Magnetism and Thermodynamics of Spin-1/2 Heisenberg Diamond Chains in a
  Magnetic Field
Magnetism and Thermodynamics of Spin-1/2 Heisenberg Diamond Chains in a
Magnetic Field
Bo Gu and Gang Su∗
College of Physical Sciences, Graduate University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China
The magnetic and thermodynamic properties of spin-1/2 Heisenberg diamond chains are investi-
gated in three different cases: (a) J1, J2, J3 > 0 (frustrated); (b) J1, J3 < 0, J2 > 0 (frustrated);
and (c) J1, J2 > 0, J3 < 0 (non-frustrated), where the bond coupling Ji (i = 1, 2, 3) > 0 stands
for an antiferromagnetic (AF) interaction, and < 0 for a ferromagnetic (F) interaction. The density
matrix renormalization group (DMRG) technique is invoked to study the properties of the system
in the ground state, while the transfer matrix renormalization group (TMRG) technique is applied
to explore the thermodynamic properties. The local magnetic moments, spin correlation functions,
and static structure factors are discussed in the ground state for the three cases. It is shown that
the static structure factor S(q) shows peaks at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5) for different
couplings in a zero magnetic field, which, however in the magnetic fields where the magnetization
plateau with m = 1/6 pertains, exhibits the peaks only at q = 0, 2π/3 and 4π/3, which are found
to be couplings-independent. The DMRG results of the zero-field static structure factor can be
nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is
observed that the six modes are closely related to the low-lying excitations of the system. At finite
temperatures, the magnetization, susceptibility and specific heat show various behaviors for different
couplings. The double-peak structures of the susceptibility and specific heat against temperature
are obtained, where the peak positions and heights are found to depend on the competition of the
couplings. It is also uncovered that the XXZ anisotropy of F and AF couplings leads the system of
case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility,
specific heat and magnetization for the compound Cu3(CO3)2(OH)2 are fairly compared with our
TMRG results.
PACS numbers: 75.10.Jm, 75.40.Cx
I. INTRODUCTION
Low-dimensional quantum spin systems with compet-
ing interactions have become an intriguing subject in the
last decades. Among many achievements in this area,
the phenomenon of the topological quantization of mag-
netization has attracted much attention both theoreti-
cally and experimentally. A general necessary condi-
tion for the appearance of the magnetization plateaus
has been proposed by Oshikawa, Yamanaka and Affleck
(OYA) [1], stating that for the Heisenberg antiferromag-
netic (AF) spin chain with a single-ion anisotropy, the
magnetization curve may have plateaus at which the
magnetization per site m is topologically quantized by
n(S − m) = integer, where S is the spin, and n is the
period of the ground state determined by the explicit
spatial structure of the Hamiltonian. As one of fasci-
nating models which potentially possesses the magne-
tization plateaus, the Heisenberg diamond chain, con-
sisting of diamond-shaped topological unit along the
chain, as shown in Fig. 1, has also gained much atten-
tion both experimentally and theoretically (e.g. Refs.
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]).
It has been observed that the compounds,
A3Cu3(PO4)4 with A = Ca, Sr[2], and Bi4Cu3V2O14[3]
can be nicely modeled by the Heisenberg diamond chain.
Another spin-1/2 compound Cu3Cl6(H2O)2·2H8C4SO2
was initially regarded as a model substance for the
spin-1/2 diamond chain [4], but a later experimental
research reveals that this compound should be described
by a double chain model with very weak bond alterna-
tions, and the lattice of the compound is found to be
Cu2Cl4·H8C4SO2 [5]. Recently, Kikuchi et al.[6] have
reported the experimental results on a spin-1/2 com-
pound Cu3(CO3)2(OH)2, where local Cu
2+ ions with
spin S = 1/2 are arranged along the chain direction, and
the diamond-shaped units consist of a one-dimensional
(1D) lattice. The 1/3 magnetization plateau and the
double peaks in the magnetic susceptibility as well as
the specific heat as functions of temperature have been
observed experimentally[6, 7], which has been discussed
in terms of the spin-1/2 Heisenberg diamond chain with
AF couplings J1, J2 and J3 > 0.
On the theoretical aspect, the frustrated diamond spin
chain with AF interactions J1, J2 and J3 > 0 was studied
by a few groups. The first diamond spin chain was ex-
plored under a symmetrical condition J1 = J3[8]. Owing
to the competition of AF interactions, the phase diagram
in the ground state of the spin-1/2 frustrated diamond
chain was found to contain different phases, in which the
magnetization plateaus at m = 1/6 as well as 1/3 are
predicted[9, 10, 11, 12, 13]. Another frustrated diamond
chain with ferromagnetic (F) interactions J1, J3 < 0 and
AF interaction J2 > 0 was also investigated theoretically,
which can be experimentally realized if all angles of the
exchange coupling bonds are arranged to be around 90◦,
a region where it is usually hard to determine safely the
coupling constants and even about their signs[14, 15].
Despite of these works, the investigations on the Heisen-
http://arxiv.org/abs/0704.0150v1
berg diamond spin chain with various competing inter-
actions are still sparse.
Motivated by the recent experimental observation on
the azurite compound Cu3(CO3)2(OH)2[6, 7], we shall
explore systematically the magnetic and thermodynamic
properties of the spin-1/2 Heisenberg diamond chain with
various competing interactions in a magnetic field, and
attempt to fit into the experimental observation on the
azurite in a consistent manner. The density matrix renor-
malization group (DMRG) as well as the transfer ma-
trix renormalization group (TMRG) techniques will be
invoked to study the ground-state properties and ther-
modynamics of the model under interest, respectively.
The local magnetic moments, spin correlation functions,
and static structure factors will be discussed for three
cases at zero temperature. It is found that the static
structure factor S(q) shows peaks in zero magnetic field
at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5) for different
couplings, while in the magnetic fields where the mag-
netization plateau with m = 1/6 remains, the peaks ap-
pear only at wavevectors q = 0, 2π/3 and 4π/3, which
are found to be couplings-independent. These informa-
tion could be useful for further neutron studies. The
double-peak structures of the susceptibility and specific
heat against temperature are obtained, where the peak
positions and heights are found to depend on the com-
petition of the couplings. It is uncovered that the XXZ
anisotropy of F and AF couplings leads the system with-
out frustration (see below) to display quite different be-
haviors. In addition, the experimental data of the sus-
ceptibility, specific heat and magnetization for the com-
pound Cu3(CO3)2(OH)2 are fairly compared with our
TMRG results.
The rest of this paper is outlined as follows. In Sec. II,
we shall introduce the model Hamiltonian for the spin-
1/2 Heisenberg diamond chain with three couplings J1,
J2 and J3, where three particular cases are identified. In
Sec. III, the magnetic and thermodynamic properties of
a frustrated diamond chain with AF interactions J1, J2
and J3 > 0 will be discussed. In Sec. IV, the physi-
cal properties of another frustrated diamond chain with
F interactions J1, J3 < 0 and AF interaction J2 > 0
will be considered. In Sec. V, the magnetism and ther-
modynamics of a non-frustrated diamond chain with AF
interactions J1, J2 > 0 and F interaction J3 < 0 will
be explored, and a comparison to the experimental data
on the azurite compound will be made. Finally, a brief
summary and discussion will be presented in Sec. VI.
FIG. 1: (Color online) Sketch of the Heisenberg diamond
chain. The bond interactions are denoted by J1, J2, and J3.
Three cases will be considered: (a) J1, J2, J3 > 0 (a frustrated
diamond chain); (b) J1, J3 < 0, J2 > 0 (a frustrated diamond
chain with competing interactions); and (c) J1, J2 > 0, J3 < 0
(a diamond chain without frustration). Note that Ji > 0
stands for an antiferromagnetic interaction while Ji < 0 for a
ferromagnetic interaction, where i = 1, 2, 3.
II. MODEL
The Hamiltonian of the spin-1/2 Heisenberg diamond
chain reads
(J1S3i−2 · S3i−1 + J2S3i−1 · S3i
+J3S3i−2 · S3i + J3S3i−1 · S3i+1
+J1S3i · S3i+1)−H ·
Sj , (1)
where Sj is the spin operator at the jth site, L is the to-
tal number of spins in the diamond chain, Ji (i = 1, 2, 3)
stands for exchange interactions, H is the external mag-
netic field, gµB = 1 and kB = 1. Ji > 0 represents the
AF coupling while Ji < 0 the F interaction. There are
three different cases particularly interesting, as displayed
in Fig. 1, which will be considered in the present paper:
(a) a frustrated diamond chain with J1, J2, J3 > 0; (b)
a frustrated diamond chain with competing interactions
J1, J3 < 0, J2 > 0; and (c) a diamond chain without
frustration with J1, J2 > 0, J3 < 0. It should be re-
marked that in the case (c) of this model, since the two
end points of the J2 bond represent the two different lat-
tice sites, it is possible that J1 and J3 can be different,
even their signs.
The magnetic properties and thermodynamics for
the aforementioned three spin-1/2 Heisenberg diamond
chains in the ground states and at finite temperatures
will be investigated by means of the DMRG and TMRG
methods, respectively. As the DMRG and TMRG tech-
niques were detailed in two nice reviews[16, 17], we
shall not repeat the technical details for concise. In the
ground-state calculations, the total number of spins in
the diamond chain is taken at least as L = 120. At finite
temperatures, the thermodynamic properties presented
below are calculated down to temperature T = 0.05 (in
units of |J1|) in the thermodynamic limit. In our calcu-
lations, the number of kept optimal states is taken as 81;
the width of the imaginary time slice is taken as ε = 0.1;
the Trotter-Suzuki error is less than 10−3; and the trun-
cation error is smaller than 10−6.
III. A FRUSTRATED HEISENBERG DIAMOND
CHAIN (J1, J2, J3 > 0)
A. Local Magnetic Moment and Spin Correlation
Function
Figure 2(a) manifests the magnetization process of a
frustrated spin-1/2 Heisenberg diamond chain with the
couplings satisfying J1 : J2 : J3 = 1 : 2 : 2 at zero tem-
perature. The plateau of magnetization per site m = 1/6
is observed. According to the OYA necessary condition
[1], the m = 1/6 plateau of a spin-1/2 Heisenberg chain
corresponds to the period of the ground state n = 3.
Beyond the magnetization plateau region, the magnetic
curve goes up quickly with increasing the magnetic field
H . Above the upper critical field, the magnetic curve
shows a s-like shape. To further look at how this magneti-
zation plateau appears, the spatial dependence of the av-
eraged local magnetic moment 〈Szj 〉 in the ground states
under different external fields is presented, as shown in
Fig. 2(b). It is seen that in the absence of external
field, the expectation value 〈Szj 〉 changes its sign at every
three sites within a very small range of (−10−3, 10−3)
because of quantum fluctuations, resulting in the magne-
tization per site m =
j=1〈Szj 〉/L = 0. 〈Szj 〉 increases
with increasing the magnetic field, and oscillates with
increasing j, whose unit of three spins is gradually di-
vided into a pair and a single, as displayed in Fig. 2(c).
At the field H/J1 = 1.5, as demonstrated in Fig. 2(d),
the behavior of 〈Szj 〉 falls into a perfect sequence such as
{..., (Sa, Sa, Sb), ...} with Sa = 0.345 and Sb = −0.190,
giving rise to the magnetization per site m = 1/6. In ad-
dition, such a sequence remains with increasing the mag-
netic field till H/J1 = 2.5, implying that the m = 1/6
plateau appears in the range of H/J1 = 1.5 ∼ 2.5, as
manifested in Fig. 2(a). When the field is promoted fur-
ther, the sequence changes into a waved succession with
smaller swing of (Sa − Sb), as shown in Fig. 2(e), which
corresponds to the fact that the plateau state ofm = 1/6
is destroyed, and gives rise to a s-like shape of M(H). It
is noting that when the plateau state of m = 1/6 is de-
stroyed, the increase of m at first is mainly attributed to
a rapid lift of Sb, and later, the double Sa start to flimsily
increase till Sa = Sb = 0.5 at the saturated field.
The physical picture for the above results could be un-
derstood as follows. For the m = 1/6 plateau state at
0 1 2 3 4 5
0 40 80
0 40 80
T = 0
 = 1 : 2 : 2,
-1E-3
T = 0, H = 0
(d) H/J
 = 1.5, 2.5
(e) H/J
FIG. 2: (Color online) For a spin-1/2 frustrated Heisenberg
diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2,
(a) the magnetization per site m as a function of magnetic
field H in the ground states; and the spatial dependence of
the averaged local magnetic moment 〈Szj 〉 in the ground states
with external field (b) H/J1 = 0, (c) 1, (d) 1.5 and 2.5, and
(e) 4.
J1 : J2 : J3 = 1 : 2 : 2, we note that if an approximate
wave function defined by[13]
(2| ↑3i−2↑3i−1↓3i〉 ± | ↑3i−2↓3i−1↑3i〉
± | ↓3i−2↑3i−1↑3i〉), (i = 1, ..., L/3) (2)
where ↑j (↓j) denotes spin up (down) on site j, is applied,
one may obtain 〈ψi|Sz3i−2|ψi〉 = 1/3, 〈ψi|Sz3i−1|ψi〉 = 1/3,
〈ψi|Sz3i|ψi〉 = −1/6, giving rise to a sequence {..., (13 ,
), ...}, andm = (1
)/3 = 1/6, which is in agree-
ment with our DMRG results {..., (0.345, 0.345, −0.190),
...}. This observation shows that the ground state of this
plateau state might be described by trimerized states.
Let Hc1 and Hc2 be the lower and upper critical mag-
netic field at which the magnetization plateau appears
and is destructed, respectively. For Hc1 ≤ H ≤ Hc2, the
magnetization m = mp = 1/6, namely, the system falls
into the magnetization plateau state. For 0 ≤ H ≤ Hc1
and J1 : J2 : J3 = 1 : 2 : 2, the magnetization curve
shows the following behavior
m(H) = mp(
)[1+α1(1−
)−α2(1−
)2/3], (3)
where Hc1/J1 = 1.44, the parameters α1 = 2/3 and α2 =
1. Obviously, when H = 0, m = 0; H = Hc1 , m = mp.
A fair comparison of Eq. (3) to the DMRG results is
presented in Fig. 3(a). For Hc2 ≤ H ≤ Hs and J1 : J2 :
J3 = 1 : 2 : 2, where Hs is the saturated magnetic field,
the magnetization curve has the form of
m(H) = mp + (H −Hc2){kc + (Hs −H)[
(H −Hc2)1/3
(Hs −H)1/3
]}, (4)
where kc = (ms −mp)/(Hs −Hc2) with ms the satura-
tion magnetization, and β1, β2 the parameters. One may
see that when H = Hc2 , m = mp; H = Hs, m = ms.
A nice fitting to the DMRG result gives the parameters
Hc2/J1 = 3.15, ms = 1/2, Hs/J1 = 4.55, β1 = 0.143,
β2 = 0.178 and kc = 0.238, as shown in Fig. 3(b).
It should be remarked that from the phenomenological
Eqs.(3) and (4), we find that, away from the plateau
region, the magnetic field dependence of the magnetiza-
tion of this model differs from those of Haldane-type spin
chains where m(H) ∼ (H −Hc1)1/2.
To explore further the magnetic properties of the frus-
trated spin-1/2 Heisenberg diamond chain in the ground
states with the couplings J1 : J2 : J3 = 1 : 2 : 2 at dif-
ferent external fields, let us look at the static structure
factor S(q) which is defined as
S(q) =
eiqj〈Szj Sz0 〉, (5)
where q is the wave vector, and 〈Szj Sz0 〉 is the spin corre-
lation function in the ground state. As demonstrated in
Fig. 4(a), in the absence of the external field, S(q) shows
three peaks: two at q = π/3, 5π/3, and one at q = π,
which is quite different from that of the spin S = 1/2
Heisenberg AF chain, where S(q) only diverges at q = π.
As indicated by Eq. (5), the peaks of S(q) reflect the
periods of the spin correlation function 〈Szj Sz0 〉, i.e., the
peaks at q = π/3 (5π/3) and π reflect the periods of 6
and 2 for 〈Szj Sz0 〉, respectively. As shown in Fig. 4(b),
in the absence of the external field, 〈Szj Sz0 〉 changes sign
every three sites, which corresponds really to the peri-
ods of 6 and 2. With increasing the magnetic field, the
small peak of S(q) at q = π becomes a round valley while
the peak at q = π/3 (5π/3) continuously shifts towards
q = 2π/3 (4π/3) with the height enhanced, indicating
the corruption of the periods of 6 and 2 but the emer-
gence of the new period ∈ (3, 6) for 〈Szj Sz0 〉, as shown
in Fig. 4(c). At the field H/J1 = 1.5, two peaks shift
to q = 2π/3 and 4π/3 respectively, and merge into the
peaks already existing there, showing the existence of
3.0 3.5 4.0 4.5
0.0 0.5 1.0 1.5
 DMRG
=1, J
=2, J
 Eq. (4)
= 1/6, k
= 0.238,
= 3.15, H
= 4.55,
= 0.143, 
= 0.178
 DMRG
 Eq.(3)
=1, J
=2, J
= 1/6, H
= 1.44, 
=2/3, 
FIG. 3: (Color online) For a spin-1/2 frustrated Heisenberg
diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2, the
DMRG results of the magnetization per site m as a function
of magnetic field H away from the plateau state can be fairly
fitted by Eqs. (3) and (4), (a) for 0 ≤ H ≤ Hc1 , and (b) for
Hc2 ≤ H ≤ Hs.
the period 3 for 〈Szj Sz0〉, as clearly displayed in Fig. 4(d).
The valley and peaks of S(q) keep intact in the plateau
state at m = 1/6. When the plateau state is destroyed
at the field H/J1 = 4, the peaks at q = 2π/3 and 4π/3
are depressed dramatically while the peaks at q = π/3,
5π/3 and π appear again with very small heights, reveal-
ing the absence of the period 3 and the slight presence of
the period 2 and 6, as shown in Fig. 4(e). At the field
H/J1 = 4.8, all peaks disappear and become zero, ex-
cept for the peak at q = 0, which is the saturated state.
Therefore, the static structure factor S(q) shows differ-
ent characteristics in different magnetic fields[18]. On the
other hand, it is known that S(q) also reflects the low-
lying excitations of the system. It is thus reasonable to
expect that the low-lying excitations of the frustrated di-
amond chain will behave differently in different magnetic
fields.
To investigate the zero-field static structure factor S(q)
in the ground state for the frustrated spin-1/2 diamond
chains with various AF couplings, the four cases with
J1 = 1, J3 > 0, and J2 = 0.5, 1, 2, and 4 are shown
in Figs. 5(a)-5(d), respectively. For J2 = 0.5, as shown
in Fig. 5(a), S(q) displays a sharp peak at q = π when
J3 < 0.5, and three peaks at q = 0, 2π/3 and 4π/3
when J3 > 0.5. It is shown from the ground state phase
-0.01
-0.01
0 40 80
0 40 80
(a) J1 = 1,
 = 2,
 = 2,
T = 0
=1.5, 
T = 0, H = 0
(c) H/J
(d) H/J
 = 1.5, 2.5
FIG. 4: (Color online) For a spin-1/2 frustrated Heisenberg
diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2,
(a) the static structure factor S(q) in the ground states under
different external fields; and the spatial dependence of the
spin correlation function 〈SzjS
0 〉 in the ground states under
external field (b) H/J1 = 0, (c) 1, (d) 1.5 and 2.5, and (e) 4.
diagram [10] that the system is in the spin fluid (SF)
phase when J1 = 1, J2 = 0.5, J3 < 0.5, and enters into
the ferrimagnetic (FRI) phase when J1 = 1, J2 = 0.5,
J3 > 0.5. For J2 = 1, as indicated in Fig. 5(b), the
incommensurate peaks exist, such as the case of J3 = 0.8,
where the system is in the dimerized (D) phase [10]. For
J2 = 2, as manifested in Fig. 5(c), S(q) has a sharp
peak at q = π and two ignorable peaks at π/3 and 5π/3
when J3 < 1; three sharp peaks at q = 0, 2π/3 and 4π/3
at J3 = 1; a round valley (J3 = 1.5) or a small peak
(J3 = 4) at q = π and two mediate peaks at π/3 and
5π/3 when J3 > 1. The system with J1 = 1 and J2 = 2
is in the D phase when 1 < J3 < 2.8, and in the SF
phase when J3 < 1 or J3 > 2.8 [10]. For J2 = 4 revealed
in Fig. 5(d), the situations are similar to that of Fig.
5(c), but here only the SF phase exists for the system
with J1 = 1, J2 = 4 and J3 > 0 [10]. It turns out that
even in the same phase, such as the SF phase, the zero-
field static structure factor S(q) could display different
characteristics for different AF couplings. In fact, we
note that the exotic peak of S(q) has been experimentally
observed in the diamond-typed compound Sr3Cu3(PO4)4
[19].
0 1 2
(a) T = 0,
H = 0,
 = 1,
 = 0.5
(b) J1 = 1,
 = 1J3=0.3
(c) J1 = 1,
 = 2J3=0.5
(d) J1 = 1,
FIG. 5: (Color online) The zero-field static structure factor
S(q) in the ground states for the spin-1/2 frustrated Heisen-
berg diamond chains with length L = 120, J1 = 1, J3 > 0
and J2 taken as (a) 0.5, (b) 1, (c) 2 and (d) 4.
If the spin correlation function for the spin-S chain can
be expressed as 〈Szj Sz0 〉 = α(−1)je−jβ , where α and β are
two parameters, its static structure factor will take the
form of
S(q) =
S(S + 1)
− α(cos q + e
cos q + coshβ
, (6)
which can recover exactly the S(q) of the spin-S AKLT
chain S(q) = S+1
1−cos q
1+cos q+2/S(S+2)
[20] with α = (S +
1)2/3 and β = ln(1+2/S). Eq.(6) has a peak at wavevec-
tor q = π. By noting that the zero-field static struc-
ture factor S(q) for the frustrated diamond chains dis-
plays peaks at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5)
for different AF couplings, the spin correlation func-
tion 〈Szj Sz0 〉 could be reasonably divided into six modes
〈Sz6m+lSz0 〉 = cl + αle−(6m+l)β or 〈Sz6m+lSz0 〉 = αl(6m +
l)−β with j = 6m+ l and l = 1, 2, ..., 6, whose contribu-
tions to the static structure factor should be considered
separately[21]. Thus, the static structure factor for the
present systems could be mimicked by a superposition of
six modes, which leads to
S(q) =
e(6−l)β cos(lq)− e−lβ cos[(6 − l)q]
cosh(6β)− cos(6q)
cos(lq)− cos[(6− l)q]
1− cos(6q)
, (7)
S(q) =
2(6m+ l)−β cos[(6m+ 1)q] +
, (8)
0 1 2
0 1 2
 DMRG
T= 0, H= 0,
= 0.5,
= 0.3
 Eq.(8)
1,3,5
= -0.17,
2,4,6
= 0.17,
 = 1.4  
 DMRG
T=0, H=0,
=0.5,
 Eq.(7)
= -0.15,
= -0.01,
= 0.16,
= 0.27,
= -0.0007,
= -0.0013,
= 0.0117
 Eq.(8)
= -0.11,
= 0.04,
= -0.21,
= 1.3
 DMRG
T= 0, H= 0,
= 0.5  
 DMRG
 Eq.(8)
= -0.04,
= -0.16,
= -0.14,
 = 1.1
T=0, H=0, J
FIG. 6: (Color online) The DMRG results of the zero-field
static structure factor as a function of wavevector for the spin-
1/2 frustrated Heisenberg diamond chains are fitted: (a) J1 =
1, J2 = 0.5, J3 = 0.3 by Eq. (8); (b) J1 = 1, J2 = 0.5, J3 = 4
by Eq. (7); (c) J1 = 1, J2 = 4, J3 = 0.5, and (d) J1 = 1,
J2 = 4, J3 = 4 by Eq. (8).
respectively, depending on which phase the system falls
into, where αl, cl and β are couplings-dependent param-
eters.
As presented in Fig. 6, the DMRG results of the static
structure factor as a function of wavevector are fitted
by Eqs. (7) and (8) for the spin-1/2 frustrated Heisen-
berg diamond chains with various AF couplings in zero
magnetic field. It can be found that the characteristic
peaks can be well fitted by Eqs. (7) and (8), with only
a slightly quantitative deviation, showing that the main
features of the static structure factor for the present sys-
tems can be reproduced by a linear superposition of six
modes. The fitting results give six different modes in
general, as shown in Fig. 4(b).
To further understand the above-mentioned behav-
iors of the zero-field static structure factor, S(q), we
have applied the Jordan-Wigner (JW) transformation
to study the low-lying excitations of the spin-1/2 frus-
trated Heisenberg diamond chain with various AF cou-
plings (see Appendix A for derivations). It can be seen
that the zero-field low-lying fermionic excitation ε(k) be-
haves differently for different AF couplings, as shown in
Figs. 7(a)-(d). Obviously, these low-lying excitations are
responsible for the DMRG calculated behaviors of S(q),
where the positions of minimums of ε(k) for different AF
couplings, as indicated by arrows in Figs. 7(a)-(c), are
exactly consistent with the locations of the peaks of zero-
field static structure factor S(q) shown in Figs. 6(a)-(c),
respectively, although there is a somewhat deviation for
Fig. 7(d) and Fig. 6(d). It also shows that the six
0 1 2
0 1 2
T=0, H=0,
 = 1, 
 = 0.5, 
 = 0.3
(b) T=0, H=0,
 = 1, 
 = 0.5, 
(c) T=0, H=0,
 = 1, 
 = 4, 
 = 0.5
(d) T=0, H=0,
 = 1, 
 = 4, 
FIG. 7: (Color online) The zero-field low-lying fermionic ex-
citation as a function of wavevector for the spin-1/2 frustrated
Heisenberg diamond chain with (a) J1 = 1, J2 = 0.5, J3 = 0.3,
(b) J1 = 1, J2 = 0.5, J3 = 4, (c) J1 = 1, J2 = 4, J3 = 0.5,
and (d) J1 = 1, J2 = 4, J3 = 4. The arrows indicate the
locations of minimums of ε(k).
modes suggested by Eqs. (7) and (8) is closely related to
the low-lying excitations of the system.
B. Magnetization, Susceptibility and Specific Heat
The magnetization process for the spin-1/2 frustrated
diamond chain with J1 = 1, J3 > 0, and J2 = 0.5 and
2 is shown in Fig. 8(a) and 8(b), respectively, where
temperature is fixed as T/J1 = 0.05. It is found that
the magnetization exhibits different behaviors for differ-
ent AF couplings: a plateau at m = 1/6 is observed, in
agreement with the ground state phase diagram [12, 13];
for J2 = 0.5, as shown in Fig. 8(a), the larger J3 is, the
larger the width of the plateau at m = 1/6 becomes; for
J2 = 2, as presented in Fig. 8(b), the width of the plateau
at m = 1/6 becomes larger with increasing J3 < 1, and
then turns smaller with increasing J3 > 1; for J3 < 1, the
larger J2, the larger the width of the plateau atm = 1/6;
for J3 = 2, the larger J2, the smaller the width of the
plateau at m = 1/6. The saturated field is obviously
promoted with increasing AF J3 and J2.
Figures 8(c) and 8(d) give the susceptibility χ as a
function of temperature T for the spin-1/2 frustrated
diamond chain with J1 = 1, J3 > 0 and J2 = 0.5
and 2, respectively, while the external field is taken as
H/J1 = 0.01. For J2 = 0.5, as shown in Fig. 8(c), the
low temperature part of χ(T ) keeps finite when J3 < 0.5,
and becomes divergent when J3 > 0.5. As clearly man-
ifested in the inset of Fig. 8(c), J3 = 0.5 is the critical
value, which is consistent with the behaviors of static
structure factor S(q) in Fig. 5(a). For J2 = 2, as shown
in Fig. 8(d), an unobvious double-peak structure at low
temperature is observed at small and large J3 such as 0.2
and 3. The temperature dependence of the specific heat
C with J1 = 1, J3 > 0 and J2 = 0.5 and 2 is shown in
Fig. 8(e) and 8(f), respectively, while the external field
is fixed as H/J1 = 0.01. For J2 = 0.5, as given in Fig.
8(e), a double-peak structure of C(T ) is observed at low
temperature for small and large J3 such as 0.3 and 1.
The case with J2 = 2 shown in Fig. 8(f) exhibits the
similar characteristics. Thus, the thermodynamics of the
system demonstrate different behaviors for different AF
couplings. As manifested in Fig. 5, the low-lying excita-
tions behave differently for different AF couplings. The
double-peak structure of the susceptibility as well as the
specific heat could be attributed to the excited gaps in
the low-lying excitation spectrum[22].
IV. A FRUSTRATED DIAMOND CHAIN WITH
COMPETING INTERACTIONS (J1, J3 < 0, J2 > 0)
A. Local Magnetic Moment and Spin Correlation
Function
Figure 9(a) shows the magnetization process of a frus-
trated spin-1/2 Heisenberg diamond chain in the ground
states with the couplings J1 : J2 : J3 = −1 : 4 : −0.5.
The plateau of magnetization per site m = 1/6 is clearly
obtained. To understand the occurrence of the magne-
tization plateau, the spatial dependence of the averaged
local magnetic moment 〈Szj 〉 in the ground states at dif-
ferent external fields is calculated. It is seen that in the
absence of the magnetic field, as presented in Fig. 9(b),
the expectation values of 〈Szj 〉 change sign every three
sites within a very small range of (−2× 10−4, 2× 10−4),
resulting in the magnetization per site m = 0. With
increasing the field, the expectation values of 〈Szj 〉 in-
crease, whose unit of three spins is gradually divided into
a pair and a single, as displayed in Fig. 9(c). At the field
H/|J1| = 0.05, as illustrated in Fig. 9(d), the behavior of
〈Szj 〉 shows a perfect sequence such as {..., (Sa, Sb, Sb), ...}
with Sa = 0.496 and Sb = 0.002, giving rise to the mag-
netization per site m = 1/6. In addition, the sequence
is fixed with increasing the field until H/|J1| = 3.2, cor-
responding to the plateau of m = 1/6. As the field is
enhanced further, the double Sb begin to rise, and the
sequence becomes a waved series with smaller swing of
(Sa − Sb) as revealed in Fig. 9(e), which corresponds to
the plateau state at m = 1/6 that is destroyed. It is
noting that the increase of m is mainly attributed to the
promotion of double Sb, as Sa is already saturated until
Sa = Sb = 0.5 at the saturated field.
As discussed above, the physical picture of them = 1/6
plateau state at J1 : J2 : J3 = −1 : 4 : −0.5 can be
0 2 4
0 2 4
0.0 0.4 0.8
0 1 2
0.0 0.2 0.4 0.6
0 1 2
0 1 2
=0.2 1 2
 = 1, J
 = 0.5,
 = 0.05
1 2J3 = 0.2
 = 1, J
 = 2,
 = 0.05
 = 1, J
 = 0.5,
        H/J
=0.01
=1, J
 = 0.01
 = 0.2
=0.5,
 = 0.01
3  = 3
=1, J
 = 0.01
 = 0.2
FIG. 8: (Color online) For the spin-1/2 frustrated Heisenberg
diamond chains with J1 = 1 and J3 > 0, the magnetization
process m(H) at temperature T/J1 = 0.05 with (a) J2 = 0.5
and (b) J2 = 2; the susceptibility χ(T ) at field H/J1 = 0.01
with (c) J2 = 0.5 and (d) J2 = 2; the specific heat C(T ) at
field H/J1 = 0.01 with (e) J2 = 0.5 and (f) J2 = 2.
understood by the following approximate wave function
(| ↑3i−2↑3i−1↓3i〉 ± | ↑3i−2↓3i−1↑3i〉).
By use of this wave function, we have 〈ψi|Sz3i−2|ψi〉 =
1/2, 〈ψi|Sz3i−1|ψi〉 = 0, 〈ψi|Sz3i|ψi〉 = 0, leading to a
sequence of {..., (1/2, 0, 0), ...}, and m = (1/2 + 0 +
0)/3 = 1/6. This is in agreement with our DMRG re-
sults {..., (0.496, 0.002, 0.002), ...}.
The static structure factor S(q) of the frustrated spin-
1/2 Heisenberg diamond chain with the competing cou-
plings J1 : J2 : J3 = −1 : 4 : −0.5 in the ground states
is considered in different external fields. At zero field,
shown in Fig. 10(a), S(q) has three peaks at q = π/3,
5π/3 and π with mediate heights, which reflects the pe-
riods of 6 and 2 for 〈Szj Sz0 〉, respectively. As shown in
0 1 2 3
0 40 80
0 40 80
T = 0
 = -1:4:-0.5,
-2E-3
T = 0, H = 0
| = 0.025
(d) H/|J
| = 0.05, 3.2
(e) H/|J
| = 3.27
FIG. 9: (Color online) For a spin-1/2 frustrated Heisenberg
diamond chain with fixed couplings J1 : J2 : J3 = −1 : 4 :
−0.5, (a) the magnetization per site m as a function of mag-
netic field H in the ground states; and the spatial dependence
of the averaged local magnetic moment 〈Szj 〉 in the ground
states with external field (b) H/|J1| = 0, (c) 0.025, (d) 0.05
and 3.2, and (e) 3.27.
Fig. 10(b), in the absence of the external field, 〈Szj Sz0 〉
changes sign every three sites, corresponding to the pe-
riods of 6 and 2. With increasing the field, the peak at
q = π becomes flat with height depressed forwardly, while
the peak at q = π/3 (5π/3) is divided into two peaks
shifting oppositely from q = π/3 (5π/3) with height de-
creased, indicating the corruption of the periods of 6 and
2 and the emergence of new periods for 〈Szj Sz0 〉, as shown
in Fig. 10(c). At the field H/|J1| = 0.05, two shift-
ing peaks have respectively reached q = 2π/3 and 4π/3,
and are merged with the existing peaks, showing the oc-
currence of period 3 for 〈Szj Sz0 〉, as clearly displayed in
Fig. 10(d). The flat and peaks keep constant during the
plateau state at m = 1/6. When the plateau state is
destroyed at field H/J1 = 3.27, the peaks at q = 2π/3
and 4π/3 are depressed sharply, revealing the decay of
the period 3, as shown in Fig. 10(e). At the saturated
field of H/J1 = 3.3, all peaks disappear and become flat
with the value zero, which is the saturated state. So, the
static structure factor S(q) shows different characteristics
in different magnetic fields. Similar to the discussions in
-0.02
0 40 80
0 40 80
 = -1,
 = 4,
 = -0.5,
T = 0
0.05, 3.2
| = 3.3
T = 0, H = 0
| = 0.025
| = 0.05, 3.2
| = 3.27
FIG. 10: (Color online) For a spin-1/2 frustrated Heisenberg
diamond chain with fixed couplings J1 : J2 : J3 = −1 : 4 :
−0.5, (a) the static structure factor S(q) in the ground states
with different external fields; and the spatial dependence of
the spin correlation function 〈Szj S
0 〉 in the ground states with
external field (b) H/|J1| = 0, (c) 0.025, (d) 0.05 and 3.2, and
(e) 3.27.
Fig. 4, the low-lying excitations of this frustrated di-
amond chain would also behave differently in different
magnetic fields.
To further investigate the zero-field static structure
factor S(q) in the ground state for the frustrated spin-1/2
diamond chains with various J1, J3 < 0 and J2 > 0, two
cases with J1 = −1, J3 < 0 and J2 = 1 and 4 are illus-
trated in Fig. 11(a) and 11(b), respectively. For J2 = 1,
S(q) shows a round peak at q = π, two sharp peaks at π/3
and 5π/3 when |J3| < 1, and a very sharp peak at q = 0
when |J3| > 1. For J2 = 4, S(q) shows three peaks at
q = π/3, π and 5π/3 as |J3| < 1, and a very sharp peak
at q = π and nearly ignorable peaks at π/3 and 5π/3
as |J3| > 1. In general, the zero-field static structure
factor S(q) shows different characteristics with different
competing couplings, whose exotic characteristics could
be experimentally observed in the related diamond-typed
compounds.
As shown in Fig. 12, the DMRG results of the static
structure factor as a function of wavevector are fitted by
Eq.(7) for the spin-1/2 frustrated Heisenberg diamond
chains with J1, J3 < 0 and J2 > 0. It can be found
0 1 2
(b) J
 = -1, J
 = 4J3 = -16
(a) T = 0, H = 0, J
 = -1, J
 = -0.1
FIG. 11: (Color online) The zero-field static structure factor
S(q) in the ground states for the spin-1/2 frustrated Heisen-
gberg diamond chains with length L = 120, J1 = −1, J3 < 0
and J2 taken as (a) 1; (b) 4.
that the characteristic behaviors can be nicely fitted by
Eq.(7), with only a slightly quantitative deviation, show-
ing that the main features of the static structure factor
for the present systems can be captured by a superposi-
tion of six modes. It is consistent with the fact that the
spin correlation function for the present systems has six
different modes, as manifested in Fig. 10(b).
Similar to the case (a) with all AF couplings in the
last section, the characteristics of zero-field S(q) for the
spin-1/2 frustrated Heisenberg diamond chain with J1,
J3 < 0 and J2 > 0 can be further undersood in terms
of the low-lying excitations of the system (see Appendix
A). By means of the JW transformation, the zero-field
low-lying fermionic excitation ε(k) of the present case is
calculated, as shown in Figs. 13(a)-(b). It is found that
the zero-field low-lying fermionic excitation ε(k) differs
for different couplings, but the positions of minimums
of ε(k) for different couplings, as indicated by arrows in
Figs. 13(a) and (b), appear to be the same. One may see
that these positions coincide exactly with the locations of
peaks of the zero-field static structure factor S(q) mani-
fested in Figs. 12(a) and (b), respectively, showing that
our fitting equation is qualitatively consistent with the
low-lying excitations of the system.
0 1 2
(b)  DMRG
T = 0, H = 0,
 = -1, 
 = 4,
 = -0.5
 Eq. (7)
= -0.07,
 = 0.3, c
1,2,...,6
= 0.0
= -0.07,
= -0.21,
(a)  DMRG
T = 0, H = 0, 
 = -1, 
 = -0.1
 Eq. (7)
= -0.03,
= -0.14,
= -0.12,
 = 0.19,
1,2,...,6
= 0.0
FIG. 12: (Color online) The DMRG results of the zero-field
static structure factor as a function of wavevector are fitted
by Eq. (7) for the spin-1/2 frustrated Heisenberg diamond
chains with (a) J1 = −1, J2 = 1, J3 = −0.1, and (b) J1 = −1,
J2 = 4, J3 = −0.5.
0 1 2
-3.35
-3.30
-3.25
(b) T=0, H=0,
 = -1, 
 = 4, 
 = -0.5
(a) T=0, H=0, 
 = -1, 
 = 1, 
 = -0.1
FIG. 13: (Color online) The zero-field low-lying fermionic
excitation as a function of wavevector for the spin-1/2 frus-
trated Heisenberg diamond chain with (a) J1 = −1, J2 = 1,
J3 = −0.1, and (b) J1 = −1, J2 = 4, J3 = −0.5. The arrows
indicate the locations of minimums of ε(k).
B. Magnetization, Susceptibility and Specific Heat
Figures 14(a) and 14(b) show the magnetization pro-
cess for the spin-1/2 frustrated diamond chain at a finite
temperature T/|J1| = 0.05 with J1 = −1, J3 < 0, and
J2 = 1 and 4, respectively. It is shown that the magneti-
zation exhibits different behaviors for different J1, J3 < 0
and J2 > 0. A plateau at m = 1/6 is observed at small
|J3|; with a fixed J2, the larger |J3|, the smaller the width
of the plateau at m = 1/6, and the plateau disappears
when |J3| exceeds the critical value; for a fixed J3, the
larger J2, the wider the width of the plateau at m = 1/6;
the saturation field is obviously depressed with the in-
crease of |J3| at a fixed J2, and is enhanced with the
increase of J2 at a fixed J3.
Figures 14(c) and 14(d) manifest the susceptibility χ
as a function of temperature T for the spin-1/2 frus-
trated diamond chain with J1 = −1, J3 < 0 and J2 = 1
and 4, respectively, where the external field is taken as
H/|J1| = 0.01. For J2 = 1, the low temperature part of
χ(T ) keeps finite when |J3| < 1, and becomes divergent
when |J3| > 1. As clearly revealed in the inset of Fig.
14(c), J3 = −1 is the critical value, which is in agreement
with the behaviors of static structure factor S(q) shown
in Fig. 11(a). For J2 = 4, a clear double-peak structure
of χ(T ) is obtained at |J3| = 8. The temperature depen-
dence of the specific heat C with J1 = −1, J3 < 0 and
J2 = 1 and 4 is shown in Figs. 14(e) and 14(f), respec-
tively, where the external field is fixed as H/|J1| = 0.01.
For J2 = 1, a double-peak structure of C(T ) is observed
for the case of |J3| = 0.5. The case with J2 = 4 shown in
Fig. 14(f) exhibits the similar characteristics. It is also
found that, owing to the competitions among J1, J3 and
J2, the thermodynamics demonstrate rich behaviors at
different couplings. As reflected in Fig. 11, the low-lying
excitations behave differently with various F interactions
J1, J3 and AF interaction J2, while the excitation gaps
could induce the double-peak structure in the suscepti-
bility as well as in the specific heat[22].
V. A DIAMOND CHAIN WITHOUT
FRUSTRATION (J1, J2 > 0, J3 < 0)
A. Local Magnetic Moment and Spin Correlation
Function
Figure 15(a) shows the magnetization process of a
non-frustrated spin-1/2 Heisenberg diamond chain in the
ground states with the couplings satisfying J1 : J2 : J3 =
1 : 2 : −0.5. The plateau of magnetization per site
m = 1/6 is observed. The appearance of the magneti-
zation plateau can be understood from the spatial de-
pendence of the averaged local magnetic moment 〈Szj 〉 in
the ground states under different external fields. In ab-
sence of the external field, the expectation values of 〈Szj 〉
change sign every one site with a waved swing within a
very small range of (−2× 10−4, 2× 10−4), giving rise to
0.0 0.4 0.8
0 1 2 3 4
0 1 2
0 1 2
0.0 0.6 1.2
0 1 2
0 1 2
-1 -0.5 J3 = -0.1
 = -1, J
 = 1,
| = 0.05
 = -8
 = -1, J
 = 4,
| = 0.05
-0.5-4
= -1.5
 = -1, J
 = 1,
|=0.01
 = -1
=-1, J
| = 0.01
 = -1
=-1, J
| = 0.01
 = -1
=-1, J
| = 0.01
 = -0.1
FIG. 14: (Color online) For the spin-1/2 frustrated Heisen-
berg diamond chains with J1 = −1 and J3 < 0, the mag-
netization process m(H) at temperature T/J1 = 0.05 with
(a) J2 = 1 and (b) J2 = 4; the susceptibility χ(T ) at field
H/J1 = 0.01 with (c) J2 = 1 and (d) J2 = 4; the specific heat
C(T ) at field H/J1 = 0.01 with (e) J2 = 1 and (f) J2 = 4.
the magnetization per site m = 0. Under a finite field,
every three successive spins have gradually cooperated
into a pair and a single, as shown in Fig. 15(c). At
H/J1 = 0.9, as given in Fig. 15(d), 〈Szj 〉 shows a perfect
sequence such as {..., (Sa, Sb, Sb), ...} with Sa = 0.393
and Sb = 0.053, resulting in the magnetization per site
m = 1/6. Moreover, the sequence is fixed with the field
increased until H/J1 = 1.8, corresponding to the plateau
state ofm = 1/6. As the field is increased further, double
Sb begin to increase, and the sequence becomes a waved
succession with a smaller swing of (Sa − Sb), as mani-
fested in Fig. 15(e), which corresponds to the fact that
the plateau state at m = 1/6 is destroyed. It is observed
that the increase of m at first is mainly attributed to the
speedy boost of double Sb, and later, Sa starts to increase
weakly until Sa = Sb = 0.5 at the saturated field.
0 1 2 3
0 40 80
0 40 80
T = 0
 = 1:2:-0.5,
-2E-4
(b) T = 0, H = 0
 = 0.7
(d) H/J
 = 0.9, 1.8
 = 2.5
FIG. 15: (Color online) For a spin-1/2 non-frustrated Heisen-
berg diamond chain with fixed couplings J1 : J2 : J3 = 1 :
2 : −0.5, (a) the magnetization per site m as a function of
magnetic field H in the ground states; and the spatial de-
pendence of the averaged local magnetic moment 〈Szj 〉 in the
ground states with external field (b) H/J1 = 0, (c) 0.7, (d)
0.9 and 1.8, and (e) 2.5.
For this non-frustrated case with couplings J1 : J2 :
J3 = 1 : 2 : −0.5, the obtained perfect sequence of
{..., (0.393, 0.053, 0.053), ...} for the m = 1/6 plateau
state could be understood by the following approximate
trimerized wave function
(2| ↑3i−2↑3i−1↓3i〉 ± 2| ↑3i−2↓3i−1↑3i〉
± | ↓3i−2↑3i−1↑3i〉). (9)
According to this function, we find 〈ψi|Sz3i−2|ψi〉 = 7/18,
〈ψi|Sz3i−1|ψi〉 = 1/18, 〈ψi|Sz3i|ψi〉 = 1/18, giving rise to a
sequence of {..., (7/18, 1/18, 1/18), ...}. It turns out that
m = (7/18 + 1/18 + 1/18)/3 = 1/6. This observation
implies that the ground state of the plateau state can
also be described by the trimerized states.
The static structure factor S(q) of the non-frustrated
spin-1/2 Heisenberg diamond chain in the ground states
with the couplings J1 : J2 : J3 = 1 : 2 : −0.5 is probed
in different external fields. As shown in Fig. 16(a), in
absence of the external field, S(q) shows a sharp peak at
q = π, similar to the behaviors of the S = 1/2 Heisen-
berg AF chain, which reflects the period of 2 for 〈Szj Sz0 〉.
0 1 2
-0.02
0 40 80
0 40 80
(a) J1 = 1,
 = 2,
 = -0.5,
T = 0
0.9, 1.8
T = 0, H = 0
 = 0.7
(d) H/J
 = 0.9, 1.8
 = 2.5
FIG. 16: (Color online) For a spin-1/2 non-frustrated Heisen-
berg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 :
−0.5, (a) the static structure factor S(q) in the ground states
with different external fields; and the spatial dependence of
the spin correlation function 〈Szj S
0 〉 in the ground states with
external field (b) H/J1 = 0, (c) 0.7, (d) 0.9 and 1.8, and (e)
As displayed in Fig. 16(b), at zero external field, 〈Szj Sz0 〉
changes sign every one lattice site, corresponding to the
period of 2. When the field is increased, the peak at
q = π becomes a flat with the height depressed greatly,
while two new peaks with small heights at q = 2π/3 and
4π/3 appear, indicating the corruption of the period of 2
and the emergence of the new period of 3 for 〈Szj Sz0 〉, as
demonstrated in Fig. 16(c). At the field H/J1 = 0.9,
the peaks at q = 0, 2π/3 and 4π/3 become sharper.
The flat and peaks of S(q) keep unchanged during the
plateau state at m = 1/6. When the plateau state is de-
stroyed at the field H/J1 = 2.5, the peaks at q = 2π/3
and 4π/3 are suppressed dramatically, revealing the de-
cay of the period 3, as shown in Fig. 16(e). At the field
H/J1 = 2.8, all peaks disappear, and become a flat with
the value zero, except for the peak at q = 0, which is the
saturated state. It can be stated that the static struc-
ture factor S(q) shows various characteristics in different
magnetic fields. Similar to what discussed in Figs. 4
and 10, the low-lying excitations of this non-frustrated
diamond chain would also behave differently in different
magnetic fields.
The zero-field static structure factor S(q) in the ground
state for the present system displays a peak at q = π with
different couplings, but whose static correlation function
〈Szj Sz0 〉 varies with the couplings. As illustrated in Fig.
17, only are the values of 〈Szj Sz0 〉 larger than zero pre-
sented for convenience. In order to gain deep insight
into physics, for a comparison we also include the static
correlation function for the S = 1/2 Heisenberg antifer-
romagnetic (HAF) chain in Fig. 17(a), whose asymptotic
behavior has the form of [23, 24]
〈Sj · S0〉 ∝ (−1)j
(2π)3/2
. (10)
Eq. (10) is depicted as solid lines in Figs. 17. To take the
finite-size effect into account, the length of the diamond
chain is taken as L = 90, 120 and 160, respectively. As
revealed in Fig. 17(a), the DMRG result of the S = 1/2
HAF chain with an infinite length agrees well with the
solid line. Fig. 17(b) shows that the static correlation
function for the spin-1/2 diamond chain with frustrated
couplings J1 : J2 : J3 = 1 : 1.2 : 0.5 decays faster than
that of the HAF chain. Compared with Figs. 17(c)-(f),
it can be found that all the static correlation functions
for the spin-1/2 non-frustrated diamond chain with AF
interactions J1, J2 and F interaction J3 fall more slowly
than that of the S = 1/2 HAF chain; for fixed AF inter-
actions J1 and J2, the static correlation functions drop
more leisurely with increasing the F interaction |J3|; for
fixed AF interaction J1 and F interaction J3, the static
correlation functions decrease more rapidly with increas-
ing the AF interaction J2.
B. Magnetization, Susceptibility and Specific Heat
Figures 18(a) and 18(b) show the magnetization pro-
cess for the spin-1/2 non-frustrated diamond chain at a
finite temperature T/J1 = 0.05 with J1 = 1, J3 < 0, and
J2 = 0.5 and 2, respectively. It is found that the magneti-
zation behaves differently with different AF interactions
J1, J2 and F interaction J3. A plateau at m = 1/6 is ob-
tained at small |J3|; for fixed J1 and J2, the larger |J3|,
the narrower the width of the plateau at m = 1/6, and
after |J3| exceeds a critical value, the plateau at m = 1/6
is eventually smeared out; for fixed J1 and J3, the larger
J2, the wider the width of the plateau at m = 1/6; the
saturated field is obviously unchanged with changing the
F interaction J3. The coupling-dependence of the spin-
1/2 non-frustrated diamond chain with AF interactions
J1, J2 and F interaction J3 is similar to that of trimerized
F-F-AF chains[25].
Figures 18(c) and 18(d) present the susceptibility χ
as a function of temperature T for the spin-1/2 frus-
trated diamond chain with J1 = 1, J3 < 0 and J2 = 0.5
and 2, respectively, where the external field is taken as
H/J1 = 0.01. A double-peak structure of χ(T ) is ob-
served at small F interaction |J3| and disappears at large
|J3|. The temperature dependence of the specific heat C
with J1 = 1, J3 < 0 and J2 = 0.5 and 2 is shown in Figs.
0 50 100 150
0 50 100 150
(a) J
=1:1:0 (HAF)
 Eq. (10)
L=90   (DMRG)
L=120 (DMRG)
L=160 (DMRG)
T = 0, H = 0
 = 1:1.2:0.5
(frustrated)
 = 1:2:-0.5
 = 1:2:-2
 = 1:0.5:-0.5
 = 1:0.5:-2
FIG. 17: (Color online) The zero-field static correlation func-
tion 〈Szj S
0 〉 versus site j in the ground state for a spin-1/2
diamond chain with different lengths and various couplings.
The couplings ratio J1 : J2 : J3 is taken as (a) 1 : 1 : 0 (HAF),
(b) 1 : 1.2 : 0.5 (frustrated), (c) 1 : 2 : −0.5, (d) 1 : 2 : −2, (e)
1 : 0.5 : −0.5, (f) 1 : 0.5 : −2. The length is taken as L = 90,
120 and 160, respectively.
18(e) and 18(f), respectively, where H/J1 = 0.01. It is
seen that, when J2 is small, C(T ) exhibits only a single
peak; when J2 is large, a double-peak structure of C(T )
is observed. In the latter case, the double-peak structure
is more obvious for small F interaction |J3|, and tend to
disappear at large |J3|. Therefore, the thermodynamics
demonstrate various behaviors with different AF interac-
tions J1, J2 and F interaction J3.
C. Effect of Anisotropy of Bond Interactions
Some magnetic materials show different behaviors un-
der longitudinal and transverse magnetic fields, showing
that the anisotropy plays an important role in the physi-
cal properties of the system. First, let us investigate the
XXZ anisotropy of the AF interaction J2 on the proper-
ties of the spin-1/2 non-frustrated diamond chain with
the couplings J1 : J2z : J3 = 1 : 2 : −0.5 for vari-
ous anisotropy parameter defined by γ2 = J2x/J2z =
J2y/J2z, where the z axis is presumed to be perpendic-
ular to the chain direction. For γ2 ≥ 1, the magneti-
zation m(H), susceptibility χ(T ) and specific heat C(T )
are presented in Figs. 19(a), (b) and (c), respectively.
0.0 0.8 1.6
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2
0 1 2
 = 0.05,
 = 1, J
 = 0.5
 = -2
 = 1, J
 = 0.05,
 = -2
 = 1, J
 = 0.5
 = 0.01,
 = -2
-0.5-0.2
 = 1, J
 = 0.01,
 = -0.2
 = 1, J
 = 0.5
 = 0.01,
 = -2
-0.2   
 = 0.01,
 = 1, J
-0.2 -0.5
 = -2
FIG. 18: (Color online) For the spin-1/2 non-frustrated
Heisenberg diamond chains with J1 = 1 and J3 < 0, the mag-
netization process m(H) at temperature T/J1 = 0.05 with
(a) J2 = 0.5 and (b) J2 = 2; the susceptibility χ(T ) at field
H/J1 = 0.01 with (c) J2 = 0.5 and (d) J2 = 2; the specific
heat C(T ) at field H/J1 = 0.01 with (e) J2 = 0.5 and (f)
J2 = 2.
With increasing γ2, it is found that when the magnetic
field H is along the z direction, the width of the mag-
netization plateau at m = 1/6 as well as the saturation
field are enlarged, while those are more increased for H
along the x direction than along the z direction; the peak
of the susceptibility χ(T ) for H along the z direction
at lower temperature side is promoted, and the second
round peak at high temperature side is depressed with a
little shift, while χ(T ) for H along the x direction shows
the similar varying trend; the peak of the specific heat
C(T ) for H along the z direction at lower temperature
side leaves almost unchanged, and the second round peak
at high temperature side moves towards the higher tem-
perature side, while C(T ) for H along the x direction
coincides with those for H along the z direction. For
0 < γ2 < 1, the anisotropy just shows very reverse effect
on the thermodynamic properties in comparison to what
we discussed above.
Now let us discuss the effect of the XXZ anisotropy of
J3 < 0 on the magnetic and thermodynamic properties
of the spin-1/2 non-frustrated diamond chain with the
couplings J1 : J2 : J3z = 1 : 2 : −0.5. Recall that as
the J2 bond connects two different lattice sites, as shown
in Fig. 1, J1 and J3 can be different, even their signs.
Define a parameter γ3 to characterize the anisotropy as
γ3 = J3x/J3z = J3y/J3z, where the z axis is perpendicu-
lar to the chain direction. For γ3 ≥ 1, the magnetization
m(H), susceptibility χ(T ) and specific heat C(T ) are de-
picted in Figs. 19(d), (e) and (f), respectively. With
increasing γ3, it is seen that the width of the plateau at
m = 1/6 for H along the z direction becomes slightly
wider, while it goes smaller for H along the x direc-
tion; the saturation field is not changed with γ3 along
both directions; the peak of χ(T ) for H along the z di-
rection at lower temperature side is promoted, and the
second round peak at higher temperature side is slightly
depressed, while the situations along the x direction are
just reverse, namely, the peak at lower temperature side
is depressed, and the second peak at higher tempera-
ture side is slightly promoted; the peak of C(T ) for H
along the z direction at lower temperature side leaves al-
most unchanged, and the second round peak at higher
temperature side moves slightly to the higher tempera-
ture side, while C(T ) along the x direction coincides with
that along the z direction. For 0 < γ3 < 1, the situation
just becomes reverse in comparison to what we discussed
above.
D. Comparison to Experimental Results
Recently, Kikuchi et al. [6] have performed a nice
measurement on a spin-1/2 diamond-chain compound
Cu3(CO3)2(OH)2, i.e., azurite. They have observed the
1/3 magnetization plateau, unambiguously confirming
the previous theoretical prediction. The two broad peaks
both in the magnetic susceptibility and the specific heat
are observed. We note that in Ref. [6], the experimen-
tal data at finite temperatures are fitted by the zero-
temperature theoretical results obtained by the exact di-
agonalization and DMRG methods, while the result of
the high temperature series expansion fails to fit the
low-temperature behavior of the susceptibility. In ac-
cordance with our preceding discussions, by using the
TMRG method, we have attempted to re-analyse the ex-
perimental data presented in Ref. [6] to fit the experi-
ments for the whole available temperature region.
Our fitting results for the temperature dependence of
the susceptibility χ of the compound Cu3(CO3)2(OH)2
are presented in Fig. 20(a). For a comparison, we
have also included the TMRG result calculated by us-
ing the parameters given in Ref. [6]. Obviously, our
TMRG results with J1 : J2 : J3z = 1 : 1.9 : −0.3 and
J3x/J3z = J3y/J3z = 1.7 fit very well the experimental
0 1 2 3 4
0 1 2 3 4
0 1 2 3
0 1 2 3
0 1 2
0 1 2
 = 0.05
=1:2:-0.5
=2, z
=2, x
 = 0.05
=1:2:-0.5
=2, z
=2, x
 = 0.01
=2, z
=2, x
 = 0.01
=2, z
=2, x
 = 0.01
 = 0.01
FIG. 19: (Color online) For the spin-1/2 non-frustrated dia-
mond chain with the couplings satisfying J1 : J2z : J3 = 1 :
2 : −0.5 for various anisotropy γ2 = J2x/J2z ≥ 1: (a) the
magnetization process m(H) at temperature T/J1 = 0.05;
(b) the susceptibility χ(T at field H/J1 = 0.01; and (c) the
specific heat C(T ) at field H/J1 = 0.01. For the spin-1/2
non-frustrated diamond chain with couplings J1 : J2 : J3z =
1 : 2 : −0.5 for various anisotropy γ3 = J3x/J3z ≥ 1: (d) the
magnetization process m(H) at temperature T/J1 = 0.05;
(e) the susceptibility χ(T ) at field H/J1 = 0.01; and (f) the
specific heat C(T ) at field H/J1 = 0.01.
data of χ, and the two round peaks at low temperatures
are nicely reproduced, while the result with J1 : J2 :
J3 = 1 : 1.25 : 0.45 obtained in Ref. [6] cannot fit the
low-temperature behavior of χ [26]. On the other hand,
the fitting results for the temperature dependence of the
specific heat C(T ) of the compound Cu3(CO3)2(OH)2
are shown in Fig. 20(b). The lattice contribution, which
is included in the raw experimental data in Ref.[6], is
subtracted according to C(T ) = CExp(T ) − αT 3, where
α is a parameter. Obviously, our TMRG result with the
same set of parameters J1 : J2 : J3z = 1 : 1.9 : −0.3
and J3x/J3z = J3y/J3z = 1.7 fits also remarkably well
the experimental data of C(T ), and the two round peaks
at low temperatures are nicely reproduced, while the re-
sult with J1 : J2 : J3 = 1 : 1.25 : 0.45 given in Ref. [6]
cannot fit the low-temperature behavior of C(T ), even
qualitatively. In addition, the sharp peak of C(T ) ex-
perimentally observed at temperature around 2K cannot
be reproduced by both sets of the coupling parameters,
which might be a three-dimensional long-range ordering
due to interchain interactions. The fitting results for the
magnetization m(H) of the compound Cu3(CO3)2(OH)2
are shown in Fig. 20(c). We would like to point out
that the quantitative fitting by our above parameters to
the width of the plateau is not so good, but the quali-
tative behavior is quite consistent with the experiments
both in the transverse and longitudinal magnetic fields,
say, H
c1 > H
c1, H
c2 < H
c2, and the saturation field is
fixed along both directions, suggesting that our fitting
parameters capture the main characteristics. It is worth
pointing out that if the anisotropy ratio is increased up
to J3x/J3z = J3y/J3z = 2.5 with the same couplings
J1 : J2 : J3z = 1 : 1.9 : −0.3, the width of the 1/3
plateau for H ‖ b will be decreased to about one-half of
that for H ⊥ b.
Therefore, our calculations show that (i) the best cou-
plings obtained by fitting the experimental data of the
susceptibility for the azurite could be J1 : J2 : J3z = 1 :
1.9 : −0.3 with the anisotropic ratio for the ferromag-
netic interaction J3x/J3z = J3y/J3z = 1.7, where z ⊥ b;
(ii) the compound may not be a spin frustrated magnet;
(iii) the double peaks of the susceptibility and the specific
heat are not caused by the spin frustration effect, but by
the two kind of gapless and gapful excitations owing to
the competition of the AF and F interactions.
One might argue that for this diamond chain com-
pound, from the point of the lattice distance it is unlikely
that J1 is AF without XXZ anisotropy while J3 is F with
strong XXZ anisotropy. We may offer another possibil-
ity to support our findings, namely, the case of J1 and
J3 with opposite signs is not excluded from the lattice
structure of the compound. A linear relationship exists
between the exchange energy and the metal-ligand-metal
bridge angle: the coupling energy, positive (ferromag-
netic) at angles near 90o, becomes increasingly smaller
(more antiferromagnetic) as the angle increases[27]. As
the ferromagnetic coupling J3 is determined by fitting
the experimental low-temperature behaviors of χ(T ) and
C(T ), this fitting coupling parameters should not be im-
possible if one considers the angle of J1 bridge to keep the
antiferromagnetic coupling while the angle of J3 bridge
to induce the ferromagnetic coupling. On the other
hand, we note that there is another compound with Cu
ions, Cu2(abpt)(SO4)2(H2O)·H2O, whose g factors in XY
plane are different from that in z direction [28]. Besides,
someone might argue that the condition J2 ≫ J1, |J3| is
necessary to explain the double peak behavior of the dia-
mond chain. In fact, such an argument is not necessarily
true, as manifested in Fig. 18(c), where the double peaks
of χ(T ) at low temperatures can also be produced with
0 10 20 30 40
0 10 20 30 40
0.020
0.025
0.030
H (T)
(c) H b (Exp.) H//b (Exp.)
 (TMRG) (Ref.[12])
 H b (TMRG) (present work)
 H//b (TMRG) (present work)
T (K)
(b) CExp- T
=0.0005
=0.00055
=0.0006
 (TMRG) (Ref.[12])
 (TMRG) (present work)
H b (Exp.)
 (TMRG)
=1:1.25:0.45,
 H b (TMRG)
 H//b (TMRG)
(a) H//b (Exp.)
=19K (Ref.[12])
=1:1.9:-0.3,
=23K, J
(present work)
FIG. 20: (Color online) A comparison of experimental re-
sults for (a) the magnetic susceptibility, (b) the specific heat
and (c) the magnetization process for the spin-1/2 diamond
compound Cu3(CO3)2(OH)2 with the TMRG results. The
experimental data are taken from Ref. [6]. See the context
for details.
the parameters J1 : J2 : J3 = 1 : 0.5 : −0.1. In other
words, the double-peak behavior of the diamond chain
may not depend on whether J2 ≫ J1, J3 or not, but may
be strongly dependent on the competition of AF and F
interactions, as discussed above.
VI. SUMMARY AND DISCUSSION
In this paper, we have numerically studied the mag-
netic and thermodynamic properties of spin-1/2 Heisen-
berg diamond chains with three different cases (a) J1,
J2, J3 > 0 (frustrated), (b) J1, J3 < 0, J2 > 0 (frus-
trated), and (c) J1, J2 > 0, J3 < 0 (non-frustrated) by
means of the DMRG and TMRG methods. In the ground
states, the local magnetic moment, spin correlation func-
tion, and static structure factor are explored. The static
structure factor S(q) at zero field shows peaks at wave
vector q = 0, π/3, 2π/3, π, 4π/3 and 5π/3 for different
couplings, in which the peaks at q = 0, 2π/3 and 4π/3
in the magnetization plateau state with m = 1/6 are ob-
served to be couplings independent. The DMRG results
of the zero-field static structure factor can be nicely fitted
by a linear superposition of six modes, where two fitting
equations are proposed. It is seen that the six modes
are closely related to the low-lying excitations of the sys-
tem. At finite temperatures, the magnetization, suscep-
tibility and specific heat are calculated, which show var-
ious behaviors for different couplings. The double-peak
structure of the susceptibility and specific heat can be
procured, whose positions and heights are found to be
dependent on competing couplings. It has been shown
that the XXZ anisotropy of F and AF couplings can have
remarkable effect on the physical behaviors of the sys-
tem. In addition, the experimental susceptibility, specific
heat and magnetization of the diamond chain compound
Cu3(CO3)2(OH)2[6] can be nicely fitted by our TMRG
results.
For the spin-1/2 frustrated Heisenberg diamond chains
with AF couplings J1, J2 and J3, the magnetization
plateau at m = 1/6 in the ground state coincides with
a perfect fixed sequence of the averaged local magnetic
moment such as {..., (Sa, Sa, Sb), ...} with 2Sa+Sb = 1/2,
which might be described by trimerized states. On the
other hand, the static structure factor S(q) shows peaks
at wave vectors q = 0, π/3 (5π/3), and 2π/3 (4π/3)
for different external fields and different AF couplings.
We note that the similar behavior of S(q) has been
experimentally observed in diamond-typed compound
Sr3Cu3(PO4)4 [19]. In addition, the DMRG results of
the zero-field static structure factor can be nicely fitted
by a linear superposition of six modes. It is observed
that the six modes are closely related to the low-lying
excitations of the present case. At finite temperatures,
the magnetizationm(H), susceptibility χ(T ) and specific
heat C(T ) demonstrate different behaviors at different
AF couplings, say, the magnetization plateau atm = 1/6
is observed whose width is found to be dependent on the
couplings; the double peak structure is observed for the
susceptibility χ(T ) and specific heat C(T ) as a function of
temperature, and the heights and positions of the peaks
are found to be dependent on the AF couplings.
For the spin-1/2 frustrated Heisenberg diamond chains
with F couplings J1, J3 and AF coupling J2, the mag-
netization plateau at m = 1/6 in the ground state cor-
responds to a perfect fixed sequence of the averaged lo-
cal magnetic moment such as {..., (Sa, Sb, Sb), ...} with
Sa+2Sb = 1/2, which could be understood by trimerized
states. The static structure factor S(q) shows peaks also
at wave vectors q = 0, π/3 (5π/3), and 2π/3 (4π/3) for
different external fields and different F couplings J1, J3
and AF coupling J2, which is expected to be experimen-
tally observed in the related diamond-type compound.
In addition, the DMRG results of the zero-field static
structure factor can be nicely fitted by a linear superpo-
sition of six modes with the fitting equations mentioned
above. The six modes are closely related to the low-
lying excitations of the system. At finite temperatures,
the magnetization m(H), susceptibility χ(T ) and spe-
cific heat C(T ) demonstrate various behaviors for differ-
ent couplings, namely, the magnetization plateau at m =
1/6 is observed whose width is found to depend on the
couplings; the double-peak structure is also observed for
the susceptibility χ(T ) and specific heat C(T ), and the
heights and positions of the peaks are found dependent
on F couplings J1, J3 and AF coupling J2.
For the spin-1/2 non-frustrated Heisenberg diamond
chains with AF couplings J1, J2 and F coupling J3, the
magnetization plateau at m = 1/6 in the ground state
coincides with a perfect fixed sequence of the averaged
local magnetic moment such as {..., (Sa, Sb, Sb), ...} with
Sa+2Sb = 1/2, which could be understood by trimerized
states. The static structure factor S(q) is observed to ex-
hibit the peaks at wave vectors q = 0 and 2π/3 (4π/3)
for different external fields and different AF couplings
J1, J2 and F coupling J3, which could be experimen-
tally detected in the related diamond-type compound. In
addition, it is found that the zero-field spin correlation
function 〈Szj Sz0 〉 is similar to that of the S = 1/2 Heisen-
berg AF chain. At finite temperatures, the magnetiza-
tion m(H), susceptibility χ(T ) and specific heat C(T )
are found to reveal different behaviors for different cou-
plings, i.e., the magnetization plateau at m = 1/6 is ob-
tained, whose width is found to depend on the couplings;
the double-peak structure is observed for the tempera-
ture dependence of the susceptibility χ(T ) and specific
heat C(T ), where the heights and positions of the peaks
depend on different AF couplings J1, J2 and F coupling
The effect of the anisotropy of the AF and F inter-
actions on the physical properties of the non-frustrated
Heisenberg diamond chain is also investigated. For the
case of the couplings satisfying J1 : J2z : J3 = 1 :
2 : −0.5, when the anisotropic ratio γ2 = J2x/J2z =
J2y/J2z 6= 1, it is found that the width of the plateau at
m = 1/6, the saturation field, and the susceptibility χ(T )
show the same tendency, but quantitatively different, un-
der the external field H along the z and x directions,
while the specific heat C(T ) for H along the z direction
coincides with that along the x direction. For the case
of the couplings satisfying J1 : J2 : J3z = 1 : 2 : −0.5,
when the anisotropic ratio γ3 = J3x/J3z = J3y/J3z 6= 1,
it is seen that the width of the plateau at m = 1/6, the
saturation field, and the susceptibility χ(T ) exhibit the
opposite trends for H along the z and x directions, while
the specific heat C(T ) for H along the z direction also
coincides with that along the x direction.
For all the three cases, plateau states of m = 1/6 are
observed during the magnetization, whose static struc-
ture factor S(q) shows peaks at wavevectors q = 0, 2π/3
and 4π/3. But in absence of the magnetic field, the static
structure factor S(q) in the ground state displays peaks
at q = 0, π/3, 2π/3, π, 4π/3, and 5π/3 for the frustrated
case with J1, J2, J3 > 0; peaks at q = 0, π/3, π, and
5π/3 for the frustrated case with J1, J3 < 0, J2 > 0;
and a peak at q = π for the non-frustrated case with J1,
J2 > 0, J3 < 0. In addition, the DMRG results of the
zero-field static structure factor can be nicely fitted by a
linear superposition of six modes, where the fitting equa-
tion is proposed. At finite temperatures, the double-peak
structure of the susceptibility and specific heat against
temperature can be obtained for all the three cases. It
is found that the susceptibility shows ferrimagnetic char-
acteristics for the two frustrated cases with some cou-
plings, while no ferrimagnetic behaviors are observed for
the non-frustrated case.
The compound Cu3(CO3)2(OH)2 is regarded as a
model substance for the spin-1/2 Heisenberg diamond
chain. The 1/3 magnetization plateau and the two broad
peaks both in the magnetic susceptibility and the spe-
cific heat have been observed experimentally[6]. Our
TMRG calculations with J1 : J2 : J3z = 1 : 1.9 : −0.3
and J3x/J3z = J3y/J3z = 1.7 capture well the main
characteristics of the experimental susceptibility, spe-
cific heat and magnetization, indicating that the com-
pound Cu3(CO3)2(OH)2 may not be a spin frustrated
magnet[26].
APPENDIX A: LOW-LYING EXCITATIONS OF
SPIN-1/2 FRUSTRATED HEISENBERG
DIAMOND CHAINS
In this Appendix, the low-lying excitations of the spin-
1/2 frustrated Heisenberg diamond chain are investigated
by means of the Jordan-Wigner (JW) transformation.
The Hamiltonian of the system reads
(J1S3i−2 · S3i−1 + J2S3i−1 · S3i + J3S3i−2 · S3i
+J3S3i−1 · S3i+1 + J1S3i · S3i+1)−H
Szj ,(A1)
where 3N is the total number of spins in the diamond
chain, Ji > 0 (i = 1, 2, 3) represent the AF coupling
while Ji < 0 the F interaction, and H is the external
magnetic field along the z direction. In accordance with
the spin configuration of the diamond chain, we start
from the Jordan-Wigner (JW) transformation with spin-
less fermions
S+j = a
j exp[iπ
a+mam],
Szj = a
j aj −
, (A2)
where j = 1, · · · , 3N . Because the period of the present
system is 3, three kinds of fermions in moment space can
be introduced through the Fourier transformations
a3i−2 =
eik(3i−2)a1k,
a3i−1 =
eik(3i−1)a2k,
a3i =
eik(3i)a3k. (A3)
Ignoring the interactions between fermions, the Hamilto-
nian takes the form of
H = E0 +
[(ω1a
1ka1k + ω2a
2ka2k + ω3a
3ka3k)
+(γ1a1ka
2k + γ2a2ka
3k + γ3a3ka
1k + h.c.)],(A4)
where E0 =
(2J1+J2+2J3−6H), ω1=−(J1+J3)−H ,
ω2=− 12 (J1+J2+J3)−H , ω3=ω2, γ1=(J1e
ik+J3e
−i2k)/2,
γ2=(J2e
ik)/2, and γ3=(J3e
ik + J1e
−ik)/2.
Via the Bogoliubov transformation
a1k = u11(k)α1k + u12(k)α2k + u13(k)α3k,
a2k = u21(k)α1k + u22(k)α2k + u23(k)α3k,
a3k = u31(k)α1k + u32(k)α2k + u33(k)α3k, (A5)
the Hamiltonian can be diagonalized as
H = Eg +
ikαik. (A6)
The coefficients of the Bogoliubov transformation can be
found through equations of motion i~ȧik = [aik, H ]:
ω1 γ1 γ3
γ∗1 ω2 γ2
γ∗3 γ
 = ǫik
 . (A7)
For a given k, the eigenvalues εik and eigenvectors
(u1i, u2i, u3i) can be numerically calculated by the driver
ZGEEV.f of the LAPACK, which is available on
the website[29]. Figs.7 show the zero-field low-lying
fermionic excitation ε(k) for the frustrated diamond
chain with different AF coupling, while Figs.13 present
the zero-field low-lying fermionic excitation ε(k) for the
frustrated diamond chain with J1, J3 < 0 and J2 > 0.
Acknowledgments
We are grateful to Prof. D. P. Arovas for useful com-
munication. This work is supported in part by the Na-
tional Science Fund for Distinguished Young Scholars
of China (Grant No. 10625419), the National Science
Foundation of China (Grant Nos. 90403036, 20490210,
10247002), and by the MOST of China (Grant No.
2006CB601102).
[] ∗Corresponding author. E-mail: gsu@gucas.ac.cn
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[29] http://www.netlib.org/lapack/
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