Motivated by the unabating interest in the spin-1/2 Heisenberg antiferromagnetic model on the kagome lattice, we investigate the energetics of projected Schwinger-boson (SB) wave functions in the J_(1)-J_(2) model with antiferromagnetic J_(2) coupling. Our variational Monte Carlo results show that Sachdev’s Q_(1)=Q_(2) SB ansatz has a lower energy than the Dirac spin liquid for J_(2) ≳ 0.08J_(1) and the q=0 Jastrow-type magnetically ordered state. This work demonstrates that the projected SB wave functions can be tested on the same footing as their fermionic counterparts

Motrunich, Olexei I.

Tay, Tiamhock

English

Caltech Authors - Main

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PHYSICAL REVIEW B 84, 020404(R) (2011)
Variational study of J1- J2 Heisenberg model on kagome lattice using projected Schwinger-boson
wave functions
Tiamhock Tay() and Olexei I. Motrunich
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 31 March 2011; published 13 July 2011)
Motivated by the unabating interest in the spin-1/2 Heisenberg antiferromagnetic model on the kagome
lattice, we investigate the energetics of projected Schwinger-boson (SB) wave functions in the J1-J2 model with
antiferromagnetic J2 coupling. Our variational Monte Carlo results show that Sachdev’s Q1 = Q2 SB ansatz has
a lower energy than the Dirac spin liquid for J2  0.08J1 and the q = 0 Jastrow-type magnetically ordered state.
This work demonstrates that the projected SB wave functions can be tested on the same footing as their fermionic
counterparts.
DOI: 10.1103/PhysRevB.84.020404 PACS number(s): 75.10.Kt, 75.10.Jm, 75.40.Mg
The Heisenberg antiferromagnet on the kagome lattice has
long been anticipated to realize a spin liquid (SL). Recently,
herbertsmithite ZnCu3(OH)6Cl2,1–4 which contains kagome
layers of spin-1/2 moments, has emerged as an experimental
candidate with no sign of any ordering down to 50 mK.
Interest in SLs on the kagome lattice has been reignited by
recent works,5–8 where density matrix renormalization group
(DMRG) studies find a spin-disordered ground state with a
small gap. A vast review of earlier literature is revisited in
Ref. 7. Preliminary DMRG data suggests that the system
moves deeper into the spin-liquid phase upon adding small
antiferromagnetic second-neighbor J2 coupling,9 and this
supports earlier exact diagonalization (ED) study that found
an increase in the gap for J2 up to 0.1.6
In an early study of the nearest-neighbor model using
large-N treatment of Schwinger-boson (SB) slave particles,
Sachdev10 found that condensation of spinons gives rise to
magnetically ordered ground states for spin S > 0.26. A
subsequent variational study by Sindzingre et al.11 using
resonating valence-bond (RVB) wave functions interpolating
between spin liquids and magnetically ordered states instead
found that the former have lower energies. However, the spin
correlations beyond first neighbors do not agree well with
ED results.11,12 More recently, the trial energy of the projected
Dirac spin liquid constructed using the fermionic slave particle
approach was found to lie very close to the ED ground-state
energy in the nearest-neighbor Heisenberg model,13,14 and a
very recent study extended this to the presence of second-
neighbor coupling J2.15 However, the observation of an energy
gap in the DMRG and ED studies suggests that the gapless
Dirac spin liquid may lose some ground in the presence ofJ2.6,9
To investigate this possibility, we study the energetics of a
class of projected SB wave functions in the J1-J2 Heisenberg
model with Hamiltonian
ˆH = J1
∑
〈i,j〉
Si · Sj + J2
∑
〈〈i,j〉〉
Si · Sj , (1)
where 〈i,j 〉 and 〈〈i,j 〉〉 denotes first- and second-neighbor
pairs. We set J1 = 1 as the unit for energy and consider only
antiferromagnetic J2  0. The SB representation of a spin S
is given by16
S = 1
2
∑
σ,σ ′
b†σσ σσ ′bσ ′ , κ =
∑
σ
b†σ bσ = 2S, (2)
where bσ is a bosonic operator, σ are Pauli matrices,
and κ is the number of bosons per site. The Hamiltonian
becomes quartic in the bosonic operators, and upon mean-field
decoupling10,16,17 leads to the following:
ˆHm.f. = 12
∑
i,j
(Aijb†i↓b†j↑ + H.c.) +
∑
i,j
|Aij |2
2Jij
−μ
∑
i
(∑
σ
b
†
iσ biσ − κ
)
, (3)
Aij = 12Jij
∑
σ,σ ′
σσ ′ 〈biσ bjσ ′ 〉,κ =
∑
σ
〈b†iσ biσ 〉. (4)
We treat the “pairing amplitudes” Aij and “chemical potential”
μ as variational parameters. Equation (4) are self-consistency
relations in the mean field. Figure 1 shows Sachdev’s ansa¨tze
for Aij , which have good mean-field energies in the J1-only
model.10 The nearest-neighbor Aij are real, and their signs
are positive if an arrow points from site i to j . Following
Sindzingre et al.,11 we label these as q = 0 and √3 ×√
3 SB ansa¨tze, which correspond to Sachdev’s Q1 = Q2
and Q1 = −Q2, respectively10,18 (also called [πHex,0Rhom]
and [0Hex,0Rhom] in Refs. 17 and 19). For the q = 0 SB
ansatz shown in Fig. 1(a), we introduce an additional real
parameter for the second-neighbor pairing, with the pattern
of arrows going clockwise in triangular loops for both first
and second neighbors. Second-neighbor Aij for the
√
3 × √3
SB ansatz are forbidden by its projective symmetry group
(PSG). In Ref. 17, Wang et al. found two more distinct
ansa¨tze for symmetric spin liquids, but they argued that the
Heisenberg model energies of those spin liquids are expected
to be considerably poorer. Our variational calculations confirm
that this is indeed true.
We first present the results of a crude study on the
accessibility of SB spin liquids at the mean-field level, by
computing the critical boson density
κc = −1 + 1
N
∑
|λα |=|μmax|
|μmax|√
μ2max − λ2α
, (5)
accessible in the mean field without Bose condensation in
the thermodynamic limit. Here {λα} are eigenvalues of the
matrix −i ˆA [cf. Eq. (6) below], μmax = −max{|λα|}, and N
020404-11098-0121/2011/84(2)/020404(4) ©2011 American Physical Society
RAPID COMMUNICATIONS
TIAMHOCK TAY AND OLEXEI I. MOTRUNICH PHYSICAL REVIEW B 84, 020404(R) (2011)
(a) q = 0 SB ansatz (b)
√
3×√3 SB ansatz
FIG. 1. (Color online) The SB ansa¨tze {Aij } from Ref. 10. The
unit cell is shaded for each ansatz. All equidistant Aij have identical
magnitudes, and Aij is positive if an arrow points from site i to j . For
the q = 0 ansatz, we extend Aij to include second-neighbor pairing.
The
√
3 × √3 ansatz has poorer energy for J2 > 0.
is the number of sites on the lattice. Below κc, the mean-field
excitation spectrum is gapped and gives rise to a stable spin
liquid. For κ  κc, the gap closes and magnetic ordering
results from spinon condensation. With only nearest-neighbor
Aij , κc ≈ 0.5 and 0.54 for the q = 0 and
√
3 × √3 SB
ansa¨tze, respectively.10,17 In analogy to Wang’s analysis for the
honeycomb lattice,21 Fig. 2 shows the critical boson density
κc vs A2/A1 for the q = 0 SB ansatz, where A1 and A2 are
the amplitudes of first- and second-neighbor Aij . We note
that κc > 1 in the parameter range −0.4 < A2/A1 < −0.18,
i.e., the second-neighbor pairing has opened up a disordered
regime relevant for S = 1/2.20
We now turn to a variational Monte Carlo (VMC) study
of the J1-J2 model on the symmetric 36-site cluster used
in previous numerical studies,11,12 which allows a direct
comparison with ED energies as well as the energies of
the Dirac SL and magnetically ordered states. We construct
projected SB wave functions as follows. Writing the real
antisymmetric matrix ˆA [see Eq. (4)] as
ˆA = i ˆM ˆ ˆM†, (6)
where ˆ is diagonal and ˆM is unitary, we solve the mean-field
Hamiltonian using Bogoliubov’s transformation and obtain the
 0
 1
 2
 3
-1 -0.5  0  0.5
κ
A2 / A1
q=0 magnetic order
q=0 spin liquid
new magnetic order
FIG. 2. (Color online) Mean-field “phase diagram” for the q = 0
SB ansatz (Ref. 20). The phase boundary separates spin liquid from
the magnetically ordered phases. A spin-liquid regime is present for
physical spin-1/2 systems with κ = 1. The new magnetic order is
complex and is not relevant for this paper.
following trial wave function:16
|	SB〉 = ˆPG exp
{∑
j,k
ujkb
†
j↑b
†
k↓
}
|0〉, (7)
ujk = i
∑
α
Mjαλα(M†)αk
−μ +√μ2 − λ2α . (8)
The Gutzwiller operator ˆPG enforces the constraint κ = 1 at
every site. Although this density may not be accessible to
a given ansatz and μ at the mean-field level (see Fig. 2),
the projected SB wave function is a valid variational state
in the physical Hilbert space. Here, ujk decays exponentially
for μ < μmax and the projected SB wave function realizes
a short-range RVB state when written in the valence-bond
basis. In the limit μ 
 μmax, Eq. (8) shows that the pattern
of ujk roughly follows that of Ajk . More generally, the PSG
of the q = 0 SB ansatz enforces the pattern of ujk postulated
in Sindzingre et al.,11 where ujk only connect sites on the
different “sublattices” A, B, C defined in the sense of the
q = 0 magnetic order in Fig. 3(a). At μ = μmax, ujk decays
in a power law ∼|rj − rk|−3; one can view μ → μmax as a
finite-size realization of magnetic orders.16,19
In our VMC simulations, the amplitude of each sampled
spin configuration is given by the permanent of theN/2 × N/2
matrix {ujk}, where j and k run over the spin-up and spin-down
sites, respectively. We make use of Ryser-Nijenhuis-Wilf
dense permanent algorithm to calculate the permanent;22 in
this way, the procedure does not have the sign problem
encountered in the valence-bond basis.11 Despite a poor
computational cost scaling ∼2N/2 with system size, it is
manageable for sizes that are already interesting.
We also consider magnetically ordered states shown in
Fig. 3 which arise from the condensation of spinons in the
respective ansatz.10 For both orderings, their classical nearest-
neighbor energies are identical, but the second-neighbor
energy is clearly lower for the q = 0 ordered state since it
has antiferromagnetic second-neighbor correlations while the√
3 × √3 state has ferromagnetic correlations. It is therefore
sufficient to consider only the former. We construct the
following trial wave function,
〈{
Szj
}∣∣	MOq=0〉 = exp
{
i
∑
j
φjS
z
j −
∑
ij
KijS
z
i S
z
j
}
, (9)
A
B C
A
B C
A
B C
C B
A
(a) q = 0 MO
A
B
C
B
C A
C
A B
B B
B
(b)
√
3×√3 MO
FIG. 3. Magnetic orderings (MO) which arise from spinon
condensation in q = 0 and √3 × √3 SB ansatze. A, B, and C are the
120◦ antiferromagnetic spin orientations. For J2 > 0, the
√
3 × √3
MO has poorer energy than the q = 0 MO.
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VARIATIONAL STUDY OF J1–J2 HEISENBERG . . . PHYSICAL REVIEW B 84, 020404(R) (2011)
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
 0  0.2  0.4  0.6  0.8  1
En
er
gy
 / 
J 1
J2 / J1
Dirac SL
q=0 SB
q=0 Jastrow MO
FIG. 4. (Color online) Comparison of trial energies per site
for Dirac SL, q = 0 SB wave function, and q = 0 Jastrow-type
magnetically ordered (MO) state. The SB state has poorer energy
than Dirac SL for J2/J1  0.08, but performs better for larger J2 and
better than the Jastrow-type MO for all J2.
where φj = {0, ± 2π/3} are the phase angles on the three
sublattices in Fig. 3(a), and Kij are two-body pseudopotentials
for the Jastrow factor. Such Jastrow-type wave functions are
widely used in VMC studies due to their simplicity. We allow
two variational parameters for the first- and second-neighbor
pseudopotentials, and two more for a power-law decay be-
tween further neighbors. We also consider a Huse-Elser23 type
of three-site phase factor allowed by the symmetry of the clas-
sical state, but it apparently does not improve the trial energy.
To get an idea of the variational energetics landscape, we
also include the Dirac SL constructed from fermionic spinons
hopping with flux π through hexagons and flux 0 through
elementary triangles.13,14 This state was extended in Ref. 15 to
include second-neighbor hopping such that triangles formed
by two nearest-neighbor bonds and one second-neighbor bond
have flux π . The amplitude of the second-neighbor hopping
provides a single variational parameter. For large sizes, we
reproduce results in Ref. 15 for J2 > 0; we perform VMC for
the present 36-site cluster and find negligible size dependence
on the scale in Fig. 4.
Figure 4 shows the variational energies of the Dirac SL, the
q = 0 SB wave function, and the q = 0 magnetically ordered
state. For the J1-only model, the Dirac SL has significantly
better energy than the SB state. However, the latter improves
quickly with J2 > 0, and becomes lowest for J2  0.08 among
the wave functions in this study. The q = 0 Jastrow-type MO
state has higher energy for all J2 values shown.
The q = 0 SB wave function has two variational parame-
ters, A2 and μ. For J2 = 0, the lowest energy of −0.420 per site
occurs at A2 ≈ −0.15. Interestingly, this A2 approaches the
spin-liquid window in Fig. 2 where we observe fairly narrow
spinon bands. We find antiferromagnetic correlations between
second-neighbor sites. These results are very close to those
obtained by Sindzingre et al. in a variational study of the
J1-only model,11 wherein they considered RVB ansa¨tze with a
few variational parameters for near neighbors ujk and a power-
law decay ∼|rj − rk|−p, with p = 5 for further neighbors. For
our projected state, μ optimizes very close to μmax, which
corresponds to formal p = 3 in the thermodynamic limit.
Despite the difference in the details of the realizations, both
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
 0  0.2  0.4  0.6  0.8  1
A 2
 
/ A
1
J2 / J1
FIG. 5. (Color online) Optimal A2/A1 vs J2/J1 for the q = 0
projected SB wave function, also optimized over μ for each J2.
are suggestive of a near-critical state at J2 = 0. Thus, Table V
in Ref. 11 indicates that such wave functions have significant
q = 0 correlations across the full 36-site cluster.
Figure 5 shows the optimal A2 against the second-neighbor
coupling J2. We find that A2 increases with J2 and is important
for improving the trial energy of the q = 0 SB wave function.
Beyond J2 ∼ 0.1, the optimal μ starts to decrease away from
μmax, e.g., it is 1.02μmax for J2 = 0.2 and moves further to
1.05μmax for J2 = 0.4.
Our energetics study reveals the q = 0 SB wave function
as a viable candidate for the J1-J2 Heisenberg model. This is
perhaps not surprising since this state is quite competitive in the
J1-only model11 and has antiferromagnetic second-neighbor
correlations which are favorable when J2 > 0 is added. The
q = 0 SB state can furthermore accommodate the J2 coupling
by varying A2. In the large J2 limit, the system breaks into three
independent kagome networks, each as difficult as the original
nearest-neighbor kagome problem. The large-A2 state in the
large J2 limit is just like the A1-only state in the J1-only model,
so while not the best, is again reasonably good in energy. Thus,
the A1-A2 ansatz provides a way to interpolate between the
small J2 and large J2 regimes and is an appealing candidate.
It wins against the Dirac SL and against the best Jastrow wave
function for the q = 0 MO, but it may also correspond to
possible MO at intermediate J2. It would be very interesting to
check our results against exact calculations in the J1-J2 model
on the 36-site cluster to assess the accuracy of the projected
q = 0 SB state.
From our SB wave-function study, we cannot address the
question whether the ground state is a spin liquid or has a
magnetic long-range order. It is known11,16 that the RVB wave
functions can realize both phases depending on the range of
the valence-bond amplitudes ujk . In the projected SB wave-
function setup, if μ is very close to μmax, this can be viewed as
a finite-size realization of the spinon condensation and hence
magnetic order. On the other hand, if μ is a finite distance away
from μmax, this gives exponentially decaying ujk and hence
short-range RVB spin liquid. While we find that μ optimizes
away from μmax for J2 > 0.1, these small-size results cannot
be used to establish the long-distance behavior, and the
ultimate phase determination must come from exact studies on
larger systems. Nevertheless, we hope that our demonstration
of the viability of the q = 0 SB wave function11,18,19,24
020404-3
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TIAMHOCK TAY AND OLEXEI I. MOTRUNICH PHYSICAL REVIEW B 84, 020404(R) (2011)
can be useful for further studies of the J1-J2 kagome
antiferromagnet.
In this work we used a dense permanent routine;22 such
calculations can in principle be pursued to 48 sites. If we
also restrict ujk to only few near neighbors, the VMC can
be scaled further due to sparseness of the matrix. Simulations
in the valence-bond basis may reach larger sizes;11 attention
to the sign problem is needed there although it is less severe
than the sign problems in QMC. An important aspect of our
work is the demonstration that projected SB wave functions
can be tested on the same footing as the slave fermion spin
liquids, for smaller but still reasonable system sizes, and can
be included in the VMC toolbox. Here we highlight our use
of permanents in a variational study of Heisenberg model on
the triangular lattice in magnetic field,25 where we obtained
excellent wave functions for Mott insulators and supersolids
of bosons with frustrated hopping. We suggest the honeycomb
spin liquid26 and Wang’s proposal21 as one context for applying
the projected SB wave functions, as well as other model
proposals in Ref. 17 for realizing unique spin liquids on the
triangular and kagome lattices.17
O.I.M. would like to acknowledge the KITP program
“Disentangling Quantum Many-body Systems: Computational
and Conceptual Approaches” and thank many participants
for discussions, and in particular to Steven White for the
inspiring informal rerun of his talk and for sharing new results.
The research is supported by the NSF through Grant No.
DMR-0907145 and the A. P. Sloan Foundation.
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020404-4


Variational study of J_(1)-J_(2) Heisenberg model on kagome lattice using projected Schwinger-boson wave functions

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Variational study of J_(1)-J_(2) Heisenberg model on kagome lattice using projected Schwinger-boson wave functions

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