#
2006 APS March Meeting

##
Monday–Friday, March 13–17, 2006;
Baltimore, MD

### Session W3: Topological Aspects of Electron Transport in Solids

2:30 PM–4:54 PM,
Thursday, March 16, 2006

Baltimore Convention Center
Room: Ballroom I

Sponsoring
Unit:
DCMP

Chair: Shoucheng Zhang, Stanford University

Abstract ID: BAPS.2006.MAR.W3.2

### Abstract: W3.00002 : Fermi Liquid Berry Phase Theory of the Anomalous Hall Effect*

3:06 PM–3:42 PM

Preview Abstract
Abstract

####
Author:

F. D. M. Haldane

(Princeton University)

Charged Fermi liquids with broken time-reversal symmetry have an
intrinsic anomalous Hall effect that derives from the Berry
phases accumulated by accelerated quasiparticles that move on the
Fermi surface. The intrinsic Hall conductivity is given by a new
fundamental geometric Fermi liquid formula that can be regarded
as the derivative with respect to magnetic flux density of the
Luttinger fomula relating the density of mobile charge carriers
to the k-space volume enclosed by the Fermi surface. This
formula can be derived by an integration-by-parts of the
Karplus-Luttinger free-electron band-structure formula to yield a
topological (QHE) part plus a geometrical part expressed
completely at the Fermi surface, and which has a natural
generalization to interacting Fermi liquid quasiparticles (QP's).
The QP Berry phases are properties of the {\it eigenstates} of
the (exact) single-particle Green's function at the Fermi
surface, which is a Hermitian matrix with Bloch-state
eigenvectors; the Berry phases derive from the variation on the
Fermi surface of the spatially-periodic factor of the QP Bloch
state that characterizes how the total QP amplitude is
distributed among the different electronic orbitals in the unit
cell. In the case of 3D ferromagnetic metals, the Berry phases
derive from the interplay of exchange splitting with spin-orbit
coupling (both must be present). Remarkably, the new formula
also applies to Fermi-liquid analogs such as the 2D composite
fermion (CF) fluid in the half-filled lowest Landau level: in
this case, the QP is a bound electron+vortex composite and not
a Bloch state. This QP structure varies on the CF Fermi surface
in a way that exactly gives the expected result $\sigma^{xy}$ =
$e^2/2h$, unaffected by any Fermi surface anisotropy, thus
explaining how a quantized value of $\sigma^{xy}$ persists even
though the CF Fermi liquid is {\it not} an incompressible FQHE
state. The geometric anomalous Hall effect formula suggests a
more intrinsic geometric description of the Fermi surface, where
the Fermi vector ${\boldmath k}_F({\bf s})$ is only one of a
number of
properties that vary on a curved $(D-1)$-dimensional Fermi
surface manifold parametrized by curvilinear coordinates ${\bf
s}$; other properties include the Berry curvature field ${\cal
F}({\bf s})$, quasiparticle mean free path $\ell({\bf s})$,{\it
etc}. The new formula also naturally takes into account
non-trivial (multiply-connected) Fermi surface topology and open
orbits.

*Supported by NSF DMR-0213706

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2006.MAR.W3.2