Quaternion
Quaternion (also known as Hamilton number), is an extension of complex number,
and the first discovered non-commutive division algebra by Hamilton.
It is spanned by three anti-communting imaginary units i, j, k, satisfying
i^2=j^2=k^2=-1, and ij=-ji=k, jk=-kj=i, and ki=-ik=j.
Quanternionic analyticity is a high dimensional generalization of the
complex analyticity, described by the Cauchy-Riemann-Feuter condition,
which is an Eucliean version of the Weyl equation.
Quaternionic analytic Landau levels
We find a deep connection between quaternionic analyticity
and high dimensional topological states [Ref. 1] .
The Landau levels are generalzed to 3D and 4D possessing the full high
dimensional rotation symmetry and flat dispersion by employing the
SU(2) Aharanov-Casher potential. In analogy to the complex analyticity
which had facilitated the study of 2D fractional quantum Hall states,
we proved that the 3D and 4D lowest Landau level wavefunctions exhibit
beautiful analytic properties, satisfying the Cauchy-Riemann-Fueter
condition of the quaternionic analyticity.
We expect that the quaternionic analyticity would provide a guiding principle
for constructing high dimensional fractional topological states, which is
a major challenging open question in the field, and the 3D LLs provide
a possible platform for this research. In contrast, most 3D topological
insulator models are based on lattice systems exhibiting complicated Bloch
wavefunctions and dispersive energy spectra, both of which are obstacles for
studying 3D fractional topological states.
The 3D lowest Landau level (LLL) wavefunctions are most intuitively
explained in the coherent state representation: Pick up an arbitrary
right-handed triad with three orthogonal axes e_1, e_2, and e_3.
The LLL wavefunctions are analytic functions in the complex plane of
e_1 + i e2 with spin polarized along the e_3 axis.
The coupling between the orbital chirality and spin polarization
forms a helicity structure, hence, the time-reversal
symmetry is maintained.
Its wavefunction analyticity exhibits in the rigid-body configuration
space, or the SU(2) group space, and is best represented by quaternions.
Each Landau level also contributes a branch of helical Dirac surface modes
on the open boundary.
We also constructed the 3D and 4D Landau levels based on the Landau-type
gauge with flat bulk spectra and topologically non-trivial surface states
[Ref 2.] .
For the 4D case, the system exhibits quantized non-linear electromagnetic
response as a spatially separated (3+1)D chiral anomaly.
High dimensional Landau levels of Dirac fermions
We further constructed the 3D LLs for Dirac electrons, whose zeroth
Landau level states are a flat band of the half-fermion Jackiw-Rebbi
zero modes
[Ref. 3] .
This can be viewed as a squre-root problem of the above high dimensional
Landau levels for non-relativistic fermions.
It is at the interface between condensed matter and high energy physics,
related to a new type of anomaly.
Unlike parity anomaly and chiral anomaly studied in field theory
in which Dirac fermions couple to gauge fields through the minimal
coupling, here Dirac fermions couple to the background
field in a non-minimal way.
References and talks
1. Yi Li, Congjun Wu, "High-Dimensional Topological Insulators
with Quaternionic Analytic Landau Levels",
Phys. Rev. Lett. 110, 216802 (2013) .
See pdf file
.
2. Yi Li, Shou-Cheng Zhang, Congjun Wu,
"Topological insulators with SU(2) Landau levels",
Phys. Rev. Lett. 111, 186803 (2013) .
See pdf file
3. Yi Li, Kenneth Intriligator, Yue Yu, Congjun Wu,
"Isotropic Landau levels of Dirac fermions in high dimensions",
Phys. Rev. B 85, 085132 (2012) , see
pdf file.
Talk:
"Quaternionic analyticity and SU(2) Landau Levels in 3D"
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Last modified: July 15, 2007.