We study the boson model on a triangular lattice interacting only via on-site hardcore repulsion by mapping to a system of spins $(S = 1/2)$. We investigate the supersolid phase of the systems which is a state matter displaying both diagonal long- range (solid) order as well as off-diagonal long-range (superfluidity) by utilizing a semionic representation for the spin-XXZ model. We show that the supersolid order is stable in the mean-field theory for a broad region of parameters. The inclusion of spin wave corrections modifies this picture, but the supersolid phase is still quite robust on the triangular lattices

Nga, Pham Thi Thanh

Thang, Nguyen Toan

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 21, No. 4 (2011), pp. 301-308
SUPERSOLIDS OF HARDCORE - BOSONS ON
A TRIANGULAR LATTICE
PHAM THI THANH NGA
Water Resources University, 175 Tay Son, Hanoi
NGUYEN TOAN THANG
Institute of Physics, VAST, 10 Dao Tan, Hanoi
Abstract. We study the boson model on a triangular lattice interacting only via on-site hardcore
repulsion by mapping to a system of spins (S = 1/2). We investigate the supersolid phase of
the systems which is a state matter displaying both diagonal long- range (solid) order as well
as off-diagonal long-range (superfluidity) by utilizing a semionic representation for the spin-XXZ
model. We show that the supersolid order is stable in the mean-field theory for a broad region of
parameters. The inclusion of spin wave corrections modifies this picture, but the supersolid phase
is still quite robust on the triangular lattices.
I. INTRODUCTION
Supersolidity where superfluid and crystalline orders coexist was firstly theoretically
conjectured by Andreev and Lifshitz in 1969 [1] and then was developed by Leggett [2]
and Chester [3]. A supersolids phase is a state of matter exhibiting simultaneously off -
diagonal (ODLRO) and diagonal long range order (DLRO). In 2004 Kim and Chan [4,5]
proved the existence of the supersolid state in solid He4 at temperatures below T = 0.2
K by measuring a tine superflow in a torsional oscillation experiment. This landmark
experiment has attracted a lot of interest in the supersolid state and subsequently many
new theories were proposed [6-11]. However, the true nature of the supersolids state is
still unclear. One of the simplest models has been applied to study the possibility of
the sufersolidity is the Quantum Gas Model (QGM) [12, 13]. Since the QGM can be
exactly mapped to s = 1/2 quantum model [14] one can use helpful methods developed
in quantum spin systems for understanding the physics underlying the sufersolidity. In
particular, upon exploiting a mapping between hardcore latlice gas models and spin - 1/2
Heisenberg model, Liu and Fisher [13] have shown that within a mean - field approximation
a supersolid phase exists in systems with a finite range of interactions between the bosons.
Recent experiments in the realisation of optical lattices in ultracold atomic system motivate
a search for a lattice supersolid, in particular in low dimensional and frustrated systems.
In the mean time interest in hardcore bosons on two dimensional lattices has been on the
rise (see [7] and Ref. therein). Recent studies in the two dimensional square lattice system
revealed that quantum fluctuations play crucial role for the sufersolid phase. Namely they
dramatically change the behavior and may even suppress the supersolid phase which is
supposed to exist within the mean field theory [7]. A model of strongly interaction hardcore
302 PHAM THI THANH NGA AND NGUYEN TOAN THANG
bosons on a triangular lattice is another open issue where the nature of the state for the
hardcore bosons with frustrated nearest neighbor hoping has not been well understood [15
- 19].
In this report we study supersolidity in the triangular lattice using a semionic rep-
resentations for spin operators, suggested by Popov and Fedotov [19]. Contrary to other
methods which are based on the use of a Lagrange multiplier [20] leading to an average
occupations the Popov - Fedotov procedure avoids this approximate treatment by intro-
ducing of an imaginary chemical potential. In the previous report we applied the Popov
- Fedotov trick for a 1/2 - spin antiferromagnetic Heisenberg system on the triangular
lattice [21 - 22]. The obtained results will be exploited on this report for considering the
supersolid phase.
This paper is organized as follows: In Sec. II we provide a Model Hamiltonian with
mapping to the XXZ spin - 1/2 model. In Sec. III and IV we describe classical and spin
- wave results. Conclusions are presented in Sec. V.
II. MODEL HAMILTONIAN
We begin with a simple model of hardcore bosons at half filling on the triangular
lattice interacting via nearest-neighbor repulsive term
H = −t
∑
〈i,j〉
(
b+i bj + b
+
j bi
)
+ V
∑
〈i,j〉
(
ni − 1
2
)(
nj − 1
2
)
, (1)
where bi (b
+
i ) annihilates (creates) a hardcore boson on site i and ni = b
+
i bi are boson
number operators. 〈i, j〉 refer to nearest neighbor links of the triangular lattice. The
transfer integral t sets the energy scale and we will shall mainly consider the frustrated
boson hopping at t < 0.
It has been proposed that bosonic lattice models can be realized by loading ultracold
bosonic atoms in regular lattice and the interaction between the bosons can be induced
by using the dipolar interaction on condensate [23]. The nearest neighbor repulsion V
promotes the formation of ”solid” order, where the boson occupations fall into regular
patterns, at special densities commensurate with the lattice. The transfer integral t favors
mobile bosons and consequently a superfluid phase at T = 0.
Since atoms cannot penetrate each other there exist only one atom at a time on a
lattice site and consequently b+i and bi are the operators of a hardcore bosons which obey
the Bose commutation relations on different lattice sites.[
b+i , b
+
j
]
=
[
bi, bj
]
=
[
bi, b
+
j
]
= 0 (i 6= j) (2)
In addition the hardcore constraint should be enforced on order to exclude the
multiple occupation of atoms at each lattice point corresponding to the atomic core. We
impose the anti - commutators on identical sites:
{b+i , b+i } = {bi, bi} = 0,
{bi, b+i } = 1 (3)
From (3) we get ni having a value either 0 or 1. Due to the unusual statistics
of hardcore bosons there does not exist a Wick’s theorem for the operators bi, b
+
i and
SUPERSOLIDS OF HARDCORE - BOSONS ON A TRIANGULAR LATTICE 303
the perturbative field theory is not applicable. Hence in the following we transform the
model (1) to an equivalent anisotropic Heisenberg spin model in order to use advanced
methods developed for quantum spin systems. It is easy to check that the spin S = 1/2
operator obey the same commutator relations (2) - (3) as hardcore bosons do. Therefore
it is feasible to replace the hardcore boson operators by spin operators:
b+i = (S
x
i + iS
y
i )
bi = (S
x
i − iSyi ) (4)
ni =
1
2
+ Szi
With this mapping the hardcore boson Hamiltonian becomes the XXZ spin - 1/2 Hamil-
tonian:
H = J
∑
〈i,j〉
[
∆
(
Sxi S
x
j + S
y
i S
y
j
)
+ Szi S
z
j
]
(5)
where J = V > 0 is antiferromagnetic longitudinal exchange, ∆ = −2tV is the anisotropy
parameter.
In the spin language the superfluid order is equivalent to the in-plane ordering of
the spins at a nonzero wave vector, while a charge density wave (solid) order implies long
range correlations of the z-component of spins also at a nonzerowave vector. Therefore
the supersolid phase corresponds to the spin having their xy-component aligned ferromag-
netically (superfluid) with the z-component also ordered at a nonzero wave vector (solid).
In the following we consider the XXZ antiferromagnetic Heisenberg model [5] in the case
of an easy axis with V > −2t > 0.
III. CLASSICAL GROUND STATE
In this section we will be following closely the analysis of Sheng and Henley [24].
The classical limit is obtained by setting S = ∞. In the ground state the triangular
lattice breaks up into three sublattices A,B,C, within each sublattice the spins are ordered
ferromagnetically and can be described by two polar angles (φi, θi) (i = A,B,C sublattice).
In the mean field solution the sublattice magnetization are co - planar, with the plane
containing the easy axis. Let us choose it as the zx - plane, i.e, φA = φB = φC = 0. By
solving the equations:
∂H(θA, θB, θC)
∂θi
= 0 (6)
we get the following equations for classical value of θi:
∆cosθA(sinθB + sinθC) = sinθA(cosθB + cosθC),
∆cosθB(sinθC + sinθA) = sinθB(cosθC + cosθA), (7)
∆cosθC(sinθA + sinθB) = sinθC(cosθA + cosθB).
The three equations in (7) are not independent of each other [24]. Thus we are free to
choose θA and parameterize the solutions of (7) as follows:
θB = arctan
(
tanθA
∆
)
+ arccos
(
∆
(1 +∆)[sin2θA +∆cos2θA]1/2
)
,
304 PHAM THI THANH NGA AND NGUYEN TOAN THANG
θC = arctan
(
tanθA
∆
)
− arccos
(
∆
(1 +∆)[sin2θA +∆cos2θA]1/2
)
. (8)
where θA ∈ [0, θo]
(
sinθo =
∆
1+∆
)
. From Eqs. (7) we can see that α1 = {θ1, θ2, θ3} is a
set of solution, then:
α2 = {θ2, θ3, θ1},
α3 = {θ3, θ1, θ2},
α4 = {pi − θ1, pi − θ2, pi − θ3}, (9)
α5 = {pi − θ2, pi − θ3, pi − θ1},
α6 = {pi − θ3, pi − θ1, pi − θ2}.
are equivalent sets of the solutions. So the classical ground state has the six - fold symmetry
parameterized by the continuous parameter θA.
The ground state energy is:
Ecolf
NJS2
= −1 +∆+∆
2
1 + ∆
(10)
The longitudinal magnetization mz is defined as:
mz = S(cosθA + cosθB + cosθC) (11)
and the transverse component is given by:
mC = S(sinθA + sinθB + sinθC) (12)
The ground state turns out to have the same absolute value of total magnetization:
|m| = 1−∆
1 +∆
S (13)
while mz = 0 Notice that (13) and (14) mean that the supersolid phase appears in a all
region of the anisotropy parameter 0 ≤ ∆ < 1.
IV. SPIN WAVE THEORY
The ground state has a non - trivial continuous degeneracy parameterized by θA
(non - trivial in the sense that the Hamiltonian of the system does not have the same
symmetries as the spin configurations), so one expects the degeneracy to be lifted by
quantum fluctuations. Here we present the spin-wave analysis by using Popov - Fedotov
trick. The main complication is that all of the classical ground states parameterized by θA
can be chosen as the starting point of a spin-wave calculation. We implement spin wave
theory in the usual way. First we perform rotation in the local z-axis to the direction of
the classical sublattice magnetization:
Sxi = S
x′
i cosθi + S
z′
i sinθi
Szi = S
z′
i cosθi + S
x′
i sinθi (14)
S
y
i = S
y′
i
SUPERSOLIDS OF HARDCORE - BOSONS ON A TRIANGULAR LATTICE 305
for i = A,B,C. This leads to:
H = −1
2
∑
〈i,j〉,α,β
J
αβ
ij S
α
i S
β
j (15)
where:
Jxxij = −2J(cosθicosθj +∆sinθisinθj)
J
yy
ij = −2J
Jzzij = −2J(sinθisinθj +∆cosθicosθj) (16)
Jxzij = J
zx
ij = −2J(cosθicosθj −∆sinθisinθj)
J
xy
ij = J
yx
ij = J
zy
ij = J
yz
ij = 0
In the Popov - Fedotov formalism the partition function of the Hamilltonian (15) is given
by the coherent state functional integral:
Z =
1
iN
∫
D[a∗iσ(τ)aiσ(τ)]exp(−S) (17)
where
S =
∫ β
α
dτ
[∑
iσ
a∗iσ(τ)∂τaiσ(τ) +H(a
∗
iσ(τ)aiσ(τ)) +
ipi
2β
∑
i,σ=1,2
a∗iσ(τ)aiσ(τ)
]
(18)
a∗iσ and aiσ(τ) are the Grassmann variables corresponding to the pseudo fermi operators
a∗iσ and aiσ(τ). The spin vectors are given as:
~Sτ =
1
2
a∗iσ~τσσ′aiσ′ (19)
where ~τ are the Pauli matrices. The last form in (18) was introduced to eliminate the
contribution of the unphysical states from the partition function [19]. Following the same
way as in [22] we get the partition function (17) in term of the auxiliary Bose fields ~ϕi:
Z =
1
Zo
∫
D~ϕe−Seff [ϕ] (20)
The effective action Seff is given by:
Seff [ϕ] =
∫ β
0
dτS0
[
~ϕ
] − ln det βKi (21)
with auxiliary field action S0:
S0[ϕ(τ)] =
1
2
∑
<ij>α,β
(J−1)αβ ~ϕi(τ)~ϕj(τ) (22)
and (J−1)αβij is the inverse of the coupling matrix J
αβ
ij . Z0 is auxiliary partition function:
Z0 =
∫
D~ϕe−i
∫ β
0
dτS0[ϕ(τ)] (23)
306 PHAM THI THANH NGA AND NGUYEN TOAN THANG
The matrix Ki in the frequency representation reads:
K̂i(ω1, ω2) =
(− iω1 − ipi
2β
)
δω1,ω2 Î +
1
2
~σ ~ϕi(ω1 − ω2) (24)
Decomposing the matrix K̂ into nonperturbation and perturbation parts:
K̂ = K̂0 + M̂ (25)
And expanding Tr(lnK̂) in a Taylor series:
Tr(lnK̂) = Tr(lnK̂0) + Tr
∞∑
n=1
(−1)n+1
n
(K−10 M̂)
n (26)
It’s straightforward to get the mean field free energy:
Fmf =
1
β
∑
ij,αβ
(J−1)αβij ϕ
α
i0 ϕ
β
j0 +
1
β
∑
i
ln2 cosh
β|ϕi0|
2
(27)
where ϕαi0 is the mean field Hubbard - Stratonovich auxiliary field:
~ϕi(Ω) = ~ϕi0(Ω = 0) + δ~ϕ(Ω) (28)
Utilizing the transformation in Eq (14) we can set ~ϕ00 = (0, 0, ϕ0). The first order fluctu-
ation δ~ϕ gives no contribution. To the second order in the fluctuations the effective action
reads:
S
(2)
eff =
∑
ijΩ
D
αβ
ij (Ω) δϕ
α
i (Ω) δϕ
β
j (−Ω) (29)
where
D
αβ
ij (Ω) =
β
2
[
(J−1)αβij + M
αβ
i (Ω)δij
]
Mxxi (Ω) =M
yy
i (Ω) = − iMxyi (Ω) = iMyxi (Ω)−
1
2
tanhϕ0
(ϕ0 − iΩ) (30)
with
Mzzi (Ω) = −
1
4
δΩ,0
cosh2 βϕ02
M
yz
i (Ω) =M
zy
i (Ω) =M
xz
i (Ω) =M
zx
i (Ω) = 0 (31)
Because of the three sublattice structure, translational invariance is valid only within
sublattices, and the Bloch vector ~k have to be chosen from the correspondingly reduced
magnetic Brillouin zone. It is therefore helpful to introduce three kind of the fluctuation
δϕA, δϕB , δϕC ; the corresponding k-space representations are δa(k), δb(k), δc(k). Per-
forming the Fourier trarnsformation, the effective action can be written in the compact
matrix form.
Seff [δΦ] =
∑
k,α,β,Ω
δΦ
α
(k,Ω)Dαβ(k,Ω)δΦα(k,Ω) (32)
where Dαβ(k,Ω) is Fourier trarnsformation of Dαβij (k,Ω) and the vector δΦ
α(k,Ω) is in-
troduced as follows:
δΦα(k,Ω) =
(
δaαk (Ω), δb
α
k (Ω), δc
α
k (Ω)
)
(33)
SUPERSOLIDS OF HARDCORE - BOSONS ON A TRIANGULAR LATTICE 307
The magnon energy is the solution of the following equation:
det
∣∣∣∣Dαβ(k,Ω)∣∣∣∣ = 0 (34)
The Dαβ(k,Ω) is 9 x 9 matrix, but the elements containing z-component are decoupled,
so only the 6 x 6 matrix remains. Since carrying though the calculation analytically is
not possible for general ∆, we omit further formal details and go over to discuss limiting
cases and numerical results.
For ∆ = 0 we regain the results for isotropic Heisenberg model, where the magnon
spectrum is given by:
ω(k) = 3JS
√(
1− γ(k))(1 + 2γ(k)) (35)
with
γ(k) =
1
3
(
coskx + 2cos
kx
2
cos
√
3
2
ky
)
(36)
For general ∆ < 1, in the small k limit we have:
ω1(k)ω2(k)ω3(k) =
9∆2J3√
61
(
1 +
1
1 + ∆
)2[
(kδ1)
2 + (kδ2)
2 + (kδ3)
2
]×
×
√
(2∆ + 1)(1 −∆) (37)
The ground state energy turns out to be lowest when θ∆ = 0 and decreases with ∆
increasing. For small anisotropy parameter ∆ << 1 the ground state energy with the spin
wave correction is:
Esω = −NS(S + 1)J 1 + ∆+∆
2
1 + ∆
− S∆J (38)
For the θ∆ = 0 ground state, the sublattice magnetizations are reduced due to spin wave
corrections. This spin reductions depend on the anisotropy parameter ∆ < 1. However,
the total z - component of magnetization doer not vanish until ∆ = 0, which means the
Neel solid order exists for 0 ≤ ∆ < 1.
V. CONCLUSIONS
We have shown that supersolid order is stable in mean - field theory for abroad region
of anisotropy parameter ∆. The spin wave corrections select the true ground state from
continuously degenerate classical ground state manifold. However the supersolid is quite
robust on the triangular lattice. The underlying reason is that the frustration prevents all
the particles from condensing into a ”solid”. Our result is complimentary to those of [15],
where one considers the unfrustrated hopping case. We also reproduce some results of the
numerical study [16 - 18]. It is not surprise because spin wave theory seems sufficient for
the triangular lattice. We close with a open important question concerning the effects of
nonzero temperature on the system. Of course, since this is a two - dimensional system,
the Bose condensate disappears for T 6= 0, but power - law ODLRO is expected to remain.
The quantum thermal fluctuation and frustration may compete to produce various phases
as the temperature is raised. It would be interesting to investigate this case by Popov -
Fedotov approach.
308 PHAM THI THANH NGA AND NGUYEN TOAN THANG
ACKNOWLEDGMENT
This research is supported by NAFOSTED (103.027.809).
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Received 10 October 2010.


Supersolids of Hard Core - Bosons on a Triangular Lattice

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