Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids

We synthesize and partly review recent developments relating the physics of the half-filled Landau level in two dimensions to correlated surface states of topological insulators in three dimensions. The latter are in turn related to the physics of certain three-dimensional quantum spin liquid states. The resulting insights provide an interesting answer to the old question of how particle-hole symmetry is realized in composite fermion liquids. Specifically the metallic state at filling $\nu =\frac{1}{2}$—described originally in pioneering work by Halperin, Lee, and Read as a liquid of composite fermions—was proposed recently by Son to be described by a particle-hole symmetric effective field theory distinct from that in the prior literature. We show how the relation to topological insulator surface states leads to a physical understanding of the correctness of this proposal. We develop a simple picture of the particle-hole symmetric composite fermion through a modification of older pictures as electrically neutral “dipolar” particles. We revisit the phenomenology of composite fermi liquids (with or without particle-hole symmetry), and show that their heat/electrical transport dramatically violates the conventional Wiedemann-Franz law but satisfies a modified one. We also discuss the implications of these insights for finding physical realizations of correlated topological insulator surfaces.

Figure 1

The standard picture of the composite fermion at $\nu =\frac{1}{2}$ regards it as an electron (of charge $e)$ bound to a $4\pi$ vortex. The vortex carries charge $-e$ and is displaced from the electron in the direction perpendicular to its momentum. The composite fermion is thus viewed as a neutral dipolar particle.

Figure 2

The proposed picture of the particle-hole symmetric composite fermion at $\nu =\frac{1}{2}$. One end of the dipole has a $2\pi$ vortex bound to charge $\frac{e}{2}$. The other end has a charge $-\frac{e}{2}$ also bound to a $2\pi$ vortex. The displacement between the two is in the direction perpendicular to their center-of-mass momentum. The positively charged end can be viewed as a $2\pi$ vortex located exactly on the electron. Thus compared to the picture in Fig. 1 only one $2\pi$ vortex is displaced from the electron.

Figure 3

When one end of the dipole of Fig. 2 is rotated in a closed loop around the other end, there is a phase of $\pi$.

Figure 4

Schematic phase diagram for the surface of the correlated $n=1$ topological insulator with $\mathrm{U}\left(1\right)\times \mathrm{C}$ symmetry. $g$ is a parameter that controls the relative strength of Cooper pairing versus the phase stiffness in the superconductor. At $B=0$, with increasing $g$ pairing is lost leading to the Dirac metal obtained within band theory. At small $g$, superconductivity is destroyed through phase fluctuations leading to the “dual” Dirac metal. Many other phases are also of course possible. Particularly interesting is a symmetry preserving surface topological order. At nonzero $B$, the composite fermi liquid emerges as one of the possible phases.

Figure 5

When a strength $\frac{mh}{e}$ monopole from the outside vacuum tunnels into the bulk insulator through a surface superconductor, it leaves behind a strength $mhe$ vortex at the surface. Understanding the properties of the monopole in the bulk constrains the properties of the corresponding surface vortex.

Figure 6

Charge-monopole lattice at $\theta =\pi$.

Figure 7

Bulk-boundary correspondence for the composite fermi liquid. The composite fermion is the surface avatar of the electrically neutral strength-2 bulk monopole, which itself is a bound state of the two $(\pm \frac{1}{2},1)$ dyons. This strength-2 monopole is a fermion, and is Kramers-doublet under $\mathrm{C}$. At the surface, the two dyons that make up this monopole correspond to the two ends of the dipole of Fig. 2.