Dynamical structure factors of the square lattice ferroquadrupolar Heisenberg model calculated using the SU(3) Schwinger bosons

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Dynamical structure factors of the square lattice ferroquadrupolar Heisenberg model calculated using the SU(3) Schwinger bosons A.S.T. Pires n Departamento de Fisica, Universidade Federal de Minas Gerais, CP 702, Belo Horizonte 3012-970, MG, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 19 November 2014 Received in revised form 9 April 2015 Accepted 11 May 2015 communicated by Xincheng Xie

We have calculated the dynamical quadrupolar and spin-spin structure factors for the ferroquadrupolar phase, at zero temperature, of the S ¼1 Heisenberg model with bilinear and biquadratic exchange interactions on the square lattice, using the SU(3) Schwinger boson formalism. The quadrupolar structure factor can, in principle, be measured using resonant inelastic x-ray scattering. & 2015 Published by Elsevier Ltd.

Keywords: A. Antiferromagnet D. Nematic phases D. Ferroquadrupolar

1. Introduction Different from quantum spin liquids, which do not show any symmetry-breaking long range order, spin nematic states exhibit long-ranged quadrupolar order without conventional dipolar magnetic order [1]. The interest in the study of spin nematic order increased with the possibility that it might occur in several real materials [2]. Systems with spin S ¼1 can exhibit nematic ground state ordering. These orderings can be induced by a spontaneous symmetry breaking as a result, for instance, of a biquadratic term in the Hamiltonian [3]. Spin nematic order occurs when fluctuations of a spin are along an axis without a direction being chosen. For instance, one can have oðSx Þ2 4 ¼ o ðSy Þ2 4 a o ðSz Þ2 4 , but with o S4 ¼0. As pointed out by Oitmaa and Hamer [4] the study of the Heisenberg antiferromagnet with biquadratic exchange interaction goes back to Refs. [5,6], where phases of both dipolar and quadrupolar nature were identified. After that, the model has been studied using several techniques [7–30]. In contrast to bilinear interaction models, quantum spin models with biquadratic interactions can have a phase diagram qualitatively different from their classical counterparts, as for instance nonmagnetic phases such as the quadrupolar phase [24]. Biquadratic interactions will in general be present in spin-1 magnets and are known to favor nematic ordering [8]. In this paper we will study the dynamics of the

n

Tel.: þ 55 31 34996224; fax: þ 55 31 34996600. E-mail address: antpires@fisica.ufmg.br

ferroquadrupolar phase of the following Hamiltonian on the square lattice. X H¼ ½J 1 Si  Sj þ J 2 ðSi  Sj Þ2 : ð1Þ i;j

It is usual to writeJ 1  J cos θ, J 2 ¼ J sin θ, where the parameter θ control the ratio of these two couplings. Recently, this model with sinθo0 was applied to the iron pnictides [11]. The two terms in the Hamiltonian (1) introduce competition between different kinds of magnetic orders [10]. For positive values of cosθ and sinθ, the first term favors the conventional antiferromagnetic order, while the second term favors an antiferro—quadrupolar order. This competition causes a strong quantum fluctuation in the regime sinθ40. For π/4ZθZ  π/2 [8] the ground state is the antiferromagnetic Néel phase. For π/24θ4π/4 a non collinear antiferroquadrupolar phase is expected. In this phase the quadrupole moments on neighboring sites take on perpendicular directions. In the ferroquadrupolar phase  3π/4oθo  π/2, where all spins are in the Sz ¼ 0state, the SU(2) rotational symmetry in spin space is broken without magnetic order. Although this state is non magnetic ð o Szn 4 ¼ 0Þit breaks the rotational symmetry because the state is defined with respect to the z-direction [8]. The order parameter here is not oSzn 4but o ðSzn Þ2 4 . Since the spin nematic states do not break time-reversal symmetry, they do not exhibit magnetic Bragg peaks, a static splitting of lines in NMR spectra, or oscillations in the μSR [29]. As pointed out by Smerald [29], the spin nematic is a “hiddenorder” that would be almost invisible to the usual probes of magnetism. However, since excitations of the spin nematic state induce a fluctuating dipole moment, spin nematic order can, in

http://dx.doi.org/10.1016/j.ssc.2015.05.006 0038-1098/& 2015 Published by Elsevier Ltd.

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principle, be detected by dynamical probes such as resonant inelastic x-ray scattering. Recently Savary and Senthil [32] have proposed a new way to experimentally probe spin nematic order in magnets using dynamic measurements. The above considerations motived us to do the calculations presented here and we hope they can be useful for comparison with future experiments. In this paper we will study the dynamics of the ferroquadrupolar phase of Hamiltonian (1), at zero temperature, using a SU (3) Schwinger boson formalism in a mean field approximation. In Section 2 we will present the formalism and in Sect. 3 our results and conclusions.

now be written as pffiffiffi S þ ¼ 2ðt zþ d þ u þ t z Þ;

S



pffiffiffi þ ¼ 2ðd t z þ t zþ uÞ;

z

þ

þ

S ¼ u u d d: ð8Þ

We can take the director of the ferroquadrupolar phase     dj ¼ dx jxi þdy y þ dz jzi; j j j

ð9Þ

in the z direction and use the approximation that the tz bosons are condensed [20,27,29], i. e. ot z 4 ¼ t. Substituting the representation (8) into the Hamiltonian (1) we obtain: H ¼ H0 þ H1 þ H2 ;

ð10Þ

where 2. SU(3) Schwinger boson formalism Traditional spin wave SU(2) theory is a very good approach for treating quantum spin models that have a magnetically (dipolar) ordered ground state. The theory however is not adequate to treat nematic, quadrupolar, octupolar, or higher multipolar ordering, because, as pointed out by Muniz et al. [3], such ground state orderings do not have a classical counterpart at zero temperature. In these cases the local fluctuations of a spin are described by the SU(2Sþ 1) group of local unitary transformation, instead of the SU (2) group of rotations. (It should be noted that the N2–1 components of the local SU(N) order parameter can be decomposed in the different irreducible representations of SU(2)). To treat quadrupolar phases, Papanicolaou [7] derived a theory which is a generalization SU(3) of the SU(2) Schwinger boson approach. In this formalism we choose the following basis jxi ¼

iðj1i  j  1iÞ pffiffiffi ; 2

  ðj1i þ j  1iÞ y ¼ pffiffiffi ; 2

jzi ¼  ij0i

ð2Þ

where jni are eigenstates of Sz. This set of basis states respect the time-reversal invariance of the spin-nematic states. The spin operators are then written via a set of three boson operators t α (α¼ x, y, z) defined by   t xþ jvi ¼ jxi; t yþ jvi ¼ y ; t zþ jvi ¼ jzi; ð3Þ where jvi is the vacuum state. To force single site occupancy on each site we impose the constraint t xþ t x þ t yþ t y þt zþ t z ¼ 1:

ð4Þ

In terms of the t operators we can write S ¼  iðt yþ t z  t zþ t y Þ; x

Sy ¼  iðt zþ t x  t xþ t z Þ;

Sz ¼  iðt xþ t y  t yþ t x Þ

ð5Þ

The states t xþ jvi and t yþ jvi, both consist of Sz ¼ 7 1 eigenstates and have the average o Sz 4 ¼ 0:This property will preserve the disorder of the ground state. To study disordered phases, such as the ferroquadrupolar phase, it is convenient to introduce another two bosonic operators u þ and d þ given by [23,30] 1 u þ ¼  pffiffiffiðt xþ þ it yþ Þ; 2

d

þ

1 ¼ pffiffiffiðt xþ  it yþ Þ; 2

ð6Þ

ð11Þ

o n;m 4

, P þ þ þ ðunþ un um um þdn dn dm dm Þ  Jð cos θ  sin θÞ H 2 ¼ J cos θ o n;m 4 X þ þ þ ðunþ un dm dm þ dn dn um um Þ þ J sin θ o n;m 4

X

o n;m 4

þ

þ

þ ðunþ dm um dn þ dn um dm un Þ:

ð12Þ

A temperature-dependent chemical potential λ has been introduced to impose the local constraintS2r ¼ SðS þ 1Þ ¼ 2. Making a mean-field decoupling to the four operator terms we obtain: P þ þ þ ½pðunþ dm þ dn um Þ þh:c H mf 2 ¼  2Jð cos θ  sin θÞ o n;m 4

zN þ ½J cos θð1  t 2 Þ2 þ4Jð cos θ  sin θÞp2 ; 2

ð13Þ

where p ¼ o un um 4 . This parameter was found to be very small and was neglected. After Fourier transforming, the Hamiltonian can be written as 1X þ H¼ Ψ k H αα Ψ k þ E0 ; ð14Þ 2 k þ

where Ψ kþ ¼ ðukþ ; dk ; u  k ;d  k Þ, and 0 1 Λk 0 0 Δk B C B 0 Λk Δk 0 C C H αα ¼ B B 0 Δk Λk 0 C; @ A 0 Λk Δk 0

ð15Þ

with E0 ¼

zN ½J cos θð1  t 2 Þ2 þ 4Jð cos θ  sin θÞp2 ; 2

ð16Þ

Λk ¼ λ þ 4t 2 γ k cos θÞ; Δk ¼ 4t 2 ð cos θ  sin θÞγ k ;

ð17Þ

1 γ k ¼ ð cos qx þ cos qy Þ; 2

ð18Þ

and I have set J ¼ 1. The Green’s function is given by Gðk; iωn Þ ¼ ðiωn σ z  I H αα Þ  1 ; ;

ð19Þ

where ωn is the Matsubara frequency. From Eq. (15), writing z ¼ iω, we obtain

and so j1i ¼ u þ jvi;

X N þ H 0 ¼ ð1 þ t 4 ÞJ sin θz þ λ ðunþ un þ dn dn þ t 2  1Þ; 2 n P þ ðunþ um þ dn dm þ h:c:Þ H 1 ¼ Jt 2 cos θ o n;m 4 X þ þ þ ðunþ dm þdn um þ h:c:Þ þ Jð cos θ  sin θÞt 2

þ

j0i ¼ t zþ jvi; þ

with the constraint u u þ d

j  1i ¼ d jvi; þ

d þ t zþ t z

ð7Þ

¼ 1. The spin operators can

det G  1 ðk; zÞ ¼ ðz2  ω2k Þ2 ; with

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ð20Þ

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0,4

160

0,3

S(q, )

S(q, )

120

80

0,2

0,1

40

0,0

0 0,00

0,05

0,10

0,15

4,0

0,20

Fig. 1. The dynamical quadrupolar structure factor as a function of ω, for θ ¼  0.6π and q¼ (0, 0). The energy ω is in unit of J. Here S(q,ω) means SQ(q,ω).

4,5

5,0

5,5

6,0

6,5

7,0

7,5

8,0

Fig. 3. The dynamical structure factor, at zero temperature, as a function of ω, for θ¼  0.6π and q ¼(π, π). Red line: quadrupolar, black line: spin. The energy ω is in unit of J. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0,30 0,25 1,2

0,20

1,0

0,15

0,8

0,10

S(q, )

S(q, )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Q4 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

3

0,05

0,6 0,4

0,00 3

4

5

6

7

8

0,2

9

0,0

Fig. 2. The dynamical structure factor as a function of ω, for θ ¼  0.6π and q¼ (0, π). Red line: quadrupolar, black line: spin. The energy ω is in unit of J. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ω2k

¼ Λ2k  Δ2k :

ð21Þ

Using the condition that the gap vanishes at k¼ (π,π) we find

λ ¼  4t 2 sin θ

t2 ¼ 2 

1 X Λq : N q ωq

ð22Þ

In term of the operators u and d, the Green’s function can be written as 0 1 þ {uk ukþ c {uk dk c {uk u  k c {uk d  k c B C þ B {dk u þ c {dk dk c {dk u  k c {dk d  k c C k B C B C: þ þ þ þ þ B {u þ C u c {u d c {u u c {u d c  k  k k k k k k k @ A þ þ þ þ þ {d  k ukþ c {d  k dk c {d  k u  k c {d  k d  k c ð23Þ We have solved Eq.(19) using MAPLE and from Eq. (23) obtained the Green’s function for each term. The dynamical structure factor is given by Z 1 1 dteiωt o Aðk; tÞAð  k; 0Þ 4 ; ð24Þ Sa ðk; ωÞ ¼ 2π  1 where A ¼ Sz for spins and A ¼ Q ¼ ðSz Þ2 for the quadrupole. In fact 2 Q n ¼ ðSzn Þ2  ; 3

ð25Þ

4

5

6

7

8

Fig. 4. The dynamical structure factor, at zero temperature, as a function of ω, for θ¼  0.5π and q¼ (π/2, π/2). Red line: quadrupolar, black line: spin. The energy ω is in unit of J. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

is the quadrupole operator of interest which describes the anisotropy of spin fluctuation, but for the dynamics we can consider only the first term in the right hand side in Eq. (25). At T ¼0 the dynamical structure factor is related to the Green's function via the fluctuations dissipation theorem: Sðk; ωÞ ¼  ImGðk; ωÞ:

Szq

Using X þ ¼ ðuqþ k uk  dq  k dk Þ;

ð26Þ

ð27Þ

k

ðSz Þzq ¼

X þ ðuqþ k uk þ dq  k dk Þ;

ð28Þ

k

we obtain

Sðq; ωÞ SQ ðq; ωÞ

) ¼

 1X 1  ðωk þ Λk ÞðΛq  k  ωq  k Þ 8 Δk Δq  k δ½ω 2 k ωk ω q  k

 ðωq  k þ ωk Þ;

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ð29Þ

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4

600 500

S(q, )

400 300 200 100 0 0,0

0,1

0,2

0,3

0,4

0,5

Fig. 5. The dynamical quadrupolar structure factor, at zero temperature, as a function of ω, for θ ¼  0.5π and q¼ (π, π). The energy ω is in unit of J. Here S(q,ω) means SQ(q,ω).

4,0 3,5 3,0 2,5

S(q, )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

2,0 1,5 1,0 0,5 0,0 3

4

5

6

7

8

Fig. 6. The dynamical structure factor, at zero temperature, as a function of ω, for θ¼  0.5π and q¼ (π/4, π/4). Red line: quadrupolar, black line: spin. The energy ω is in unit of J. Here S(q,ω) means SQ(q,ω). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Results and conclusions Long range order implies a divergence of the static correlation function at a wave vector q0 characterizing the type of order. The function o Szq Sz q 4shows no such divergence, consistent with the absence of ordinary magnetic order in the ferroquadrupolar phase. On the other side the divergence in o ðSzq Þ2 ðSz q Þ2 4 implies ferroquadrupolar long-range correlations. We can understand the different kind of behavior of these two functions looking at Eq. (29). The divergences occur when the frequency ωq in the denominator vanishes. For the spin-spin correlation function the numerator vanishes at the same points and so the divergence does not occur. The low energy modes of the dipolar ordering are magnons. However nematic waves, which are bound states of two magnons, become part of the spectrum of excitations. In Figs. 1–6, we show the dynamical structure factor, obtained from Eq. (29), for some values of θ and the wave vector q. In Figs. 1 and 5 the spin–spin dynamical structure factor vanishes, consistent with the absence of long range order.

We have contribution for wave vectors for which ω ¼ ωq  k þ ωk ;representing the two-magnon creation scattering. As it is expected, SQ ðq; ωÞvanishes for ωo Ωmin and ω4Ωmax, where Ωðq; kÞ ¼ ωq  k þ ωk : In Fig. 1 we have θ¼ 0.6π and q ¼(0,0). As it should be expected, SQ ðq; ωÞ diverges at ω¼ 0. Fig. 2 shows the quadrupolar and spin dynamical structure factors for θ¼  0.6π and q ¼(0,π). The quadrupolar term is slight above the spin term. In Fig. 3, we depict both terms for θ¼  0.6π and q ¼(π,π). For small values of ω, the quadrupolar term is now stronger than the spin term as compared with Fig. 2. Fig. 4 shows both terms for θ ¼  0.5π and q¼ (π/2, π/2). They have the same values at ω¼ Ωmax and ω¼ Ωmin. Fig. 5 shows SQ ðq; ωÞ for θ ¼  0.5π and q ¼(π,π). As expected, it diverges at ω¼0. Finally, in Fig. 6 we present both terms for θ¼ 0.5π and q ¼(π/4,π/4). The behavior is qualitatively similar to the one in Fig. 3, although the intensities, mainly for the quadrupolar component, are larger. As we can see, bound states of two magnons are already captured by the SU(3) theory at the linear level. We hope our results can give, to the experimentalist, an indication to where to look for the signature of the nematic order. In conclusion, using the SU(3) Schwingr bosons formalism in a mean field approximation we have studied the zero temperature dynamics of the ferroquadrupolar phase of the S¼ 1 Heisenberg model with bilinear and biquadratic exchange interactions on the two dimensional square lattice. Dynamic quadrupolar susceptibility measurements probes directly the order parameter and can therefore provides evidence for the existence of spin-nematic order.

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