Novel ground states in an antiferromagnetic double-exchange model on a triangular lattice

Science and Technology of Advanced Materials 7 (2006) 26–30 www.elsevier.com/locate/stam

Novel ground states in an antiferromagnetic double-exchange model on a triangular lattice Yoshihiro Shimomura, Shin Miyahara *, Nobuo Furukawa Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara 229-8558, Japan Received 10 September 2005; received in revised form 25 November 2005; accepted 25 November 2005 Available online 3 February 2006

Abstract As a typical example of frustrated itinerant electron systems, ground states of an antiferromagnetic double-exchange model on a triangular lattice have been investigated. In the model, an electron transfer depends on pseudo-spins configurations: a magnitude of the hopping amplitude t for an antiparallel configuration is bigger than that of t 0 for a parallel configuration. Thus, there is an effective antiferromagnetic interaction between pseudo-spins, which leads frustration on geometrically frustrated lattices. The energy depends on not only the nearest-neighbor bonds structure but also the global local spin structures, since the interactions are caused by the kinetics of electrons. Therefore, in the ground state, the degeneracy expected in the antiferromagnetic Ising model is lifted and a novel state is stabilized. For t 0 =t( 0:15, a non-degenerate state with a ~ commensurate periodic structure is a ground state. On the other hand, a highly degenerate antiferromagnetic chain stacked state appears for 0 t =tT 0:15. q 2006 Elsevier Ltd. All rights reserved. Keywords: Double-exchange model; Triangular lattice; Frustration; Monte Carlo simulation; Antiferromagnetic chain stacked state

1. Introduction The effects of geometrical frustration suppress simple classical order states and often produce novel ground states. Such an effect has been studied well, especially in spin systems, and it is well known that geometrical frustration induces unusual ground states, e.g. macroscopically degenerate states on antiferromagnetic Ising models [1–4], and spin gapped singlet states in spin-1/2 Heisenberg systems [5–7]. Recently, the authors have found a dodecamer ordered ground state in a double exchange spin ice model on a kagome´ lattice as a first example of a spin cluster ordered state in the frustrated electron systems [8,9]. In the model, pseudo-spins are coupled to electron transfer integrals in such a way that antiparallel configurations of pseudo-spins gain kinetic energies, which leads an effective antiferromagnetic interactions between pseudo-spins. Since, the effective interactions originate from the electron hoppings, the feature of the interactions is qualitatively different from that in the Ising model. In the Ising model on a kagome´ lattice, the ground state * Corresponding author. Tel./fax: C81 42 759 6288. E-mail address: [email protected] (S. Miyahara).

1468-6996/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.stam.2005.11.014

is macroscopically degenerate [2,3]. On the other hand, in the electron model, the degeneracy is lifted and the cluster ordered state is stabilized due to the kinetics of electrons. In this way, the realization of non-trivial ground state is expected in such frustrated electron systems. In the double exchange spin ice model on a kagome´ lattice, effective antiferromagnetic interactions can be realized due to the realization of the spin ice mechanism. In the model, a uniaxial anisotropy forces the localized spin to point inward or outward for a triangle in a unit cell within the kagome´ plane. An inward spin is energetically preferable next to an outward spin on each triangle due to the double exchange mechanism. Once we consider pseudo spins t, which are defined as tZ1 (K1) for the inward (outward) spins, the effective ferromagnetic interactions between the localized spins are regarded as the effective antiferromagnetic interactions between the pseudo spins. However, the inward and outward spins can be defined only on the special lattices, e.g. kagome´ and pyrochlore lattices. To investigate non-trivial ground states in the frustrated itinerant electron systems on a general lattice [10], we defined an antiferromagnetic double-exchange model as a generalized extension of the double exchange spin ice model. The Hamiltonian is written as X X † H ZK tðti ;tj Þðc†i cj C hcÞKm ci ci ; hi;ji

i

(1)

Y. Shimomura et al. / Science and Technology of Advanced Materials 7 (2006) 26–30

where

(

tðti ;tj Þ Z

t

ðti ZKtj Þ;

t0

ðti Z tj Þ:

(2)

Here t and t 0 are strong and weak transfer integrals, respectively. It is obvious that, on a geometrically frustrated lattice, the model has frustration. Note that the doubleexchange spin ice model is equivalent to the antiferromagnetic pffiffiffi double-exchange model on a kagome´ lattice with t 0 =tZ 1= 3. In this paper, we treat the model on a triangular lattice for an arbitrary value of t 0 /t assuming the condition mZ0 as a typical example of a frustrated lattice and see the nature of the ground states. 2. Ground state It is well known that a ground state of the Ising model on a triangular lattice is macroscopically degenerate. In the ground state, the number of bonds which consist of antiferromagnetically coupled spins is maximized. Even in the the ground state of the antiferromagnetic double-exchange model, it is expected that the number of such bonds is maximized. This indicates that the spin configuration for the ground state of the Ising model may be a good candidate for that of the Hamiltonian (1). However, in the antiferromagnetic double-exchange model, the degeneracy observed in the Ising model should be lifted, because the energy depends on not only the nearest-neighbor bond structure but also on the global local spin structure. In this section, we start from reviewing the ground states of the Ising model on a triangular lattice [1]. After that, we discuss the nature of the ground states of the antiferromagnetic doubleexchange model. For t 0 =t( 0:15, a non-degenerate commensurate state is a ground state, and for t 0 =tT 0:15, an antiferromagnetic chain stacked state appears. In Section 2.3, we prove that the antiferromagnetic chain stacked state is highly degenerate. 2.1. Ground state of Ising model on triangular lattice

27

chain stacked state (Fig. 1) and the other is a three sublattice state (Fig. 2). In the antiferromagnetic chain stacked state, the state is constructed by stacking antiferromagnetic chains as shown in Fig. 1, where arrows indicate antiferromagnetic chains. It is obvious that flipping an antiferromagnetic chain does not change the number of bonds with antiferromagnetic spin configurations. Thus, states made by flipping some antiferromagnetic chains are also ground state. A three sublattice state consists of up-spin, down-spin and A sublattices, where the direction of the spins on A sublattice is arbitrary. Because the spins on A sublattice always have three up-spins and three down-spins at nearest neighbor sites, the direction of spins on A sublattices does not affect the energy of the system. Moreover, there are contingent freedoms. If three neighboring ringed spins on A sublattices happen to have equal spin, their center can be chosen at random. The examples of the contingent spin are indicated by the dashed triangle in Fig. 3(a). In this way, the three sublattice states are macroscopically degenerate states, which lead to finite entropy. 2.2. Ground state of antiferromagnetic double-exchange model We have performed Monte Carlo calculations to find a pseudo-spin configuration for the ground state of the antiferromagnetic double-exchange model on the triangular lattice. In each Monte Carlo step, the kinetic energy of electrons has been calculated using the twisted boundary conditions on a finite supercell of local spins. Here we have averaged over phases to extrapolate the energy in the thermodynamic limit. Monte Carlo calculations have been done for t 0 /tZ0, 0.2, 0.4 and 0.6 within 6!6 supercells. For t 0 /tZ0, the spin configuration of the ground state is a contingent state as shown in Fig. 3(a), and for t 0 /tZ0.2, 0.4 and 0.6, the ground state spin structures are antiferromagnetic chain stacked states (Fig. 3(b)). Both spin configurations are in the degenerate ground states of the Ising model. In addition, the spin configurations of the metastable states shown in Fig. 4, which appear in the Monte Carlo calculation, are also the

We can classify the ground states of the Ising model on the triangular lattice into two types: one is an antiferromagnetic

A A

A

A

t t’

AF-chain

A A

A

A

A

A A

A A

A

x

A

A

y

A

A

A A

a Fig. 1. One of the antiferromagnetic chain stacked states. Here antiferromagnetic chains are indicated by arrows. Open (closed) circles represent up(down)spins. Solid (dashed) lines are the bonds with antiferromagnetic (ferromagnetic) spin configurations.

Fig. 2. Three sublattice structure, which consists of up-spin (open circle), down-spin (closed circle) and A sublattices. Here the direction of the spins on A sublattices is arbitrary, since such spins always have three up-spins and three down-spins at nearest neighbor sites.

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Y. Shimomura et al. / Science and Technology of Advanced Materials 7 (2006) 26–30

(a) contingent state

(a) metastable state at t'=0

(b) AF chain stacked state

(b) metastable state at t'=0.2

Fig. 3. The ground state spin configurations calculated by Monte Carlo calculations on 6!6 supercells. (a) A ground state for t 0 /tZ0, which is a contingent state of a three sublattice state. A sublattice is shown by dashed circle and a contingent spin is a center of the dashed triangle. (b) Antiferromagnetic chain stacked state, which is ground states for t 0 /tZ0, 0.2, 0.4 and 0.6.

ground states of the Ising model. Thus, we conclude that in the antiferromagnetic double-exchange model, the degeneracy of the Ising model is lifted due to the kinetics of electrons. Using the six spin configurations shown in Figs. 3 and 4, the electron energy on each superstructure has been calculated for various values of hopping amplitudes of t 0 /t. The energy differences DExa between the antiferromagnetic chain stacked state and the other states are shown in Fig. 5. The energy differences are defined as DExa ðt 0 =tÞ Z Exa ðt 0 =tÞKE0 ðt 0 =tÞ;

(c) metastable state at t'=0.4

(3)

where E0(t 0 /t) is an energy on the antiferromagnetic chain stacked state and Exa(t 0 /t) is an energy on the local spin configurations in Fig. x (a). For t 0 =tT 0:15, the lowest energy state among the calculated states is an antiferromagnetic chain stacked state. On the other hand, for small t 0 /t, the lowest energy state is a contingent state shown in Fig. 3(a). The density of states (DOS) of the ground state for t 0 /tZ0 and 0 t /tZ0.6 are shown in Fig. 6(a) and (b). As shown in Fig. 6(b), antiferromagnetic chain stacked states has a metallic band structure. On the other hand, the ground state for t 0 /tZ0 is a zero-gap insulator. Here, the zero-gap originates from the periodic spin structure. Such a ground state is non-degenerate. However, the lowest energy configuration strongly depends on the parameters m and the size of supercell, since it is stabilized by opening the band gap with commensurate periodic structure. Because of such a feature, we cannot neglect the possibility of the existence of the spin structure, which realizes the lower energy

(d) metastable state at t'=0.6 Fig. 4. The metastable configurations obtained by Monte Carlo calculations on 6!6 supercells. A sublattices in a three sublattice structure are shown by dashed circles. (a) The metastable state for t 0 /tZ0, which is a three-sublattice state. (b) The metastable state for t 0 /tZ0.2, which is one of contingent states. (c) The metastable state for t 0 /tZ0.4, which is one of contingent states. (d) The metastable state for t 0 /tZ0.6, which is one of contingent states.

within larger supercells. In fact, in Ref. [10], the spin structure shown in Fig. 4(a) was obtained as a ground state for t 0 /tZ0 based on the calculation on the smaller supercells and 6!6 three sublattice supercells except for contingent states. In this way, we cannot determine the configuration of the local spin precisely, but it is likely that the ground state for t 0 =t( 0:15 is commensurate and non-degenerate. On the other hands, in the DOS of the

Y. Shimomura et al. / Science and Technology of Advanced Materials 7 (2006) 26–30

0.003

29

(a)

DOS

0.002

∆E

0.001

0

-0.001

-4

-2

0 E

2

4

6

-6

-4

-2

0 E

2

4

6

(b)

-0.002

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t'/t

1

DOS

-0.003

-6

Fig. 5. The t 0 /t dependence of kinetic energy differences between the energy of the antiferromagnetic chain stacked state and the other five states in Figs. 3 and 4. DExa indicates the difference between the kinetic energy for the antiferromagnetic chain stacked spin configurations and that for the configurations shown in Fig. x (a).

antiferromagnetic chain stacked state, there is no gap and the ground state is metallic. For larger value of t 0 /t, electrons gain kinetic energy on both t and t 0 bonds. In such a case, the regular spin configuration is favorable, and, as a result, the antiferromagnetic chain stacked state is a ground state. Thus, the stability of such a state hardly depends on the size pffiffiffi of supercells and chemical potential m. In fact, for t 0 =tZ 1= 3, the antiferromagnetic chain stacked state is the ground state for wide range of chemical potentials around mZ0. In the antiferromagnetic chain stacked states, there is a freedom how to stack antiferromagnetic chains. However, all antiferromagnetic chain stacked states in the antiferromagnetic double-exchange model are degenerate as well as the Ising models, which will be proved in Section 2.3.

Fig. 6. (a) DOS of the ground state at t 0 /tZ0 (Fig. 3(a)). (b) DOS of the antiferromagnetic chain stacked state at t 0 /tZ0.6 (Fig. 3(b)).

where tyi ðkx Þ Z

(

t0 ðkx ÞeifCðkx Þ Z teiðkx =2Þ C t 0 eKiðkx =2Þ t0 ðkx ÞeifKðkx Þ Z teKiðkx =2Þ C t 0 eiðkx =2Þ

1 X cxi ;yi Z pffiffiffi ck ;y eKikx xi ; L kx x i the Hamiltonian (1) can be written as, o Xn tyi ðkx Þðc†kx ;yiC1 ckx ;yi C hcÞ C 3kx c†kx ;yi ckx ;yi H ZK kx ;yi

(4)

(5)

(6)

and 3kx Z 2t cos kx :

(7)

Using the Unitary transformation, 0 1 Y Kif ðk Þ e yj x Ackx ;yi c~kx ;yi Z Uckx ;yi Z @

(8)

yj!yi

2.3. Highly degenerate antiferromagnetic spin stacked state In the antiferromagnetic double-exchange model, most of the degeneracy in the Ising model is lifted. However, the antiferromagnetic chain stacked states are still degenerate. This is obvious in the case t 0 Z0, since all states are topologically equivalent to the square lattice as long as the electron hoppings t are concerned. Even for t 0 s0, all states are still degenerate, though the hopping configurations are not topologically equivalent. It is proved in the following way. Let us consider the case that antiferromagnetic chains are stacked to y direction as shown in Fig. 1. Using the Fourier transformation along x-direction,

;

the Hamiltonian is rewritten as o Xn U † HU ZK t0 ðkx Þðc~†kx ;yiC1 c~kx ;yi C hcÞ C 3kx c~†kx ;yi c~kx ;yi ; kx ;yi

(9) which is equivalent to a free fermion system. Thus, the nature of the system does not depend on the spin configurations for the antiferromagnetic chain stacked states, and the ground states for t 0 =tT 0:15 are highly degenerate. 3. Conclusion In the antiferromagnetic double-exchange model, nondegenerate ground state is realized for t 0 =t( 0:15 and a highly degenerate antiferromagnetic chain stacked state is stable for t 0 =tT 0:15. In the latter state, the entropy due to the degeneracy is pffiffiffiffi proportional to N , and, therefore, in the thermodynamic limit, the residual entropy per site vanishes. This may indicate the possibility of the existence of the liquid like state of the

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antiferromagnetic chains. Although the antiferromagnetic state is not macroscopic degenerate state, it has still high degeneracy. It is expected that such a high degeneracy leads a novel behavior of the electrons, e.g. in the transport of electrons. To clarify the electron behaviors in the highly degenerate ground state is strongly desired, and it might delight the observation of the novel phenomena in frustrated electron systems. In the frustrated electron systems, the kinetics of electrons lifts the degeneracy in the Ising model, and leads to novel ground states, i.e. highly degenerate state and cluster ordered state. Acknowledgements This work was partially supported by a Grant-in-Aid for 21st COE program from the Ministry of Education, Culture, Sports, Science and Technology. References [1] G.H. Wannier, Antiferromagnetism. The triangular Ising net, Phys. Rev. 79 (1950) 357–364.

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