Reconciling field-theoretical and semiclassical approaches to quantum 2D antiferromagnets

Reconciling field-theoretical and semiclassical approaches to quantum 2D antiferromagnets

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 272–276 (2004) 892–893 Reconciling field-theoretical and semiclassical approaches to qua...

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ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 272–276 (2004) 892–893

Reconciling field-theoretical and semiclassical approaches to quantum 2D antiferromagnets Bernard B. Bearda, Alessandro Cuccolib,c,*, Ruggero Vaiad,c, Paola Verrucchic a

Department of Physics and Mechanical Engineering, Christian Brothers University, Memphis, TN 38104, USA b Dipartimento di Fisica dell’Universita" di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy c Istituto Nazionale di Fisica della Materia, Unita" di Ricerca di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy d Istituto di Fisica Applicata ‘Nello Carrara’ del C.N.R., via Madonna del Piano, I-50019 Sesto Fiorentino (FI), Italy

Abstract We show that, starting from the field-theoretical expression for the correlation length x of the 2D quantum Heisenberg antiferromagnet corrected for cut-off effects and our knowledge of the classical thermodynamics of the model, the observed temperature dependence of x can be reproduced. We find also that the cut-off effects contains the same pure-quantum renormalizations of the pure-quantum self-consistent harmonic approximation semiclassical framework: the gap between quantum field theoretical and semiclassical results is thus bridged. r 2003 Elsevier B.V. All rights reserved. PACS: 05.30.d; 05.50.+q; 75.10.b; 75.10.Jm Keywords: 2D antiferromagnet; Non-linear sigma model; Semiclassical methods; Effective Hamiltonian; Field theory

The 2D isotropic quantum Heisenberg antiferromagnet, described by the Hamiltonian X # ¼ 1J H ð1Þ S# i  S# iþd 2 id with J > 0; i ¼ ði1 ; i2 Þ; d ¼ ð71; 71Þ; has been the subject of intense research in the last two decades. In many papers [1–7], the attention was mainly focused on the temperature and spin dependence of the staggered correlation length xðT; SÞ; defined by the asymptotic behaviour limjrj-N GðrÞpejrj=x of the staggered spin– spin correlation function GðrÞ  ð1Þr1 þr2 /S# zi S# ziþr S: On the theoretical side, the most successful results in interpreting the experimental [1] and numerical simulation [2,3] outcomes were those of two apparently completely different approaches: (i) the semi-classical *Corresponding author. Dipartimento di Fisica dell’Universit"a di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy. Tel.: +39-0554572045; fax: +390554572121. E-mail address: cuccoli@fi.infn.it (A. Cuccoli).

method known as pure-quantum self-consistent harmonic approximation (PQSCHA) [5], which gives xðT; SÞ for the quantum model in terms of the corresponding quantity of the classical model xcl and of an effective exchange constant Jeff renormalized by pure quantum fluctuations xðT; SÞ ¼ xcl ðT=Jeff ðT; SÞÞ with Jeff ðT; SÞ ¼ equations: zi ðT; SÞ ¼ 1 þ

JS 2 z21 ðT; SÞ

ð2Þ given by the self-consistent

1 1 X ð1  g2k Þ1=2 Lk ðT; SÞ;  2S 2SN k ð1  gk Þ1i

Lk ðT; SÞ ¼ coth

ok ðSÞ 2T ;  2T ok ðSÞ

ok ðSÞ ¼ 4JSz1 ð0; SÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1  g2k

with sums over wavevectors k ¼ ðk1 ; k2 Þ in the first Brillouin zone, and gk ¼ ðcos k1 þ cos k2 Þ=2; (ii) the nonlinear s-model field-theoretical approach [6], giving the

0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1149

ARTICLE IN PRESS B.B. Beard et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 892–893

1.0

0.8

0.6

0.4

0.2 0.0

0.5

1.0

t = T/JS

1.5

2.0

2

Fig. 1. Ratio x=x3l vs. T=JS2 for S ¼ 52: Dash-dotted line: Hasenfratz’s result (4); solid line: Eq. (5) with xclð3lÞ replaced by the exact xcl ; thin dashed line: Eq. (2) with JeffðHÞ ; dashed line: original PQSCHA result; symbols: QMC data.

3-loop expression x3l ðT; SÞ ¼

  2  e c ð2prS =TÞ 1 T T e 1 þO 4 ; 8 2prS 2 2prS S

ð3Þ

recently corrected by Hasenfratz [7] for cut-off effects to obtain xH ðT; SÞ ¼ x3l ðT; SÞeCðT;SÞ ;

ð4Þ

where CðT; SÞ; defined in Eq. (14) of Ref. [7], is an integral of familiar spin-wave quantities over the first Brillouin zone. While the former theory has proven to be able to quantitatively describe the behaviour of xðT; SÞ for different values of S in the full experimentally accessible range of temperatures, but was up to now unable to reproduce the asymptotic T-0 critical regime, the field-theoretical approach well describes the asymptotic region, but in its original form Eq. (3) holds only for astronomically large values of x for SX1 and still gives numerically accurate results only down to x\103 for all S also when cut-off effects are included as in Eq. (4). Starting from the original expression of CðT; SÞ given in Ref. [7] as a first step we have been able to show that by reclassifying the different contributions according to their order in 1=S; Eq. (4) can be recast in a form equivalent to Eq. (2):   T xH ðT; SÞ ¼ xclð3lÞ aðT; SÞ; ð5Þ JeffðHÞ ðT; SÞ

893

where aðT; SÞ ¼ 1 þ OðS2 Þ þ OðT=S 2 Þ contains higherorder corrections, and xclð3lÞ is the 3-loop result for the classical correlation length. JeffðHÞ ðT; SÞ ¼ JS 2 z0 ðT; SÞz1 ðT; SÞ remarkably contains renormalization of pure-quantum origin only, as Jeff ðT; SÞ of the PQSCHA. The similarity of Eqs. (5) and (2) suggests as a further step the substitution of xclð3lÞ with the known exact xcl ; thus making Eq. (5) able to reproduce QMC data in the whole temperature range as shown in Fig. 1: the low- and intermediate-T regime are thus connected, and it is also made apparent that the intermediate temperature behaviour is essentially due to classical nonlinear effects, unaccounted by the low-temperature perturbative treatment of the non-linear s-model. The procedure leading from Eq. (4) to (5) gives also further insight into the relevance of the contribution of different phase-space regions to the thermodynamics of the system at different temperatures allowing us to understand that the main reason of the failure of the PQSCHA in the asymptotic region stays in a very subtle detail of the derivation of Eq. (2): a more careful analysis shows that the original definition of Jeff is indeed the most suitable one for intermediate and high temperatures, while at low T one of the factors z1 ðT; SÞ must be more appropriately replaced by z0 ðT; SÞ getting agreement with JeffðHÞ ðT; SÞ: The two different theoretical approaches are thus reconciled, and both are able to explain the main observed properties of x:

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