arXiv:1001.0266v2  [cond-mat.str-el]  4 May 2010
Realization of the Exactly Solvable Kitaev Honeycomb Lattice Model in a Spin
Rotation Invariant System
Fa Wang 1
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
The exactly solvable Kitaev honeycomb lattice model is real ized as the low energy eﬀect Hamil-
tonian of a spin-1/2 model with spin rotation and time-rever sal symmetry. The mapping to low
energy eﬀective Hamiltonian is exact, without truncation e rrors in traditional perturbation series
expansions. This model consists of a honeycomb lattice of cl usters of four spin-1/2 moments, and
contains short-range interactions up to six-spin(or eight -spin) terms. The spin in the Kitaev model
is represented not as these spin-1/2 moments, but as pseudo-spin of the two-dimensional spin singlet
sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin corre-
lations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this
model. This exact construction is quite general and can be us ed to make other interesting spin-1/2
models from spin rotation invariant Hamiltonians. We discu ss two possible routes to generate the
high order spin interactions from more natural couplings, w hich involves perturbative expansions
thus breaks the exact mapping, although in a controlled mann er.
PACS numbers: 75.10.Jm, 75.10.Kt
Contents
I. Introduction. 1
II. Formulation of the Pseudo-spin-1/2 from
Four-spin Cluster. 2
III. Realization of the Kitaev Model. 3
IV. Generate the High Order Physical Spin
Interactions by Perturbative Expansion. 5
A. Generate the High Order Terms by Coupling
to Optical Phonon. 5
B. Generate the High Order Terms by Magnetic
Interactions between Clusters. 7
V. Conclusions. 8
Acknowledgments 8
A. Coupling between Distortions of a
Tetrahedron and the Pseudo-spins 8
B. Derivation of the Terms Generated by
Second Order Perturbation of Inter-cluster
Magnetic Interactions9
References 10
I. INTRODUCTION.
Kitaev’s exactly solvable spin-1/2 honeycomb lattice
model1 (noted as the Kitaev model hereafter) has in-
spired great interest since its debut, due to its exact
solvability, fractionalized excitations, and the potential
to realize non-Abelian anyons. The model simply reads
HKitaev = −
∑
x−links <jk>
Jxτx
j τx
k −
∑
y−links <jk>
Jyτy
j τy
k
−
∑
z−links <jk>
Jzτz
j τz
k
(1)
where τx,y,z are Pauli matrices, and x, y, z -links are de-
ﬁned in FIG. 1. It was shown by Kitaev 1 that this spin-
1/2 model can be mapped to a model with one Majo-
rana fermion per site coupled to Ising gauge ﬁelds on the
links. And as the Ising gauge ﬂux has no ﬂuctuation, the
model can be regarded as, under each gauge ﬂux conﬁg-
uration, a free Majorana fermion problem. The ground
state is achieved in the sector of zero gauge ﬂux through
each hexagon. The Majorana fermions in this sector have
Dirac-like gapless dispersion resembling that of graphene,
as long as|Jx|, |Jy|, and |Jz| satisfy the triangular rela-
tion, sum of any two of them is greater than the third
one1. It was further proposed by Kitaev 1 that opening of
fermion gap by magnetic ﬁeld can give the Ising vortices
non-Abelian anyonic statistics, because the Ising vortex
will carry a zero-energy Majorana mode, although mag-
netic ﬁeld destroys the exact solvability.
Great eﬀorts have been invested to better understand
the properties of the Kitaev model. For example, sev-
eral groups have pointed out that the fractionalized Ma-
jorana fermion excitations may be understood from the
more familiar Jordan-Wigner transformation of 1D spin
systems2,3. The analogy between the non-Abelian Ising
vortices and vortices in p + ip superconductors has been
raised in serveral works 4–7. Exact diagonalization has
been used to study the Kitaev model on small lattices 8.
And perturbative expansion methods have been devel-
oped to study the gapped phases of the Kitaev-type
models9.