Magnons versus free spinons in finite quantum frustrated antiferromagnets

ARTICLE IN PRESS Physica B 404 (2009) 2858–2860

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Magnons versus free spinons in finite quantum frustrated antiferromagnets I.J. Hamad, L.O. Manuel, A.E. Trumper  Instituto de Fı´sica Rosario (CONICET) and Universidad Nacional de Rosario, Boulevard 27 de Febrero 210 bis, 2000 Rosario, Argentina

a r t i c l e in f o

PACS: 71.10.Fd 75.50.Ee Keywords: Quantum magnetism Frustration

a b s t r a c t We have investigated the validity of doping with a vacancy the J 1 2J 3 frustrated Heisenberg model on a finite square lattice as a way to test the existence of fractional spin excitations. Using a generalized t2J1 2J 3 model we have computed the vacancy spectral functions in the self-consistent Born approximation. We have found that by including spiral fluctuations in the magnetic ground state, the spectral functions on finite systems agree very well with the unbiased exact ones. In contrast to the recent proposal that the quasiparticle weight reduction could be a signal of a spinon free excitation in finite systems, we have found strong evidence that such a reduction is due to the existence of spiral fluctuations. & 2009 Elsevier B.V. All rights reserved.

In the last two decades frustrated magnetism has been intensively studied. One of the main interests was focused on the realization of spin liquid phases, with neutral S ¼ 12 excitations that preserve all symmetries of the underlying microscopic Hamiltonian [1], and valence bond solid (VBS) phases with S ¼ 1 excitations which preserve spin rotation but break translational symmetries [2]. In the literature [2,3] it has been argued that the frustrated AF J1 2J3 Heisenberg model (with interaction to first and third neighbors on a square lattice, see Fig. 1) is one of the candidates to show the possible realization of spin liquid or valence bond solid states. Spin wave studies [4,5], high temperature series expansions [6] and the nonlinear sigma model [7] predict a disordered ground state between Ne´el and spiral phases for intermediate values of J3 =J1 . These studies can determine the range of parameters where the semiclassical spin excitations are valid but they can state nothing about the nature of the disordered ground state. On the other hand, large-N expansion [3], exact diagonalization on finite clusters sizes [8,9], classical Monte Carlo [10], and series expansions [11] give some evidence for the presence of plaquette states with gapped S ¼ 1 excitations between Ne´el and spiral orders. This succession of magnetic transitions found by different techniques lend support to the J1 2J 3 model as an ideal microscopic model to investigate the possible realization of a fractionalization scenario. In practice, spin liquid phases in frustrated microscopic models have been elusive and it is even more difficult to compute its magnetic excitations. So, it is important to look for alternative ways, though artificially created, to study the behavior of spin 12 excitations. Recently it has been hypothesized that the efficient coupling of a nonmagnetic vacancy to a strongly fluctuating

 Corresponding author. Fax: +54 341 4821772.

E-mail address: trumper@ifir-conicet.gov.ar (A.E. Trumper). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.06.102

magnetic background may shed light on the competing orders and the magnetic excitations of the clean frustrated magnetic ground state [2]. In particular, it is believed that, even in finite systems, doping such paramagnetic states with a static or dynamic nonmagnetic vacancy could liberate a S ¼ 12 spinon from its vicinity [12]. Recently, this kind of study has been performed with exact diagonalization up to cluster sizes of N ¼ 32 in the disordered regime of the J 1 2J 3 model [12,13]. Technically, the observable that have been computed is the static version of the quasiparticle weight z ¼ jhCgs jCbare ij2 , where jCgs i is the magnetic many-body ground state of the system with one vacancy at site 0 and jCbare i ¼ c0;s jC0 i which is the magnetic state that result from removing one bare electron with spin s on site 0 from the clean magnetic state. These authors assumed that z ¼ 1 can be interpreted as spinon confinement while a reduction of about 20% of the weight as spinon deconfinement. In fact, this assumption is based on a reasonable physical picture where the magnetic excitations above such paramagnetic phases can be envisaged by breaking a singlet bond and generating two unpaired neutral S ¼ 12 spinons. Then, for the VBS state it is expected that once a vacancy is introduced the liberated spinon excitation remains confined to the vacancy due to the linearly increasing potential generated by the long range crystalline order, while in the spin liquid phase the lack of a particular bond order associated to the ground state would allow the spinon to move freely without a confining force to the vacancy. Furthermore, for a mobile vacancy it has also been observed a further reduction of z and, by assuming a smooth connection between the static and the dynamic vacancy cases, it has been interpreted as a support of the spinon deconfinement found in the static case. In this article we will address the reliability of using static and mobile vacancies to study the possible realization of fractionalization in the J 1 2J3 model. In fact, a fractionalization scenario in the

ARTICLE IN PRESS I.J. Hamad et al. / Physica B 404 (2009) 2858–2860

the undoped magnetic ground state in the SWA. In the SCBA, the vacancy self-energy is given by the following self-consistent equation:

(Q, Q)

(π, π) J1 J3

Sk ðoÞ ¼ J3/J1

Fig. 1. Antiferromagnetic exchange interactions corresponding to the J 1 2J 3 Heisenberg model (left). Schematic magnetic phase diagram (right). Between Ne´el ðp; pÞ and spiral ðQ ; Q Þ phases, valence bond solid phases has been proposed.

J 1 2J3 model on finite systems is a very important issue, but a delicate one, since in effective field theories the fractionalization can only be explained in terms of Berry phases and emergent gauge fields [14], which are just not explicitly present in the microscopic Hamiltonian. Furthermore, sometimes the ground state in a finite system may be strongly fluctuating between competing orders. For instance, in the J 1 2J 3 model it has been shown that spiral fluctuations compete with plaquette fluctuations in the disordered regime [8]. We will show that if such spiral magnetic fluctuations are considered in the ground state the predicted vacancy spectral functions agree very well with unbiased exact diagonalization spectra. Furthermore, the reduction of the spectral weight z in the critical region of the J1 2J 3 model can simply be explained by the coupling of the mobile vacancy with the magnonic excitations of the underlaying spiral fluctuations of the finite system. Our strategy is to compute the spectral function of the mobile vacancy solving the generalized t2J 1 2J3 model in the selfconsistent Born approximation (SCBA). Following a standard procedure [15], we obtain the effective Hamiltonian for the mobile vacancy, X X X k hyk hk þ oq ayq aq þ ðMkq hyk hq ayqk þ H:c:Þ, Heff ¼ k

q

k;q

where hk is the Fourier transform of the spinless fermion operator and aq are the Bogoliubov operators that diagonalize the J1 2J 3 Heisenberg part in the spin wave approximation (SWA). The effective Hamiltonian comprises three terms: (i) a free-like hopping term that takes into account the possibility of the vacancy to move without disturbing the underlying magnetic background. It is characterized by the vacancy dispersion P P k ¼ d td cos Q  d=2 cos k  d, where d is the summation over neighbors d of a given site, connected by the hopping term t d (first neighbors in our case). Q is the magnetic wave vector, ðp; pÞ for the Ne´el phase, and ðQ ; Q Þ for the spiral phase, with Q being the vector that minimizes the classical energy Eclas ¼ J1 ½cosðkx Þ þ cosðky ÞÞþ 2J2 cosðkx Þ cosðky Þ þ J 3 ðcosð2kx Þ þ cosð2ky Þ. It should be noticed that a dispersive bare vacancy band will exist only for magnetic backgrounds with spiral phases; (ii) a free magnon term, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 characterized by the dispersion in the SWA oq ¼ ðgq  GÞ2  xq , P P where gk ¼ 12 d Jd cos2 ðQ  d=2Þ cos k  d, G ¼ 12 d J d cos Q  d, and P xk ¼ 12 d J d sin2 ðQ  d=2Þ cos k  d; and (iii) a vacancy–magnon interaction term, which incorporates the magnon-assisted mechanism for the vacancy motion, characterized by the vacancy–magnon vertex interaction i M kq ¼ pffiffiffiffi ðbq vk2q  bk uq2k Þ, N P where bk ¼ d t d sinðQ  d=2Þ sin k  d, and the Bogoliubov coefficients are uq ¼ððgq  G þ oq Þ=2oq Þ1=2 , vq ¼ sgnðxq Þððgq  G  oq Þ= 2oq Þ1=2 . We compute the vacancy spectral function as Ak ðoÞ ¼ ð1=pÞ Im Ghk ðoÞ, where Ghk ðoÞ ¼ hAFjhk ½1=ðo þ iZþ  y Heff Þhk jAFi is the retarded vacancy Green function, and jAFi is

X

jM kq j2

q

o þ iZþ  oq  k2q  Sk2q ðo  ok2q Þ

.

In Fig. 2 we plot the SCBA spectral function for a vacancy with momentum k ¼ ðp; 0Þ in a cluster of N ¼ 32 sites. For each value of frustration J3 =J1 , instead of using the magnetic structure that results from the minimum of Eclas we have used the magnetic wave vector Q corresponding to the maximum of the exact magnetic structure factor of the cluster [8]. Surprisingly, our approximated SCBA spectra agree very well for the whole range of energy and frustration with the unbiased exact ones obtained by Poilblanc et al. (see Fig. 5 of Ref. [12]). In particular, at the low energy sector it can be observed that the suppression of the quasiparticle weight z as the transition from Ne´el to spiral fluctuations takes place. For the pure Heisenberg case it has been argued that a plaquette order should be stable for the range 0:33  J 3 =J 1  0:63 [9]. However, in an N ¼ 32 cluster size it has also been observed that spiral fluctuations compete with the plaquette ones at around J3 =J1 ¼ 0:5 [8]. In the light of these results the reduction of the quasiparticle weight can be traced back to the coupling of the vacancy to the spiral fluctuations which in our approximation are represented by magnons of S ¼ 1. It should be pointed out, however, that our results do not preclude the existence of spinons even in finite systems. In fact there exists the possibility that spinons are present in the finite system but they do not couple to the vacancy. So, from our results we can only conclude that the reduction of the quasiparticle weight is not a reliable observable to detect a fractionalization scenario since it does not imply necessary the existence of free spinons in doped finite systems as was recently argued in the literature [12,13]. In Fig. 3 we show the dependence of zav with frustration extrapolated to the thermodynamic limit, where zav is the Brillouin zone averaged quasiparticle weight. The computation has been performed in the weak coupling regime, J 1 =t ¼ 10, in order to smoothly connect to a static vacancy case, as much as possible in our approximation. As J 3 =J 1 increases there is an

J3/J1 = 0.05

1.5

J3/J1 = 0.5 J3/J1 = 1 1 A (k. ω)

Disorder

2859

0.5

0 -2

0

2

4

ω/t Fig. 2. Vacancy spectral function Aðk; oÞ for k ¼ ðp; 0Þ and t=J1 ¼ 2:5 predicted by the SCBA in a cluster size of N ¼ 32 sites for different values of frustration. To be consistent with the magnetic background of the 32 cluster we have selected the magnetic structure Q according to the maximum of the exact structure factor taken from Ref. [8]. These spectra are in excellent agreement with Fig. 5 of Ref. [12].

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I.J. Hamad et al. / Physica B 404 (2009) 2858–2860

ample range of the Brillouin zone the vacancy spectral functions are completely incoherent, thus leaving place to nonconventional scenarios. In particular, the vanishing of the quasiparticle weight has been related to a kind of spin orthogonality catastrophe which is thought to occur in the vicinity of a quantum critical point [16]. If we would like to extend our theory to the proximity of a quantum critical point besides transverse spin wave fluctuations new magnetic degrees of freedom, such as longitudinal fluctuations, should be included. Work in this direction is in progress.

1

ZAV

0.9

0.8

0.7 We acknowledge useful discussions with C.J. Gazza and A. Dobry. This work was supported by the ANPCYT under PICT2005 no. 38045. I.J.H. thank Fundacio´n J. Prats for partial support.

0.6 0

0.25

0.5

0.75 J3/J1

1

1.25

1.5

Fig. 3. Dependence of the averaged quasiparticle weight zav with frustration predicted by the SCBA in the thermodynamic limit for the weak coupling regime J 1 =t ¼ 10. The region between dashed lines represents the disordered regime predicted by spin wave theory for the J1 2J 3 model. On the left of this region the ground state is Ne´el ordered while on the right the ground state is spiral ordered.

important reduction of weight in the regime where spin wave theory predicts a disordered ground state (between dashed lines around J 3 =J 1 ¼ 0:25 of Fig. 3). In this region we have introduced a gapped magnetic dispersion in order to take into account the assumed short range magnetic order. Such a weight reduction can be attributed to the interaction of the vacancy with the proliferating magnons, but it can be observed that the reduction of zav is even stronger when long range spiral order is induced by frustration (see the minimum at J3 =J1 0:4 in Fig. 3). Actually, for the strong coupling regime t=J 1 ¼ 2:5, we have found that for an

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[16]

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