Spin models of quasi-1D quantum ferrimagnets with competing interactions
We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong competing interactions, such as the frustrated J₁ - J₂ chain with alternating (1, 1/2)...
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irk-123456789-1200162017-06-11T03:05:20Z Spin models of quasi-1D quantum ferrimagnets with competing interactions Ivanov, N.B. We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong competing interactions, such as the frustrated J₁ - J₂ chain with alternating (1, 1/2) spins, the spin-1/2 diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric frustration. As a rule, the discussed models exhibit rich quantum phase diagrams and provide some interesting examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems in the reviewed research area are discussed. Ми представляємо короткий огляд недавньої теоретичної роботи, пов’язаної з загальними спiновими моделями Гайзенберга, що описують квазiодновимiрнi квантовi ферiмагнетики. Наголос зроблено на квантових ланцюжках i драбинках з сильною конкуренцiєю взаємодiй, таких, як фрустрований J₁ - J₂ ланцюжок зi змiнними (1, 1/2) спiнами, спiн-1/2 ромбiчний ланцюжок з чотириспiновими циклiчними взаємодiями i деякi загальнi типи драбинок з геометричними фрустрацiями з рiзними значеннями спiна. Як правило, обговорюванi моделi виявляють багатi квантовi фазовi дiаграми i становлять цiкавi приклади одновимiрних квантових фазових переходiв магнетик – парамагнетик. Оговорюються деякi вiдкриттi проблеми у цiй областi дослiджень. 2009 Article Spin models of quasi-1D quantum ferrimagnets with competing interactions / N.B. Ivanov // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 435-447. — Бібліогр.: 88 назв. — англ. 1607-324X PACS: 75.10.Jm, 75.10Pq, 75.45.+j DOI:10.5488/CMP.12.3.435 http://dspace.nbuv.gov.ua/handle/123456789/120016 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing
quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong
competing interactions, such as the frustrated J₁ - J₂ chain with alternating (1, 1/2) spins, the spin-1/2
diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric
frustration. As a rule, the discussed models exhibit rich quantum phase diagrams and provide some interesting
examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems
in the reviewed research area are discussed. |
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Article |
| author |
Ivanov, N.B. |
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Ivanov, N.B. Spin models of quasi-1D quantum ferrimagnets with competing interactions Condensed Matter Physics |
| author_facet |
Ivanov, N.B. |
| author_sort |
Ivanov, N.B. |
| title |
Spin models of quasi-1D quantum ferrimagnets with competing interactions |
| title_short |
Spin models of quasi-1D quantum ferrimagnets with competing interactions |
| title_full |
Spin models of quasi-1D quantum ferrimagnets with competing interactions |
| title_fullStr |
Spin models of quasi-1D quantum ferrimagnets with competing interactions |
| title_full_unstemmed |
Spin models of quasi-1D quantum ferrimagnets with competing interactions |
| title_sort |
spin models of quasi-1d quantum ferrimagnets with competing interactions |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120016 |
| citation_txt |
Spin models of quasi-1D quantum ferrimagnets with competing interactions / N.B. Ivanov // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 435-447. — Бібліогр.: 88 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ivanovnb spinmodelsofquasi1dquantumferrimagnetswithcompetinginteractions |
| first_indexed |
2025-07-08T17:04:54Z |
| last_indexed |
2025-07-08T17:04:54Z |
| _version_ |
1837099172015112192 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 3, pp. 435–447
Spin models of quasi-1D quantum ferrimagnets with
competing interactions
N.B. Ivanov1,2
1 Institute of Solid State Physics, Bulgarian Academy of Sciences,
72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria
2 Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany
Received June 19, 2009
We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing
quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong
competing interactions, such as the frustrated J1 − J2 chain with alternating (1, 1/2) spins, the spin-1/2
diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric
frustration. As a rule, the discussed models exhibit rich quantum phase diagrams and provide some interesting
examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems
in the reviewed research area are discussed.
Key words: quantum spin chains, ferrimagnets, frustration
PACS: 75.10.Jm, 75.10Pq, 75.45.+j
1. Introduction
Ferrimagnets are unsaturated magnetic materials exhibiting a net ferromagnetic (FM) mo-
ment, as a rule resulting from magnetic sublattices with different magnetic ions and/or different
number of magnetic sites [1]. During the past two decades it has become possible to synthesize a
large variety of quasi-one-dimensional (1D) mixed-spin compounds with ferrimagnetic properties
[2]. Most of these materials are bimetallic molecular magnets containing two different transition
metal ions per unit cell, which are alternatively distributed on the lattice (see figure 1). MnCu
(dto)2(H2O)3·4.5H2O (dto = dithiooxalato) is the first structurally characterized ferrimagnetic
chain [3,4]. Two families of ferrimagnetic chains are described by ACu(pba)(H2O)3· nH2O (pba =
1,3-propylenebis) and ACu(pbaOH)(H2O)3· nH2O (pbaOH = 2-hydroxo-1,3 - propylenebis), where
A=Ni, Fe, Co, and Mn [5–7]. The cited pioneer studies have stimulated further extensive experi-
mental research on heterometallic and homometallic ferrimagnetic chains [8–13]. Another class
of ferrimagnetic materials are the so-called topological ferrimagnets. The homometallic magnetic
compound A3Cu3(PO4)4 (A=Ca,Sr,Pb) [11] is a quasi-1D example of such materials. In this com-
pound, the Cu2+ ions form diamond chains with strongly coupled trimers bridged by oxygen ions.
Similar magnetic structures appear in the magnetic compound Cu3(CO3)2(OH)2 [12,13] known
as azurite. In addition, there are a number of experimental works on quasi-1D organic and inor-
ganic magnetic materials with similar structures and ferrimagnetic properties [14]. Apart from pure
scientific interest, the discussed systems possess a potential for various technological applications.
The discussed experimental research has established the basis for future theoretical studies on
1D quantum spin models exhibiting quantum ferrimagnetic states. Nowadays, this is one of the
areas in the framework of the intensive research on 1D magnetism [15]. Below we present a brief
survey of the theoretical results in this hot area, the emphasis being on some basic quantum spin
models of quasi-1D quantum ferrimagnets with competing interactions. A generic spin model of
the 1D quantum ferrimagnet is the Heisenberg spin chain with AFM nearest-neighbor exchange
interactions and two different alternating spins S1 and S2 (S1 > S2). The extreme quantum variant
c© N.B. Ivanov 435
N.B. Ivanov
(a) (b)
Figure 1. (a) Structure of the quasi-one-dimensional bimetallic compound
NiCu(pba)(H2O)3·2H2O with alternating site spins S1 = SNi = 1 and S2 = SCu = 1
2
along the axis b. Hydrogen atoms are omitted for clarity [9]. (b) The classical Néel state of the
mixed-spin AFM chain model (1) describing the bimetallic compound NiCu(pba)(H2O)3·2H2O.
of this model (S1 = 1, S2 = 1/2) is used in section 2 to survey some basic ground-state and ther-
modynamic properties of 1D quantum ferrimagnets. The emphasis in section 3 is on the quantum
phase diagrams of a few generic Heisenberg spin models describing 1D quantum ferrimagnets with
strong competing interactions, such as the frustrated J1 − J2 chain with alternating (1,1/2) spins,
the spin-1/2 diamond chain with four-spin cyclic couplings, and some basic types of mixed-spin
ladders with geometric frustration. The last section contains some conclusions.
2. The antiferromagnetic mixed-spin Heisenberg chain
A generic spin model of the 1D quantum ferrimagnet is the Heisenberg spin chain with AFM
nearest-neighbor exchange interaction (J > 0) and alternating spins S1 and S2 (S1 > S2), described
by the Hamiltonian
H = J
L
∑
n=1
(S1,n + S1,n+1) · S2,n − µBH
L
∑
n=1
(
g1S
z
1,n + g2S
z
2,n
)
. (1)
The integers n number the unit cells, each containing two lattice spacings and two kinds of quantum
spin operators S1,n and S2,n characterized by the quantum spin numbers S1 and S2 (S1 > S2),
respectively. g1 and g2 are the g-factors of the site magnetic moments, µB is the Bohr magneton,
and H is an external uniform magnetic field applied along the z direction. As an example, the
following parameters of the Hamiltonian (1) have been extracted from magnetic measurements
on the recently synthesized quasi-1D bimetallic compound NiCu(pba)(D2O)3·2D2O [9] with a
structure which is similar to the one shown in figure 1: (S1, S2) ≡ (SNi, SCu) = (1, 1/2), J/kB =
121K, g1 ≡ gNi = 2.22, g2 ≡ gCu = 2.09. Henceforth we suggest g1 = g2 ≡ g and use the Planck
constant ~, the Boltzmann constant kB, and the lattice spacing a0 as unites.
2.1. Ground-state properties and excitations
2.1.1. H = 0
Let us start with the case H = 0. According to Lieb-Mattis’ theorem [16], for H = 0 the ground
state of the bipartite model (1) has a total spin ST = (S1 −S2)L, L being the number of unit cells,
so that it is necessarily long-range ordered. Such ferrimagnetic ground states may also be referred
436
Spin models of quasi-1D quantum ferrimagnets
to as quantized ferrimagnetic states since the FM moment is quantized in integral (or half-integral)
multiples of the number of unit cells L. As explicitly demonstrated below (section 3), strong enough
competing interactions can suspend this quantization rule in some regions of the parameter space
where the long-range ferrimagnetic order still survives. Since the model has a magnetically ordered
ground state, this makes the 1D problem amenable to the spin-wave theory (SWT) approach [17].
Valuable qualitative information about the ground state and low-lying excitations can be ex-
tracted already from the linear spin-wave theory (LSWT) [18–21]. The on-site magnetizations
m1 = 〈Sz
1,n〉 and m2 = −〈Sz
2,n〉 (m1 − m2 = S1 − S2) are parameters of the quantum ferrimag-
netic phase keeping information about the long-range spin correlations. LSWT implies that in the
extreme quantum case (S1, S2) = (1, 1/2) the quantum spin fluctuations reduce substantially the
classical on-site magnetizations (m2/S2 ≈ 0.39). Notice, however, that due to the broken sublattice
symmetry, in the mixed-spin models there appear important first-order corrections to m1 and m2
resulting from two-boson interactions in the bosonic Hamiltonian [22]. Up to second order in 1/S2
(in respect to the linear approximation), SWT series gives the precise result m2 = 0.29388 [23],
to be compared with the density-matrix renormalization-group (DMRG) estimate m2 = 0.29248
[20]. Another interesting peculiarity of the mixed-spin chain is the extremely small correlation
length (smaller than the unit cell) of the short-range spin fluctuations [19,20]. This observation
explains the good quantitative description of the model achieved with the variational approach
using matrix-product states [24,25].
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
E
π
k
k
+
−
2k/
E
E
xc
ita
tio
n
en
er
gi
es
+ l l
l
l
l
l
l
l
l
l
l
l
l l l l
l
l
l
l
l
l
l
l
l l l l
l
Figure 2. Dispersions of the one-magnon excitations (E±
k
) in the system (S1, S2) = (1, 1/2)
calculated up to second order in the 1/S2 relative to the linear approximation [27]. The points
in E±
k
show numerical ED results for periodic chains (for E+
k
) and QMC results (for E−
k
) [21].
Turning to the excitation spectrum, SWT predicts two types of low-lying magnon excitations
E±
k , respectively, in the subspaces with z components of the total spin Sz
T = ST ± 1 (see figure 2).
In the long wavelength limit k � 1, the FM mode E−
k takes the Landau-Lifshitz form
E−
k =
%s
M0
k2 + O(k4) , (2)
where %s = JS1S2 is the spin stiffness constant [26] and M0 = (S1 − S2)/2 is the linear density of
the net FM moment. This form of the Goldstone modes is typical of Heisenberg ferromagnets, and
reflects the fact that the FM order parameter is itself a constant of the motion. As demonstrated
below, the parameters %s and M0 play a basic role in the low-temperature thermodynamics of
the model. On the other hand, the AFM branch E+
k is gapped, with a minimum at k = 0 given
by ∆+ = 2(S1 − S1)J in a LSWT approximation. Note, however, that the LSWT estimates for
%s and the AFM gap ∆+ are further renormalized by the boson-boson interactions [27]: In the
extreme quantum system (S1, S2) = (1, 1/2), already the first order corrections in 1/S2 give the
results %s/(Js1s2) = 0.761 and ∆(+) = 1.676J . The quantum Monte-Carlo (QMC) result for this
437
N.B. Ivanov
parameter 1.759 [21] clearly indicates the importance of the 1/S2 corrections. Precise estimates for
the parameters of the (1, 1/2) chain have also been obtained by using cluster series expansions [28]:
m2 = 0.292487(6), ∆+/J = 1.7591(6) and %s/(Js1s2) = 0.831(5).
2.1.2. H 6= 0
The introduction of a Zeeman term in equation (1) leaves all eigenstates with a given z compo-
nent of the total spin Sz
T invariant, while shifting the eigenenergies by hSz
T, where h = gµBH . In
particular, the one-magnon states E±
k , carrying magnetizations ±1, change to E±
k (h) = E±
k ∓h, i.e,
the Zeeman term introduces a gap ∆−(h) = h for the FM branch and reduces the gap of the AFM
branch, ∆+(h) = ∆+ − h. If the magnetic field exceeds the critical value h = hc1 = ∆+, the AFM
gap closes and the system enters a critical phase which is expected to be a kind of Luttinger liquid,
in analogy with the behavior of other gapped spin models like the spin-1 chain and the spin-1/2
ladder [29–31]. The critical phase terminates at a second critical field h = hc2 = 3J at which the
system becomes fully polarized [32]. An accurate description of the mixed-spin (1, 1/2) Heisenberg
chain (H 6= 0) in the critical phase hc1 < h < hc2 has been achieved by a mapping to an effective
spin-1/2 XXZ chain in external magnetic field which uses variational matrix-product states [25]. In-
terestingly, such a critical phase seems to be characteristic of the entire class of mixed-spin (S1, S2)
anisotropic XXZ Heisenberg chains with easy-plane anisotropy and short-range interactions. An
extensive numerical analysis, using exact-diagonalization (ED) results for (1, 1/2) and (3/2, 1/2)
periodic chains, suggests universal critical properties for these chains in the entire interval from
the FM to the ferrimagnetic isotropic points [33]. Along this phase, the critical fluctuations are
ruled by a conformal field theory of Gaussian type with a topological charge c = 1.
The discussed one-magnon excitations control the magnetization process of the mixed-spin
chain. Since the lowest excitation increasing the FM moment M = 〈Sz
1,n +Sz
2,n〉 exhibits an energy
gap, the magnetization curve M(H) has a plateau at M = S1 − S2. This plateau phase, however,
appears in the related classical system as well. In this connection, it is interesting to examine the
role of different anisotropies which may appear in real materials. As to the exchange anisotropy,
it occurred that the quantum effects simply stabilize the plateau at M = S1 − S2 against the XY
anisotropy, i.e., the effect is reduced to quantitative changes of the classical result [34]. A clear
indication for the quantum origin of the discussed magnetization plateau was provided for the
(1, 1/2) chain with a single-ion anisotropy for spin-1 sites, D
(
Sz
1,n
)2
, which is even more realistic
for the existing mixed-spin materials [35]. It was revealed, in particular, that the mechanism of
the M = 1/2 plateau can change from a Haldane to a large-D type through a Gaussian quantum
critical point, estimated as D/J = 1.114, which is a kind of justification for the quantum origin
of the M = 1/2 plateau in this system. Two plateau phases (M = 1/2) with different parities
have also been studied in a frustrated mixed-spin chain with different next-nearest-neighbor AFM
bonds (see figure 4a) [36].
2.2. Thermodynamics
Most of the experiments on quasi-1D mixed-spin systems carried out in the past concerned
the magnetic susceptibility of ferrimagnetic chains with rather big S1 spins (typically, S1 = 5/2),
making the system rather classical [37]. As a matter of fact, quantum effects are most pronounced
in the extreme quantum case (S1, S2) = (1, 1/2), realized, for example, in the quasi-1D bimetallic
material NiCu(pba)D2O)3· 2D2O [9]. As mentioned above, this material has an exchange interac-
tion of about 121 K, and shows a three dimensional AFM ordering transition at 7 K (see figure 3)
which somewhat obscures the very low temperature (T ) features of the 1D ferrimagnet. Still, low-
field measurements are in good agreement with the theoretically expected results. On the other
hand, the magnetic fields necessary to reach the theoretically most interesting critical phase would
be for this compound well beyond 100 T, making the search for other compounds with weaker
interactions an interesting challenge for experimentalists.
Low-temperature thermodynamics (T � %sM0) of the mixed-spin chains is controlled by the
FM magnons, so that in zero magnetic field it can be described by using Takahashi’s constrained
438
Spin models of quasi-1D quantum ferrimagnets
(a) (b)
Figure 3. (a) Experimental results (χ · T vs. T ) for the (1, 1/2) mixed-spin chain
NiCu(pba)D2O)3· 2D2O [9]. (b) Cv vs. T of the (1/1/2) mixed-spin chain (DMRG) compared
to Cv(T ) of the spin-1/2 FM chain (DMRG) at h = 0.05J . The difference of both specific heats
can be identified as AFM contribution [32]. Cv and T are presented in the unites of J .
SWT for ferromagnets [40,41]. A few variants of this theory have also been applied to 1D ferrimag-
nets [38,42,43]. In terms of the ground-state parameters %s and M0, the explicit form of the series
in powers of t ≡ T/%sM0 for the uniform susceptibility χ and the specific heat Cv read as [44]
χT
M0
=
2
3
t−1 − ζ( 1
2 )√
π
t−
1
2 +
ζ2( 1
2 )
2π
+ O
(
t
1
2
)
,
Cv
M0
=
3
8
ζ( 3
2 )√
π
t
1
2 − t
2
+
[
const − 15ζ( 1
2 )
32
√
π
]
t
3
2 + O
(
t2
)
. (3)
Here, ζ(z) is Riemann’s zeta function, and const = 15(S2
1 + S1S2 + S2
2)ζ( 5
2 )/128
√
π.
As may be expected, the above expansions reproduce Takahashi’s original expansions for the
Heisenberg FM chain characterized by the parameters M0 = S and %s = JS2. Apart from the
coefficient of t3/2 in the expansion for Cv , the above expressions reproduce precisely the thermo-
dynamic Bethe-ansatz calculations for the spin-1/2 FM Heisenberg chain [45,46]. It is interesting
to note that without the factor const in the expression for Cv , both expansions fulfill the general
hypothesis according to which in 1D Heisenberg ferromagnets all observables should be universal
functions of the bare couplings M0, %s, and h, realizing a no-scale-factor universality [47,48]. In
particular, this means that the free-energy density should have the generic form
F
N
= TM0ΦF
(
ρsM0
T
,
h
T
)
, (4)
where ΦF (x, y) is a universal scaling function with no arbitrary scale factors: for example, the site
spin S enters only indirectly, through %s and M0. The appearance of the factor violating the above
scaling hypothesis is probably an artifact of Takahashi’s SWT. As far as we know, the universal
low-T thermodynamic properties of quasi-1D quantum ferrimagnets have not been studied so far.
Whereas low-T thermodynamics reflects the dispersion relation of the lower gapless FM branch
E−
k , the intermediate-T behavior is dominated by that of the upper AFM branch E+
k . Therefore,
when the condition T � ∆+ is violated, all the thermodynamic parameters are expected to demon-
strate a crossover from a FM to a AFM behavior. In particular, for the uniform susceptibility χ one
may expect χ(T ) ∝ T−2 (as T → 0) and the Curie law behavior χ(T ) ∝ T−1 in the high-T region.
Figure 3a displays a typical experimental curve (χT vs. T ) which is usually used to determine
the character of the short-range interactions. For a paramagnetic system, Curie’s law implies the
439
N.B. Ivanov
behavior χT = const over the whole T range. If the magnetic system has dominant FM (AFM)
interactions, χT increases (decreases) when T is decreased. Thus, in the interval 70 K< T < 300 K
the dominating coupling between nearest-neighbor spins is AFM, whereas the increase of χT below
the minimum around T ≈ 70 K implies that the system behaves like a FM chain at low T . The
steep decrease below 10 K may be attributed to an established 3D AFM long-range order.
Turning to the behavior of the specific heat, it changes from Cv(T ) ∝ T 1/2 in the extreme
low-T region [see equation (3)], through an AFM Schottky-like peak (for intermediate T ) to a
paramagnetic T−2 decay for high T (see figure 3b). In addition, Cv(T ) acquires a characteristic
double-peak structure for 0 < h < hc1, which is related to the gapped modes E±
k [38]. As a matter
of fact, both gapped antiferromagnets as well as ferromagnets in an external magnetic field exhibit
a pronounced peak whose position is related to the gaps ∆±. Such a double-peak structure has
also been predicted for the (1, 1/2) chain with a FM exchange constant (J < 0) [39].
3. Mixed-spin chains and ladders with magnetic frustration
Over the last years there has been an increasing interest in quantum spin systems with com-
peting exchange interactions [49]. Quasi-1D quantum spin systems with geometric frustration, both
for half-integer and integer spins [50–54] (see also the review article [15]) set up an important part
of this research. It is remarkable that up to now a relatively small amount of research on 1D
frustrated mixed-spin models has been published. These models typically exhibit ground states
with a net FM moment, so that a number of intriguing issues like the nature of the new phases as
well as the character of the ferrimagnetic-paramagnetic transitions may be studied. What makes
such transitions interesting is the fact that the order parameter is a conserved quantity: As is
known, in systems with a Heisenberg SO(3) symmetry this conservation law is expected to lead
to strong constraints on the critical field theories [55,56]. Below we survey the available results for
some generic Heisenberg spin models relevant to quasi-1D quantum ferrimagnets with competing
interactions.
3.1. Mixed-spin J1 − J2 Heisenberg chains
The mixed-spin Heisenberg chain composed of two types of alternating spins (S1 > S2), which
are connected via competing nearest-neighbor (J1 > 0) and next-nearest-neighbor (J2, J
′
2 > 0)
AFM exchange bonds (see figure 4a), is one of the simplest realistic models of frustrated 1D
Figure 4. Some generic 1D Heisenberg spin models with AFM exchange bonds exhibiting fer-
rimagnetic ground states: (a) The mixed-spin J1 − J2 chain with AFM nearest- (J1) and next-
nearest-neighbor (J2, J
′
2) exchange bonds; (b) The distorted (J1 6= J
′
1) diamond chain with
frustrated vertical bonds (J
′
) and four-spin cyclic exchange interactions (K); (c,d,e) Three
generic types of mixed-spin ladders with geometric frustration.
440
Spin models of quasi-1D quantum ferrimagnets
quantum ferrimagnets [22,36]. This model may also be considered as a ferrimagnetic analogue of
the frustrated FM Heisenberg chain with FM nearest-neighbor- (J1 < 0) and AFM next-nearest-
neighbor (J2 > 0) exchange bonds. The spin-1/2 frustrated J1−J2 FM chain has recently attracted
much attention, as it is supposed to describe a number of quasi-1D edge-sharing cuprates, such as
Rb2Cu2Mo3O12 [57] and LiCuVO4 [58]. The latter material exhibits multiferroic properties [59]
as well as an interesting specific phase transition in a magnetic field from an ordered spiral to an
ordered modulated-collinear magnetic phase [60]. On the theoretical side, the latter FM model was
shown to exhibit a vector chiral long-range oder (in a moderate magnetic field) as well as a rich
variety of exotic quantum phases with different kinds of multipolar spin correlations (in a larger
magnetic field) [61,62].
Turning to the mixed-spin J1 − J2 chain, let us begin with a comment on its classical limit.
The ground state of the model can be described by the ansatz Si,n = Si [u cos(Qn) + v sin(Qn)]
(i = 1, 2), where u ⊥ v are unit vectors in the spin space. The classical ferrimagnetic state
with a pitch angle between neighboring spins Q/2 = π is stable up to the phase transition point
J2c = S1S2J1/[2(S2
1 + S2
2)]. In the strongly frustrated region J2 > J2c, the stable state is a spiral
with an ordering wave vector given by cos(Q/2) = −S1S2J1/[2J2(S
2
1 + S2
2)]. In the limit J2 → ∞,
Q = π and the system is composed of two decoupled AFM chains with site spins S1 and S2.
Next, let us discuss the extreme quantum case (S1, S2) = (1, 1/2). Already a qualitative semi-
classical analysis implies that near the classical transition point at J2 = 0.2J1 the FM magnon
branch E−
k (see figure 2) is strongly flattened, whereas the optical AFM branch shows only a
smooth increase of the AFM gap ∆+. This means that the optical magnons do not play any im-
portant role in the mechanism of the transition. On approaching the classical transition point, the
frustration and quantum spin fluctuations play somewhat different roles: Whereas the magnetic
frustration strongly reduces the short-range spin correlation length in the ferrimagnetic phase, the
quantum fluctuations stabilize the ferrimagnetic order up to the point J2 = 0.231J1 beyond the
classical transition point at J2 = 0.2J1. Another effect of the quantum fluctuations is related to
the change of the character of the classical transition: A detailed DMRG analysis of the low-energy
levels around the classical transition clearly indicates a level crossing, i.e., a first-order quantum
phase transition to a singlet ground state at J2 = 0.231J1. Further, DMRG has also indicated that
at least from J2 = 0.25J1 upwards the discussed model exhibits a singlet ground state [22].
A number of open issues related to the mixed-spin J1 − J2 Heisenberg model can be indicated.
One of the important open problems concerns the nature of the singlet ground states established
beyond the ferrimagnetic phase. Since in 1D systems the classically broken SO(3) symmetry in
the spiral state is generally expected to be restored by quantum fluctuations, one is enforced to
look for possible magnetically disordered states. A valuable information can be obtained from
the Lieb-Schultz-Mattis theorem [63] adapted to mixed-chain spin models [64]. The theorem is
applicable to systems with a half-integer cell spin (S1 + S2) and says that the model either has
gapless excitations or else has degenerate ground states. Therefore, one may look for phases with
presumably some broken discrete symmetry. In particular, the long-ranged chiral phase found in
the frustrated spin-1/2 FM J1 − J2 model in the small magnetic field region could be a possible
candidate [61]. However, the variety of possible non-magnetic quantum phases may be considerably
enlarged due to the existence of two kinds of site spins in the mixed variant of the parent model.
3.2. Frustrated diamond chains
The diamond Heisenberg chain is another generic model of 1D quantum ferrimagnets con-
structed from one kind of spins living on sublattices with different number of sites (see figure 4b).
The frustrated symmetric diamond chain (SDC) (J1 = J
′
1, J = 0) with AFM vertical bonds J⊥ > 0
is probably the first studied model of 1D quantum ferrimagnet with competing interactions [65,66].
A particular variant of the frustrated model, the distorted (J1 6= J
′
1) spin-1/2 diamond chain, has
received an increasing theoretical [67,68] as well as experimental interest in the past decade due
to its rich quantum phase diagram [69] (see figure 5a) and the relevance to the real material
Cu3(CO3)2(OH)2 (azurite) [70]. Without external magnetic field, three quantum phases in the pa-
rameter space (J1/J⊥, J
′
1/J⊥) have been discussed for the distorted model: For J1/J⊥, J
′
1/J⊥ � 1,
441
N.B. Ivanov
the low-energy sector is governed by an effective spin-1/2 AFM Heisenberg model, which indicates
the formation of a gapless spin-fluid phase (SF) with some additional high-energy modes related
to local excitations of the vertical dimers. For intermediate J1/J⊥ and J
′
1/J⊥, the ground state
dimerizes, forming a twofold degenerate sequence of alternating tetramers and dimers (TD1 phase).
Finally, for both J1/J⊥ and J
′
1/J⊥ sufficiently large, the ground state is a 1D ferrimagnet. These
phases can be clearly identified already in the SDC: SF (J1 < 0.5J⊥), TD1 (0.5J⊥ < J1 < 1.10J⊥),
and 1D ferrimagnet (J1 > 1.10J⊥).
Figure 5. (a) Quantum phase diagram (T = 0) of the distorted diamond chain in the parameter
space J1 vs. J
′
1 (J⊥ = 1) [69]. SF indicates the gapless spin-fluid phase, whereas TD1 marks the
tetramer-dimer phase constructed from alternating dimers and tetramers in local singlet states.
(b) Quantum phase diagram (T = 0) of a frustrated SDC with competing four-spin cyclic
interactions in the parameter space J⊥ vs. K (J
′
1 = J
′
1 = 1) [71]. In this system, there appear at
least five new phases denoted as follows: FM1 (fully-polarized FM phase), FM2 (another phase
with a net FM moment), DM (fourfold degenerate singlet phase), TD2 (another tetramer-dimer
phase with tetramers in local triplet states), and an exotic non-Lieb-Mattis ferrimagnetic state
denoted by a question mark. The dashed lines trace the phase boundaries of the single-diamond
system.
The diamond Heisenberg chain is also one of the simplest quantum spin models admitting a
four-spin cyclic exchange coupling. Below we discuss the impact of this competing interaction on
the quantum phase diagram (T = 0) of the frustrated SDC [71]. The schematic Hamiltonian of
the model is presented in figure 4b, the standard four-spin cyclic exchange coupling [72] in a single
diamond being controlled by the parameter K. It is important to notice that the cyclic coupling
does not violate the local symmetry of the frustrated (J⊥ > 0) SDC model under the exchange of
pairs of off-chain spins S1a,n and S1b,n in the diamonds. Thus, in the important case of spin-1/2 off-
chain operators, the system is characterized (as in the frustrated model without cyclic interactions)
by L local good quantum numbers sn = 0, 1 (n = 1, 2, . . . , L) related to the composite off-chain
spins S1,n = S1a,n + S1b,n: S
2
1,n = sn(sn + 1). Using this local symmetry, the Hamiltonian of the
SDC can be represented in the compact form
Hc = E0 +
L
∑
n=1
[
J1S1,n · (S2,n + S2,n+1) + JnS2,n · S2,n+1 +
K
2
{S1,n · S2,n,S1,n · S2,n+1}
]
. (5)
Here E0 = J⊥
∑L
n=1[sn(sn +1)−3/4] is a fixed number for every sector defined as a sequence of the
local quantum numbers [s1, s2, . . . , sL], Jn = J + K/4 − snK, and {A, B} is the anticommutator
of the operators A and B.
Let us briefly comment the phase diagram for the spin-1/2 SDC (figure 5b), by using some
symmetries of the Hamiltonian (5). In the parameter space where the ground state is characterized
by sn = 0 (n = 1, 2, . . . , L), the first and third terms in the square brackets vanish and the model
442
Spin models of quasi-1D quantum ferrimagnets
is equivalent to the spin-1/2 Heisenberg chain with an effective exchange parameter J +K/4. This
explains the presence of a fully polarized FM1 (J +K/4 < 0) and a gapless spin-fluid (J +K/4 > 0)
phases for large enough J⊥/J1. Note that besides the well-documented collective modes, these
phases exhibit specific additional single-particle modes which are related to local triplet excitations
on the vertical dimers. Being eigenstates of the Hamiltonian, these excitations are completely
localized in the SDC.
For intermediate values of the parameter J⊥/J1, the ground state lies in the sector [1, 0, · · · , 1, 0].
The tetramer-dimer phase TD1 studied in the DDC model (see figure 5a) survives in the region with
relatively small cyclic interactions. This doubly degenerate singlet state may roughly be thought
of as a product of plaquette singlet states on every second diamond, as depicted in figure 5a.
For larger K at fixed J⊥, there appears another tetramer-dimer phase (TD2), with every second
diamond approximately in a triplet state. The numerical ED analysis shows relatively strong AFM
correlations between the neighboring triplet diamonds, as opposed to the TD1 state where the
diamonds are weakly correlated. Clearly, both tetramer-dimer phases are gapped and doubly-
degenerated.
Finally, for J⊥/J1 < 1 the ground state lies in the sector sn = 1 (n = 1, 2, . . . , L), so that the
low-energy region of equation (5) describes a mixed-spin (1, 1/2) chain with competing three-spin
exchange interactions. To the best of our knowledge, realistic 1D mixed-spin Heisenberg models
with multiple-spin exchange interactions have not been discussed in the literature, although these
interactions may play an important role in some recently synthesized mixed-spin magnetic materials
and nanomagnets [73,74]. Here we restrict ourselves to a general overview of the spin phases of
this interesting mixed-spin model with cyclic interactions in the extreme quantum case (S1, S2) =
(1, 1/2). We also suppose a FM effective exchange interactions between the in-chain spins (i.e.,
Jn ≡ J − 3K/4 < 0). In the region K > 1.2J1, a detailed numerical ED analysis indicates at
least two additional phases, denoted by DM and FM3 in figure 5b [71]. DM is a dimerized singlet
phase stabilized approximately in the region 1.5J1 6 K 6 2.3J1, whereas FM3 is a magnetic phase
with a net FM moment. Finally, in the narrow interval 1.2J1 < K < 1.5J1 a specific ferrimagnetic
phase characterized by a reduced magnetic moment per cell, M < M0 = S1 −S2, seems to appear.
Below, other mixed-spin models exhibiting similar exotic (non-Lieb-Mattis) ferrimagnetic phases
are surveyed.
3.3. Mixed-spin ladders with geometric frustration
Some typical examples of mixed-spin ladder structures are shown in figure 4. The first two struc-
tures reproduce, e.g., arrangements of the magnetic atoms Mn (S1 = 5/2) and Cu (S2 = 1/2) along
the a-axis in the compounds MnCu(pbaOH)(H2O)3 (pbaOH = 2-hydroxy-1,3- propylenebisoxam-
ato) and MnCu(pba)(H2O)3·2H2O (pba = 1,3-propylenebisoxamato), respectively. Along the c-
axis, the magnetic ions in both mixed-spin compounds are arranged as shown in figure 4(e) [2]. A
very recently synthesized quasi-1D mixed-valent-iron material [FeIIFeIII (trans-1,4-cyclohexanedi-
carboxylate)1.5] exhibiting ferrimagnetic properties, seems to be related to the frustrated mixed-
spin structure shown in figure 4d [75]. We are not aware of any real ferrimagnetic compound related
to the ladder structure shown in figure 4e. Nevertheless, in view of the easy control of the molecular-
unit positions in the molecular chemistry, it may be expected that other materials, related to the
generic mixed-spin models presented in figure 4, will be synthesized in the near future.
On the theoretical side, unfrustrated variants of the ladder models presented in figure 4 have
already been analyzed in a number of publication [76–81]. The unfrustrated checkerboard ladder
(figure 4c) with AFM bonds (J, J1 > 0) exhibits a ferrimagnetic ground state. Its low-energy prop-
erties closely reproduce the properties of the generic mixed-spin chain discussed in section 2 [79].
Note that a variant of this structure with FM legs (J1 < 0) demonstrates completely different
features: The FM leg coupling drives the system into a gapless spin-fluid ground state which is
characteristic of the spin-1/2 AFM Heisenberg chain [81]. Similar gapless phase appears in the un-
frustrated variant of the model displayed in figure 4e [77,78], whereas the model shown in figure 4d
possesses a gapped non-degenerate rung-singlet ground state [80], which is the characteristic phase
of the uniform spin-1/2 Heisenberg ladder.
443
N.B. Ivanov
(a) (b)
Figure 6. (a) Phase diagram of the checkerboard mixed-spin (1/1/2) ladder (figure 4c) in the
(J2/J1, h/J1) plane [82]. The phase boundary of the fully polarized phase (M0 = 3/2) is exact.
Here M0 is the FM moment per rung. The boundary (ABCD) of the ferrimagnetic M0 = 1/2
plateau phase is obtained from numerical ED of periodic clusters with L = 10 (squares) and
L = 12 (crosses) rungs. The points B, C, and D mark, respectively, the change of the lowest
excited states, the tip of the lobe, and the h = 0 transition point at J2/J1 = 0.399 from the
M0 < 1/2 ferrimagnetic phase to a gapless spin-fluid phases. The latter phase occupies the rest
of the phase diagram. (b) M0 vs. J2/J1 for the non-Lieb-Mattis (i.e., M0 < 1/2) ferrimagnetic
phase, as obtained from numerical ED of L = 12 clusters (dashed line). The solid line connects
the midpoints of the steps [83].
To understand the role of the geometric frustration in such mixed-spin models, let us turn
to the phase diagram in figure 6a presenting the phases of the frustrated (1, 1/2) checkerboard
ladder (figure 4c) in the parameter space (J2/J1, h/J1) [82]. We are not aware of any publications
studying the other two frustrated models displayed in figures 4d, and e. As a function of the
frustration parameter J2/J1, the classical phase diagram of the frustrated checkerboard model
exhibits three phases, which can be described by the angles (θ, φ) fixing the directions of the
classical spins S1,n and S2,n in respect to the classical ferrimagnetic configuration (θ, φ) = (0, 0)
with up-S1,n and down- S2,n spins. In the special case (S1, S2) = (1, 1/2), the classical canted
state shown in figure 4c is stable in the interval 0.3219J1 < J2 < 0.4606J1. For larger J2, a
collinear configuration with (θ, φ) = (π/2, π/2) is stabilized. In the classical limit, the transitions
between the canted state and the other two phases are continuous. Turning to the quantum model,
the following changes in the quantum phase diagram (line h = 0 in figure 6a) can be indicated.
Whereas the classical ferrimagnetic phase survives quantum fluctuations, the collinear magnetic
state is completely destroyed. Instead, for J2 > 0.399J1 there appears a gapless spin-fluid phase.
In the general chase, the quantum paramagnetic state is either critical (for half-integer S1 + S2),
or gapped (for integer S1 + S2).
The most interesting changes appear in the classical canted state. In figure 6b we show numerical
ED results for the FM moment per rung M0 as function of the frustration parameter J2/J1. We
see that the quantum phase which substitutes the classical canted phase is characterized by a finite
FM moment which is, however, less than the quantized FM moment per rung (S1 − S2 = 1/2)
of a standard Lieb-Mattis ferrimagnet. Notice that this phase exists only at h = 0. For h > 0
it merges into the Luttinger liquid phase (see figure 6a). An extrapolation of the ED results for
L = 8, 10 and 12 rungs definitely indicates the presence of this phase in the interval 0.341J1 <
J2 < 0.399J1. Already a qualitative semiclassical analysis (supported by ED results) implies that
on approaching the phase transition point at J2 = 0.341J1 from the ferrimagnetic phase, the
lower magnon branch E−
k softens in the vicinity of k = π, and the gap at k = π vanishes at
the transition point to the canted phase. Thus, there appears a linear Goldstone mode which is
444
Spin models of quasi-1D quantum ferrimagnets
characteristic of the classical canted phase. It may be suggested that this critical mode survives
quantum fluctuations [55], whereas the spin rotation symmetry U(1) in the xy plane should be
restored, i.e., 〈Sx
1,n〉 = 〈Sx
2,n〉 = 0. This scenario with a power-law decay of the transverse spin-spin
correlations is supported by the renormalization-group analysis of similar phases in quantum rotor
models [55] as well as by a recent DMRG analysis of a similar phase found in a generalized SDC
model with an additional competing AFM interaction between the off-chain spins in figure 4b [84].
The first studied quantum spin model exhibiting such an exotic quantum state seems to be the
spin-1/2 two-leg ladder constructed from different (one FM and another AFM) legs and AFM
rungs [85]. Quite recently, there has been a number of reports indicating similar 1D quantum
magnetic phases in some decorated quantum spin chains [86,87] as well as in a mixed-spin (2, 1)
Heisenberg chain with competing single-ion anisotropies [88].
4. Conclusion
In conclusion, we have surveyed the available theoretical results related to some generic quan-
tum spin models displaying 1D ferrimagnetic ground states. The stress was put on the interplay
between quantum fluctuations and the competing interactions. The discussed models of quasi-1D
ferrimagnets with competing interactions exhibit a rich variety of magnetic and paramagnetic quan-
tum phases and provide unique examples of 1D magnetic-paramagnetic quantum phase transitions.
Finally, a great deal of important open issues deserving further studies have been debated.
5. Acknowledgements
The author thanks the staff of the Fakultät für Physik, Universität Bielefeld for hospitality and
Jürgen Schnack for the critical reading of the manuscript. This work was supported by Deutsche
Forschungsgemeinschaft (Grant SCHN 615/13–1) and Bulgarian Science Foundation (Grant D002–
264/18.12.08).
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Спiновi моделi квазiодновимiрних квантових ферiмагнетикiв
з конкуруючими взаємодiями
Н.Б. Iванов1,2
1 Iнститут фiзики твердого тiла, Болгарська академiя наук,
72 Царiградско шосе, 1784 Софiя, Болгарiя
2 Факультет фiзики, Унiверситет Бiлєфельда, D–33501 Бiлєфельд, Нiмеччина
Отримано 19 червня 2009 р.
Ми представляємо короткий огляд недавньої теоретичної роботи, пов’язаної з загальними спiнови-
ми моделями Гайзенберга, що описують квазiодновимiрнi квантовi ферiмагнетики. Наголос зробле-
но на квантових ланцюжках i драбинках з сильною конкуренцiєю взаємодiй, таких, як фрустрова-
ний J1 − J2 ланцюжок зi змiнними (1, 1/2) спiнами, спiн-1/2 ромбiчний ланцюжок з чотириспiновими
циклiчними взаємодiями i деякi загальнi типи драбинок з геометричними фрустрацiями з рiзними
значеннями спiна. Як правило, обговорюванi моделi виявляють багатi квантовi фазовi дiаграми i
становлять цiкавi приклади одновимiрних квантових фазових переходiв магнетик – парамагнетик.
Оговорюються деякi вiдкриттi проблеми у цiй областi дослiджень.
Ключовi слова: квантовi спiновi ланцюжки, ферiмагнетики, фрустрацiї
PACS: 75.10.Jm, 75.10Pq, 75.45.+j
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