Neel to staggered dimer order transition in a generalized honeycomb lattice Heisenberg model

Abstract

We study a generalized honeycomb lattice spin-1/2 Heisenberg model with nearest-neighbor antiferromagnetic 2-spin exchange, and competing 4-spin interactions which serve to stabilize a staggered dimer state which breaks lattice rotational symmetry. Using a combination of quantum Monte Carlo numerics, spin wave theory, and bond operator theory, we show that this model undergoes a strong first-order transition between a Neel state and a staggered dimer state upon increasing the strength of the 4-spin interactions. We attribute the strong first order character of this transition to the spinless nature of the core of point-like Z(3) vortices obtained in the staggered dimer state. Unlike in the case of a columnar dimer state, disordering such vortices in the staggered dimer state does not naturally lead to magnetic order, suggesting that, in this model, the dimer and Neel order parameters should be thought of as independent fields as in conventional Landau theory. Comment: 13 pages, 10 figs

Full text

Page 1

arXiv:1012.4546v2 [cond-mat.str-el] 22 Dec 2010
N´ eel to staggered dimer order transition in a generalized honeycomb lattice
Heisenberg model
Argha Banerjee and Kedar Damle
Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Mumbai, India 400 005
Arun Paramekanti
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada and
Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
We study a generalized honeycomb lattice spin-1/2 Heisenberg model with nearest-neighbor anti-
ferromagnetic 2-spin exchange, and competing 4-spin interactions which serve to stabilize a staggered
dimer state which breaks lattice rotational symmetry. Using a combination of quantum Monte Carlo
numerics, spin wave theory, and bond operator theory, we show that this model undergoes a strong
first-order transition between a N´ eel state and a staggered dimer state upon increasing the strength
of the 4-spin interactions. We attribute the strong first order character of this transition to the
spinless nature of the core of point-like Z3 vortices obtained in the staggered dimer state. Unlike
in the case of a columnar dimer state, disordering such vortices in the staggered dimer state does
not naturally lead to magnetic order, suggesting that, in this model, the dimer and N´ eel order
parameters should be thought of as independent fields as in conventional Landau theory.
I.INTRODUCTION
Phase transitions between two phases of matter distinguished by symmetry properties are usually well-described by
Landau theory. In its simplest form, Landau theory expresses the free energy of the system as an analytic function
of the order parameter whose value captures the symmetry breaking inherent in the ordered state. All analytic
terms consistent with the symmetries of the microscopic Hamiltonian are included in this free energy function, with
coefficients that are undetermined functions of the microscopic interactions in the system. The phase of the system in
this description is obtained by minimizing this free energy function over different values of the order parameter. In this
description, phase transitions to the broken symmetry long-range ordered phase are driven by changes in the values of
the coefficients of various terms, which change the position of the minimum to a non-zero value of the order parameter
at the phase transition. Symmetry considerations, which dictate the form of the terms allowed in the Landau free
energy function, then allow one to decide whether a particular phase transition is generically a first order transition or
second-order in nature. For instance, for systems with a global Z2symmetry and a scalar order parameter (such as the
Ising model), this Landau theory approach correctly predicts that the transition to the symmetry breaking long-range
ordered phase is generically a second-order transition, with the order parameter growing continuously from zero at
the transition. On the other hand, when the two phases on either side of the transition break different symmetries,
and thereby possess different order parameters, the prediction of Landau theory in the generic case is that the change
from one phase to the other proceeds either by via multiple transitions (going through an intermediate phase with
coexisting orders or with both orders being absent), or via a direct first-order transition with one order parameter
abruptly jumping to zero and the other abruptly becoming non-zero at precisely the same transition point.
Recent work by Senthil and co-authors1,2suggests that such a Landau theory approach is misleading for a class
of quantum phase transitions between Neel ordered antiferromagnets and valence-bond ordered paramagnets, most
notably in two-dimensional square-lattice antiferromagnets. In this case, Landau theory would proceed by writing
down the free energy function as an expansion in the Neel order parameter n and the valence-bond order parameter ψ.
Since one of these lives in spin-space, and the other represents order in real-space, Landau theory considerations would
predict a first-order transition or an intermediate phase in the generic case. Such an intermediate phase appears likely
for instance on the honeycomb lattice, when an additional next-nearest neighbour exchange coupling destroys the Neel
ordering of the nearest neighbour Heisenberg antiferromagnet3. However, Ref 1 argues that the transition can in fact
be a generically continuous transition, and is better described in terms of ‘deconfined’ spinon variables rather than the
order parameter fields of Landau theory. This failure of Landau theory has been ascribed to the presence of crucial
Berry phase terms in the action written in terms of the order parameter fields, and it is conjectured that when these
are correctly taken into account, one arrives naturally at a description in terms of ‘deconfined’ spinons interacting
with a U(1) gauge field. This continuum description leads to a prediction of a direct second order transition between
the two phases if certain monopole operators allowed in the theory are actually irrelevant at the fixed point describing
the transition, rendering the emergent gauge field effectively non-compact. This was conjectured to be the case for the
transition from the Neel ordered antiferromagnet to a four-fold symmetry breaking valence bond solid phase in square
lattice antiferromagnets. Various numerical works on a particular spin model with multiple-spin interactions appears

Page 2

2
to lend support to these field theoretic ideas. Furthermore, even in cases where these monopole operators are not
irrelevant, this approach suggests that the transition would be weakly-first order, allowing one to use this continuum
description to describe the physics at all but the largest length scales at which the weakly-first order nature of the
transition asserts itself.
A more intuitive view of ‘deconfined criticality’ was provided in subsequent work by Levin and Senthil4who came up
with a simple picture for the spin-1/2 spinon variables and the U(1) gauge field that make up the basic ingredients for
the correct description of a generically continuous transition between a Neel ordered antiferromagnet and a columnar
valence-bond solid (VBS) ordered paramagnet on the square lattice. The basic idea was that a Z4vortex formed by
breaking up the sample into four domains of solid order meeting at the vortex core necessarily contains a free spin-1/2
variable localized at its core. The core energy of these vortices is expected to decrease upon approaching the vicinity
of the transition to the antiferromagnetic phase. Under the assumption that the Z4anisotropy is irrelevant, the phase
of the order parameter winds continuously around the core (as in U(1) vortices). It is then natural to think of this
transition to the antiferromagnet in terms of the proliferation of these U(1) vortices, and write down a theory for the
transition in these vortex variables. Since solid order is destroyed by the proliferation of these vortices, the destruction
of the solid order is accompanied by establishment of spin ordering corresponding to the condensation of these spin-1/2
degrees of freedom in the cores of the proliferating vortices. Reasoning in this manner, Levin and Senthil were able
to deduce the form of the continuum theory for the resulting phase transition from symmetry arguments and these
simple intuitive considerations.
Motivated by this simple argument, it seems reasonable to make the following supposition: If the lowest core-energy
vortices in the valence-bond order have a free spin in their cores, then the transition to the adjacent antiferromagnetic
phase may be expected to admit a natural description in spinon variables, of the type developed in Ref. 1. This would
l
i
k
j
l
ij
k
m
n
FIG. 1: A four-spin operator in the JQ model on the honeycomb lattice (left panel) and a six-spin operator (right panel)in the
staggered JQ3 model on the square lattice.
imply either a generically continuous phase transition if the Znspatial anisotropy (with n = 4 for the square lattice)
is irrelevant at the critical fixed point, or a weakly-first order transition if this anisotropy turn out to be relevant. On
the other hand, if the lowest core-energy vortices have no free spin in their core, there is no natural way to obtain
magnetic order by proliferating vortices in the dimer order; one may then expect standard Landau theory to be valid,
and the transition to a nearby antiferromagnetic state would then proceed via an intermediate phase or be strongly
first order.
Some evidence for this point of view has already been provided by recent work on models that exhibit a strongly
first order transition between a ‘staggered’ valence bond solid order and Neel order on the square lattice5,6in a model
with nearest neighbour exchange and competing 6 spin interactions on the square lattice:
H = −J
?
?ij?
Pij− Q3
?
?ij,kl,mn?
PijPklPmn,(1)

Page 3

3
FIG. 2: Cartoon of vortices in staggered valence bond nematic on the honeycomb lattice (left panel) and the staggered valence
bond solid (right panel) on the square lattice. Note that one is not forced to have a free spin in the core of these vortices.
where Pijdenotes a bipartite singlet projector, Pij=1
are represented pictorially in Fig. 1 (right panel).
As is clear from Fig 2 (right panel), the simplest caricatures of vortices in the staggered valence bond solid order on
the square lattice are indeed without any free spins at their core. In the staggered case, it is also possible to construct
vortices with spins at their cores, but these are expected to cost more energy due to additional singlet formation
cost associated with leaving one spin free in the core, as is clear from Fig 3 (right panel). This suggests that the
transition would be strongly first order, which is consistent with recent numerical works5,6. This should be contrasted
with studies of very similar “JQ” models on the square lattice, with the plaquette interactions chosen to favour
columnar order7. Extensive numerical work on these models has led to the conclusion that the transition from N´ eel to
columnar valence bond solid order is indeed a continuous transition,8–10although the critical point exhibits logarithmic
violations of standard finite size scaling8,11. These violations of scaling have also been interpreted as possible evidence
of first order behaviour12. SU(3) and SU(4) versions of the same transition have also been studied9,13,14and found
to be continuous in nature.
Here we focus on a related system on the honeycomb lattice: On the honeycomb lattice, the staggered ordering
of near-neighbour valence bonds does not break lattice translation symmetry, but does break symmetry of three-fold
lattice rotations. As is clear from Fig 2 (left panel), the simplest construction of elementary Z3 vortices in this
valence-bond nematic order naturally yields vortices with spinless cores. Again, vortices with spinful cores are also
possible, but are expected to cost more core-energy due to two reasons. First, forming a spinful vortex core shown
in Fig 3 (left panel) involves the breaking of additional singlet bonds, and would cost an additional core energy of
the order of the spin gap. At the same time, the spinless vortex core, shown in Fig. 2 (left panel), allows for dimer
resonances on the core plaquette which is expected to lead to core energy lowering.
The transition between such a valence-bond nematic and a nearby antiferromagnetic phase on the honeycomb lattice
is thus expected to be strongly first order based on our supposition above. Here we focus on a spin Hamiltonian with
nearest neighbour exchange couplings that compete with four-spin interaction terms that stabilize a valence-bond
nematic state. We study this system numerically by a projector Monte Carlo algorithm in the valence-bond basis, as
well as analytically by spinwave and bond-operator expansions. Our numerics finds a strongly first order transition
between a valence-bond nematic and a antiferromagnet, consistent with the intuitive picture outlined above. In
addition, we use our approximate analytical calculations to provide a reasonable semi-quantitative account of our
numerical results. We note that an exactly soluble model with such a valence bond nematic ground state has also
been proposed recently.15
The remainder of the paper is organized as follows: In section II, we define the model studied, and describe the
approximate analytical methods that we use for their study. In section III, we describe our numerical approach and
summarize the results obtained. Finally, we conclude with a brief discussion in section IV.
4−Si·Sj. The plaquette interactions Q3in the formula above

Page 4

4
FIG. 3: Vortices with free spins in their cores are also possible in staggered valence bond nematic on the honeycomb lattice
(left panel) and the staggered valence bond solid (right panel) on the square lattice, but are expected to cost higher core energy
than ones with spinless cores.
II.MODEL AND ANALYTICAL TREATMENT
The honeycomb lattice staggered JQ model Hamiltonian is defined as folows,
H = −J
?
?ij?
Pij− Q
?
?ij,kl?
PijPkl,(2)
where the four spin plaquette operators are products of two bipartite singlet projectors acting on two parallel bonds
of the hexagons in the honeycomb lattice (see Fig. 1(left panel)). This Hamiltonian can be written explicitly as a sum
of two-spin and four-spin interactions as follows,
HJQ = H2+ H4
H2 = ˜J
(3)
?
?i,j?
Si· Sj
(4)
H4 = −Q
?
?
?
(S?
1· S?
2)(S?
4· S?
5) + (S?
2· S?
3)(S?
5· S?
6) + (S?
3· S?
4)(S?
6· S?
1)
?
(5)
where?
labelling the 6 sites clockwise around the hexagon, and˜J = J + Q/2.
At a heuristic level, we expect that the Q term drives a transition to a valence bond nematic state with staggered
ordering of near-neighbour singlet bonds, while the˜J term favours a Neel state with spins on the A-sublattice (B-
sublattice) of the honeycomb lattice aligned (anti-aligned) along a spontaneously chosen axis in spin space. A simple
approximate description of the N´ eel phase is obtained by doing a standard spin wave analysis. On the other hand, to
describe the valence-bond nematic, we use a description in terms of bond-operators that becomes very accurate in the
limit of strong nematic order. By comparing the predictions for the ground state energy from these two approximate
calculations, we also estimate the value of Q at which we expect a transition between the two phases.
?denotes a sum over all hexagons of the lattice, S?
i
denotes the i-th spin on a hexagon, with i = 1-6
A.Spin wave theory of the Neel state
In the N´ eel ordered phase of the JQ model defined above, let us assume we can decouple the four spin interactions
as
(S?
1· S?
2)(S?
4· S?
5) = −α(S?
1· S?
2+ S?
4· S?
5) − α2
(6)

Page 5

5
where α ≡ −?Si· Sj? is the nearest neighbor spin correlation which is assumed to be the same on all neighboring
bonds and which needs to be determined self-consistently. Such a decomposition can alternatively be obtained within
a path integral formalism by using a Hubbard Stratonovich (HS) decoupling of the quartic spin operators followed by
a static mean field theory of the HS field. In this case, the Hamiltonian reduces to the form
Hmf= Jeff
?
?i,j?
Si· Sj+ 3Qα2NH
(7)
where we have defined Jeff =˜J + 2Qα, and NH is the number of unit cells on the lattice (the number of spins is
NS= 2NH).
We treat this effective Hamiltonian using the standard spin wave theory by defining Holstein Primakoff bosons via
(Sz
respectively. Retaining terms to quadratic order in the bosons leads to the Hamiltonian
A,S+
A,S−
A) ≡ (1
2− a†a ,a ,a†) and (Sz
B,S+
B,S−
B) ≡ (b†b −1
2,b†,b ) for sites on the A-sublattice and B-sublattice
Hmf=Jeff
2
?
k
?
a†
kb−k
??3Γk
3
Γ∗
k
? ?ak
b†
−k
?
−3
4JeffNH+ 3Qα2NH−Jeff
2
?
k
3(8)
where Γk≡ (1 + e−ikb+ e−ika−kb) with ka = k · ˆ a and kb = k ·ˆb. Setting Γk= |Γk|eiγk, we can diagonalize this
Hamiltonian via a Bogoliubov transformation
?ak
b†
−k
?
=
?coshθkeiγk
−sinhθk
−sinhθk
coshθke−iγk
??ck
d†
−k
?
(9)
where sinh2θk= |Γk|/Ωk, cosh2θk= 3/Ωk, and Ωk=?9 − |Γk|2. This yields the diagonalized Hamiltonian
Hmf = −3
4JeffNH+ 3Qα2NH+Jeff
2
?
k
(Ωk− 3) +Jeff
2
?
k
Ωk(c†
kck+ d†
kdk)(10)
with a ground state energy
ESW= −3
4JeffNH+ 3Qα2NH+Jeff
2
?
k
(Ωk− 3)(11)
Applying the Feynman-Hellman theorem to the mean field Hamiltonian, Hmfin Eq.7, yields
α = −
1
3NH
∂ESW
∂Jeff
=1
4−
1
6NH
?
k
(Ωk− 3)(12)
Eqns. (11) and (12), together, determine the ground state energy of this model in the N´ eel ordered phase. A feature
of our mean-field spin-wave analysis is that α is independent of Q/˜J. Numerically, we find α ≈ 0.3549 within this
spin wave approach which agrees with earlier spin wave results16. This value is quite close to the value, α ≈ 0.3627,
deduced from a recent optimized valence bond trial wave function study17of the nearest neighbor Heisenberg model,
and series expansion results18on the Heisenberg model which give α ≈ 0.3659. This spin wave result for α must be
used as input to compute the ground state energy per spin of the J-Q model which is given by eSW= −3
this is plotted in Fig.4. Within this approach, the ordered moment is also independent of Q/˜J, and is given by
2(˜J +Qα)α;
m =1
2−
1
2NH
?
k
(3
Ωk
− 1)(13)
and takes on a value m ≈ 0.2420, whereas series expansion studies of the nearest neighbor Heisenberg model18yield
m ≈ 0.266. In the next section, we will check to what extent the numerical results bear out our prediction of a nearly
Q-independent value of m in the N´ eel phase.
B.Bond operator theory of the staggered dimer state
In the other limit, when the Q term dominates over˜J Hamiltonian at large Q has spontaneous staggered dimer
order in the ground state, we can use a bond operator formalism to compute the ground state energy and correlations

Page 6

6
in this phase. We label the spins by their unit cell position r = mˆ a + nˆb and a sublattice index p = 1,2. In the
staggered dimer state, let us assume that the spins at sites (r,1) and (r,2) in every unit cell form a singlet. This dimer
pattern then spontaneously breaks the lattice rotational symmetry but leaves the translational symmetry intact. We
define the singlet and triplet states on this pair of sites in terms of bond operators via
|s? =
1
√2(↑1↓2− ↓1↑2) ≡ s†|0?
1
√2(↑1↓2+ ↓1↑2) ≡ t†
−1
√2(↑1↑2− ↓1↓2) ≡ t†
i
√2(↑1↑2+ ↓1↓2) ≡ t†
(14)
|z? =
z|0?(15)
|x? =
x|0? (16)
|y? =
y|0? (17)
where we must satisfy the constraint s†
the bond operators as
rsr+ t†
rαtrα= 1 at each site. The spin operators can be rewritten in terms of
Sα
r,p=1
2(−1)p?s†
rtrα+ t†
rαsr
?−i
2ǫαβγt†
rβtrγ
(18)
where α = x,y,z. Assuming the staggered dimer state corresponds to a uniform singlet condensate, we can replace
s†
r→ ¯ s and sr→ ¯ s. Furthermore, if the singlet condensate is robust, so that the dimer order is strong as observed in
the QMC numerics, we expect the triplet density to be small and a Bogoliubov-type theory, where the triplet density
plays the role of a small parameter, to be a reasonable starting point. Expanding out the Hamiltonian in terms of
these bond operators, and retaining leading quadratic terms in the triplet operators, we find
HBO= (
˜J
4−λ)
?
r
t†
rαtrα+3Q
16¯ s2?
r
(t†
rαtrα+ t†
r+ˆ a,αtr+ˆ a,α)
− ˜J¯ s2
4
?
r
(t†
rα+trα)(t†
r−ˆb,α+tr−ˆb,α+t†
r−ˆ a−ˆb,α+tr−ˆ a−ˆb,α)
−
3
4
˜J¯ s2NH−9
16Q¯ s4NH− λNH(¯ s2− 1)(19)
where λ is a chemical potential which serves to satisfy the constraint equation on average by fixing ?t†
Going to momentum space, the Hamiltonian takes the form
rαtrα? = 1− ¯ s2.
HBO= E(0)+
′?
k
?
t†
kαt−kα
??A + Bk
Bk
Bk
A + Bk
??tkα
t†
−kα
?
−3
2ANH
(20)
where
E(0)= −(3˜J
4¯ s2+9Q
4+3Q
˜J
2¯ s2(coskb+ cos(ka+ kb))
16¯ s4)NH− λNH(¯ s2− 1) (21)
A = (
˜J
8¯ s2− λ)(22)
Bk = −
(23)
and the prime on the momentum sum indicates that we only sum over half the Brillouin zone (keeping states with
ka> 0 for example). Diagonalizing this via a Bogoliubov transform
?tkα
t†
−kα
?
=
?
coshθk
−sinhθk
−sinhθk
coshθk
??ckα
d†
−kα
?
(24)
we find that the diagonal Hamiltonian takes the form
H = E(0)+
′?
k
ωk(c†
kαckα+ d†
kαdkα+ 3) −3
2NH(
˜J
4+3Q
8¯ s2)(25)

Page 7

7
Dimer (BO)
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
0.00.5 1.01.5
Q / J~
2.0 2.53.0
EI
JNs
Neel (SW)
~
FIG. 4: (Color online) Energy per spin (in units of˜J) of the N´ eel state obtained in spin wave theory (SW, red-solid line)
compared with the staggered dimer ordered state obtained using bond operator theory (BO, blue-dotted line). There is a
strong first order transition at Q/˜J ≈ 1.35 (Q/J ≈ 4.2).
where ωk≡?A(A + 2Bk). This leads to a ground state energy
EBO= E(0)+ 3
′?
k
ωk−3
2NH(
˜J
4+3Q
8¯ s2).(26)
This energy must be minimized with respect to ¯ s once λ is determined by the number equation
3
2NH
?
k
(A + Bk
ωk
− 1) = 1 − ¯ s2
(27)
which follows from demanding ∂EBO/∂λ = 0 in order to satisfy the constraint relation ?t†
Solving these equations numerically, we self-consistently determine ¯ s and λ. Using these, we compute the ground
state energy and excitation spectrum of the valence bond nematic state. As seen from Fig. 4, we find that the energy
of this state lies below the antiferromagnetically ordered state for large Q, as we expect, but there is an energy level
crossing at Q/˜J ≈ 1.35 (Q/J ≈ 4.2), so that the N´ eel state has lower energy at smaller Q. (Within the bond operator
approach, we find that the triplons condense around Q/˜J ≈ 0.2 which is well below the first order transition point.)
Close to the transition, in the spin gapped phase, we find the spin-spin correlation on the dimerized bond to be
≈ −0.73 which corresponds to nearly complete dimerization, so that this is a very strong first order transition. We
next turn to a numerically exact quantum Monte Carlo study of this model.
rαtrα? = 1 − ¯ s2.
III. NUMERICAL STUDY
We use the valence bond projector Monte Carlo technique19to study various ground state properties of the staggered
version JQ model on honeycomb lattice as a function of coupling Q/J. Taking advantage of the improved loop-update
technique developed in Ref. 20, we scan the phase diagram of this model for systems with upto 2 × 32 × 32 sites. A
Monte Carlo projection length of 6L3is used to ensure the convergence of observables to the ground state expectation
values (we have checked the convergence by comparing with the results of exact diagonalization studies at small sizes).
Also, to counter ergodicity problems in the Monte Carlo simulations in the valence bond nematic phase and in the

Page 8

8
vicinity of the first order transition, we employ many different Monte Carlo moves to update the bond and plaquette
operators in a Monte Carlo configuration. These updates include attempts to reflect a plaquette operator about its
diagonal bond (Fig.5a), to rotate the diagonal plaquette operators about a site (Fig.5b), to transform a plaquette
operator into a bond operator by deleting a bond and vice versa (Fig.5c), and so on.
z z
i
(1/4 − S . S )j
z z
i
(1/4 − S . S )j
or
ij
i
j
x x
y
(S . S + S. S )
y
−
a)
b)
c)
FIG. 5: Monte Carlo moves to update plaquette operators: a) reflecting a plaquette operator about a bond with a diagonal
operator acing on it, b) rotating a diagonal plaquette operator about a site, and c) transforming a diagonal or off-diagonal
bond operator into a plaquette operator and vice versa.
We find it convenient to define the valence bond nematic order parameter, ψ(r) = S(r)·S(r+e1)+exp(i2π/3)S(r)·
S(r+e2)+exp(i4π/3)S(r)·S(r+e3), where e1,2,3denote the three nearest neighbor bond vectors of the honeycomb
lattice. By monitoring the correlations of ψ simultaneously with those of the spins, we are able to distinguish easily
between the Neel antiferromagnet, and the rotation-symmetry breaking valence bond nematic. To do this we measure
C(l) =
1
2Ns
?
r∈A
?n(r) · n(r + l)?, (28)

Page 9

9
where n(r) = S(r) − S(r + e1), and the sum over r runs over all A-sublattice sites of the lattice. We also define
1
Ns
r
D(l) =
?
?ψ∗(r)ψ(r + l)?, (29)
where r = (x,y), and l = (L/2,L/2) with L being the linear dimension of the system and Ns= 2L2being the number
of sites in the system. In the presence of long range antiferromagnetic order, the asymptotic large size limit of C(l)
tends to a finite value, which equals the square of the staggered magnetisation
M2=
1
N2
s
?
r∈A,r′∈A
?n(r) · n(r′)?. (30)
In the absence of long range antiferromagnetic order, we expect this quantity to vanish in the large size limit. Similarly,
D(l) serves as a order parameter for long range valence bond nematic order.
A. Results for honeycomb lattice model
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
C(l) |D|(l)+0.15
Q/2J
16
24
32
FIG. 6: Simultaneous first order jump in N´ eel order parameters C(l) and valence bond nematic order parameter D(l), as Q/J
is varied for the JQ model on the honeycomb lattice.
From our numerical study of these observables, we deduce that a transition from the Neel phase to the valence bond
nematic phase takes place as Q/J is increased to the vicinity of Q/J ∼ 7. In this vicinity, there are clear first order
jumps in both the order parameters as shown in Fig.6. At first sight, the data suggests that the transition point, as
determined from the location of the jump, may have a tendency to drift with increasing system size. However, this is

Page 10

10
actually an artifact of the slowness of the code near the first order transition region at larger sizes, as is clear from an
alternate more precise analysis which we now summarize. In this alternative approach, we determine the transition
point by locating the kink in the plot of energy as function of tuning parameter at smaller sizes, where there are no
such issue of slowness of the Monte Carlo update procedure used for calculating ground state properties. As shown
in Fig. 7, the energy per site shows a linear behaviour, but with slopes of the line changing discontinuously near
the transition. This can be understood as an avoided level crossing taking place at the first order transition point.
Identifying the crossing point of these two linear extrapolations for the energy on is a convenient way to determine the
transition point5with reasonable accuracy. Using this method, with data from three different system sizes L = 12,16
and 24, we identify the first order transition to be at Q/J = 6.4 ± 0.2.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
4.2 5.2 5.6 6 6.4 6.8 7.2 7.6
Q/J
8
energy + a + b Q/J
L=12
L=16
L=24
FIG. 7: An avoided level crossing of the two competing candidate ground states, showing up as a kink in the ground state
energy of the honeycomb lattice JQ model, when plotted as a function of the tuning parameter Q. This serves as clear signature
of the first order transition taking place in this model system. Note that the energy difference from the fitted linear behaviour
on the antiferromagnetic side is plotted, not the absolute ground state energies on both sides - this is just to highlight the
discontinuity of slopes of the two curves at the transition point. To obtain linear fits to energy values on the valence bond
nematic side, the data points at Q/J < 6.8 have not been considered.
First order transitions are associated with two coexisting free energy minima corresponding to the two phases.
This leads to the appearance of hysteresis effects in the vicinity of the transition for all but the most efficient Monte
Carlo algorithms when a configuration from one of the phases is evolved by tuning the coupling constant across the
critical value at finite rate (i.e without allowing for an extremely large number of equilibriation steps between each
successive change of coupling constant). For example, starting with a configuration deep in the valence bond nematic
phase, the system remains stuck in the metastable valence bond nematic state even when Q is reduced to less than
Qcand conversely, the system remains in the Neel phase upon ramping Q up beyond Qcif one starts with a typical
configuration obtained from the projection algorithm deep in the Neel phase. We present such a hysteresis plot for
antiferromagnetic order parameter (Fig.8) near the transition to emphasize the first order nature of the transition.
Our QMC simulations also yield a characteristic double peaked order parameter histogram near Qc as shown in

Page 11

11
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
3 3.2 3.4 3.6 3.8 4
<M2>
Q/2J
L = 24
FIG. 8: Hysteresis of ?M2? around transition point for honeycomb lattice JQ model.
Fig. 9. However, the ‘time’ (number of Monte Carlo steps) taken for the system to ‘tunnel’ between the two phases
is extremely large, and it is therefore prohibitively expensive to perform runs that could encompass a large number
of such tunneling events. As a result, it is not possible with our limited computational resources to obtain accurate
estimates of the relative heights of the two peaks in the histogram.
Finally, we note that in a variant of staggered JQ2model on honeycomb lattice, where parallel bonds from neigh-
bouring hexagons make up the plaquette interactions, we find that increasing Q fails to destroy AF order. For this
model, as shown in Fig. 10, there is no hint of a transition even at large Q/J.
IV. DISCUSSION
We have thus studied an example of a transition between a Neel ordered antiferromagnet and a three-fold rotation
symmetry breaking valence bond nematic on the honeycomb lattice. Our numerical results showed that the transition
is strongly first order, consistent with the general scenario outlined in the introduction. The transition point obtained
from our analytical considerations is at Q/˜J ≈ 1.35 (Q/J ≈ 4.2), which is in reasonable agreement with the results
of our QMC simulations, which give a strong first order transition at Q/˜J ≈ 1.5 (Q/J ≈ 6.4). Furthermore, the
nearly Q independent values of the staggered magnetization M in the N´ eel phase, and of the nematic order |D| in
the staggered dimer state, are also consistent with our analytic considerations. These observations suggest that the
bond operator approach provides a reasonable account of the dimer phase in situations where the kinetic fluctuations
of dimers are not important in the vicinity of the transition. This appears to be different from the N´ eel to columnar
dimer transition in the square lattice case where such singlet fluctuations appear to be important in driving the dimer
order to zero at the quantum phase transition.21

Page 12

12
0
0.1
0.2
0.3
0.4
0.5
0.6
0
1/12 1/6 1/4
|D|
1/3 5/12 1/2
P(|D|)
L=24, Q/2J=3.75
Q/2J=4.00
Q/2J=3.00
FIG. 9: Histogram of valence-bond-nematic order parameter |D| ≡ |D|(l) for L=24 at three different Q values for honeycomb
lattice JQ model. Presence of double peak structure at Q/J = 7.5 indicates first order nature of transition.
V.ACKNOWLEDGEMENTS
We acknowledge computational resources at TIFR, grant support (KD) from the Indian DST (DST SR/S2/RJN-
25/2006), NSERC of Canada (AP), and an Early Researcher Award from the Government of Ontario (AP), and thank
Arnab Sen and Anders Sandvik for useful discussions.
1T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004).
2T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004).
3A. Mulder, R. Ganesh, L. Capriotti, and A. Paramekanti, Phys. Rev. B 81, 214419 (2010).
4M. Levin, and T. Senthil, Phys. Rev. B 70, 220403 (2004).
5A. Sen, and A. W. Sandvik, arXiv:1009.0691.
6A. Banerjee, Ph.D thesis, unpublished.
7A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).
8A. W. Sandvik, Phys. Rev. Lett. 104, 177201 (2010).
9J. Lou, A. W. Sandvik, and N. Kawashima, Phys. Rev. B 80, 180414 (2009)
10R. G. Melko and R. K. Kaul, Phys. Rev. Lett. 100, 017203 (2008)
11A. Banerjee, K. Damle, and F. Alet, Phys. Rev. B 82, 155139 (2010).
12F. J. Jiang, M. Nyfeler, S. Chandrasekharan, and U. J. Wiese, J. Stat. Mech.: Theory Exp. (2008) P02009.
13R. K. Kaul, arXiv:1010.1937, unpublished (2010).
14A. Banerjee, K. Damle, and F. Alet, arXiv:1012:3691.

Page 13

13
0
0.025
0.05
0.075
0.1
0.125
0 1 2 3 4
Q
5 6 7 8
<M2>
L = 16
FIG. 10: Order parameters ?M2? as Q/J is varied for the JQ honeycomb model with plaquettes made up of parallel bonds
from neighbouring hexagons, showing no sign of a transition with increasing Q/J.
15R. Kumar, D. Kumar, B. Kumar, Phys. Rev. B 80, 214428 (2009).
16J. D. Reger, J. A. Riera, and A. P. Young, J. Phys. C 1, 1855 (1989).
17Z. Nourbakhsh, F. Shahbazi, S. A. Jafari, and G. Baskaran, J. Phys. Soc. Jpn. 78, 054701 (2009).
18J. Oitmaa, C. J. Hamer. and Z. Weihong, Phys. Rev. B 45, 9834 (1992).
19A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).
20A. W. Sandvik, and H. G. Evertz, Phys. Rev. B 82, 024407 (2010).
21V. N. Kotov, D. X. Yao, A. H. Castro-Neto, D. K. Campbell, Phys. Rev. B 80, 174403 (2009).