Strongly interacting phases of metallic wires in strong magnetic field

We investigate theoretically an interacting metallic wire with a strong magnetic field directed along its length and show that it is a highly tunable one-dimensional system. By considering a suitable change in spatial geometry, we build an analogy between the problem in the zeroth Landau level with Landau level degeneracy $N$ to one-dimensional fermions with an $N$-component pseudospin degree of freedom and $SU\left(2\right)$-symmetric interactions. This analogy allows us to establish the phase diagram as a function of the interactions for small $N$ (and make conjectures for large $N$) using renormalization group and bosonization techniques. We find pseudospin-charge separation with a gapless $U\left(1\right)$ charge sector and several possible strong-coupling phases in the pseudospin sector. For odd $N$, we find a fluctuating pseudospin-singlet charge density wave phase and a fluctuating pseudospin-singlet superconducting phase which are topologically distinct. For even $N>2$, similar phases exist, although they are not topologically distinct, and an additional novel pseudospin-gapless phase appears. We discuss experimental conditions for observing our proposals.

Figure 1

(a) Energy spectrum of noninteracting electrons in a wire ($\mathbb{R}\times D^2$) geometry and strong magnetic field. (b) Energy spectrum of noninteracting electrons in an $\mathbb{R}\times S^2$ geometry and strong magnetic field. The dark curve is $(n_{\varphi }+1)$-fold degenerate and the light one is $(n_{\varphi }+3)$-fold degenerate. In both cases, the dark levels are in the $n=0$ Landau level and the light ones are in $n=1$.

Figure 2

Nonchiral interactions which are marginal at the free fermion fixed point. The labels $S_0$ and $m,m^{\prime },n,n^{\prime }$ indicate the ${\mathbf{L}}^2$ and the $L_3$ eigenvalue, respectively. The interaction is decomposed according to the angular momentum transfer $(L^2,L_3)=(S,p)$ from the left mover to the right mover.

Figure 3

RG flow for $N=2$.

Figure 4

RG flows for $N=3$. (a) The full RG flow; the only finite-coupling fixed point is at the origin. (b) Corresponding phase diagram. (c) RG for ${\tilde{g}}_2=g_2/g_1$ with $g_1>0$. Points B and C are stable fixed “rays” corresponding to 45-degree lines in (a).

Figure 5

(a) Two cuts, at $g_2=0$ and a schematic cut as $g_1\to +\infty$, of the RG flow as a function of $g_1, g_2/g_1$, and $g_3/g_1$ for $N=4$. The sphere at the origin is the free pseudospin fixed point. (b) RG flows for the ratios of coupling constants for $N=4$ and $g_1>0$. The labeled gray circles are fixed rays where the ratio of the couplings remains fixed but all couplings become strong. (b) Phase diagram corresponding to the flows in (a).