In general, I am interested in strongly-interacting quantum many-body condensed-matter systems, explored by non-perturbative methods:
Geometry of the fractional quantum Hall effect (FQHE): The FQHE occurs when the kinetic energy of electrons bound to a 2D surface becomes quantized in the presence of a high magnetic fields when they move in Landau orbits. The "guiding centers" of their orbits have non-commuting components and define a "quantum geometry". "Incompressible quantum fluid" states of the electrons at fractional Landau-level filling p/q can be modeled as a "condensate" of "composite bosons" formed from p particles and q "flux quanta" (guiding-center orbitals). I recently found that the shape (geometry) of the "composite boson" was the previously-unrecognized collective degree of freedom of the FQHE, and is described by a "unimodular" spatial metric-tensor field. This leads to a new field-theoretical description of FQHE states in terms of Chern-Simons fields plus the metric-tensor field that (unlike the c. 1990 "Chern-Simons + Ginzburg-Landau" description that it replaces) successfully describes the FQHE collective mode. The new picture appears to provide the previously-missing ingredient for a quantitative description of FQHE incompressibility.
"Entanglement spectrum" of quantum states: With my student Hui Li, we investigated the spectrum of "pseudo-energies" of the Schmidt singular-value bipartite decomposition of the support of a quantum state (a FQHE fluid) into two spatial regions. This is usually used to obtain the "entanglement entropy" of the state, but we found that plotting the spectrum it as a function of momentum parallel to the translationally-invariant boundary between the two regions revealed a gapless "pseudo-energy" excitation spectrum from which the topological order could be identified. This new notion of examining the "entanglement spectrum" as a function of its quantum numbers, rather than just an "entropy", has turned into a powerful tool for detecting and identifying topological order.
Model wavefunctions for the FQHE: The Laughlin, Moore-Read, and other model wavefunctions for FQHE states can be represented as Jack Polynomials, which have the key property they only contain occupation number configurations that can be "squeezed" from a root state. The connection to Jack polynomials exposes so-far unexploited algebraic properties of such states, which are in some sense fractional-statistics analogs of free-particle states, with extra simplicity in their properties, as compared to "generic" FQHE states. With A. Bernevig, I have been interested in whether these properties might lead to ways to carry out exact calculations of properties of these model states.
Topological Insulators, and "Chern Insulators": In 1988, I proposed a graphene-like model of a topologically non-trivial band structure, with broken time-reversal symmetry but regular Bloch states, that exhibited an (integer) "quantum Hall effect without Landau levels" in the absence of a net magnetic fiux density. This would today be called a "Chern Insulator", the original "topological insulator". A simple generalization of the model lead to Kane and Mele's 2004 model for a time-reversal-invariant 2D topological insulator, which in turn lead to the discovery of the 3D topological insulators. As a different generalization, with my student S. Raghu, I used this model to demonstrate theoretically that topologically-non-trivial photonic band structures could be made that would have "unidirectional" edge modes that could propagate around corners an obstructions without backscattering. More recently, it was found that models of Chern insulators with "flat bands" may support an FQHE without Landau levels, which is a current interest.
- F. D. M. Haldane,
Geometric description of the fractional quantum Hall effect,
arXiv:1106.3375 (preprint, June 2011)
- H. Li and F. D. M. Haldane,
Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in fractional quantum Hall states.
Phys. Rev. Lett. 101, 010504 (2008).
- F. D. M. Haldane,
Quantum Hall effect without Landau levels: a condensed-matter realization of the parity anomaly,
Phys. Rev. Lett. 61, 2015 (1988).
- F. D. M. Haldane and S. Raghu,
Possible realization of unidirectional waveguides in photonic crystals with broken time-reversal symmetry,
Phys. Rev. Lett. 100, 013904, (2008)
- B. A. Bernevig and F. D. M. Haldane,
Clustering properties and model wavefunctions for non-Abelian fractional quantum Hall quasielectrons,
Phys. Rev. Lett. 102, 066802 (2009).