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Quantum effects in the 2D XY model
Journal of Magnetism and Magnetic Materials 226}230 (2001) 403}404
Quantum e!ects in the 2D XY model C. Schindelin , H. Fehske *, H. BuK ttner , D. Ihle Physikalisches Institut, Universita( t Bayreuth, D-95440 Bayreuth, Germany Institut fu( r Theoretische Physik, Universita( t Leipzig, D-04109 Leipzig, Germany
Abstract Ground-state and thermodynamic properties of the S"1/2 X> model are calculated by a Green's-function approach and compared with recent quantum Monte Carlo data. The maxima in the temperature dependences of the longitudinal and transverse uniform static spin susceptibilities are explained as e!ects of magnetic short-range order. 2001 Elsevier Science B.V. All rights reserved. Keywords: X> model; Quantum #uctuations; Spin correlations; Susceptibility, temperature dependent
In spite of many studies of the S"1/2 X> model on the square-lattice (cf. Refs. [1}3]), 1 H" (SVSV#SWSW) (1) G H G H 2 6G H7 (i, j denote nearest neighbors), analytical approaches to the spin susceptibility which take into account the magnetic short-range order (SRO) over the whole temperature region are still lacking. Quantum Monte Carlo (QMC) simulations have shown [3] that there is a Kosterlitz}Thouless (KT) transition at ¹ K0.3427 [3] )2 with the same qualitative critical scaling behavior as in the classical model which is determined by long ranged #uctuations [4]. In this paper we focus on short-ranged #uctuations and distinctive quantum e!ects due to the di!erences in spin space as compared with the classical analogue, in particular on the longitudinal spin correlators and static susceptibilities. To this end, we adapt the Green's-function method recently developed for the easy-plane XXZ-model [5,6]. We calculate the spin susceptibilities >\ ()"!S> and q q ;S\q ! \ S XX by the projection method for q ()"!SX q ;SX q \ S the two-time retarded Green's functions in a generalized
* Corresponding author. Tel.: #49-921-55-3212; fax: #49921-55-2991. E-mail address: [email protected] (H. Fehske).
mean-"eld approximation. Choosing the two-operator basis (S> q , iSQ > q ) and (SX q , iSQ X q ) we obtain MJq Jq ()"! ; "#!, zz, !(Jq )
(2)
with M>\ "!4(C>\!2CXX q ) , q
(3)
MXX q "!4C>\(1!q ) ,
(4)
"S>S\ , r"ne #me , and q " CJ ,CJr , C>\ r r V W LK (cos q #cos q )/2. The spectra are calculated in the V W approximations !S$ > )S> $ Xq "(XX q "(>\ q q and !S q )SX q introducing vertex parameters J (i"1,2). We get G (>\ )"1#2>\(C>\#2C>\) q #2>\[2CXX (4q !1)!3C>\q ],
(5)
(XX q )"2(1!q )[1#2XX(C>\#2C>\) !2XXC>\(1#4q )]
(6)
with XX"lim XX. Here, the results for the XXZ model (anisotropy parameter ) are taken in the limit P0, where XX(¹"0) turns out to be "nite [6]. The long-range order at ¹"0 is re#ected by
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 9 3 - 8
404
C. Schindelin et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 403}404
Fig. 1. Uniform susceptibility T and nearest-neighbor longituO dinal spin correlation function (inset) vs. temperature.
Fig. 2. Speci"c heat and internal energy (inset) for the 2D X> model.
Table 1 Longitudinal spin correlators at ¹"0.2
QMC result [1]), so that is mainly determined by CXX (0. The increase of (¹) and the crossover to the high-temperature Curie}Weiss law are caused by the decrease of magnetic SRO with temperature (cf. inset of Fig. 1), i.e., of the spin sti!ness against the orientation along a homogeneous external "eld in z-direction. Similarly, the temperature dependence of >\(¹) may be explained as an e!ect of transverse magnetic SRO (sti!ness against the orientation of the transverse spin components along a homogeneous "eld perpendicular to the z-direction). Considering the short-ranged transverse correlations, the speci"c heat C " / ¹ (with the internal energy per 4 site "2C>\) shown in Fig. 2 is in qualitative agreement with the QMC data [1], e.g., we get a weaker increase of C at ¹K¹ and a reduced asymmetry near the peak 4 )2 above ¹ . )2
n
CXX (theory) L
CXX (QMC data [1]) L
1 2 3 4 5
!4.197;10\ !1.838;10\ !5.88;10\ !1.51;10\ !4.7;10\
!4.118;10\ !1.828;10\ !6.78;10\ !1.68;10\ (No data)
>\ Q "0[Q"( , )]; the magnetization m is calculated as 1 m" C>\ e\ Qr"C r Nr
(7)
with the condensation part Ce Qr separated from C>\ . r The parameters J (¹) are determined from the sum rules C>\"1/2 and CXX "1/4. To get J (¹) we take the QMC data [1] for CJ (¹"0) and assume, as additional conditions, [ (¹)!1]/[ (¹)!1]"const. and XX(¹)/ [XX(¹)!1]"const. First let us emphasize that the ground-state results obtained for the magnetization, m"0.4248, and for the susceptibility, ,XX"0.1954, agree very well with the QMC data [2], m"0.437 and "0.2096. Concerning the temperature dependence of the uniform susceptibilities depicted in Fig. 1, (¹) is in reasonable agreement with the QMC results [2], in contrast to the chiral perturbation theory [7] which yields a temperature-independent susceptibility. The maximum in (¹) can be explained as follows. The correlators CXX are r found to be negative for all r and ¹ and very short ranged (cf. Table 1; at ¹"0.2, CXX for n*2 is two orders of L magnitude smaller than CVV in accordance with the L
To conclude, our Green's-function theory of magnetic SRO allows the calculation of ground-state and thermodynamic properties of the 2D X> model in reasonable agreement with QMC data. In particular, the quantum e!ects re#ected in the longitudinal spin correlators and susceptibilities are well described.
References [1] [2] [3] [4]
H.-Q. Ding, Phys. Rev. B 45 (1992) 230. A.W. Sandvik, C.J. Hamer, Phys. Rev. B 60 (1999) 6588. K. Harada, N. Kawashima, J.Phys. Soc. Jpn. 67 (1998) 2768. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. B 51 (1995) 12840. [5] S. Winterfeldt, D. Ihle, Phys. Rev. B 56 (1997) 5535. [6] H. Fehske, C. Schindelin, A. Wei{e, H. BuK ttner, D. Ihle, cond-mat/10006272, Brazilian J.Phys., to appear. [7] P. Hasenfratz, F. Niedermayer, Z. Phys. B 92 (1993) 91.
Quantum e!ects in the 2D XY model C. Schindelin , H. Fehske *, H. BuK ttner , D. Ihle Physikalisches Institut, Universita( t Bayreuth, D-95440 Bayreuth, Germany Institut fu( r Theoretische Physik, Universita( t Leipzig, D-04109 Leipzig, Germany
Abstract Ground-state and thermodynamic properties of the S"1/2 X> model are calculated by a Green's-function approach and compared with recent quantum Monte Carlo data. The maxima in the temperature dependences of the longitudinal and transverse uniform static spin susceptibilities are explained as e!ects of magnetic short-range order. 2001 Elsevier Science B.V. All rights reserved. Keywords: X> model; Quantum #uctuations; Spin correlations; Susceptibility, temperature dependent
In spite of many studies of the S"1/2 X> model on the square-lattice (cf. Refs. [1}3]), 1 H" (SVSV#SWSW) (1) G H G H 2 6G H7 (i, j denote nearest neighbors), analytical approaches to the spin susceptibility which take into account the magnetic short-range order (SRO) over the whole temperature region are still lacking. Quantum Monte Carlo (QMC) simulations have shown [3] that there is a Kosterlitz}Thouless (KT) transition at ¹ K0.3427 [3] )2 with the same qualitative critical scaling behavior as in the classical model which is determined by long ranged #uctuations [4]. In this paper we focus on short-ranged #uctuations and distinctive quantum e!ects due to the di!erences in spin space as compared with the classical analogue, in particular on the longitudinal spin correlators and static susceptibilities. To this end, we adapt the Green's-function method recently developed for the easy-plane XXZ-model [5,6]. We calculate the spin susceptibilities >\ ()"!S> and q q ;S\q ! \ S XX by the projection method for q ()"!SX q ;SX q \ S the two-time retarded Green's functions in a generalized
* Corresponding author. Tel.: #49-921-55-3212; fax: #49921-55-2991. E-mail address: [email protected] (H. Fehske).
mean-"eld approximation. Choosing the two-operator basis (S> q , iSQ > q ) and (SX q , iSQ X q ) we obtain MJq Jq ()"! ; "#!, zz, !(Jq )
(2)
with M>\ "!4(C>\!2CXX q ) , q
(3)
MXX q "!4C>\(1!q ) ,
(4)
"S>S\ , r"ne #me , and q " CJ ,CJr , C>\ r r V W LK (cos q #cos q )/2. The spectra are calculated in the V W approximations !S$ > )S> $ Xq "(XX q "(>\ q q and !S q )SX q introducing vertex parameters J (i"1,2). We get G (>\ )"1#2>\(C>\#2C>\) q #2>\[2CXX (4q !1)!3C>\q ],
(5)
(XX q )"2(1!q )[1#2XX(C>\#2C>\) !2XXC>\(1#4q )]
(6)
with XX"lim XX. Here, the results for the XXZ model (anisotropy parameter ) are taken in the limit P0, where XX(¹"0) turns out to be "nite [6]. The long-range order at ¹"0 is re#ected by
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 9 3 - 8
404
C. Schindelin et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 403}404
Fig. 1. Uniform susceptibility T and nearest-neighbor longituO dinal spin correlation function (inset) vs. temperature.
Fig. 2. Speci"c heat and internal energy (inset) for the 2D X> model.
Table 1 Longitudinal spin correlators at ¹"0.2
QMC result [1]), so that is mainly determined by CXX (0. The increase of (¹) and the crossover to the high-temperature Curie}Weiss law are caused by the decrease of magnetic SRO with temperature (cf. inset of Fig. 1), i.e., of the spin sti!ness against the orientation along a homogeneous external "eld in z-direction. Similarly, the temperature dependence of >\(¹) may be explained as an e!ect of transverse magnetic SRO (sti!ness against the orientation of the transverse spin components along a homogeneous "eld perpendicular to the z-direction). Considering the short-ranged transverse correlations, the speci"c heat C " / ¹ (with the internal energy per 4 site "2C>\) shown in Fig. 2 is in qualitative agreement with the QMC data [1], e.g., we get a weaker increase of C at ¹K¹ and a reduced asymmetry near the peak 4 )2 above ¹ . )2
n
CXX (theory) L
CXX (QMC data [1]) L
1 2 3 4 5
!4.197;10\ !1.838;10\ !5.88;10\ !1.51;10\ !4.7;10\
!4.118;10\ !1.828;10\ !6.78;10\ !1.68;10\ (No data)
>\ Q "0[Q"( , )]; the magnetization m is calculated as 1 m" C>\ e\ Qr"C r Nr
(7)
with the condensation part Ce Qr separated from C>\ . r The parameters J (¹) are determined from the sum rules C>\"1/2 and CXX "1/4. To get J (¹) we take the QMC data [1] for CJ (¹"0) and assume, as additional conditions, [ (¹)!1]/[ (¹)!1]"const. and XX(¹)/ [XX(¹)!1]"const. First let us emphasize that the ground-state results obtained for the magnetization, m"0.4248, and for the susceptibility, ,XX"0.1954, agree very well with the QMC data [2], m"0.437 and "0.2096. Concerning the temperature dependence of the uniform susceptibilities depicted in Fig. 1, (¹) is in reasonable agreement with the QMC results [2], in contrast to the chiral perturbation theory [7] which yields a temperature-independent susceptibility. The maximum in (¹) can be explained as follows. The correlators CXX are r found to be negative for all r and ¹ and very short ranged (cf. Table 1; at ¹"0.2, CXX for n*2 is two orders of L magnitude smaller than CVV in accordance with the L
To conclude, our Green's-function theory of magnetic SRO allows the calculation of ground-state and thermodynamic properties of the 2D X> model in reasonable agreement with QMC data. In particular, the quantum e!ects re#ected in the longitudinal spin correlators and susceptibilities are well described.
References [1] [2] [3] [4]
H.-Q. Ding, Phys. Rev. B 45 (1992) 230. A.W. Sandvik, C.J. Hamer, Phys. Rev. B 60 (1999) 6588. K. Harada, N. Kawashima, J.Phys. Soc. Jpn. 67 (1998) 2768. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. B 51 (1995) 12840. [5] S. Winterfeldt, D. Ihle, Phys. Rev. B 56 (1997) 5535. [6] H. Fehske, C. Schindelin, A. Wei{e, H. BuK ttner, D. Ihle, cond-mat/10006272, Brazilian J.Phys., to appear. [7] P. Hasenfratz, F. Niedermayer, Z. Phys. B 92 (1993) 91.