Microscopic foundation of Haldane's geometric description of fractional quantum Hall states: intrinsic metric elasticity, geometry across the compressible transition, and quasiparticles as curvature defects

Haldane's geometrical description of the fractional quantum Hall effect (Phys. Rev. Lett. 107, 116801 (2011)) reorganizes incompressible fractional quantum Hall (FQH) liquids around an emergent unimodular intrinsic metric $\bar{g}_{ab}(\mathbf{r},t)$ — the local shape of guiding-center correlations — whose quantum fluctuations are governed by a topologically quantized guiding-center spin $\bar{s}$ and whose Gaussian curvature couples to electric charge. The effective-field-theory side of this program now exists: the bimetric theory of Gromov and Son couples Chern-Simons hydrodynamics to a dynamical unimodular metric, and the resulting spin-2 graviton has been observed experimentally by two-photon inelastic light scattering. The microscopic side remains incomplete. The problem asks for its completion: (a) a quantitative determination, from lowest-Landau-level guiding-center Hamiltonians with generic two-body interactions, of the equilibrium intrinsic metric and of the elastic coefficients of the correlation energy $E(\bar{g})$; (b) a characterization of how $\bar{s}$ and $\bar{g}_{ab}$ evolve across the transition to a compressible phase, where the quasiparticle gap and the topological quantization of $\bar{s}$ are lost; and (c) an explicit microscopic construction of quasiparticles as localized, quantized curvature defects (rational cone singularities with $K=\pm 4\pi/(2\bar{s})$) of the intrinsic metric.

Background

In a two-dimensional electron gas in a strong perpendicular magnetic field, the kinetic energy is quantized into Landau levels (the quantized cyclotron orbitals). In the fractional quantum Hall effect (FQHE), the lowest Landau level (LLL) is partially filled and electrons condense into strongly correlated incompressible liquids with a fractionally quantized Hall conductance. Within a single Landau level the electron coordinates reduce to the guiding centers $\mathbf{R}$ — the centers of the cyclotron orbits — whose two components do not commute ($[R^x,R^y]=-i\ell_B^2$, where $\ell_B$ is the magnetic length), so the LLL liquid is effectively a noncommutative geometry; its density operators obey the Girvin-MacDonald-Platzman (GMP) commutation relations, a closed commutator algebra of the Landau-level-projected density operators. The archetypal state, Laughlin's wavefunction at filling $\nu=1/q$, is the exact zero-energy ground state of special repulsive pseudopotential interactions (rotationally invariant two-body repulsions whose strength depends on the relative angular momentum of an electron pair) and exhibits a correlation hole around each electron.

Haldane's 2011 reformulation (the source of this problem) begins with the observation that an LLL problem with anisotropic band mass or anisotropic interactions contains two inequivalent physical metrics: the Galilean metric $g_{ab}$ (set by the mass tensor, fixing the shape of cyclotron orbits) and the Coulomb metric $\tilde{g}_{ab}$ (set by the dielectric tensor, fixing the shape of the interaction). Only the unimodular part of a metric (the part with unit determinant, encoding pure shape at fixed area) is meaningful in a magnetic field. Nothing requires the correlation hole to be circular in either metric: the liquid has its own intrinsic unimodular metric $\bar{g}_{ab}$, first introduced in Haldane's 2009 preprint on Hall viscosity. The source paper argues that (i) $\bar{g}_{ab}$ minimizes the correlation energy $E(\bar{g})$ — in the source's variational definition, the energy $E(\bar{g})=\langle\Psi^L_{q,\alpha}(\bar{g})|H_R|\Psi^L_{q,\alpha}(\bar{g})\rangle$ of the metric-deformed $\nu=1/q$ Laughlin trial family defined by $P_m(\bar{g})|\Psi^L_{q,\alpha}(\bar{g})\rangle=0$ for relative angular momenta $m<q$, with $H_R$ the guiding-center Hamiltonian — and is generally 'a compromise intermediate between the two physical metrics'; (ii) $E(\bar{g})$ is quadratic in slow local deformations $\bar{g}_{ab}(\mathbf{r},t)$ around its minimum, making the incompressible liquid an elastic medium of shape; (iii) the quantum fluctuations of $\bar{g}_{ab}$ are governed by the guiding-center spin $\bar{s}$ — a topologically quantized angular momentum per flux quantum, related to the shift (the characteristic particle-to-flux ratio of a Hall fluid) and to the Hall viscosity (a nondissipative geometric viscosity of quantized Hall fluids); and (iv) the local elementary-droplet (composite-boson) density $\bar{\rho}$, from which the electric charge density is $pe\,\bar{\rho}$, couples to the Gaussian curvature $K$ of the intrinsic metric, $\bar{\rho}(\mathbf{r},t)=\frac{1}{2\pi pq}\left[\frac{pe}{\hbar}B(\mathbf{r})+\bar{s}K(\mathbf{r},t)\right]$ at filling $\nu=p/q$ — an internal-geometry analogue of the Wen-Zee coupling of charge to externally imposed curvature.

The neutral collective mode of FQH liquids — the GMP mode or magnetoroton, the density-wave excitation originally computed by Girvin, MacDonald and Platzman — carries angular momentum 2 at long wavelength, exactly the signature of a quadrupolar (spin-2) deformation: in the geometric picture it is the quantum of $\bar{g}_{ab}$, a 'graviton' in the condensed-matter sense. Two major later developments delivered the effective-theory side of the program. First, Gromov and Son constructed the bimetric effective field theory: Chern-Simons theory (the topological field theory encoding the quantized Hall conductance, the standard long-wavelength description of Hall fluids) coupled to a dynamical unimodular metric, i.e. a nonrelativistic analogue of bimetric massive gravity; it reproduces the GMP algebra, the graviton mode, the Hall viscosity, and Ward identities, but its dimensionful couplings are fixed phenomenologically by matching to the single-mode approximation (SMA — a variational treatment of the GMP mode as a small quadrupolar deformation of the ground state). Second, the graviton was observed: Liang et al. detected long-wavelength magnetorotons of well-defined chirality by inelastic scattering of circularly polarized light at fillings $\nu=1/3$, $2/5$, $2/3$, $3/5$, confirming the spin-2 quantum of the intrinsic metric. Microscopic calculations by exact diagonalization (numerical diagonalization of finite systems), infinite-cylinder density matrix renormalization group (DMRG, a variational tensor-network method), and variational trial states have confirmed the existence of the intrinsic metric and its compromise behavior in specific models. What remains missing is the microscopic quantitative layer: a derivation of the equilibrium metric and of the elastic coefficients of $E(\bar{g})$ from the underlying interaction, a theory of the geometry across the transition to compressible phases, and an explicit construction of the fractionally charged quasiparticles as curvature defects of $\bar{g}_{ab}$, which the source describes as rational cone singularities — defects at which the Gaussian curvature of the metric is concentrated at a point, as at the tip of a cone — with localized Gaussian curvature $K=\pm 4\pi/(2\bar{s})$.

Problem Statement

Haldane's geometrical description of the fractional quantum Hall effect (the source paper) identifies the unimodular intrinsic metric $\bar{g}_{ab}(\mathbf{r},t)$ as the fundamental collective degree of freedom of an incompressible FQH liquid, with topologically quantized fluctuations (guiding-center spin $\bar{s}$) and charge coupled to its Gaussian curvature; the continuum field theory of this degree of freedom now exists (the bimetric theory) and its spin-2 quantum has been observed experimentally. The problem is the microscopic completion of this description. For lowest-Landau-level guiding-center Hamiltonians with generic translationally invariant two-body interactions $V(q)$ and independent Galilean and Coulomb metrics $g_{ab}$ and $\tilde{g}_{ab}$:

(a) determine the equilibrium intrinsic metric $\bar{g}_{ab}$ that minimizes the correlation energy $E(\bar{g})$ — in the source's variational definition at filling $\nu=1/q$, the energy $E(\bar{g})=\langle\Psi^L_{q,\alpha}(\bar{g})|H_R|\Psi^L_{q,\alpha}(\bar{g})\rangle$ of the metric-deformed Laughlin family defined by $P_m(\bar{g})|\Psi^L_{q,\alpha}(\bar{g})\rangle=0$ for $m<q$ — as a function of the interaction and of the two physical metrics (the source conjectures 'a compromise intermediate between the two physical metrics'), and compute the elastic coefficients of the quadratic expansion of $E(\bar{g})$ in local deformations $\bar{g}_{ab}(\mathbf{r},t)$;

(b) characterize the behavior of $\bar{s}$ and $\bar{g}_{ab}$ across the transition from an incompressible FQH liquid to a compressible phase, where the quasiparticle gap collapses and $\bar{s}$ becomes unquantized or indeterminate: what replaces the topological quantization, and how do the intrinsic metric and its elastic coefficients evolve through the transition? Any representative incompressible-to-compressible transition may be studied, provided it is specified explicitly: the system (filling factor and interaction family), the tuning parameter that drives the transition, and the target compressible phase.

(c) construct the fractionally charged quasiparticles explicitly, within microscopic guiding-center calculations, as localized curvature defects of the intrinsic metric — rational cone singularities (defects at which the Gaussian curvature of $\bar{g}_{ab}$ is concentrated at a point, as at the tip of a cone) carrying the quantized localized Gaussian curvature $K=\pm 4\pi/(2\bar{s})$ demanded by the curvature-charge relation, understood as the curvature integrated over the defect (the deficit angle of the cone) — and establish their relation to the quasiparticle charge and statistics.

These three components jointly constitute the microscopic quantitative foundation of the geometric description: they share one objective (deriving the geometric degree of freedom, its elasticity, and its defects from the microscopic Hamiltonian).

References

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