Phase transition of spin–orbital models with a four-spin ring exchange

Solid State Communications 149 (2009) 2102–2105

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Phase transition of spin–orbital models with a four-spin ring exchange Sha-Sha Ke, Hai-Feng Lü ∗ , Hua-Jun Yang, Xiao-Tao Zu, Huai-Wu Zhang Department of Applied Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China

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info

Article history: Received 8 April 2009 Received in revised form 2 August 2009 Accepted 6 August 2009 by S. Miyashita Available online 12 August 2009 PACS: 75.10.Jm 75.40.Gb 71.27.+a 74.20.Mn

abstract We extend the bond-operator mean-field theory to study the rung singlet phase and its phase boundary, the triplet excitation, and the spin gap of the spin–orbital models with four-spin exchanges. The theory gives a well description of the rung singlet phase and phase boundaries in two-dimensional (2D) and three-dimensional (3D) cases are predicted. It is shown that consideration of the ring exchange suppresses the excitation spectrum and decreases the spin gap. For 2D and 3D spin–orbital models, positive ring and leg coupling tend to collaborate with each other to break the rung singlet phase. On the boundary line Jleg = Jring < Jrung /4, the rung singlet density is one and a second-order phase transition occurs. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved.

Keywords: D. Phase transition D. Spin-orbital model D. Bond-operator

1. Introduction The multiple-spin exchange interactions have drawn steady research interest from both theoretical and experimental points of view over the years [1–4]. These four-spin exchange terms, such as (Si · Sj )(Sk · Sl ), are usually derived from fourth-order perturbation in the strong coupling limit (small t /U) of the Hubbard model and can induce many exotic ground states [4]. The most important mechanism is the so-called ring exchange [1–3], which is introduced first to describe the magnetic properties of solid 3 He. Recently it was suggested that inclusion of a small ring exchange can induce obvious effect on some strongly correlated systems like spin ladders and the high-Tc cuprates [1,5,6]. Among lowdimensional spin systems, two-leg antiferromagnetic spin-ladder systems provide an important playground for studying the effect of ring exchange terms. The motivation to study the effect of ring exchange is that the magnon dispersion at the zone boundary obtained by inelastic neutron scattering experiments cannot be described within a nearest-neighbor Heisenberg model. It is also found that the inclusion of a cyclic spin exchange term of about 20% can reproduce the dispersion [5]. The Hamiltonian of the spin–orbital chain (also called spin ladder) with extended four-spin exchange interactions (as Fig. 1(a)) is

∗

Corresponding author. Tel.: +86 13880583640. E-mail addresses: [email protected] (H.-F. Lü), [email protected] (H.-W. Zhang).

usually defined as H = Jrung

X

S1,i S2,i + Jleg

i

i

+

X (S1,i S1,i+1 + S2,i S2,i+1 )

Jring X 2 hijkli

−1 (Pijkl + Pijkl ),

(1)

where Jrung and Jleg denote the rung and leg coupling constants, the index i refers to the rungs, and 1, 2 label the two legs. The cyclic permutation operator Pijkl for four spins on a plaquette is given by −1 Pijkl + Pijkl = 4(Si Sj )(Sk Sl ) + 4(Si Sl )(Sj Sk ) − 4(Si Sk )(Sj Sl )

+ Si Sj + Sk Sl + Si Sl + Sj Sk + Si Sk + Sj Sl .

(2)

In Fig. 1(b), (c), the extended coupled spin-ladder models are also illustrated in two-dimensional (2D) and three-dimensional (3D) cases. Consideration of the four-spin ring exchange has been widely studied [5–15] in recent years, which can induce the gapped staggered dimer phase, scalar chiral phase, rung singlet phase and other exotic phases. Especially the scalar chiral phase can be realized by ring exchange [2,7], which is considered to be difficult to realize in SU(2)-symmetric systems in the past. In most cases we understand the ground state of model (1) with the help of numerical techniques, such as the density–matrix renormalization group (DMRG) [7,8,12], numerical exact diagonalization of small clusters [11,14]. Analytical investigations of the influence of this exchange have employed perturbative approaches [9], spin-wave analysis [13], exact diagonalization in combination with conformal field theory [5].

0038-1098/$ – see front matter Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.08.010

S.-S. Ke et al. / Solid State Communications 149 (2009) 2102–2105

a

c Jring

Jrung

Jleg + Jring X

H2 =

4

b

(3)

i

= Jrung

Jring

Jrung

Jring Jleg

Jleg

Fig. 1. Schematic structure of spin–orbital models with additional ring exchange of (a) one dimension, (b) two dimension, and (c) three dimension.

The bond-operator mean-field approach [16,17] is an important method to discuss the disordered and frustrated spin systems and superior than SU(2) mean-field approximation in many cases, which is proposed firstly by Sachdev [16]. Its basic idea is to introduce a set of bond operators to create and annihilate singlet and triplet bonds between a pair of spins. The effect of the local hardcore constraint is handled by Brueckner approximation as introduced by Kotov et al. [18]. Until now, this method has been applied widely and successfully to many spin systems, such as spin-ladder system [17,19], the bilayer model [18,20], J1 − J2 model [21], and so on. For the spin-ladder system, bond-operator approach reproduces the dispersion and the spin gap in the rung singlet phase [17], which ensures us to extend the discussion with taking the ring exchange into account. In the present paper, we use the bond-operator formalism to study the spin–orbital model with a four-spin ring exchange. The starting point is the dimer limit of uncoupled rungs. In Section 2 we give the theoretical framework and the self-consistent equations for the present model. In Section 3 we study the effect of the fourspin ring exchange on the elementary excitations, the energy gap and other physical quantities. The limitation of the present meanfield approach is discussed. We also present the exactly soluble condition for generalized spin-ladder system with rung singlet ground state, while a similar result is also proposed by variational ansatz. Finally, Section 4 gives the summary. 2. Theoretical framework Ď

Ď

Ď

For a pair of spins, four operators sĎ , tx , ty , tz are introduced to create the four states in Hilbert space. A representation of the spins between two ladders in terms of these singlet and triplet operators Ď is given by S1α = 12 (sĎ tα + tαĎ s − iαβγ tβ tγ ), S2α = 21 (−sĎ tα − tαĎ s − Ď

iαβγ tβ tγ ), where α , β , and γ represent components along the x, y, and z axes, respectively, and  is the antisymmetric Levi-Civita tensor. The s and tα operators satisfy the bosonic commutation relations with a local constraint sĎ s + tαĎ tα = 1. Substituting the operator representation into the original Hamiltonian (1), we get H = H0 + H1 + H2 , H0 =

X  Jrung 4

i

−

X

3

+ Jring − (Jrung +

µi (sĎi si

8

Ď 2Jring )si si



+ tiĎ,α ti,α − 1),

i

H1

(1 − δα,β )

i,α,β

  × tiĎ,α tiĎ+1,β ti+1,α ti,β − tiĎ,α tiĎ+1,α ti+1,β ti,β + h.c . X Ď Ď + 2Jring si si si+1 si+1 ,

Jleg

2103

 Jleg + Jring X  Ď Ď = si si+1 ti,α ti+1,α + h.c . 2 i,α +

 Jleg − Jring X  Ď Ď si si+1 ti,α ti+1,α + h.c . , 2 i,α

where a site-dependent chemical potential µi has been introduced to impose the local constraint. For a relatively large Jrung , we take hsi i = s¯, which means that s bosons are condensed. It can be seen that the consideration of Jring renormalizes the coupling parameters of the individual terms in the original Heisenberg Hamiltonian. We also define two mean fields P and Q as P = htiĎ,α ti+1,α i, Q = htiĎ,α tiĎ+1,α i to decouple the Hamiltonian. Then the Hamiltonian can be solved by a mean-field approach. Ď Ď Ď Ď Defining a generalized Nambu spinor Ψk = (tk,1 , tk,2 , tk,3 , t−k,1 , t−k,2 , tk,3 ) and after Fourier transformation, the decoupled Hamiltonian can be written in a compact form: H =

1X 2

k

 +

Ď

Ψk Hαα Ψk + N [−(Jrung + 2Jring )¯s2 + 2Jring s¯4

5 2

2

− s¯



µ − 3(Jleg + Jring )(P 2 − Q 2 )],

(4)

where Hαα = Λk Ω1 + ∆k Ω2 with Ω1 = σx ⊗ σ0 , Ω2 = σ00 ⊗ σ0 , σx is the Pauli matrix, σ00 is the 2 × 2 unit matrix and σ0 is the 3 × 3 unit matrix, Λk = −µ + (Jleg + Jring )(¯s2 + 2P ) cos k, ∆k = (Jleg − Jring )¯s2 cos k − 2(Jleg + Jring )Q cos k. Here Ω1(2) are expressed in the form of direct product of σx(00 ) and σ0 . The corresponding Matsubara Green’s function (GF) is thus deduced to G−1 (k, iωn ) = iωn σz ⊗ I − Hmf (k), where ωn is the bosonic Matsubara frequency. The poles q of the GF matrix give rise to the quasiparticle spectra:

(k) =

Λ2k − ∆2k . From the free energy of the system, the saddlepoint equations at T = 0 K are derived as Z 3 π dk 1 5 q = − s¯2 , 2 −π 2π 2 2 1 − Γk Z 1 π dk cos k q = P, 2 −π 2π 1 − Γk2 Z 1 π dk −Γk cos k q = Q, 2 −π 2π 1 − Γk2 3Jleg (P + Q ) + 3Jring (P − Q ) = µ + Jrung + 2Jring (1 − 2s¯2 ),

(5)

where Γk = ∆k /Λk . For the 2D and 3D cases, one can obtain the self-consistent equations directly by replacing cos k with (cos kx + cos ky )/2 and (cos kx + cos ky + cos kz )/3 in the dispersion relation in Eq. (5). For a given value of Jleg /Jrung and Jring /Jrung , we have a set of solutions for µ, s¯2 , P and Q . It has be shown in numerous former studies that most rungs occupy the singlet state for a relatively large Jrung and so the effect of P and Q are small [17,19]. The bond-operator mean-field theory has been applied successfully to the spin-ladder system without taking the ring exchange into account [19]. In 2D case, the model reduces to the well-known bilayer spin model for Jring = 0. It is shown that the zero-temperature quantum phase transition occurs at a critical ratio of Jleg /Jrung ≈ 0.86. Without taken the four-spin exchange terms into account, the 2D model considered here is also called net spin model, where a set of exactly soluble net spin models for any spin S have been presented [22]. In the following discussion, Jrung is taken as the energy unit. For small Jleg and Jring , the spectrum is real and positive everywhere in the Brillouin zone. From the expression of

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S.-S. Ke et al. / Solid State Communications 149 (2009) 2102–2105

ω(k)

b

ω(k)

a

Fig. 2. Effect of the ring exchange on the dispersions of the spin-triplet excited states of the spin ladders for different coupling values: (a) Jring = −0.15, 0.0, 0.15, and 0.25 for Jleg = 0.25, the result from the perturbation theory (Ref. [7]) is given for comparison (dotted); (b) Jring = −0.2, 0.0, and 0.2 for Jleg = 1.0, the result from the Lanczos method (Ref. [9]) is given for comparison (circle).

quasiparticle spectra (k) =

q

Λ2k − ∆2k , it can be deduced that the band minimum occurs at k = π , so the spin gap is deduced as q ∆ = (µ + 2Jleg s¯2 )(µ + 2Jring s¯2 ). (6) At the critical points, it can be proved that the energy gap reduces to zero at (π , π ) (2D) or (π , π , π ) (3D), µ = −2Jleg s¯2 for Jleg > Jring , and µ = −2Jring s¯2 for Jleg < Jring . 3. Results and discussion Firstly we study the effect of the four-spin ring exchange on the elementary excitations for a spin ladder. For comparing with the former results obtained by the cluster expansion of the perturbation theory and numerical results [9,11], we plot the excitation spectra for different coupling Jring = −0.15, 0.0, 0.15, and 0.25 at Jleg = 0.25 and Jleg = 1.0 in Fig. 2. It can be seen that consideration of the ring exchange suppresses the triplet dispersion and the shape of the dispersion slightly changed. The data of the dispersions agree well with the perturbative results, especially for a small Jring . The reason is that the results obtained by the present theory are more accurate when more spins per rung occupy the singlet state. As the increase of Jleg and Jring , the singlet density decreases and effect of the quartic interactions in the Hamiltonian (3) becomes more important. In the case that s¯2 decreases to a small value, the element assumption that s operator condensed is not suitable any more, and the results obtained by the present treatment become worse even taking high-order correction into account [19]. So the bond-operator mean-field theory only gives a reliable description of the rung dimer phase for a spin-ladder system. It is also shown in Fig. 2(b) that the dispersions obtained by bond-operator theory agree better with the numerical results for a large k while those do worse at small k when the coupling parameters Jleg and Jring become large. On the so-called dimer line where Jleg = Jring , the self-consistent equations (5) can be solved exactly with Γk = 0, s¯2 = 1 and P , Q = 0, which means the ground state is a product of singlet dimers placed on the ladder rungs. The triplet spectrum reduces to a pure cosine form:

(k) = Jrung − 2Jring + 2Jring cos k,

(7)

which is consistent with the exact results [9,10]. It also can be seen that the energy gap vanishes at k = π for Jring = 14 Jrung . The similar exactly solution is also proposed by variational ansatz [10]. It is interesting that the present mean-field treatment reproduces the exact results. The possible reason is that all rungs are condensed to the singlet state (s¯2 = 1) is just considered as the starting point of the bond-operator theory. In the limit of s¯2 = 1, the

approximated results thus reduce to the exact ones. Above results are also applicable for the higher-dimensional spin–orbital models. From the above discussion it can be seen that bond-operator theory can be considered as a perturbation treatment away from the rung singlet limit, whose accuracy is determined by the singlet density s¯2 . In one dimension, it is hard to get the critical coupling values where a phase transition occurs from the rung dimer phase to other phases in the present mean-field framework except for the special exactly soluble cases mentioned above. This is quite different from the case of higher dimensions [20,21] where a phase transition point can be well determined by the mean-field theory for lower quantum fluctuations. Fig. 3 presents the phase diagram in (Jring , Jleg ) plane for 2D and 3D spin–orbital models. The gap vanishes at the critical points, where the transition from rung singlet phase to other phases occurs. For the cubic lattice the results are similar to two dimensions. The gapped phase regions (I) and (II) in two dimensions is slightly larger than that in three dimensions. In the presence of the four-spin ring exchange, lower values of the leg field are required to enter into the rung singlet phase as the increase of Jring . It also shows Jleg = Jring is a special boundary line: the density of s¯2 always equal to one in the singlet phase and the propagating triplet is an exact excitation which softens at Jleg = Jring = Jrung /4. Actually in regions (III) and (IV), the system belongs to the dimerized phase [9]. It seems that no matter in the rung singlet phase or below the region of dimerized phase, Jleg = Jring is always a critical boundary line and the phase transition on this line is the second-order type in the singlet phase. Finally we discuss the limitation of the present MF treatment. The most important condition that the approximation is valid here is that the density of s¯2 is still not far away to one at the phase transition point. This condition is necessary but not sufficient and another condition is that the phase transition occurs at a relatively large Jrung . The reason is that our starting point is the large Jrung limit. If the phase transition occurs at a small Jrung , this means effect of the term causes rung singlet on the ground state energy is small and the present approximation is not applicable. The bondoperator MF theory works well when the singlet density is near to one. If the singlet density is small, the results are much worse even if taking the higher-order interactions into account. 4. Conclusion In summary, a bond-operator mean-field theory is proposed to discuss the effect of the ring exchange on spin-ladder systems in the rung singlet phase. It is shown that the excitation spectrum is suppressed and the spin gap decreases with the increase of fourspin ring exchange. A kind of exactly soluble cases for general spin–orbital models is also obtained in the rung singlet phase. It shows that the singlet density s¯2 is still near to one at the phase

S.-S. Ke et al. / Solid State Communications 149 (2009) 2102–2105

a

2105

Jring

b

Jleg

Jleg

Fig. 3. Phase diagrams in 2D and 3D spin–orbital models in (Jring , Jrung ) parameter space. Regions (I) and (II) correspond to the gapped rung singlet phase, while (III) and (IV) are the dimerized phases.

transition point in two and three dimensions. The phase diagram of the 2D and 3D spin–orbital models are discussed. For positive ring and leg coupling strength, it shows that they tend to collaborate with each other to break the rung singlet phase. On the boundary line Jleg = Jring < Jrung /4, all rungs occupy the singlet state and a second-order phase transition occurs on this line. Acknowledgements We acknowledge the financial support from the Foundation for Innovative Research Groups of the NSFC under Grant (No. 60721001) and the NSFC (No. 10774083). References [1] E. Dagotto, T.M. Rice, Science 271 (1996) 618. [2] A. Läuchli, G. Schmid, M. Troyer, Phys. Rev. B 67 (2003) 100409(R). [3] U. Schollwock, J. Richter, D.J.J. Farnell, R.F. Bishop, Quantum Magnetism, Springer press, 2004.

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