arXiv:cond-mat/0606806v1 [cond-mat.str-el] 30 Jun 2006
Finite-temperature order-disorder phase transition in a frustrated bilayer quantum
Heisenberg antiferromagnet in strong magnetic fields
Johannes Richter1, Oleg Derzhko1,2,3and Taras Krokhmalskii2
1Institut f¨ ur Theoretische Physik, Universit¨ at Magdeburg, P.O. Box 4120, D-39016 Magdeburg, Germany
2Institute for Condensed Matter Physics, National Academy of
Sciences of Ukraine, 1 Svientsitskii Street, L’viv-11, 79011, Ukraine
3National University “Lvivska Politechnika”, 12 S. Bandera Street, L’viv, 79013, Ukraine
(Dated: February 6, 2008)
We investigate the thermodynamic properties of the frustrated bilayer quantum Heisenberg an-
tiferromagnet at low temperatures in the vicinity of the saturation magnetic field. The low-energy
degrees of freedom of the spin model are mapped onto a hard-square gas on a square lattice. We
use exact diagonalization data for finite spin systems to check the validity of such a description.
Using a classical Monte Carlo method we give a quantitative description of the thermodynamics of
the spin model at low temperatures around the saturation field. The main peculiarity of the consid-
ered two-dimensional Heisenberg antiferromagnet is related to a phase transition of the hard-square
model on the square lattice, which belongs to the two-dimensional Ising model universality class. It
manifests itself in a logarithmic (low-)temperature singularity of the specific heat of the spin system
observed for magnetic fields just below the saturation field.
PACS numbers: 75.10.Jm, 75.45.+j
Keywords: bilayer antiferromagnet, geometric frustrations, high magnetic fields, hard-square problem
The quantum Heisenberg antiferromagnet (HAFM) on
geometrically frustrated lattices has attracted much at-
tention during last years.1,2Besides intriguing quan-
tum ground-state phases at zero magnetic field those
systems often show unconventional properties in finite
magnetic fields like plateaus and jumps in the magne-
tization curve, see e.g.Ref.
that a wide class of geometrically frustrated quantum
spin antiferromagnets (including the kagom´ e, checker-
board and pyrochlore lattices) has quite simple ground
states in the vicinity of the saturation field,4namely in-
dependent localized-magnon states, has further stimu-
lated studies of the corresponding frustrated quantum
antiferromagnets at high magnetic fields.5,6,7,8,9In par-
ticular, the low-temperature high-field thermodynamics
of various one- and two-dimensional frustrated quan-
tum antiferromagnets which support localized-magnon
states, can be discussed from a quite universal point
of view by mapping the low-energy degrees of freedom
of the quantum HAFM onto lattice gases of hard-core
objects.6,7,8,9,10For instance, the kagom´ e (checkerboard)
HAFM in the vicinity of the saturation field can be
mapped onto a gas of hard hexagons (squares) on a tri-
angular (square) lattice.7,8,9,10The exactly soluble hard-
hexagon model exhibits an order-disorder second-order
phase transition.11The hard-core lattice-gas model cor-
responding to the checkerboard HAFM consists of large
hard squares on the square lattice with edge vectors
? a1 = (2,0) and ? a2 = (0,2) (i.e.
neighbor and next-nearest-neighbor exclusion). For the
latter model no exact solution is available, but most likely
there is also an order-disorder phase transition.12The ex-
istence of a phase transition in the hard-hexagon (large-
3.The recent finding
there is a nearest-
hard-square) model would imply a corresponding finite-
temperature transition of the corresponding spin model
near saturation provided the low-temperature physics is
correctly described by the hard-core lattice-gas model.
However, at the present state of the investigations no
conclusive statements for the kagom´ e and checkerboard
antiferromagnets are available, since both models ad-
mit additional degenerate eigenstates not described by
more for these spin models precise statements on the gap
between the localized-magnon ground states and the ex-
citations are not available. Therefore, the effect of addi-
tional ground states and the excited states on the low-
temperature thermodynamics remains unclear.
The motivation for the present paper is to find
and discuss another two-dimensional frustrated quantum
HAFM, for which a hard-core lattice gas completely cov-
ers all low-energy states of the spin model in the vicinity
of the saturation field and where all excitations are sep-
arated by a finite energy gap. For such a spin model
one can expect that an order-disorder phase transition
inherent in the hard-core lattice-gas model can be ob-
served as a finite-temperature phase transition in the spin
model. It might be worth to note that such a phase tran-
sition of course does not contradict the Mermin-Wagner
theorem15that forbids magnetic long-range order (break-
ing the rotational symmetry) for the two-dimensional
Heisenberg model at any non-zero temperature and at
A spin model which satisfies these requirements is a
frustrated bilayer quantum HAFM. The investigation of
the bilayer quantum HAFM was initially motivated by bi-
layer high-Tcsuperconductors16and has been continued
till present time, see e.g. Ref. 17 and references therein.
Below we will illustrate that the corresponding hard-core
FIG. 1: (Color online) The frustrated bilayer antiferromagnet
(black lines). The vertical bonds have the strength J2whereas
all other bonds have the strength J1. The trapping cells (ver-
tical bonds) occupied by localized magnons are shown by fat
lines. The auxiliary square lattice is a simple square lattice
filled by hard squares (indicated as red squares) which corre-
spond to localized magnons.
lattice-gas model is a model of hard squares on a square
lattice, however, in difference to the checkerboard lattice
with smaller hard squares with edge vectors ? a1= (1,1)
and ? a2= (−1,1) (i.e. there is a nearest-neighbor exclu-
sion, only, cf. Fig. 1). This hard-square model exhibits
an order-disorder phase transition.11,18,19In the context
of different universality classes discussed in Ref. 9 the
frustrated bilayer quantum HAFM is the first example of
a spin system which belongs to the universality class of
(small) hard squares.
The paper is organized as follows. First, we specify the
frustrated bilayer model and illustrated the correspond-
ing localized-magnon states (Sec. II). Then in Sec. III we
calculate the contribution of the independent localized-
magnon states to the thermodynamic quantities using the
hard-square model. We compare our results with exact
diagonalization data for finite spin-1/2 Heisenberg sys-
tems of up to N = 32 sites. Finally, in Sec. IV we report
the low-temperature high-field thermodynamic quanti-
ties obtained on the basis of Monte Carlo simulations for
hard squares focusing on the (low-)temperature depen-
dence of the specific heat in the vicinity of the saturation
II.THE FRUSTRATED BILAYER
ANTIFERROMAGNET AND INDEPENDENT
To be specific, we consider the nearest-neighbor
Heisenberg antiferromagnet in an external magnetic field
on the lattice shown in Fig. 1. This lattice may be viewed
as a two-dimensional version of the frustrated two-leg lad-
der considered in Refs. 20,21,22 (for some similar models
see Ref. 23). The Heisenberg Hamiltonian of N quantum
spins of length s reads
Here the sum runs over the bonds (edges) which connect
the neighboring sites (vertices) on the spin lattice shown
in Fig. 1. Jnm > 0 are the antiferromagnetic exchange
constants between the sites n and m which take two val-
ues, namely, J2for the vertical bonds and J1for all other
bonds. ∆ ≥ 0 is the exchange interaction anisotropy pa-
rameter, h is the external magnetic field, and Sz=?
is the z-component of the total spin. In our exact diago-
nalization studies reported below we will focus on s = 1/2
and ∆ = 1.
We note that the Hamiltonian (1) commutes with the
operator Szand hence we may consider the subspaces
of its eigenstates with different values of Szseparately.
Evidently, the fully polarized state |0? = |s,...,s? is the
eigenstate of the Hamiltonian (1) with Sz= Ns and can
be considered as the vacuum state with respect to the
number of excited magnons. This state is the ground
state for high magnetic fields.
Consider next the one-magnon subspace with Sz=
Ns − 1.The one-particle
−s(J2+ ∆(8J1+ J2)) + h and Λ(2)
s(4J1(coskx+ cosky) + J2− ∆(8J1+ J2)) + h (here
kα= 2πnα/Mα, nα= 1,2,...,Mα, α = x,y, MxMy =
N/2). Obviously, the excitation branch Λ(1)
sionless and it becomes the lower one when J2 > 4J1.
Throughout this paper we assume J2≥ 4J1. Then the
saturation field is given by h1= s(J2+ ∆(8J1+ J2)).
The N/2 dispersionless one-magnon excitations can
be written as localized excitations on the N/2 vertical
bonds, i.e. |1? = |lm?v|s,...,s?eis an eigenstate of (1) in
the subspace with Sz= Ns−1 with the zero-field eigen-
value EFM− ǫ1, where EFM = 4N∆s2J1+ N∆s2J2/2
and ǫ1 = s(J2+ ∆(8J1+ J2)). In |1? the first part is
the localized one-magnon excitation on the vertical bond
number v, i.e. |lm?v= 2−1/2(|s,s − 1? − |s − 1,s?)vand
the second part |s,...,s?eis the fully polarized environ-
We pass to the subspaces with Sz= Ns − 2,Ns −
3,...,Ns − nmax, where nmax = N/4. We can easily
construct many-particle states in these subspaces using
the localized-magnon states. Explicitly the wave function
of n independent localized magnons has the form
energy isgiven by
|n? = |lm?v1...|lm?vi...|lm?vn|s,...,s?e. (2)
It is important to note that any two vertical bonds vi
and vj in Eq. (2) where localized magnons live are not
allowed to be direct neighbors. The energy of the n inde-
pendent localized-magnon state (2) in zero field h = 0 is
En= EFM−nǫ1. Since n independent localized magnons
can be put on the bilayer in many ways the eigenstates
(2) are highly degenerate. We denote this degeneracy by
gN(n), that is the number of ways to put n hard squares
on a lattice of N = N/2 sites (see Fig. 1). According to
Refs. 4,24 the independent localized-magnon states (2)
are the states with the lowest energy in the correspond-
ing subspaces. Moreover, they are linearly independent
(orthogonal type in the nomenclature of Ref. 14) and
form an orthogonal basis in each subspace.14Due to their
linear independence they all contribute to the partition
function of the spin system.
In the presence of an external field the eigenstates (2)
have the energy En(h) = EFM− hsN − n(ǫ1− h). At
the saturation field, h = h1 = ǫ1, all they are ground
states and the ground-state energy En(h1) does not de-
pend on n. As a result the ground-state magnetization
curve exhibits a jump at the saturation field. This jump
is accompanied by a preceding wide plateau, where the
width of this plateau can be obtained following the ar-
guments given in Ref. 22 and from finite-size data. We
find for s = 1/2, ∆ = 1 a plateau width of h1−h2= 4J1.
This plateau belongs to the two-fold degenerate ground
state with maximum density nmax = N/4 of localized
magnons, the so-called magnon crystal,4where all local-
ized magnons occupy only one of the two sublattices of
the underlying square lattice.
Furthermore, the degeneracy of independent localized-
magnon states at the saturation field W =?nmax
grows exponentially with the system size N that im-
plies a nonzero ground-state residual entropy S
k limN→∞(lnW/N) = 0.2037...k (see Ref. 19 and also
below). Due to their high degeneracy the independent
localized-magnon states are also dominating the ther-
modynamic properties at low temperatures for magnetic
fields around the saturation field as we will dicuss in de-
III. HARD-SQUARE MODEL
We want to calculate the contribution of the indepen-
dent localized magnons to the canonical partition func-
tion of the spin system,
where µ = ǫ1− h = h1 − h.
gN(n) is the canonical partition function Z(n,N) of n
hard squares on a square lattice of N = N/2 sites,
whereas Ξ(T,µ,N) =
grand canonical partition function of hard squares on
a square lattice of N = N/2 sites and µ is the chemi-
cal potential of the hard squares. As a result we arrive
at the basic relation between the localized-magnon con-
tribution to the canonical partition function of the spin
model and the grand canonical partition function of hard-
It is apparent that
n=0gN(n)exp(µn/kT) is the
Zlm(T,h,N) = exp
the spin system Flm(T,h,N) = −kT lnZlm(T,h,N),
whereas the entropy S,
the magnetization M=
usual formulas, Slm(T,h,N) = −∂Flm(T,h,N)/∂T,
Clm(T,h,N) = T∂Slm(T,h,N)/∂T, Mlm(T,h,N) =
sN − kT∂ lnΞ(T,µ,N)/∂µ.
We use exact diagonalization data to check this picture
for s = 1/2 isotropic Heisenberg system (1) of N = 16
and N = 20 sites (full diagonalization) and N = 32 (only
in the subspaces with Sz= 16,...,11) imposing periodic
boundary conditions. We fix the energy scale by putting
J1 = 1. For the vertical exchange bonds we consider
J2= 4, 5, 10.
First we compare the degeneracies gN(n) of the local-
ized n-magnon states calculated for spin systems of sizes
N = 16,20,32 with the corresponding values Z(n,N/2)
of the hard-square model.
plete agreement between both models for J2 > 4J1.
As an example we give here the numbers for N = 20:
g20(n) = 1, 10, 25, 20, 10, 2 for n = 0, 1, 2, 3, 4, 5.
For J2 = 4J1 the spin system has one extra state for
n = 1, i.e. in the one-magnon sector, since the dispersive
and the dispersionless mode Λ(1)
ate at k = (π,π).
To estimate the relevance of excited states, not de-
scribed by the localized-magnon scenario, we have de-
termined the thermodynamically relevant energy sepa-
ration ∆DOS between the localized-magnon states and
the other states of the spin system by calculating the
integrated low-energy density of states at saturation
field. We define ∆DOS as that energy value above the
localized-magnon ground-state energy, where the contri-
bution of the higher-energy states to the integrated den-
sity of states becomes as large as the contribution of the
localized-magnon states. For both values J2 = 5 and
J2= 10 we find ∆DOS≈ 1 independent of the size of the
system N. We can expect that the contribution of the
localized-magnon states to the partition function is dom-
inating for temperatures kT significantly smaller than
In Fig. 2 we present our results for the magnetization,
the specific heat and the entropy (panels from top to
bottom) for the spin system of size N = 20 (triangles
and diamonds). We compare those data with the corre-
sponding data taking into account only the contribution
of independent localized-magnon states, described by the
finite hard-square model of N = 10 sites (thin dotted
lines). Up to kT ≈ 0.1 both data sets coincide, demon-
strating that the hard-square description perfectly works
at low temperatures. But we observe good agreement
also for higher temperatures up to kT ≈ 0.3, indicating
that the independent localized magnons still dominate
the thermodynamic quantities. Further increasing kT,
(4) yieldsthe Helmholtzfree energyof
the specific heat C and
?Sz? are given by the
As expected we find com-
HS, 10 sites
HS, 64 sites
(h - h1)/(kT)
FIG. 2: The magnetization (sN−M(T,h,N))/N (a), the spe-
cific heat C(T,h,N)/kN (b) and the entropy S(T,h,N)/kN
(c) as functions of (h − h1)/kT. Symbols correspond to the
exact diagonalization data (N = 20, J2 = 5, kT = 0.2 (tri-
angles), kT = 0.3 (diamonds)). Thin lines correspond to the
results for a small hard-square system N = 10 (thin dotted
lines) and N = 64 (thin solid lines). The exact diagonaliza-
tion data for the temperatures up to kT = 0.1 coincide with
the corresponding data for the N = 10 hard-square system.
We also show by thick lines the Monte Carlo simulation results
for the magnetization and the specific heat for a hard-square
system of sizes up to 800 × 800 sites.
the high-energy states more and more contribute to the
partition function and the hard-square description loses
its validity. Note that an identical statement can be made
for N = 16 (N = 8).
THERMODYNAMICS AROUND THE
Now we discuss the low-temperature high-field thermo-
dynamics of the frustrated bilayer quantum HAFM using
the results for the hard-square model. From Fig. 2b it
is obvious, that fixing the temperature to a sufficiently
low value the spin system can be driven through a phase
transition by increasing the magnetic field h towards the
saturation field h1. On the other hand, we can fix the
magnetic field slightly below the saturation field and vary
the temperature. Then the phase transition is driven by
the temperature and the specific heat exhibits a singu-
larity at a critical temperature Tc(h), see Fig. 3, where
we show the results of a Monte Carlo simulation for large
hard-square systems (periodic cells with up to 800×800
sites and 3 · 106steps) for the specific heat for two val-
ues of the magnetic field. The data clearly indicate a
phase transition which occurs in the hard-square model
at zc= 3.7962..., i.e. at ((h1−h)/kT)c= lnzc≈ 1.3340
which yields kTc(h) ≈ (h1− h)/1.3340. The correspond-
ing order parameter is the difference of the density (of
hard squares or localized magnons in hard-square or spin
language, respectively) on the A- and B-sublattices of the
underlying square lattice.11For ((h1− h)/kT)c < lnzc
(i.e. for T > Tc(h), h < h1) both sublattices are
equally occupied, but for ((h1− h)/kT)c> lnzc(i.e. for
T < Tc(h), h < h1) one of two sublattices is more occu-
pied than the other. Therefore in the ordered phase the
translational symmetry of the spin (hard-square) system
is broken. Finally, at T = 0 only one sublattice is occu-
pied and the other is empty and the ground state of the
spin system is a magnon-crystal state, see Sec. II.
We can estimate the critical temperature for a fixed
deviation of the field from the saturation value using the
above given expression for kTc(h). For 1 − h/h1= 0.02
we find kTc/h1 ≈ 0.0150, whereas for 1 − h/h1 = 0.01
we have kTc/h1 ≈ 0.0075 and for the spin system
with J2 = 5 we find for h = 8.91J1 (h = 8.82J1)
kTc ≈ 0.0675J1 (kTc ≈ 0.1349J1). Such temperatures
are within a temperature range where the hard-square
description for finite systems works perfectly well (see
Figs. 2, 3). However, one may expect that the scenario
of the phase transition may hold also at temperatures,
for which the localized-magnon states are still dominant
but also higher-energy states of the spin system not de-
scribed by the hard-square model contribute to the par-
We mention that the hard-square model belongs to
the two-dimensional Ising model universality class11,25,26
with the critical exponents β = 1/8 for the order param-
J2=5, N=20, 1.02h1
J2=5, N=20, 1.01h1
HS, 10 sites, 1.02h1
HS, 10 sites, 1.01h1
HS MC, 1.02h1
HS MC, 1.01h1
FIG. 3: The temperature dependence of the specific heat
C(T,h,N)/kN for h = 1.01h1 (broken lines, diamonds) and
h = 1.02h1 (solid lines, triangles) (upper panel) and for
h = 0.99h1 (broken lines, diamonds) and h = 0.98h1 (solid
lines, triangles) (lower panel). The results of Monte Carlo
simulations for a hard-square system of sizes up to 800 × 800
sites are shown by lines. We also report the corresponding re-
sults for the hard-square system with N = 10 (open symbols)
and the exact diagonalization data for the finite spin system
with N = 20, J2 = 5 (filled symbols).
eter and α = 0 for the specific heat, i.e. the specific heat
shows a logarithmic singularity at the critical point. Note
that this universality class is different from the one of the
hard-hexagon model.11Thus, the low-temperature peak
(singularity) in the temperature dependence of the spe-
cific heat in the vicinity of the saturation field is a spec-
tacular sign of highly degenerate independent localized-
magnon states of the frustrated bilayer quantum HAFM.
Their ordering leads to the hard-square type peculiarity
just below the saturation field.
Another thermodynamic quantity of interest is the
curve of constant entropy as a function of magnetic field
and temperature. Since hard-square description implies
the dependence of the entropy only on (h1− h)/kT this
curve is similar to an ideal paramagnet. Therefore, the
considered spin system is expected to exhibit a large
magnetocaloric effect in the vicinity of the saturation
Finally we note, that the effect of the localized
magnons is a pure quantum effect which disappears as
s increasing approaches the classical limit s → ∞.4
The numerical calculations were performed using
J. Schulenburg’s spinpack. The present study was sup-
ported by the DFG (Project No. 436 UKR 17/13/05).
O. D. acknowledges the kind hospitality of the Magde-
burg University in the summer of 2006.
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