- Email: [email protected]
Ring exchange in low-dimensional spin systems
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 904–905
Ring exchange in low-dimensional spin systems M. Muller, . T. Vekua, H.-J. Mikeska* Institut fur . Theoretische Physik, Universitat . Hannover, Appelstrasse 2, Hannover 30167, Germany
Abstract The isotropic S ¼ 12 antiferromagnetic spin ladder with additional four-spin ring exchange is studied perturbatively in the strong coupling regime with the cluster expansion technique, and by bosonization in the weak coupling limit. It is found that a sufficiently large amount of ring exchange leads to a second-order phase transition into a spontaneously dimerized phase. The shape of the phase boundary in the vicinity of the exactly known transition point is obtained. The relevance of ring exchange for materials with 2D CuO2 planes is discussed. r 2003 Elsevier B.V. All rights reserved. Keywords: Ring exchange; CuO2 plaquette; Quantum phase transition
The cyclic four-spin (ring) exchange emerges from a t=U expansion of the Hubbard Hamiltonian [1]. Recently it was suggested that some strongly correlated electron systems like cuprates [2] and spin ladders [3] are expected to exhibit ring exchange. The analysis of the low-lying excitation spectrum of the p–d model [2,4] as well as inelastic neutron scattering [3], NMR [5] and optical conductivity [6] measurements on the (Sr,Ca,La)14 Cu24 O41 system shows that the Hamiltonians describing CuO2 planes should contain a finite amount of ring exchange. Real two-leg ladder cuprates are always in the rung-singlet phase. Ring exchange drives these systems close to critical point. Therefore it is important to understand the nature of the phase transition with increasing frustrating ring interaction. Hamiltonian of our model has the form H ¼ Jrung
N X i
þ
S1;i S2;i þ Jleg
N X
Sa;i Sa;iþ1
a¼1;2;i
Jring X Pijkl þ P1 ijkl ; 2 /ijklS
ð1Þ
known [7] that at Jring =Jleg ¼ 1; the exact ground state is the direct product of rung dimers and the exact lowest excited state is the propagating triplet with following dispersion: et ðqÞ ¼ Jrung 2Jring þ 2Jring cosðqÞ:
ð2Þ 1 4Jrung :
We have The q ¼ p mode becomes soft for Jring ¼ obtained the phase boundary from the softening of the q ¼ p mode using the cluster expansion approach up to 14th order. The result is given in Fig. 1. Using the Pade! approximation technique we determined that the excitation gap is increasing linearly upon deviating from the critical line. To clarify the nature of the ground state and low energy excitations with increasing ring exchange we applied weak-coupling bosonization approach. Close to the limit of decoupled chains (Jleg cJrung ; Jring ) the Hamiltonian in Majorana representation becomes similar to the one considered in Ref. [8]: X ivt H¼ ðxaR @x xaR xaL @x xaL Þ imt xaR xaL 2 a¼1;2;3
ivs 0 ðx @x x0 x0L @x x0L Þ ims x0R x0L ; 2 R R
ð3Þ
where Pijkl permutes spin over the elementary plaquettes of the ladder. From the matrix product approach it is
mt BðJrung aJring Þ; ms Bð3Jrung aJring Þ:
*Corresponding author. Tel.: +49-511-762-3665; fax: +49511-762-3023. E-mail address: [email protected] (H.-J. Mikeska).
It follows, that mt ¼ 0 for Jrung ¼ aJring : The universality class of the quantum phase transition is that of the critical, exactly integrable, S ¼ 1 spin chain with central charge c ¼ 32: At the criticality relative spin of two chains
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.444
ð4Þ
ARTICLE IN PRESS M. Muller et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 904–905 .
0.4
Jring/Jrung
0.3
dimerized phase
0.2
exact line
rung-singlet phase 0.1
0 0
0.1
0.2
0.3
0.4
0.5
Jleg/Jrung
Fig. 1. Results from the 14th order of perturbation theory in Jring =Jrung and Jleg =Jrung
as well as relative dimerization decay algebraically according to SU2 ð2Þ universality class /S ðrÞS ð0ÞSB/e ðrÞe ð0ÞSBr3=4 :
ð5Þ
The phase with dominant ring exchange was established to be spontaneously dimerized using Ising variables representation of spin operators, the result later confirmed by DMRG studies [9]. Next we discuss the effects of the cyclic interactions to models and materials with CuO2 planes since the basic plaquette of those is the same as in the ladder model discussed above. In fact, such contributions have been discussed for experimental results from IR absorption [10] and from neutron scattering in pure La2 CuO4 [4]. The signature of cyclic exchange in the 2D Heisenberg model which is usually assumed for materials with CuO2 planes is a nonzero difference in the energies of two elementary excitations at the boundary of the Brillouin zone, p p D ¼ o qx ¼ p; qy ¼ 0 o qx ¼ ; qy ¼ : ð6Þ 2 2 For the 2D Heisenberg antiferromagnet with its LRO, elementary excitations are described to lowest order in the Holstein Primakoff (HP) spin wave approximation. In this approximation DpJring results, i.e. D vanishes for
905
the Heisenberg model with only bilinear exchange. Higher order corrections to the HP result as calculated in Ref. [11] lead to DE 1:4 102 J; in agreement with the experimental result on copper deuteroformate tetradeuterate (CFTD), another 2D Heisenberg magnet. Neutron scattering experiments in pure La2 CuO4 ; however, find DE þ 3 102 and thus require finite cyclic exchange. The two materials considered appear to differ in nothing but their energy scale (JE1400 K for La2 CuO4 and JE70 K for CFDT) and experimental results would be contradictory when bilinear and biquadratic exchange scale with the same factor. However, in terms of the basic Hubbard model with hopping amplitude t and on site Coulomb energy U one has Jp7t72 =U and Jring p7t74 =U 3 : Comparing two materials with identical single ion Coulomb energies, any differences result from different hopping rates. This leads to Jring pJ 2 =U: Thus in materials with high energy scale such as La2 CuO4 cyclic exchange is enhanced and it is therefore observable whereas cyclic exchange goes unnoticed in materials with low energy scale such as CFTD. We thank A. Kolezhuk for fruitful discussions.
References [1] M. Takahashi, J. Phys. C 10 (1977) 1289. [2] H.J. Schmidt, Y. Kuramoto, Physica C 167 (1990) 263; Y. Honda, Y. Kuramoto, T. Watanabe, Phys. Rev. B 47 (1993) 11329. [3] M. Matsuda, et al., J. Appl. Phys. 87 (2000) 6271; R.S. Eccleston, et al., Phys. Rev. Lett. 81 (1998) 1702. [4] R. Coldea, et al., Phys. Rev. Lett. 86 (2001) 5377. [5] T. Imai, et al., Phys. Rev. Lett. 81 (1998) 202. [6] M. Windt, et al., Phys. Rev. Lett. 87 (2001) 127002; T. S. Nunner, et al., cond-mat /0203472. [7] S. Brehmer, et al., Phys. Rev. B 60 (1999) 329. [8] A. Nersesyan, A. Tsvelik, Phys. Rev. Lett. 78 (1997) 3939. [9] A. Laeuchli, G. Schmid, M. Troyer, cond-mat/0206153. [10] J. Lorenzana, et al., Phys. Rev. Lett. 83 (1999) 5122. [11] H.M. R^nnow, et al., Phys. Rev. Lett. 89 (2002) 079702.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 904–905
Ring exchange in low-dimensional spin systems M. Muller, . T. Vekua, H.-J. Mikeska* Institut fur . Theoretische Physik, Universitat . Hannover, Appelstrasse 2, Hannover 30167, Germany
Abstract The isotropic S ¼ 12 antiferromagnetic spin ladder with additional four-spin ring exchange is studied perturbatively in the strong coupling regime with the cluster expansion technique, and by bosonization in the weak coupling limit. It is found that a sufficiently large amount of ring exchange leads to a second-order phase transition into a spontaneously dimerized phase. The shape of the phase boundary in the vicinity of the exactly known transition point is obtained. The relevance of ring exchange for materials with 2D CuO2 planes is discussed. r 2003 Elsevier B.V. All rights reserved. Keywords: Ring exchange; CuO2 plaquette; Quantum phase transition
The cyclic four-spin (ring) exchange emerges from a t=U expansion of the Hubbard Hamiltonian [1]. Recently it was suggested that some strongly correlated electron systems like cuprates [2] and spin ladders [3] are expected to exhibit ring exchange. The analysis of the low-lying excitation spectrum of the p–d model [2,4] as well as inelastic neutron scattering [3], NMR [5] and optical conductivity [6] measurements on the (Sr,Ca,La)14 Cu24 O41 system shows that the Hamiltonians describing CuO2 planes should contain a finite amount of ring exchange. Real two-leg ladder cuprates are always in the rung-singlet phase. Ring exchange drives these systems close to critical point. Therefore it is important to understand the nature of the phase transition with increasing frustrating ring interaction. Hamiltonian of our model has the form H ¼ Jrung
N X i
þ
S1;i S2;i þ Jleg
N X
Sa;i Sa;iþ1
a¼1;2;i
Jring X Pijkl þ P1 ijkl ; 2 /ijklS
ð1Þ
known [7] that at Jring =Jleg ¼ 1; the exact ground state is the direct product of rung dimers and the exact lowest excited state is the propagating triplet with following dispersion: et ðqÞ ¼ Jrung 2Jring þ 2Jring cosðqÞ:
ð2Þ 1 4Jrung :
We have The q ¼ p mode becomes soft for Jring ¼ obtained the phase boundary from the softening of the q ¼ p mode using the cluster expansion approach up to 14th order. The result is given in Fig. 1. Using the Pade! approximation technique we determined that the excitation gap is increasing linearly upon deviating from the critical line. To clarify the nature of the ground state and low energy excitations with increasing ring exchange we applied weak-coupling bosonization approach. Close to the limit of decoupled chains (Jleg cJrung ; Jring ) the Hamiltonian in Majorana representation becomes similar to the one considered in Ref. [8]: X ivt H¼ ðxaR @x xaR xaL @x xaL Þ imt xaR xaL 2 a¼1;2;3
ivs 0 ðx @x x0 x0L @x x0L Þ ims x0R x0L ; 2 R R
ð3Þ
where Pijkl permutes spin over the elementary plaquettes of the ladder. From the matrix product approach it is
mt BðJrung aJring Þ; ms Bð3Jrung aJring Þ:
*Corresponding author. Tel.: +49-511-762-3665; fax: +49511-762-3023. E-mail address: [email protected] (H.-J. Mikeska).
It follows, that mt ¼ 0 for Jrung ¼ aJring : The universality class of the quantum phase transition is that of the critical, exactly integrable, S ¼ 1 spin chain with central charge c ¼ 32: At the criticality relative spin of two chains
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.444
ð4Þ
ARTICLE IN PRESS M. Muller et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 904–905 .
0.4
Jring/Jrung
0.3
dimerized phase
0.2
exact line
rung-singlet phase 0.1
0 0
0.1
0.2
0.3
0.4
0.5
Jleg/Jrung
Fig. 1. Results from the 14th order of perturbation theory in Jring =Jrung and Jleg =Jrung
as well as relative dimerization decay algebraically according to SU2 ð2Þ universality class /S ðrÞS ð0ÞSB/e ðrÞe ð0ÞSBr3=4 :
ð5Þ
The phase with dominant ring exchange was established to be spontaneously dimerized using Ising variables representation of spin operators, the result later confirmed by DMRG studies [9]. Next we discuss the effects of the cyclic interactions to models and materials with CuO2 planes since the basic plaquette of those is the same as in the ladder model discussed above. In fact, such contributions have been discussed for experimental results from IR absorption [10] and from neutron scattering in pure La2 CuO4 [4]. The signature of cyclic exchange in the 2D Heisenberg model which is usually assumed for materials with CuO2 planes is a nonzero difference in the energies of two elementary excitations at the boundary of the Brillouin zone, p p D ¼ o qx ¼ p; qy ¼ 0 o qx ¼ ; qy ¼ : ð6Þ 2 2 For the 2D Heisenberg antiferromagnet with its LRO, elementary excitations are described to lowest order in the Holstein Primakoff (HP) spin wave approximation. In this approximation DpJring results, i.e. D vanishes for
905
the Heisenberg model with only bilinear exchange. Higher order corrections to the HP result as calculated in Ref. [11] lead to DE 1:4 102 J; in agreement with the experimental result on copper deuteroformate tetradeuterate (CFTD), another 2D Heisenberg magnet. Neutron scattering experiments in pure La2 CuO4 ; however, find DE þ 3 102 and thus require finite cyclic exchange. The two materials considered appear to differ in nothing but their energy scale (JE1400 K for La2 CuO4 and JE70 K for CFDT) and experimental results would be contradictory when bilinear and biquadratic exchange scale with the same factor. However, in terms of the basic Hubbard model with hopping amplitude t and on site Coulomb energy U one has Jp7t72 =U and Jring p7t74 =U 3 : Comparing two materials with identical single ion Coulomb energies, any differences result from different hopping rates. This leads to Jring pJ 2 =U: Thus in materials with high energy scale such as La2 CuO4 cyclic exchange is enhanced and it is therefore observable whereas cyclic exchange goes unnoticed in materials with low energy scale such as CFTD. We thank A. Kolezhuk for fruitful discussions.
References [1] M. Takahashi, J. Phys. C 10 (1977) 1289. [2] H.J. Schmidt, Y. Kuramoto, Physica C 167 (1990) 263; Y. Honda, Y. Kuramoto, T. Watanabe, Phys. Rev. B 47 (1993) 11329. [3] M. Matsuda, et al., J. Appl. Phys. 87 (2000) 6271; R.S. Eccleston, et al., Phys. Rev. Lett. 81 (1998) 1702. [4] R. Coldea, et al., Phys. Rev. Lett. 86 (2001) 5377. [5] T. Imai, et al., Phys. Rev. Lett. 81 (1998) 202. [6] M. Windt, et al., Phys. Rev. Lett. 87 (2001) 127002; T. S. Nunner, et al., cond-mat /0203472. [7] S. Brehmer, et al., Phys. Rev. B 60 (1999) 329. [8] A. Nersesyan, A. Tsvelik, Phys. Rev. Lett. 78 (1997) 3939. [9] A. Laeuchli, G. Schmid, M. Troyer, cond-mat/0206153. [10] J. Lorenzana, et al., Phys. Rev. Lett. 83 (1999) 5122. [11] H.M. R^nnow, et al., Phys. Rev. Lett. 89 (2002) 079702.