure 4

The zigzag correlation function, normalized by the average wire width; $C_{\mathrm{zz}}(|i-j|)/⟨y^2⟩=⟨(-1^i)y_i(-1^j)y_j⟩/⟨y^2⟩$, plotted for various values of $r_s$ at (a) $\omega =0.1$ and (b) $\omega =0.6$ for $N=30$ electrons. ($y\equiv r-\overline{r}$ is measured in units of the effective Bohr radius $a_0^{*}$.) In the linear phase (dashed lines), $C_{\mathrm{zz}}$ decays to 0; in the zigzag phase (solid lines), $C_{\mathrm{zz}}$ saturates to a finite value, indicating long-range zigzag order. In the liquid phase (dotted lines) $C_{\mathrm{zz}}$ again decays. [Note that since our system is periodic, $C_{\mathrm{zz}}(k)=C_{\mathrm{zz}}(N-k)$, thus $C_{\mathrm{zz}}(k)$ must be flat at $k=N/2$.]

Figure 5

The “Zigzag Amplitude” order parameter $M_{\mathrm{zz}}$, as a function of $r_s$ at $\omega =0.1$ (blue circles) and $\omega =0.6$ (red squares). As $r_s$ is decreased beyond the critical value and the system enters the zigzag regime, $M_{\mathrm{zz}}$ increases sharply; at lower $r_s$, $M_{\mathrm{zz}}$ decreases gradually as the system enters the liquid regime. The behavior deviates significantly from the $\sqrt{n-n_{\mathrm{critical}}}$ behavior for classical electrons (black dashed line). Dotted lines are a guide to the eye. (Lengths are scaled by $r_0$. We note that there are points for both values of $\omega$ where $M_{\mathrm{zz}}$ seems to be lower than one would expect by drawing a curve through the other points; we believe that this is a systematic error from the VMC optimization step of our QMC calculation.)