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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

I. INTRODUCTION

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

the hard-hexagon/large-hard-squaremodel.13,14Further-

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

zero field.

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

Page 2

2

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

field.

II.THE FRUSTRATED BILAYER

ANTIFERROMAGNET AND INDEPENDENT

LOCALIZED MAGNONS

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

H =

?

(nm)

Jnm

?1

2

?s+

ns−

m+ s−

ns+

m

?+ ∆sz

nsz

m

?

− hSz.(1)

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

Λ(1)

k

=

−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-

ment.

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

nsz

n

energy isgiven by

k

=

k

is disper-

|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

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3

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-

tail below.

n=0gN(n)

=

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,

Zlm(T,h,N) =

nmax

?

n=0

?nmax

gN(n)exp

?

−En(h)

kT

?

= exp

?

−EFM− hsN

kT

?

n=0

gN(n)exp

?µ

kTn

?

, (3)

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) =

?nmax

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

square model,

Zlm(T,h,N) = exp

?

−EFM− hsN

kT

?

Ξ(T,µ,N). (4)

Eq.

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

mode Λ(2)

k

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

∆DOS.

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-

k

are degener-

Page 4

4

0

0.1

0.2

(sN -M(T,h,N))/N

HS, 10 sites

HS, 64 sites

HS, MC

T=0.2

T=0.3

0

0.1

0.2

C(T,h,N)/(kN)

0

0.1

0.2

-3 -2-1

(h - h1)/(kT)

0123

S(T,h,N)/(kN)

a

b

c

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).

IV. LOW-TEMPERATURE

THERMODYNAMICS AROUND THE

SATURATION FIELD

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-

tition function.

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-

Page 5

5

0

0.1

0.2

C(T,h,N)/(kN)

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

0

0.1

0.2

00.020.04

C(T,h,N)/(kN)

kT/h1

0.98h1

0.99h1

0.98h1

0.99h1

0.98h1

0.99h1

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

field.6,7,9

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

Acknowledgments

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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