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
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