A family of spin-S chain representations of SU(2)kWess–Zumino–Witten models
Ronny Thomale1, Stephan Rachel2, Peter Schmitteckert3,4, and Martin Greiter4,5
1Department of Physics, Stanford University, Stanford, CA 94305, USA
2Department of Physics, Yale University, New Haven, CT 06520, USA
3Institute f¨ ur Nanotechnologie, KIT, 76344 Eggenstein-Leopoldshafen, Germany
4Theorie der Kondensierten Materie, KIT, Campus S¨ ud, D 76128 Karlsruhe and
5Institut f¨ ur Festsk¨ orperphysik, Postfach 3640, KIT, D 76021 Karlsruhe
We investigate a family of spin-S chain Hamiltonians recently introduced by one of us . For
S = 1/2, it corresponds to the Haldane–Shastry model. For general spin S, we find indication that
the low–energy theory of these spin chains is described by the SU(2)k Wess–Zumino–Witten model
with coupling k = 2S. In particular, we investigate the S = 1 model whose ground state is given by
a Pfaffian for even number of sites N. We reconcile aspects of the spectrum of the Hamiltonian for
arbitrary N with trial states obtained by Schwinger projection of two Haldane–Shastry chains.
PACS numbers: 75.10.Jm,75.10.Pq,75.10.D
Introduction— Conformal field theory (CFT) has sig-
nificantly deepened our understanding of quantum spin
chains and phase transitions in two-dimensional space-
time [2–4]. In particular, it has provided a platform to
distinguish different universality classes of critical spin
chains according to e.g. the power law parameters of
multi-point spin correlation functions.
central charge parameter c of the CFT emerges in experi-
mentally connected quantities such as specific heat [5, 6].
For critical SU(2) spin-S chains, the conformal symme-
try is supplemented by a Lie group symmetry and yields
a low energy field theory description in terms of Wess-
Zumino-Witten (WZW) models [7, 8]. The WZW action
consists of a non-linear σ term plus the Wess-Zumino
term, whose topological coupling factor k is constrained
to be integer and defines the level k of the WZWk
model . In the absence of broken continuous symme-
tries, WZW1 is the generic low energy theory for half-
odd-integer antiferromagnetic spin chains , while inte-
ger spin chains show a Haldane gap . Spin chains
associated with WZWk>1 models only appear at rare
multi-critical points, and are of particular interest for in-
teger spin chains where they can mark phase transitions
between different gapped phases [9, 11, 12].
The 1/r2Heisenberg spin-1/2 chain independently
found by Haldane and Shastry [13, 14] plays a special
role among all other models related to WZW1. While
any finite size chain generally exhibits logarithmic cor-
rections to the WZW long-wavelength limit depending
on the system length L = aN where N is the number of
sites and a is the lattice constant, the Haldane–Shastry
model (HSM) exactly obeys the WZW1scalings for any
finite L. (This is related to its enlarged Yangian quan-
tum group symmetry .) Moreover, the exact HSM
ground state wave function corresponds to the Laughlin
wave function  in terms of bosonic spin flip particles
at ν = 1/2 filling, and establishes the notion of fractional
quantization and statistics of spinons as the fundamental
excitations of quantum spin chains . In this respect,
it is the one-dimensional analogue of the chiral spin liq-
uid [18, 19], where the concept of topological order was
introduced in spin models .
It was realized recently that not only the Laughlin
state, but a subset of the bosonic Read-Rezayi quan-
tum Hall series  can be generalized to a series of
singlet spin S wave functions at spin flip particle fill-
ing ν = S . For one spatial dimension, these states
are constructed by a Schwinger boson projection tech-
nique  of k copies of HSM ground states. The possi-
bility of defining SU(2) invariant spin S = k/2 states
related to the parafermionic CFT construction of the
Read-Rezayi states is intuitive, as their current algebra
reduces to the SU(2)k Kac-Moody algebra  for the
relevant fillings. As one particularly interesting mem-
ber, the k = 2 state corresponds to an S = 1 Pfaffian
spin state, which promises to establish the basis at which
manifestations of non-Abelian spinons in spin chains can
From the combined view of state properties and low en-
ergy theory, it is then natural to ask whether these states
may establish finite size representations of spin chains as
the conformally invariant fixed point of WZWk>1in the
same way as the HSM does for WZW1. A step in this di-
rection has been recently accomplished by one of us ,
who introduced a family of Hamiltonians which singles
out the spin S chain states obtained by projection from
k = 2S HS ground states as exact ground states. In
this Letter, we further investigate these Hamiltonians.
We numerically show that these S = k/2 spin chains are
critical and indeed connected to the WZWkin the long
wavelength limit. In contrast to the HSM, however, we
find that the S > 1/2 models exhibit logarithmic correc-
tions and hence do not describe the conformally invariant
fixed point of WZWk>1. For the S = 1 chain, we further
analyze the excited states of the model, and generalize
the Schwinger boson projection method to trial states
for the simplest excitations.
Hamiltonians and ground states— The general Hamil-
arXiv:1110.5956v1 [cond-mat.str-el] 27 Oct 2011
fit c = 1.60(2)
fit c = 1.46(2)
for S = 1 and S = 3/2 vs. the conformal length d(x) =
L/π sinπx/L. Data points are given for L = 20 (red), L = 24
(blue), and L = 30 (green). The thermodynamic WZW re-
sults are sketched by dashed lines. The fitted central charges
(solid lines) agree well with SU(2) WZW2S field theory within
reasonable error bars.
(color online) Entanglement entropy S(x) of HS
tonian for the spin-S chain  consists of a bilinear and
biquadratic as well as a three-site Heisenberg term
FIG. 2. (color online) Scaling dimensions for the Haldane
Shastry model (left) and H1(right). Left: The finite size data
(crosses) exactly matches the estimated CFT result x =
(dashed line).Right: The finite size linear fit (solid line)
shows a mismatch with the estimated CFT result x ∼3
2(S + 1)(2S + 3)
(SαSβ)(SαSγ) + (SαSγ)(SαSβ)
(¯ ηα− ¯ ηβ)(ηα− ηγ)
where Sα is a spin-S operator acting on site α and
periodic boundary conditions are imposed by sites
parametrized as complex roots of unity ηα= exp(i2π
α ∈ 1,2,...,N. (Note that the biquadratic two-site term
is contained in the three-site term in (1) as the special
case β = γ.)For S = 1, this Hamiltonian has very
recently been obtained independently through field theo-
retical methods by Nielson, Cirac, and Sierra . H1/2
is the HSM [13, 14] (the three-site term trivially simpli-
fies with the biquadratic term in this case). In Holstein-
Primakoff representation, the ground state, which is the
Gutzwiller wave function , takes the ν = 1/2 Laugh-
lin form ΨHS
M = νN and the zi’s denote the coordinates of the spin
flip operators acting on a spin polarized vacuum. The
1st degree homogeneous polynomial multiplied with the
squared Jastrow factor ensures that the state is both real
and a spin singlet.
is the building block for all other spin-S ground
states. The spin-S state is constructed by symmetriza-
tion of the spin flip coordinates of k = 2S identical copies
0. Technically, this can be accomplished conve-
niently in terms of auxiliary Schwinger boson creation
operators a†and b†, where the symmetrization effectively
reduces to a multiplication of k = 2S copies
The S = 1 ground state of (1) is hence computed
from the Schwinger projection of two HSM ground
states and takes the explicit form ΨPfaff
i=1zi, where M = N and
antisymmetrization operator. Note that as the HSM sin-
glet ground state requires N even, the ground states for
all S ≥ 1 of (1) require N even as well. The case of N
odd will be analyzed below.
0[a†,b†]?2S|0? . The ground state energy yields
?with E0= −2π2
i<j(zi − zj)?M
zi−zj) = A
, and A is the
Conformal field theory— Let us now determine the
long wavelength universality classes of HS. We employ
the density matrix renormalization group (DMRG) 
to compute the von Neumann entanglement entropy (EE)
for large system sizes. To begin with, we checked the an-
alytic formula for the ground state energy up to large
values of S and N. For gapped models, the EE saturates
to a constant value; for gapless models, the EE diverges
logarithmically, and, for finite chains takes the analytic
FIG. 3. (color online) S = 1 energy spectra scaled to units of unity for the Sz= 0 sector for N = 16 sites. Left: S = 1
Takhtajan-Babudjan Hamiltonian. Right: HS for S = 1.
The computation of the ground states in (1) turns out to
be challenging. One reason are the long-range terms in
the Hamiltonian which adds to the enlarged local basis
for higher spin S. For S = 1 (S = 3/2) we have kept up
to 800 (1200) states of the reduced density matrix and
performed up to 12 (14) sweeps, which enabled us to keep
the discarded entropy below 10−7(10−5). For (1) with
S = 1/2, 1, and 3/2, we confirm critical behavior due
to the log-divergent behavior of the EE. For S = 1/2,
1, and 3/2, we find central charges c = 1, c = 1.46(2)
and c = 1.60(2) from chains up to L = 30 [Fig. 1], which
are approximately consistent with the expected values
cWZW = 3k/(k + 2) for WZWk. The discrepancy be-
tween the numerically obtained c and the asymptotically
expected values is absent for S = 1/2 but sets in for
higher S. From finite size scaling, we cannot exclude
the existence of non-monotonous corrections for small L
spin-S chains. Still, over all we find numerical indication
that the low–energy theory of HSis the SU(2) WZW2S
model within reasonable error bars.
Logarithmic corrections— Having established that the
CFT’s related to the spin chains in (1) are of WZWk
type, we now address the question of logarithmic correc-
tions. As stated before, the HSM as the S = 1/2 realiza-
tion of (1) shows no finite size corrections as compared
to the long wavelength limit. To analyze this issue in
more detail, we calculate the scaling dimension x of the
WZWkprimary fields, as this quantity is highly sensitive
to finite size corrections . Specifically, the value for
the primary field at momentum π can be extracted from
E1−E0= 2πvx/L, where v is the velocity parameter and
E0and E1are the energies of the ground state and first
excited state. Only small system sizes are needed to de-
termine whether a spin chain resides at the conformally
invariant fixed point or not. In Fig. 2 we have plotted
E1− E0for H1/2and H1as computed by exact diago-
nalization (ED) for chain lengths up to N = 16. (As H1
is not very sparse, the number of scattering elements is
already of O(1013) for N = 16.) With x =1
S = 1/2 and 1 as predicted from CFT, we nicely observe
the absence of finite size corrections for the HSM, but
its presence for the S = 1 chain [Fig. 2]. This suggests
that logarithmic corrections are present for S >1
as a consequence that the models (1) do not represent the
conformally invariant fixed points of WZWkfor k > 1.
Spectrum of the S = 1 chain— We investigate the
structure of the finite size spectrum of (1) for S = 1
as obtained from ED. For this purpose, we compare
it with the spectrum of the Takhtajan-Babudjan (TB)
model [11, 12], which is likewise connected to WZW2and
describes the critical point between the dimerized phase
and the Haldane gap phase of the bilinear-biquadratic
spin-1 chain. The low energy spectra for N = 16 are
plotted in Fig. 3. Aside from the singlet ground state at
momentum k = 0, the spin multiplet quantum numbers
of the lowest energy modes in the different momentum
subsectors are related. The lowest level at momentum
k = π determines the finite size gap. Both spectra look
very similar, showing a two-lobe feature [Fig. 3]. Still,
the overlap of the Pfaffian and TB ground state is be-
low .85 already for small system sizes, indicating that
the models have different finite size structures. This be-
comes explicit as we develop a specific trial state Ansatz
for the S = 1 model in the following, which does not
similarly apply to the TB model.
Generalized Schwinger projection scheme— Up to now,
we have only used our Schwinger projection scheme for
HSM ground states to generate the exact N even ground
states for the spin-S models in (1). We now investigate
what we can achieve when we employ the same approach
for HSM excited states before projection. To begin with,
we use the Schwinger projection scheme to obtain a suit-
able trial state for the N odd ground state of the S = 1
chain [Fig. 4]. Consider first the two S = 1/2 chains be-
fore projection. For N odd, the low energy modes of the
HSM are given by the single spinon branch which only
covers one half of the momenta (−π/2 < ksp≤ π/2 and
π/2 < ksp≤ 3π/2 for N = 1 mod 4 and N = 3 mod 4,
N = 15
0 0.05 0.1
FIG. 4. (color online) (a) Construction of the S = 1 trial state ψtrial (k = 0) for N odd out of two Haldane-Shastry one-spinon
states with momenta k = ±π/2. (b) Spectrum of the S = 1 chain for N = 15. The ground state at k = 0 is ψodd
. (c) Overlap
? vs. 1/N. For N → ∞, the overlap is 0.9983(1).
respectively) [Fig. 4a]. The ground state for N odd in
the S = 1 chain is located at k = 0. (N = 15 is shown in
Fig. 4b.) As phases of the wave function are preserved
in Schwinger boson notation, they multiply under pro-
jection, and the total momentum of the projected wave
function is given by the sum of single chain momenta be-
fore projection. In order to find a good trial state Ψodd
for the S = 1 model (1), we choose the lowest energy
one-spinon states in two HSM chains at −π/2 and π/2
before projection. To fully match the quantum numbers
of the target state, we also project the spin part onto the
singlet component of the projected two one-spinon wave
functions. With this construction, we obtain an excellent
overlap with the N odd ground state of (1) with S = 1,
which is of the order of 0.999 and hardly changes with
system size [Fig. 4c]. Good overlaps can also be achieved
for trial states to match other modes such as the lowest N
even eigenstates of the S = 1 model in different momen-
tum sectors [Fig. 3]. There, we project the lowest lying
two-spinon HS eigenstate for different momenta and a
HS ground state together. This correspondence explains
how the lobe features for the S = 1 model are connected
to the two-spinon levels of the HSM before projection.
From there, a unified picture emerges: we can inter-
pret the Schwinger boson projection of Haldane-Shastry
states as the creation of spinon product states. This is
a peculiar property of the Schwinger boson projection
of Haldane-Shastry chains, as we found elsewhere that
taking two copies of a generic S = 1/2 nearest neighbor
Heisenberg model ground state as building blocks, for ex-
ample, generates a trial ground state for a gapped spin-1
model, where the spinons become confined . In con-
trast to this, the HSM projection still provides gapless
states, which is somewhat intuitive as the HSM is a free
spinon gas. As the Hilbert space after projection is signif-
icantly smaller than the product space of the individual
Haldane-Shastry chains before projection, this yields an
overcomplete basis, and gives rise to selection rules that
specify which many-spinons states before projection map
onto each other after projection. This can be one way of
defining the notion of a ”blocking mechanism” connected
to the non-Abelian statistics of the spinons . Compar-
ing the trial states we constructed here via Schwinger
boson projection with the actual eigenstates of (1) for
finite systems, we see that this construction yields rea-
sonable approximations to the low energy modes of the
system, even though it does not provide us with exact
eigenstates. The trial state we have constructed for the
S = 1 model with N odd illustrates this point: since
the spinons before projection have been chosen such that
they cannot decay any further as they are located at the
outer edges of the dispersion branches, we suppose that
the ”ideal” finite size WZW2 spin chain model in the
sense of spinon product states should exactly correspond
to this construction and give an overlap of unity. The ob-
served deviation from that is another way to interpret the
logarithmic corrections we found for the Hamiltonian (1)
for S > 1/2.
RT thanks J.S. Caux for comments and D. P. Arovas,
B. A. Bernevig, and F. D. M. Haldane for discussions
as well as collaborations on related topics. MG thanks
A. W. W. Ludwig and K. Schoutens for discussions. RT is
supported by an SITP fellowship by Stanford University.
SR acknowledges support from DFG under Grant No.
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