Phase transitionsin anisotropic two-dimensional quantum antiferromagnets

T. Roscilde*,1, A. Cuccoli2, and P. Verrucchi2

1 Dipartimento di Fisica “A. Volta”, Universit� di Pavia, via A. Bassi 6, I-27100 Pavia, Italy,and INFM

2 Dipartimento di Fisica, Universit� di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy,and INFM

Received 1 July 2002, accepted 15 October 2002Published online 11 March 2003

PACS 02.70.S, 75.10.J, 75.30.K

We study the thermodynamic properties of anisotropic two-dimensional quantum antiferromagnets, withboth easy-plane and easy-axis anisotropy. We make use of the Quantum Monte Carlo (QMC) methodbased on the continuous-time loop algorithm. For S ¼ 1=2 and very small anisotropies our QMC datareveal that both the Ising and Berezinskii–Kosterlitz–Thouless universality class (for the easy-axis andeasy-plane models, respectively) are well observed and that the critical temperature remains finite evenfor anisotropies as small as 10�3, i.e. comparable to the anisotropies measured in real layered com-pounds; this result rules out the possibility of quantum fluctuations to destroy the finite-temperaturetransition before the isotropic limit is reached. The phase diagram of the S ¼ 1=2 2d-XXZ antiferro-magnet in the region of small anisotropies is then presented.

Two-dimensional antiferromagnets (AFMs) have received in the last decade an increasing attention.On the experimental side, investigations on a large class of layered AFMs [1], some of them beingparent compounds of high-temperature superconductors, lead to the fundamental question of how two-dimensional intra-layer antiferromagnetic correlations affect the magnetic behaviour of these systems,and, in particular, to which extent the observed magnetic transitions to 3d long-range order (LRO) areto be attributed to the onset of 2d long-range correlations. On the theoretical side, it is well knownthat the isotropic Heisenberg model on the square lattice, which is the simplest candidate to describean isolated layer, cannot display a finite-temperature phase transition [2], and that it keeps its correla-tion length finite at any temperature T > 0. Nevertheless, a non-diverging intra-layer correlationlength can lead to 3d-LRO, due to the existence of a residual inter-layer coupling, typically orders ofmagnitude smaller than the inter-layer one, but effectively amplified of a factor � x2ð2dÞ by the presenceof intra-layer correlations on the length scale xð2dÞ [3]. A phase transition due to such mechanism,being uniquely due to the presence of the inter-layer coupling, belongs to the 3d-Heisenberg universal-ity class. However the critical temperatures measured in many layered compounds seem to be too highto be accounted for by the very weak inter-layer coupling only, and the observed critical exponentsshow better consistency with 2d-Ising or 2d-XY universality classes. This suggests anisotropies in theintra-layer coupling to be better candidates to explain the fundamental mechanism of ordering inlayered AFMs, leading to finite-temperature divergence of xð2dÞ in each layer independently which inturn triggers the transition to 3d-LRO of the whole system, due to the inter-layer coupling.

* Corresponding author: e-mail: roscilde@fi.infn.it

# 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0370-1972/03/23603-0433 $ 17.50þ.50/0

phys. stat. sol. (b) 236, No. 2, 433–436 (2003) / DOI 10.1002/pssb.200301697

Measured anisotropies in real layered compounds are very small, ranging in the interval 10�4–10�2

[1], so that their ordering effect can be strongly weakened in presence of quantum fluctuations, en-hanced by the low dimensionality: the study of quantum effects on weakly anisotropic 2d AFMshence becomes of major importance for the understanding of the magnetic transition in real layeredAFMs.

We consider the two-dimensional S ¼ 1=2 XXZmodel on the square lattice, whose hamiltonian reads:

HH ¼P

i;dJXY SSxi SS

xiþd þ SSyi SS

yiþd

� �þ JZSSzi SS

ziþd

� �ð1Þ

where i ¼ ði1; i2Þ runs over the sites of a square lattice, d connects each site to its four nearest neigh-bours, JXY ; JZ > 0 are the antiferromagnetic exchange integrals. The spin operators SSai (a ¼ x; y; z)are such that jSSj2 ¼ SðSþ 1Þ with S ¼ 1=2 and obey ½SSai ; SS

bj � ¼ ieabgdijSS

gi . The XXZ model interpo-

lates continuously between the quantum Heisenberg, quantum XY and Ising model upon varying theexchange integrals JXY ; JZ . The anisotropic character of the model will be hereafter expressed interms of the ratio between JXY and JZ : the ratio m ¼ JXY=JZ < 1 represents the easy-axis case, whilethe ratio l ¼ JZ=JXY < 1 represents the easy-plane case. The actual amount of anisotropy is expressedby Dm ¼ 1� m in the easy-axis case and Dl ¼ 1� l in the easy-plane case. The limit m ¼ 0 repro-duces the conventional 2d Ising model, while the limit l ¼ 0 reproduces the 2d S ¼ 1=2 XY-model.

In the classical limit, exact proofs [4] and numerical evidence [5] suggest that an arbitrarily smallanisotropy of easy-axis and easy-plane type, perturbing the isotropic Heisenberg model in two dimen-sions, can induce a finite-temperature phase transition of Ising and Berezhinskii–Kosterlitz–Thouless(BKT) type, respectively. The transition temperature Tc is expected to vanish continuously with van-ishing anisotropy as Tc � ½ln ðA=Dm;lÞ��1, where A is a constant.

In the case of the quantum AFM, exact proofs are of more limited validity, and only numericalevidence [6, 7] is given, in the case S ¼ 1=2, for the existence of a transition at very small anisotro-pies; moreover the existing quantum simulations are based on algorithms which are not suited toexplore the critical region, and in the easy-axis case their results have been recently cast into doubt byanalitical studies based on real-space renormalization group techniques [8], predicting the existence ofa critical value of the anisotropy DðcÞ

m � 0:2 below which the critical temperature is seen to vanishunder the disordering effect of quantum fluctuations.

To investigate the critical properties of the model (1) with S ¼ 1=2 in the case of realistically smallanisotropies, we make use of the Quantum Monte Carlo (QMC) method based on the loop algorithm[9] in continuous imaginary time [10]. This modern QMC scheme allows to treat big lattices near acritical point without any complication arising from critical slowing down, so that, supplemented withfinite-size scaling analysis, it stands as an ideal tool to investigate phase transitions.

We have considered two values of the easy-axis anisotropy, m ¼ 0:990 and m ¼ 0:999, for differentlattice sizes L2 with L = 16, 32, 64, 128. In the case of an Ising-like transition the order parameter isgiven by the (staggered) magnetization, and, without invoking the Ising universality class, the phasetransition is shown to exist if the order parameter converges to a finite value at finite temperature asthe size of the system grows. This kind of behaviour is observed for both anisotropies considered (inFig. 1a we report only the most interesting case of the smallest anisotropy considered); therefore wecan conclude that the Ising transition indeed exists.

Estimates of the Ising critical temperature TI are obtained by conventional finite-size scaling analy-sis [11]. The most general strategy is to look for the temperature at which the magnetic correlationlength scales linearly with the size of the system; this kind of analysis does not require any priorassumption on the universality class. Remarkably, the resulting estimate is consistent with that foundby assuming the Ising universality class, i.e. looking for the temperature at which the staggered mag-netization scales as hszi � L�b=n and the staggered susceptibility scales as cðzzÞ ¼ bL2hðszÞ2i � Lg=n,with b ¼ 1=8, g ¼ 7=4 and n ¼ 1 [15]. Therefore, having found consistent results for two criticalexponents (b and g), the Ising universality class is fully verified, and this is a remarkable featurewhen considering that the model is essentially an isotropic Heisenberg model with a small Ising per-turbation, and that strong quantum fluctuations are switched on.

434 T. Roscilde et al.: Phase transitions in anisotropic 2D quantum antiferromagnets

As for the easy-plane anisotropy, we have considered the following values: l ¼ 0:98; 0:999, andlattice sizes L2 with L = 32, 64, 128, 200. In the case of BKT transition, the order parameter (stag-gered magnetization) is rigorously zero, but the topologically ordered phase is characterized by a non-zero value of the helicity modulus U, i.e. the response function to an uniform twist of the order param-eter; this quantity is easily measured in Quantum Monte Carlo simulations, and even admits a veryefficient improved estimator [12].

In close analogy to the easy-axis case, the existence of the phase transition is proven if the helicitymodulus scales to a finite value at finite temperature, as suggested by Fig. 1b. The BKT criticaltemperature TBKT is then estimated by looking for the temperature where the critical scaling of thestaggered in-plane susceptibility cðxyÞ ¼ bL2hðsxðyÞÞ2i � L�7=4 is best verified, together with the scalingof the helicity modulus UðLÞ � 2TBKT=pJXY � ½1þ 1=ð2 log ðL=L0ÞÞ�, according to Kosterlitz scalingequations [13]. Consistency of the critical temperature estimate coming from the two scaling analysis[15] remarkably verifies Kosterlitz–Thouless theory in a model whose easy-plane character stands as asmall perturbation to the isotropic Heisenberg character.

The critical temperature estimates, drawn as described in the previous sections, are collected inFig. 2, where the phase diagram of the weakly anisotropic 2d S ¼ 1=2 XXZ model is presented, com-pared with the existing QMC results by Ding [6]. Where direct comparison is made with Ding’sresults, we find them to be slightly overestimated; this is possibly due to the fact that these estimateshave not been obtained by finite-size scaling analysis, but by fitting the temperature behaviour ofrelevant quantities to critical laws in the disordered phase. Our results are well described by a logarith-

phys. stat. sol. (b) 236, No. 2 (2003) 435

102

1/L

0.06

0.11

0.16

0.21

<sz >

T/JZ

= 0.18

0.20

0.21

0.22

0.23

0.24

0.25

a)

0.00 0.01 0.02 0.03

1/L

0.08

0.09

0.10

0.11

0.12

0.13

0.14

Υ(L

)

0.17

0.18

0.19

0.20

T/JXY

=0.16

b)

Fig. 1 Results of simulations for the 2d S ¼ 1=2 XXZ model on the square lattice: (a) Scaling of the staggeredmagnetization for the easy-axis model with m ¼ 0:999; (b) Scaling of the helicity modulus for the easy-planemodel with l ¼ 0:999.

104

103

102

101

∆µ , ∆λ

0.1

0.2

0.3

0.4

0.5

TI(∆

µ) /

JZ, T

BK

T(∆

λ) /

JXY

2.22/ln(330/∆λ)

2.49/ln(70/∆µ)

easy-axis

easy-plane

Fig. 2 Phase diagram of the weakly anisotropic 2dS ¼ 1=2 XXZ-model on the square lattice; full symbolsare our original QMC results, open symbols are previousQMC results by Ding [6]. Dashed and dash-dotted linesare logarithmic fits to our results; the fitting functionsare given in the figure.

mic function, suggesting that the critical temperature remains finite for every finite value of the aniso-tropy. As in Ref. [6] the two-parameter logarithmic fit is only apparently performed on two points,since the fitting curve is constrained to pass through the “Heisenberg point” (TI;BKT ¼ 0 for Dm;l ¼ 0).

When compared to the classical limit [14], the S ¼ 1=2 critical temperatures appear to be stronglyreduced by quantum fluctuations. Nevertheless, quantum fluctuations are not able to wash out thetransition at finite temperature even in the limit of very small anisotropy. As in the classical case, thedependence of critical temperatures on anisotropy in this limit is logarithmic, thus keeping the criticaltemperatures away from the very-low temperature range. Finally, our results highlight the importanceof a tiny anisotropy in the thermodynamic behaviour of strongly fluctuating low-dimensional magnets,inducing to reconsider many experimental results on layered AFMs which were mainly discussed interms of the isotropic Heisenberg model.

References

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P. Verrucchi, and R. Vaia, Eur. Phys. J. B 20, 55 (2001).[15] T. Roscilde et al., in preparation.

436 T. Roscilde et al.: Phase transitions in anisotropic 2D quantum antiferromagnets