2
y
x
zz z
z zzz
z z
y y
y
x
xx
x x x
y yy
y
x
y
x
z
z
y y
x x x
y
z
x
x
FIG. 1: The honeycomb lattice for the Kitaev model. Filled
and open circles indicate two sublattices.x, y, z label the links
along three diﬀerent directions used in (1).
derived as well. There have been several proposals to
open the fermion gap for the non-Abelian phase without
spoiling exact solvability4,6. And many generalizations
to other(even 3D) lattices have been developed in the
last few years10–16. All these eﬀorts have signiﬁcantly
enriched our knowledge of exactly solvable models and
quantum phases of matter.
However, in the original Kitaev model and its later
generalizations in the form of spin models, spin rotation
symmetry is explicitly broken. This makes them harder
to realize in solid state systems. There are many pro-
posals to realized the Kitaev model in more controllable
situations, e.g. in cold atom optical lattices17,18, or in
superconducting circuits 19. But it is still desirable for
theoretical curiosity and practical purposes to realize the
Kitaev-type models in spin rotation invariant systems.
In this paper we realize the Kitaev honeycomb lattice
model as the low energy Hamiltonian for a spin rotation
invariant system. The trick isnot to use the physical spin
as the spin in the Kitaev model, instead the spin-1/2 in
Kitaev model is from some emergent two-fold degener-
ate low energy states in the elementary unit of physical
system. This type of idea has been explored recently by
Jackeli and Khaliullin20, in which the spin-1/2 in the Ki-
taev model is the low energy Kramers doublet created by
strong spin-orbit coupling oft2g orbitals. In the model
presented below, the Hilbert space of spin-1/2 in the Ki-
taev model is actually the two dimensional spin singlet
sector of four antiferromagnetically coupled spin-1/2 mo-
ments, and the role of spin-1/2 operators(Pauli matrices)
in the Kitaev model is replaced by certain combinations
ofSj · Sk [or the spin-chirality Sj · (Sk × Sℓ)] between the
four spins.
One major drawback of the model to be presented is
that it contains high order spin interactions(involves up
to six or eight spins), thus is still unnatural. However it
opens the possibility to realize exotic (exactly solvable)
models from spin-1/2 Hamiltonian with spin rotation in-
variant interactions. We will discuss two possible routes
to reduce this artiﬁcialness through controlled perturba-
tive expansions, by coupling to optical phonons or by
magnetic couplings between the elementary units.
The outline of this paper is as follows. In Section II
we will lay out the pseudo-spin-1/2 construction. In Sec-
4
2 3
1
z z
x
x
y
y
23
4
1
FIG. 2: Left: the physical spin lattice for the model (8). The
dash circles are honeycomb lattice sites, each of which is ac-
tually a cluster of four physical spins. The dash straight li nes
are honeycomb lattice bonds, with their type x, y, z labeled.
The interaction between clusters connected by x, y, z bonds
are the Jx,y,z terms in (8) or (9) respectively. Note this is not
the 3-12 lattice used in Ref. 9,10. Right: enlarged picture of
the clusters with the four physical spins labeled as 1 , . . . , 4.
Thick solid bonds within one cluster have large antiferromag-
netic Heisenberg coupling Jcluster.
tion III the Kitaev model will be explicitly constructed
using this formalism, and some properties of this con-
struction will be discussed. In Section IV we will discuss
two possible ways to generate the high order spin in-
teractions involved in the construction of Section III by
perturbative expansions. Conclusions and outlook will
be summarized in Section V.
II. FORMULATION OF THE PSEUDO-SPIN-1/2
FROM FOUR-SPIN CLUSTER.
In this Section we will construct the pseudo-spin-1/2
from a cluster of four physical spins, and map the phys-
ical spin operators to pseudo-spin operators. The map-
ping constructed here will be used in later Sections to
construct the eﬀective Kitaev model. In this Section we