Collective excitations of hard-core Bosons at half filling on square and
  triangular lattices: Development of roton minima and collapse of roton gap
Collective excitations of hard-core Bosons at half filling on square and triangular
lattices: Development of roton minima and collapse of roton gap
Tyler Bryant and Rajiv R. P. Singh
Department of Physics, University of California, Davis, CA 95616, USA
(Dated: November 28, 2018)
We study ground state properties and excitation spectra for hard-core Bosons on square and tri-
angular lattices, at half filling, using series expansion methods. Nearest-neighbor repulsion between
the Bosons leads to the development of short-range density order at the antiferromagnetic wavevec-
tor, and simultaneously a roton minima in the density excitation spectra. On the square-lattice, the
model maps on to the well studied XXZ model, and the roton gap collapses to zero precisely at the
Heisenberg symmetry point, leading to the well known spectra for the Heisenberg antiferromagnet.
On the triangular-lattice, the collapse of the roton gap signals the onset of the supersolid phase.
Our results suggest that the transition from the superfluid to the supersolid phase maybe weakly
first order. We also find several features in the density of states, including two-peaks and a sharp
discontinuity, which maybe observable in experimental realization of such systems.
PACS numbers:
I. INTRODUCTION
A microscopic theory for rotons in the excitation spec-
tra of superfluids was first developed by Feynman, where
he showed that the roton minima was related to a
peak in the static structure factor.1 This study has had
broad impact in condensed matter physics ranging from
Quantum Hall Effect2 to frustrated antiferromagnets.3,4,5
In recent years considerable interest has also centered
on Supersolid phases of matter.6 While the existence
of such homogeneous bulk phases in Helium remains
controversial,7,8 in case of lattice models such phases have
been clearly established. One such example is that of
hard-core Bosons hopping on a triangular-lattice, where
a large enough nearest-neighbor repulsion leads to su-
persolid order.9,10,11,12,13,14 The nature of the excitation
spectra in the superfluid phase and on approach to the
supersolid transition has not been addressed for the spin-
half model.
Here we use series expansion methods to study the
ground state properties and excitation spectra of hard-
core Bosons, at half filling, on square and triangular lat-
tices, with nearest neighbor repulsion. On the square-
lattice, the model is equivalent to the antiferromagnetic
XXZ model, and we present the elementary excitation
spectra for the XXZ model with XY type anisotropy. To
our knowledge this calculation has not been done before.
It should be useful for experimental studies of antiferro-
magnetic materials with XY anisotropy. We set the XY
coupling to unity and study the spectra as a function of
the Ising coupling Jz. For the XY model, the spectra is
gapless at q = 0 (the Goldstone mode of the superfluid)
and has a maximum at the antiferromagnetic wavevector
(π, π). As the Ising coupling is increased a roton minima
develops at the antiferromagnetic wavevector, which goes
to zero at the point of Heisenberg symmetry (Jz = 1), as
expected for the system with doubled unit cell.
For the triangular-lattice, the hard-core Boson model
maps onto a ferromagnetic XY model, which is unfrus-
trated. The nearest-neighbor repulsion, on the other
hand corresponds to an antiferromagnetic Ising coupling,
which is frustrated. This model cannot be mapped onto
an antiferromagnetic XXZ model on the triangular lat-
tice. For this model, we calculate the equal-time struc-
ture factor S(q) as well as the excitation spectra, ω(q).
Once again, we find that in the absence of nearest-
neighbor repulsion, the excitation spectra is gapless at
q = 0 and has a maximum at the antiferromagnetic
wavevector ((4π/3, 0) and equivalent points). As the re-
pulsion is increased, a pronounced peak develops in S(q)
at these wavevectors and simultaneously a sharp roton
minima develops in the spectra. Series extrapolations
suggest that the roton gap vanishes when the repulsion
term (Jz) reaches a value of ≈ 4.5. However, we are un-
able to estimate any critical exponents for the vanishing
of the gap or for the divergence of the structure factor. A
comparison of our structure factor data with the Quan-
tum Monte Carlo data of Wessel and Troyer, leads us to
suggest that the transition to the supersolid phase maybe
weakly first order and occurs for a value of Jz slightly less
than 4.5.
Our calculations also show a near minimum and flat re-
gions in the spectra at the wavevectors (π, π/
3), which
correspond to the midpoint of the faces of the Bril-
louin zone. These are points where the antiferromagnetic
Heisenberg model has a well defined minima.4 In our case
the dispersion is very flat along some directions and a
minimum along others. There are several distinguishing
features in the density of states (DOS) of the excitation
spectra. The largest maximum in the DOS is close to the
maximum excitation energy and is not unlike many other
antiferromagnets. But, here, in addition, we get a second
maximum in the DOS from the flat regions in the spectra
at the midpoint of the faces of the Brillouin zone and a
sharp drop in the DOS at the roton energy. It maybe
possible to engineer such hard-core Boson systems on a
triangular-lattice in cold atomic gases. It should, then,
be possible to excite these collective exciations either op-
http://arxiv.org/abs/0704.1642v2
tically or by driving the system out of equilibrium. A
measurement of the energies associated with the charac-
teristic features in the density of states can be used to
accurately determine the microscopic parameters of the
system.
II. METHOD
The linked-cluster series expansions performed here in-
volve writing the Hamiltonian of interest as
H = H0 + λH1 (1)
where the eigenstates of H0 define the basis to be used
and H1 is the perturbation to be applied in a linked clus-
ter expansion. Ground state properties are then obtained
as a power series in λ using Raleigh-Schrodinger pertur-
bation theory.
Excited state properties are obtained following the pro-
cedure outlined in22, in which a similarity transformation
is obtained in order to block diagonalize the Hamiltonian
where the ground state sits in a block by itself and the
one-particle states form another block.
Heff = S−1HS (2)
where Heff is an effective Hamiltonian for the states
which are the perturbatively constructed extensions of
the single spin-flip states. The effective Hamiltonian
is then used to obtain a set of transition amplitudes
r=0 λ
rcr,m,n that describe propagation of the excita-
tion through a distance (mx̂+ nŷ) for the square lattice
and (1
nŷ) with m and n both even or both odd
for the triangular lattice.
These transition amplitudes are used to obtain the
transition amplitudes for the bulk lattice by summing
over clusters. Fourier transformation of the bulk transi-
tion amplitudes then gives the excitation energy in mo-
mentum space.
∆(qx, qy) =
cr,m,nfm,n(qx, qy) (3)
where fm,n(qx, qy) is given by the symmetry of the lattice,
f sqrm,n(qx, qy) = [cos(mqx + nqy) + cos(mqx − nqy)
+cos(nqx +mqy) + cos(nqx −mqy)] /4 (4)
for the square lattice and
f trim,n(qx, qy) =
qx)cos(
+ cos(
3(m+ n)
qy)cos(
m− 3n
qx) (5)
+ cos(
3(m− n)
qy)cos(
m+ 3n
for the triangular lattice.
In order to access values of the expansion parameter λ
up to and including λ = 1, we use standard first order
integrated differential approximants18 (IDAs) of the form
QL(x)
+RM (x)f + ST (x) = 0 (6)
where QL,RM ,ST are polynomials of degree L,M,and T
determined uniquely from the expansion coefficients.
When gapless modes are present, estimates of the spin-
wave velocity are made using the technique of Singh and
Gelfand15. For small q = |q| the spectrum is assumed
to have the form ∆(q) ∼ [A(λ) + B(λ)q2]1/2. To calcu-
late the spin-wave velocity, we expand ∆(q) in powers
of q, ∆(q) = C(λ) +D(λ)q2 + ... and identify C = A1/2
and D = B/2A1/2. Thus the series 2C(λ)D(λ) provides
an estimate for B, which is the square of the spin-wave
velocity.
III. SQUARE LATTICE
On the square lattice we perform two distinct types
of expansions. For Jz ≥ J⊥, one can expand directly in
J⊥/Jz by choosing
<i,j>
Szi S
<i,j>
(Sxi S
j + S
j ) (7)
In this case λ in (1) is J⊥ (setting Jz = 1). Since H1
conserves the total Sz, one can perform the computation
to high order by restricting the full Hilbert space to the
total Sz sector of interest, which in this paper will be
restricted to total Sz = 0 (half filling).
Series expansion studies of the excitation spectra by
Singh et al.15 and subsequently extended by Zheng et
al.16 have been performed for Jz ≥ J⊥, with expansions
involving linked clusters of up to 11 sites (λ10) and 15
sites (λ14) respectively.
Fig. 1 shows the results of the spin-wave disper-
sion analysis for J⊥ from the dispersionless Ising model
J⊥ = 0 to the Heisenberg model J⊥ = 1. One can see the
development of minima at (0, 0) and (π, π) with increas-
ing J⊥, with the gap completely closing at J⊥ = 1. Since
IDAs are not accurate near the gapless points, the dot-
ted line shows the estimated spin-wave velocity v = 1.666
when J⊥ = 1.
To obtain the spectra with XY anisotropy (Jz ≤ J⊥),
we need to develop a different type of expansion. We
consider the following break up of the Hamiltonian: (for
Jz ≤ J⊥)
<i,j>
Sxi S
<i,j>
j + JzS
j ) (8)
(π/2,π/2)(π,0)(0,0)(π,π)(π,0)
FIG. 1: (Color online) The spin-wave dispersion of the XXZ
model on the square lattice for various values of J⊥ (Jz = 1).
The error bars give an indication of the spread of various
IDAs. The lines around the gapless points for J⊥ = 1 show
the calculated spin-wave velocity.
where J⊥ = 1. Now, a new series is obtained for each
value of Jz, and the XXZ model is only obtained upon
extrapolation to λ = 1. In contrast to the first type of
expansion, H1 does not conserve total Sz, and so the
entire Hilbert space must be used, limiting the order of
computation of the series to λ10 (11 sites).
Fig. 2 shows the results of the spin-wave dispersion
analysis for several values of Jz from the XY model
(Jz = 0) to the Heisenberg model (J⊥ = 1). We find
that for the pure XY model, there is gapless excitations
at q = 0 (Goldstone modes of the superfluid phase),
but there is no roton minima at the antiferromegnetic
wavevector. As Jz is increased, the spin-wave veloc-
ity increases and a clear roton-minima develops at the
antiferromagnetic wavevector. This minima collapses to
zero as the Heisenberg point is approached. In fact, the
doubling of the unit cell implies that for the Heisenberg
limit, the spectra at q and at q+(π, π) become identi-
cal. Another point of interest is that along the direction
(π, 0) to (π/2, π/2), which corresponds to the antiferro-
magnetic zone boundary, the dispersion is very flat for
the pure XY model. A weak minimum develops at (π, 0)
as the Heisenberg symmetry point is reached. These re-
sults should be useful in comparing with spectra of two-
dimensional antiferromagnets, where there is significant
exchange anisotropy.
IV. TRIANGULAR LATTICE
There has been much recent interest in the XXZ model
on the triangular lattice. The spin- 1
XXZ model with
ferromagnetic in-plane coupling J⊥ < 0 and antiferro-
magnetic coupling in the z direction Jz > 0 can be
mapped to a hard-core boson model with nearest neigh-
(π/2,π/2)(π,0)(0,0)(π,π)(π,0)
FIG. 2: (Color online) The spin-wave dispersion of the XXZ
model on the square lattice for various values of Jz (J⊥ = 1).
The error bars give an indication of the spread of various
IDAs. The lines around the gapless points show the calculated
spin-wave velocities.
TABLE I: Series coefficients for the ground state energy per
site E0/N and M
n E0/N for Jz=0 M for Jz=0
0 -5.000000e-01 1.250000e-01
2 -4.166667e-02 -6.944444e-03
4 -4.282407e-03 -2.267072e-03
6 -1.251190e-03 -1.141688e-03
8 -5.538567e-04 -7.184375e-04
10 -2.990401e-04 -5.039687e-04
12 -1.823004e-04 -3.784068e-04
14 -1.015895e-04 -2.494459e-04
bor repulsion.
Hb = −t
<i,j>
ibj + bib
j) + V
<i,j>
ninj (9)
where b
i is the bosonic creation operator, ni = b
ibi. The
parameters are related by t = −J⊥/2 and V = Jz . For
the rest of this section, we let J⊥ = −1, and so V/t =
−2Jz/J⊥ = 2Jz.
We will continue to use the spin language as it is natu-
ral for our study. For Jz = 0, the ferromagnetic in-plane
coupling is unfrustrated. As Jz is increased, the com-
peting interaction leads to an emergence of a supersolid
order.
We have performed expansions for the triangular lat-
tice XXZ model of the form
H0 = −
<i,j>
Sxi S
<i,j>
(−Syi S
j + JzS
j ) (10)
where J⊥ = −1. Series are obtained for each value of
Jz, and the XXZ model is obtained upon extrapolation
to λ = 1.
The static structure factor
S(k) =
eik·r〈Sz0Szr 〉 (11)
is shown in Fig. 3 along contours shown in Fig. 4. As
Jz increases, a peak forms at wavevector q=(4π/3, 0). A
plot of this point is shown in fig. 5 along with QMC data
from Wessel and Troyer.10
EBQPCOBA
FIG. 3: (Color online) The static structure factor of the
XXZ model on the triangular lattice for various values of Jz
(J⊥=−1). The error bars give an indication of the spread of
IDAs.
2π/31/2
-2π/31/2
4π/32π/30-2π/3-4π/3
O P Q
FIG. 4: The hexagonal first Brillouin zone of the triangu-
lar lattice and the path ABOCPQBE along which the static
structure factor and spin-wave dispersion have been plotted
in Figs. 3 and 6.
 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5
Series
12x12
24x24
36x36
FIG. 5: (Color online) The static structure factor at
q=(4π/3, 0) of the XXZ model on the triangular lattice ver-
sus Jz (J⊥=−1). The error bars for the series data give
an indication of the spread of IDAs. Also shown are QMC
data for 12x12,24x24,and 36x36 site clusters from Wessel and
Troyer.10
Fig. 6 shows the results of the spin-wave dispersion
analysis for various values of Jz (J⊥ = −1). The error
bars give an indication of the spread of IDAs. The lines
around the gapless points show the calculated spin-wave
velocities. One can see the development of minima at Q
with increasing Jz, with the gap completely closing at
Jz ∼ 4.5. Since IDAs are not accurate near the gapless
point (q = 0), the dotted line shows the estimated spin-
wave velocities.
We have been unable to get any consistent estimates
for the critical exponents characterizing the divergence
of the antiferromagnetic structure factor and the van-
ishing of the roton gap as the supersolid phase is ap-
proached. Furthermore, the comparison with the QMC
data of Wessel and Troyer show that the QMC data begin
to show deviations from our series expansion results be-
fore Jz = 4.5. We believe, this implies that the superfluid
to supersolid transition is weakly first order. Wessel and
Troyer estimate the transition to be at Jz ≈ 4.3 ± 0.2
(|t/V | = 0.115 ± 0.005). Note that the spin-wave the-
ory gives the transition point to be at Jz = 2,
9 so that
quantum fluctuations play a substantial role here. Addi-
tional QMC studies, should provide further insight into
the nature of the transition.24
The calculations also show that near the midpoint of
the faces of the Brillouin Zone (point B in Fig. 4), the dis-
persion is a minima in the direction perpendicular to the
zone face QB and is very flat in other directions. This be-
havior is reminiscent of the dispersion in the Heisenberg
antiferromagnet on the traingular lattice where there is
a true minimum at this point.4,5 Note that this behav-
ior is unrelated to any peak in the static structure factor
and thus, as in case of the Heisenberg model, is more
quantum mechanical in nature.
In Fig. 7, we show the density of states for the spectra
for Jz = 2. There are several distinguishing features
in the density of states. First the largest peak in the
density of states occurs close to the highest excitation
energies. This is not unlike what is found in many other
antiferromegnets. However, here, there is a second peak
that corresponds to the flat regions in the spectra near
the point B. Finally, at the roton energy there is a sharp
drop in the density of states. The only contributions to
the density of states below the roton gap comes from the
Goldstone modes near q = 0. Since the latter have very
small density of states, there is a discontinuity in the
density of states at the roton energy.
EBQPCOBA
FIG. 6: (Color online) The spin-wave dispersion of the XXZ
model on the triangular lattice for various values of Jz (J⊥ =
−1). The error bars give an indication of the spread of IDAs.
The lines around the gapless points show the calculated spin-
wave velocities.
 0  0.2  0.4  0.6  0.8  1
FIG. 7: The density of states for the XXZ model on the tri-
angular lattice for Jz = 2 (J⊥ = −1).
V. SUMMARY AND CONCLUSIONS
In this paper, we have studied the excitation spectra of
hard-core Boson models at half-filling on square and tri-
angular lattices. The calculations show the development
of the roton minima at the antiferromagnetic wavevector,
due to nearest-neighbor repulsion. In accord with Feyn-
man’s ideas, the development of the minima is correlated
with the emergence of a sharp peak in the static structure
factor. The case of triangular-lattice is clearly more in-
teresting as one has a phase transition from a superfluid
to a supersolid phase, where the roton gap goes to zero.
Our series results suggest that the roton-gap vanishes at
Jz ≈ 4.5. However, there maybe a weakly first order
transition slightly before this Jz value. A more careful
finite-size scaling analysis of the QMC data should pro-
vide further insight into this issue.
Our results of the spectra suggest two peaks in the
density of states and a sharp drop in the density of states
at the energy of the roton minima. If such a hard-core
Boson system on a triangular-lattice is realized in cold-
atom experiments, a measurement of the two peaks in
the density of states and the roton minima can be used
to determine independently the hopping parameter t and
the nearest-neighbor repulsion V .
Acknowledgments
This research is supported in part by the National Sci-
ence Foundation Grant Number DMR-0240918. We are
greatful to Stefan Wessel for providing us with the QMC
data for the structure factors and to Marcos Rigol and
Stefan Wessel for discussions.
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TABLE II: Series coefficients for the magnon dispersion on the square lattice for Jz = 0 (XY model), nonzero coefficients up
to r=9 are listed for compactness (the complete series can be found in Ref. 23)
(r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n
(0,0,0) 2.000000e+00 (3,2,1) -7.291667e-02 (8,3,3) -1.715283e-03 (9,5,2) -1.334928e-03
(2,0,0) -4.166667e-02 (5,2,1) -1.968093e-02 (4,4,0) -2.712674e-03 (8,5,3) -7.225832e-04
(4,0,0) -1.023582e-02 (7,2,1) -8.897152e-03 (6,4,0) -2.980614e-03 (9,5,4) -3.993759e-04
(6,0,0) -5.390283e-03 (9,2,1) -4.845897e-03 (8,4,0) -1.932294e-03 (6,6,0) -1.156309e-04
(8,0,0) -2.781363e-03 (4,2,2) -1.627604e-02 (5,4,1) -5.303277e-03 (8,6,0) -3.216877e-04
(1,1,0) -1.000000e+00 (6,2,2) -5.758412e-03 (7,4,1) -4.614353e-03 (7,6,1) -3.803429e-04
(3,1,0) 4.340278e-02 (8,2,2) -2.558526e-03 (9,4,1) -3.055177e-03 (9,6,1) -6.801940e-04
(5,1,0) 1.811921e-02 (3,3,0) -1.215278e-02 (6,4,2) -3.468926e-03 (8,6,2) -3.612916e-04
(7,1,0) 7.679634e-03 (5,3,0) -7.265535e-03 (8,4,2) -2.946432e-03 (9,6,3) -2.662506e-04
(9,1,0) 4.056254e-03 (7,3,0) -3.368594e-03 (7,4,3) -1.901715e-03 (7,7,0) -2.716735e-05
(2,1,1) -2.500000e-01 (9,3,0) -1.895272e-03 (9,4,3) -1.807997e-03 (9,7,0) -1.043517e-04
(4,1,1) -2.314815e-02 (4,3,1) -2.170139e-02 (8,4,4) -4.516145e-04 (8,7,1) -1.032262e-04
(6,1,1) -5.841368e-03 (6,3,1) -1.033207e-02 (5,5,0) -5.303277e-04 (9,7,2) -1.141074e-04
(8,1,1) -1.566143e-03 (8,3,1) -4.852573e-03 (7,5,0) -1.004891e-03 (8,8,0) -6.451636e-06
(2,2,0) -1.250000e-01 (5,3,2) -1.060655e-02 (9,5,0) -9.122910e-04 (9,8,1) -2.852685e-05
(4,2,0) -3.067130e-02 (7,3,2) -6.456544e-03 (6,5,1) -1.387570e-03 (9,9,0) -1.584825e-06
(6,2,0) -9.598676e-03 (9,3,2) -3.716341e-03 (8,5,1) -1.771298e-03
(8,2,0) -3.989385e-03 (6,3,3) -2.312617e-03 (7,5,2) -1.141029e-03
TABLE III: Series coefficients for the ground state energy per site E0/N and M
n E0/N for Jz=0 M for Jz=0 E0/N for Jz=1 M for Jz=1 E0/N for Jz=2 M for Jz=2
0 -7.500000e-01 -2.500000e-01 -7.500000e-01 -2.500000e-01 -7.500000e-01 -2.500000e-01
1 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
2 -3.750000e-02 7.500000e-03 -1.500000e-01 3.000000e-02 -3.375000e-01 6.750000e-02
3 -7.500000e-03 3.000000e-03 6.750000e-02 -2.700000e-02
4 -3.102679e-03 1.989902e-03 -6.428572e-04 3.271769e-03 -3.081696e-02 3.263208e-02
5 -1.557668e-03 1.356060e-03 4.457109e-02 -4.706087e-02
6 -9.211778e-04 1.018008e-03 -1.686432e-03 2.253907e-03 -5.708928e-02 7.245005e-02
7 -5.949646e-04 7.975125e-04 7.181401e-02 -1.117528e-01
8 -4.102048e-04 6.468307e-04 -7.027097e-04 1.498661e-03 -1.001587e-01 1.839365e-01
9 -2.965850e-04 5.380214e-04 1.440002e-01 -3.025679e-01
10 -2.225228e-04 4.565960e-04 -3.752974e-04 1.029273e-03 -2.164303e-01 5.141882e-01
11 -1.719314e-04 3.937734e-04 3.343757e-01 -8.849178e-01
12 -1.360614e-04 3.441169e-04 -2.322484e-04 7.779323e-04 -5.294397e-01 1.545967e+00
TABLE IV: Series coefficients for the magnon dispersion on the triangular lattice Jz = 0, J⊥ = −1, nonzero coefficients up to
r=9 are listed for compactness (the complete series can be found in Ref. 23)
(r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n
(0,0,0) 3.000000e+00 (6,5,1) -6.186994e-03 (5,8,0) -2.318836e-03 (9,10,6) -3.135408e-05
(2,0,0) -3.125000e-02 (7,5,1) -4.204807e-03 (6,8,0) -2.496675e-03 (9,10,8) -4.493212e-07
(3,0,0) 6.927083e-03 (8,5,1) -3.031857e-03 (7,8,0) -2.214348e-03 (6,11,1) -9.073618e-05
(4,0,0) -5.786823e-03 (9,5,1) -2.274746e-03 (8,8,0) -1.849982e-03 (7,11,1) -2.637766e-04
(5,0,0) -2.071746e-03 (4,5,3) -3.466797e-03 (9,8,0) -1.528161e-03 (8,11,1) -3.828143e-04
(6,0,0) -2.263644e-03 (5,5,3) -4.350420e-03 (5,8,2) -1.098718e-03 (9,11,1) -4.387479e-04
(7,0,0) -1.418733e-03 (6,5,3) -3.723124e-03 (6,8,2) -1.554950e-03 (7,11,3) -7.642054e-05
(8,0,0) -1.158308e-03 (7,5,3) -2.959033e-03 (7,8,2) -1.579331e-03 (8,11,3) -1.651312e-04
(9,0,0) -8.857566e-04 (8,5,3) -2.305863e-03 (8,8,2) -1.427534e-03 (9,11,3) -2.316277e-04
(1,2,0) -1.500000e+00 (9,5,3) -1.817374e-03 (9,8,2) -1.241335e-03 (8,11,5) -1.824976e-05
(2,2,0) -2.500000e-01 (3,6,0) -7.265625e-03 (6,8,4) -2.268405e-04 (9,11,5) -4.925111e-05
(3,2,0) 2.236979e-02 (4,6,0) -1.113487e-02 (7,8,4) -4.458355e-04 (9,11,7) -1.797285e-06
(4,2,0) -3.470238e-03 (5,6,0) -7.839022e-03 (8,8,4) -5.558869e-04 (6,12,0) -1.512270e-05
(5,2,0) 6.223272e-03 (6,6,0) -5.440344e-03 (9,8,4) -5.861054e-04 (7,12,0) -9.486853e-05
(6,2,0) 2.258837e-03 (7,6,0) -3.834575e-03 (7,8,6) -1.528411e-05 (8,12,0) -1.917140e-04
(7,2,0) 2.729869e-03 (8,6,0) -2.794337e-03 (8,8,6) -6.113630e-05 (9,12,0) -2.589374e-04
(8,2,0) 1.793769e-03 (9,6,0) -2.110041e-03 (9,8,6) -1.121505e-04 (7,12,2) -4.585233e-05
(9,2,0) 1.431100e-03 (4,6,2) -5.200195e-03 (5,9,1) -5.493588e-04 (8,12,2) -1.209317e-04
(2,3,1) -1.875000e-01 (5,6,2) -5.161886e-03 (6,9,1) -1.094328e-03 (9,12,2) -1.836224e-04
(3,3,1) -5.677083e-02 (6,6,2) -4.187961e-03 (7,9,1) -1.233659e-03 (8,12,4) -2.281220e-05
(4,3,1) -2.743217e-02 (7,6,2) -3.218811e-03 (8,9,1) -1.182811e-03 (9,12,4) -5.685542e-05
(5,3,1) -1.275959e-02 (8,6,2) -2.454043e-03 (9,9,1) -1.066119e-03 (9,12,6) -4.193665e-06
(6,3,1) -8.214468e-03 (9,6,2) -1.905072e-03 (6,9,3) -3.024539e-04 (7,13,1) -1.528411e-05
(7,3,1) -4.959005e-03 (5,6,4) -5.493588e-04 (7,9,3) -5.232866e-04 (8,13,1) -6.113630e-05
(8,3,1) -3.456161e-03 (6,6,4) -1.094328e-03 (8,9,3) -6.255503e-04 (9,13,1) -1.121505e-04
(9,3,1) -2.443109e-03 (7,6,4) -1.233659e-03 (9,9,3) -6.426095e-04 (8,13,3) -1.824976e-05
(2,4,0) -9.375000e-02 (8,6,4) -1.182811e-03 (7,9,5) -4.585233e-05 (9,13,3) -4.925111e-05
(3,4,0) -4.916667e-02 (9,6,4) -1.066119e-03 (8,9,5) -1.209317e-04 (9,13,5) -6.290497e-06
(4,4,0) -2.605934e-02 (4,7,1) -3.466797e-03 (9,9,5) -1.836224e-04 (7,14,0) -2.183444e-06
(5,4,0) -1.289340e-02 (5,7,1) -4.350420e-03 (8,9,7) -2.607109e-06 (8,14,0) -1.878307e-05
(6,4,0) -8.269582e-03 (6,7,1) -3.723124e-03 (9,9,7) -1.376709e-05 (9,14,0) -4.937808e-05
(7,4,0) -5.280757e-03 (7,7,1) -2.959033e-03 (5,10,0) -1.098718e-04 (8,14,2) -9.124881e-06
(8,4,0) -3.753955e-03 (8,7,1) -2.305863e-03 (6,10,0) -4.726374e-04 (9,14,2) -3.135408e-05
(9,4,0) -2.735539e-03 (9,7,1) -1.817374e-03 (7,10,0) -7.110879e-04 (9,14,4) -6.290497e-06
(3,4,2) -2.179687e-02 (5,7,3) -1.098718e-03 (8,10,0) -7.825389e-04 (8,15,1) -2.607109e-06
(4,4,2) -1.596136e-02 (6,7,3) -1.554950e-03 (9,10,0) -7.669912e-04 (9,15,1) -1.376709e-05
(5,4,2) -9.470639e-03 (7,7,3) -1.579331e-03 (6,10,2) -2.268405e-04 (9,15,3) -4.193665e-06
(6,4,2) -6.186994e-03 (8,7,3) -1.427534e-03 (7,10,2) -4.458355e-04 (8,16,0) -3.258886e-07
(7,4,2) -4.204807e-03 (9,7,3) -1.241335e-03 (8,10,2) -5.558869e-04 (9,16,0) -3.685726e-06
(8,4,2) -3.031857e-03 (6,7,5) -9.073618e-05 (9,10,2) -5.861054e-04 (9,16,2) -1.797285e-06
(9,4,2) -2.274746e-03 (7,7,5) -2.637766e-04 (7,10,4) -7.642054e-05 (9,17,1) -4.493212e-07
(3,5,1) -2.179687e-02 (8,7,5) -3.828143e-04 (8,10,4) -1.651312e-04 (9,18,0) -4.992458e-08
(4,5,1) -1.596136e-02 (9,7,5) -4.387479e-04 (9,10,4) -2.316277e-04
(5,5,1) -9.470639e-03 (4,8,0) -8.666992e-04 (8,10,6) -9.124881e-06
TABLE V: Series coefficients for the magnon dispersion on the triangular lattice Jz = 1, J⊥ = −1, nonzero coefficients up to
r=9 are listed for compactness (the complete series can be found in Ref. 23)
(r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n
(0,0,0) 3.000000e+00 (8,4,2) 1.641818e-02 (8,7,3) -3.898852e-03 (6,10,2) -1.174079e-03
(2,0,0) -1.250000e-01 (4,5,1) -7.072545e-02 (6,7,5) -4.696316e-04 (8,10,2) -2.957178e-03
(4,0,0) 8.773202e-02 (6,5,1) -2.266297e-02 (8,7,5) -2.220857e-03 (8,10,4) -1.051498e-03
(6,0,0) -3.747008e-02 (8,5,1) 1.641818e-02 (4,8,0) -3.632812e-03 (8,10,6) -5.518909e-05
(8,0,0) 2.658737e-02 (4,5,3) -1.453125e-02 (6,8,0) -1.296471e-02 (6,11,1) -4.696316e-04
(2,2,0) -1.000000e-00 (6,5,3) -1.709052e-02 (8,8,0) -2.634458e-03 (8,11,1) -2.220857e-03
(4,2,0) 3.067262e-01 (8,5,3) -1.306500e-05 (6,8,2) -8.548979e-03 (8,11,3) -1.051498e-03
(6,2,0) -1.478629e-01 (4,6,0) -4.674479e-02 (8,8,2) -3.898852e-03 (8,11,5) -1.103782e-04
(8,2,0) 1.192248e-01 (6,6,0) -2.107435e-02 (6,8,4) -1.174079e-03 (6,12,0) -7.827194e-05
(2,3,1) -7.500000e-01 (8,6,0) 8.083746e-03 (8,8,4) -2.957178e-03 (8,12,0) -1.178397e-03
(4,3,1) 8.683532e-02 (4,6,2) -2.179687e-02 (8,8,6) -3.787294e-04 (8,12,2) -7.614846e-04
(6,3,1) -7.198726e-02 (6,6,2) -1.775987e-02 (6,9,1) -5.919357e-03 (8,12,4) -1.379727e-04
(8,3,1) 6.383529e-02 (8,6,2) 1.136255e-03 (8,9,1) -4.075885e-03 (8,13,1) -3.787294e-04
(2,4,0) -3.750000e-01 (6,6,4) -5.919357e-03 (6,9,3) -1.565439e-03 (8,13,3) -1.103782e-04
(4,4,0) -2.357440e-02 (8,6,4) -4.075885e-03 (8,9,3) -3.182356e-03 (8,14,0) -1.146580e-04
(6,4,0) -4.667627e-02 (4,7,1) -1.453125e-02 (8,9,5) -7.614846e-04 (8,14,2) -5.518909e-05
(8,4,0) 4.476721e-02 (6,7,1) -1.709052e-02 (8,9,7) -1.576831e-05 (8,15,1) -1.576831e-05
(4,4,2) -7.072545e-02 (8,7,1) -1.306500e-05 (6,10,0) -2.499303e-03 (8,16,0) -1.971039e-06
(6,4,2) -2.266297e-02 (6,7,3) -8.548979e-03 (8,10,0) -3.646508e-03
TABLE VI: Series coefficients for the magnon dispersion on the triangular lattice Jz = 2, J⊥ = −1, nonzero coefficients up to
r=9 are listed for compactness (the complete series can be found in Ref. 23)
(r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n
(0,0,0) 3.000000e+00 (6,5,1) -5.113269e-01 (5,8,0) 1.114660e-01 (9,10,6) 3.094665e-02
(2,0,0) -2.812500e-01 (7,5,1) 3.541550e-01 (6,8,0) -2.789923e-01 (9,10,8) 4.332352e-04
(3,0,0) -6.234375e-02 (8,5,1) -5.927875e-01 (7,8,0) 3.889189e-01 (6,11,1) -1.020493e-02
(4,0,0) 3.427127e-01 (9,5,1) 1.818953e+00 (8,8,0) -5.937472e-01 (7,11,1) 5.778696e-02
(5,0,0) 1.248868e-01 (4,5,3) -9.659180e-02 (9,8,0) 1.024914e+00 (8,11,1) -1.791293e-01
(6,0,0) -3.138557e-01 (5,5,3) 2.102338e-01 (5,8,2) 5.215530e-02 (9,11,1) 3.824414e-01
(7,0,0) -2.065582e-01 (6,5,3) -3.793451e-01 (6,8,2) -1.818856e-01 (7,11,3) 1.646341e-02
(8,0,0) 2.679692e-01 (7,5,3) 4.434905e-01 (7,8,2) 3.088576e-01 (8,11,3) -8.048695e-02
(9,0,0) 6.130976e-01 (8,5,3) -6.470161e-01 (8,8,2) -5.178979e-01 (9,11,3) 2.165711e-01
(1,2,0) 1.500000e+00 (9,5,3) 1.205017e+00 (9,8,2) 8.731811e-01 (8,11,5) -8.618625e-03
(2,2,0) -2.250000e+00 (3,6,0) 6.539063e-02 (6,8,4) -2.551233e-02 (9,11,5) 4.893766e-02
(3,2,0) -2.013281e-01 (4,6,0) -3.105654e-01 (7,8,4) 9.867811e-02 (9,11,7) 1.732941e-03
(4,2,0) 1.349036e+00 (5,6,0) 3.819521e-01 (8,8,4) -2.507655e-01 (6,12,0) -1.700822e-03
(5,2,0) 3.245835e-01 (6,6,0) -4.794666e-01 (9,8,4) 4.867085e-01 (7,12,0) 2.057252e-02
(6,2,0) -9.772865e-01 (7,6,0) 4.227344e-01 (7,8,6) 3.292682e-03 (8,12,0) -9.194406e-02
(7,2,0) -1.324972e+00 (8,6,0) -6.511353e-01 (8,8,6) -2.934241e-02 (9,12,0) 2.400559e-01
(8,2,0) 1.736288e+00 (9,6,0) 1.561357e+00 (9,8,6) 1.091019e-01 (7,12,2) 9.878045e-03
(9,2,0) 1.998149e+00 (4,6,2) -1.448877e-01 (5,9,1) 2.607765e-02 (8,12,2) -5.858223e-02
(2,3,1) -1.687500e+00 (5,6,2) 2.496909e-01 (6,9,1) -1.264872e-01 (9,12,2) 1.748291e-01
(3,3,1) 5.109375e-01 (6,6,2) -4.028414e-01 (7,9,1) 2.546730e-01 (8,12,4) -1.077328e-02
(4,3,1) 1.438694e-01 (7,6,2) 4.542205e-01 (8,9,1) -4.585226e-01 (9,12,4) 5.664036e-02
(5,3,1) 4.631412e-01 (8,6,2) -6.635572e-01 (9,9,1) 7.791699e-01 (9,12,6) 4.043528e-03
(6,3,1) -7.686440e-01 (9,6,2) 1.278006e+00 (6,9,3) -3.401644e-02 (7,13,1) 3.292682e-03
(7,3,1) -2.673657e-01 (5,6,4) 2.607765e-02 (7,9,3) 1.163420e-01 (8,13,1) -2.934241e-02
(8,3,1) 2.424913e-01 (6,6,4) -1.264872e-01 (8,9,3) -2.770943e-01 (9,13,1) 1.091019e-01
(9,3,1) 2.292894e+00 (7,6,4) 2.546730e-01 (9,9,3) 5.242482e-01 (8,13,3) -8.618625e-03
(2,4,0) -8.437500e-01 (8,6,4) -4.585226e-01 (7,9,5) 9.878045e-03 (9,13,3) 4.893766e-02
(3,4,0) 4.425000e-01 (9,6,4) 7.791699e-01 (8,9,5) -5.858223e-02 (9,13,5) 6.065292e-03
(4,4,0) -3.406189e-01 (4,7,1) -9.659180e-02 (9,9,5) 1.748291e-01 (7,14,0) 4.703831e-04
(5,4,0) 5.419257e-01 (5,7,1) 2.102338e-01 (8,9,7) -1.231232e-03 (8,14,0) -8.933341e-03
(6,4,0) -6.661367e-01 (6,7,1) -3.793451e-01 (9,9,7) 1.347384e-02 (9,14,0) 4.854225e-02
(7,4,0) 3.675761e-02 (7,7,1) 4.434905e-01 (5,10,0) 5.215530e-03 (8,14,2) -4.309313e-03
(8,4,0) -2.016564e-01 (8,7,1) -6.470161e-01 (6,10,0) -5.380211e-02 (9,14,2) 3.094665e-02
(9,4,0) 2.293150e+00 (9,7,1) 1.205017e+00 (7,10,0) 1.541458e-01 (9,14,4) 6.065292e-03
(3,4,2) 1.961719e-01 (5,7,3) 5.215530e-02 (8,10,0) -3.358010e-01 (8,15,1) -1.231232e-03
(4,4,2) -4.619167e-01 (6,7,3) -1.818856e-01 (9,10,0) 6.055844e-01 (9,15,1) 1.347384e-02
(5,4,2) 4.570889e-01 (7,7,3) 3.088576e-01 (6,10,2) -2.551233e-02 (9,15,3) 4.043528e-03
(6,4,2) -5.113269e-01 (8,7,3) -5.178979e-01 (7,10,2) 9.867811e-02 (8,16,0) -1.539040e-04
(7,4,2) 3.541550e-01 (9,7,3) 8.731811e-01 (8,10,2) -2.507655e-01 (9,16,0) 3.577909e-03
(8,4,2) -5.927875e-01 (6,7,5) -1.020493e-02 (9,10,2) 4.867085e-01 (9,16,2) 1.732941e-03
(9,4,2) 1.818953e+00 (7,7,5) 5.778696e-02 (7,10,4) 1.646341e-02 (9,17,1) 4.332352e-04
(3,5,1) 1.961719e-01 (8,7,5) -1.791293e-01 (8,10,4) -8.048695e-02 (9,18,0) 4.813724e-05
(4,5,1) -4.619167e-01 (9,7,5) 3.824414e-01 (9,10,4) 2.165711e-01
(5,5,1) 4.570889e-01 (4,8,0) -2.414795e-02 (8,10,6) -4.309313e-03
