Pairing in Luttinger Liquids and Quantum Hall States

We study spinless electrons in a single-channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire, the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity coupled to an external superconductor, for an isolated wire there is no separate phase of a topological, weakly paired superconductor. Rather, this phase is adiabatically connected to the Luttinger liquid phase. The properties of the one-dimensional topological superconductor emerge when the number of channels in the wire becomes large. The quantum Hall states that may be formed by an array of single-channel wires depend on the Landau-level filling factors. For odd-denominator fillings $\nu =1/(2n+1)$, wires at the Luttinger phase form Laughlin states, while wires in the strongly paired phase form a bosonic fractional quantum Hall state of strongly bound pairs at a filling of $1/(8n+4)$. The transition between the two is of the universality class of Ising transitions in three dimensions. For even-denominator fractions $\nu =1/2n$, the two single-wire phases translate into four quantum Hall states. Two of those states are bosonic fractional quantum Hall states of weakly and strongly bound pairs of electrons. The other two are non-Abelian quantum Hall states, which originate from coupling wires close to their critical point. One of these non-Abelian states is the Moore-Read state. The transitions between all of these states are of the universality class of Majorana transitions. We point out some of the properties that characterize the different phases and the phase transitions.

Popular Summary

The theory of Luttinger liquids is a successful model for understanding how electrons interact in one-dimensional conductors such as carbon nanotubes and semiconducting wires. Theories for many two-dimensional electronic systems, however, lack the clarity and rigor of Luttinger theory. Such systems include what are known as fractional quantum Hall states, wherein the electrical conductance of a two-dimensional gas of electrons becomes quantized when subject to a strong magnetic field. The peculiar properties of some of these states might be useful for quantum computation. Using the Luttinger theory as a starting point, we develop a unified mathematical framework that not only describes two-dimensional systems such as these quantum Hall states but also offers new insight into certain one-dimensional electronic systems.

Our approach views the two-dimensional plane as an array of coupled wires. We start by considering a single wire of attractively interacting electrons and extend Luttinger theory to account for the case of strong interaction. This leads us to distinguish between two phases of the single wire: the superconducting phase, where pairs of electrons are the low-energy degrees of freedom, and the Luttinger liquid phase, where single electrons are low-energy degrees of freedom as well. We then consider an array of such wires in a magnetic field and show how controlling the phases of the individual wires, as well as the coupling between them, leads to distinct types of quantized Hall states.

We believe that the method we developed will be useful for analyzing other forms of exotic fractional quantum Hall states as well.

Figure 1

Critical behavior of the edge tunneling conductance $G$ as a function of temperature $T$ and tuning parameter $\delta$ near the 3D Ising critical point. Panel (a) shows $G$ as a function of $\delta$ for several values of $T$, highlighting the zero-temperature behavior in which $G(0,\delta )\sim \theta (\delta ){\delta }^{2\beta }$. Panel (b) shows $G$ as a function of $T$ for several values of $\delta$, highlighting the behavior at the critical point $G(T,0)\sim T^{m+\eta }$.

Figure 2

Phase diagram for $m$ odd as a function of parameters ${\epsilon }_0$ and ${\mathrm{\Delta }}_2$ in Eq. (58).

Figure 3

Schematic diagrams for quantum Hall states at $\nu =1/2$. Each shaded region depicts a wire at the pairing transition with right- and left-moving Majorana modes (dashed lines) as well as right- and left-moving charge modes (solid lines) describing ${\varphi }_{\rho ,i}^{R/L}={\phi }_{\rho ,i}\pm m{\theta }_{\rho ,i}$ with $m=2$. The charge modes are coupled by $t_2$ in Eq. (46), leaving a single unpaired chiral charge mode on each edge. The Majorana modes are coupled in different ways as described in the text, resulting in topologically distinct states with different numbers of chiral Majorana edge modes. In panel (d), C-SP refers to the particle-hole conjugate of the strong-paired state.