Phase transitions and order-by-quantum disorder for the antiferromagnetic XY model in the checkerboard lattice

Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

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Phase transitions and order-by-quantum disorder for the antiferromagnetic XY model in the checkerboard lattice A.R. Moura Departamento de Física, Universidade Federal de Viçosa, 36570-900, Viçosa, Minas Gerais, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 30 April 2015 Received in revised form 7 June 2015 Accepted 15 June 2015 Available online 18 June 2015

In this work we have studied the phases of the XY antiferromagnetic model in the checkerboard lattice (the two-dimensional analog of the pyrochlore lattice). Using the traditional linear spin–wave approximation, we obtain a transition from an order to a disordered phase in some critical value rc = J ′ /J . The phase transition occurs due to quantum fluctuations. However, when we consider higher order perturbations the scenario is entirely different. We have applied the Self-Consistent Harmonic Approximation method and the results show that quantum perturbations induce long-range spin–spin correlations even above the critical point rc. This is a typical feature of order-by-quantum disorder when the system chooses one of the many classical ground states to occupy. We have also determined the thermal phase transitions similar to the Berezinskii–Kosterlitz–Thouless phase transition and the action of a single-ion anisotropy, responsible for a Quantum Phase Transition at zero temperature. & 2015 Elsevier B.V. All rights reserved.

PACS: 73.43.Nq 75.30.Ds 74.62.En 75.10.Kt 75.50.Ee Keywords: Checkerboard lattice Antiferromagnetic XY model Self-Consistent Harmonic Approximation Order-by-disorder

1. Introduction Frustrated models have received a lot of attention recently. There are still some objections but it is believed that frustration can be the key to obtain disorder in high dimensions. In one-dimensional space, the disordered phase is well documented. The Heisenberg antiferromagnetic chains (HAFC), for instance, are known to present the quantum spin liquid phase, characterized by the absence of any kind of order even at zero temperature. Such chains have a correlation function with power-law decay if the spin is half-odd-integer while for integer spin the correlation function shows an exponential decay due to the famous Haldane gap [1–3]. Since in antiferromagnetic frustrated models it is impossible to align anti-parallel neighbors’ spins at the same time, the ground state should be classically disordered at zero temperature. Despite this, thermo-fluctuations, and occasionally quantum perturbations, raise the degeneracy. Therefore, the system chooses one configuration as a ground state, the so-called order-by-disorder (or order-by-quantum disorder when the transition is caused by quantum fluctuations). The pyrochlore lattice is the most studied lattice when we are interested in frustrated three-dimensional models. The lattice is http://dx.doi.org/10.1016/j.jmmm.2015.06.027 0304-8853/& 2015 Elsevier B.V. All rights reserved.

composed by corner sharing tetrahedra with ions in the tetrahedron corners. Materials as Ho2Ti2O7 and Dy2Ti2O7 are Heisenberg antiferromagnets that show mainly the effects of frustration besides the exotic magnetic monopoles [4,5]. Already the Er2Ti2O7 (the most studied XY antiferromagnet pyrochlore) is one of the rare compounds which has been experimentally proved to be the order-by-disorder phenomenon [6–9]. Actually, there are many experimental and computational data but theoretical studies are relatively less due to the complexity of the model. Models in lower dimensions have the advantage of the simplicity while keeping mainly the physical aspects of the theory. In this work we have studied the two-dimensional version of the antiferromagnetic XY pyrochlore lattice. The checkerboard lattice can be obtained as the planar projection of the pyrochlore lattice and it is composed by a square lattice with diagonal crossing in half of the sites. The lattice can be represented by two interconnected sublattices as shown in Fig. 1. Similar to the threedimensional pyrochlore model, an ordered phase also emerges when we consider higher order quantum perturbations. Canals and Zhitomirsky [10] have determined the order-by-disorder phenomenon in the presence of a magnetic field for the XY checkerboard antiferromagnetic. Here we are interested in other

A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

61

integer spin). Already for 0 < J ′ ≤ J , Canals found an ordered ground state (at least for integer spin) if quantum fluctuations beyond the first order are considered [13]. Curiously, if only first order spin–wave approximation is taken into account, above a critical ratio rc = J ′ /J the system does not show magnetic ordering. In this case it is important to determine whether the disordered phase is a consequence of lattice frustration (due to the lattice geometry) or quantum fluctuations over the ground state. A similar behavior is observed in the kagomé lattice where nonmagnetic phase prevails above rc ≈ 0.9. The size system has also a

Fig. 1. The checkerboard lattice is composed of two interconnected square lattices, each one represented by a different color (only in the digital version). The solid line represents the J interaction while the dotted line is the J′ interaction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

thermodynamics properties and phase transitions besides the possible order-by-quantum disorder. Considering different square and crossing interactions, the Hamiltonian is given by

H=

J 2

∑ (Six S jx + Siy S jy ) + 〈i, j〉

J′ ∑ (Six S jx + Siy S jy ), 2 〈〈i, j〉〉

(1)

where the first sum is over nearest neighbors and the second one is over half of the next-nearest neighbors. The 1/2 factor is included to avoid a double counting in spin interactions. In the limit

J = J ′, the model may be solved as a sum of squares (indicated by the ⋄ symbol) in a square lattice and the Hamiltonian is written as

H=

J 2

⎡⎛

⎞2

⎛ ⎞2 + ⎜⎜∑ Siy ⎟⎟ + ⎠ ⎝i ∈⋄ ⎠

∑ ⎢⎢ ⎜⎜∑ Six ⎟⎟ ⋄

⎣ ⎝i ∈⋄

⎤

∑ (Siz )2⎥⎥ − 2NJS (S + 1). i ∈⋄

⎦

decisive factor. For J ′ < J , finite models present singlets as ground state configurations while in the thermodynamic limit we obtain a Néel state [14]. Here we shall consider only infinite systems and, in many situations the classical ground state is disordered but an ordered phase arises when we consider thermo as well as high order quantum perturbation. Our objective is to investigate the ground state and lower excitations at zero and finite temperature as well as the phase transitions. We have adopted the interval 0 ≤ J ′ ≤ J and consider perturbations of order higher than one. We have showed that, similar to the Heisenberg model, the XY model presents a critical ratio rc which separates a magnetic from a non-magnetic phase when we consider a linear spin–wave approximation. Otherwise, using a Self-consistent Harmonic Approximation (SCHA) the magnetic ordering extends over the entire range 0 ≤ rc ≤ 1 at zero temperature. In the SCHA, higher order perturbations are considered by a temperature dependent renormalized parameter. Therefore, it is natural to consider that the magnetic ordering is caused by spin–wave quantum interactions, neglected in the linear spin–wave approximation. We have also studied the thermodynamic of the model and determined the Berezinskii–Kosterlitz– Thouless (BKT) transition as a function of rc. Finally, we have considered the inclusion of a single-ion anisotropy responsible for a known Quantum Phase Transition (QPT).

2. Spin–wave approximation

(2)

The classical ground state is obtained imposing the constraint

( ∑i∈⋄ Six )2 = ( ∑i∈⋄ Siy )2 = ∑i∈⋄ (Siz )2 = 0. Obviously there are many possible configurations obeying the constraint and this implies an energy spectrum with a flat region in the momentum space besides a residual finite entropy at zero temperature. Locally, the checkerboard lattice in the isotropic limit ( J = J ′) has a structure similar to the pyrochlore and both lattices share many features. For

J ′ = 0, we recovery the traditional XY model in a square lattice whose properties are well known. At zero temperature, the XY model shows magnetic ordering and the lower excitations are collective modes of spin-0, the magnons (Goldstone modes associated to the spontaneous symmetry-breaking as dictated by the Mermin–Wagner theorem). Furthermore, at a finite temperature TBKT, a thermal phase transition occurs due to the emergence of magnetic vortices. Already in the other limit J ¼0, we have a set of uncoupled XY antiferromagnetic chains disordered at zero temperature with spinons (collective modes of spin-1/2) as the lower spin excitations. The exact nature of the checkerboard lattice is surrounded by uncertainty. In the limit J ′⪡J , Singh et al. [11] have used a spin– wave theory to demonstrate the possibility of a spin liquid state in the checkerboard lattice. Yet considering weakly uncoupled chains, Oleg et al. [12] have obtained a nondimerized spin liquid ground state for the Heisenberg antiferromagnetic model (for half-

In this section we have adopted a linear spin–wave approximation to determine the ordering of the ground state as a function of the spin S. The representation of the spin operators is done by the traditional Holstein–Primakoff bosons [15] considering only first order terms. How we will show in the next sections, the exclusion of spin–wave interactions has a dramatic consequence and distinct results, when considering higher order perturbations, are obtained by the implementation of the SCHA. We have assumed a Néel order as the ground state for 0 ≤ J ′ ≤ J since it is true for J ′⪡J . Gomez-Santos and Joannopoulos [16] suggest to consider the quantized axis in the plane, instead of z-axis, to guarantee coherent results. We have adopted the z-axis as the quantized one with the additional introduction of a constraint ∑i ∈ A Siz + ∑ j ∈ B S jz = 0 in order to keep a zero staggered magnetization [17]. The checkerboard lattice has been treated as two coupled sublattices (A and B) with two spins (SA and SB) for each cell. We have used the traditional linear Holstein–Primakoff representation: Si+A = 2S ai , Si−A = 2S ai† and SizA = S − ai† ai for the sublattice A (after a rotation, we obtain a similar representation for the sublattice B). Through a Bogoliubov transform in the momentum space, the diagonal Hamiltonian is given by

H = Eg +

q

where

⎛

⎞

⎝

⎠

∑ ⎜⎜ωqα αq† αq + ωqβ βq† βq ⎟⎟, the

ground

(1/2) ∑q (ωqα + ωqβ )

state and

the

(3) energy

E g = − Nλ (2S + 1)+

spin–wave

energies

are

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A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

ωqα =

1 [ω q 2

+ 2J ′ S (γqA − γqB )] and ωqβ =

1 [ω q 2

− 2J ′ S (γqA − γqB )] . For

J ′ < J , both spin–wave energies are gapless and increase linearly close to q¼ 0. The condition:

1 N

∑

2J ′ S (γqA + γqB ) + 2λ ωq

q

= (2S + 1), (4)

fixes the parameter λ = 4JS − 2J ′ S . For comparison, if J ′ = 0 we obtain E g /N = − 0.63J (for S = 1/2), slightly smaller than that obtained from Ref. [16] which provides E g /N = − 0.54J . The magnetization is given by

M=S+

1 1 − 2 8π 2

∫zB

2J ′ S (γqA + γqB ) + 2λ ωq

d2q,

Since a linear approximation is insufficient to entirely describe the scenario, we have used the SCHA [18–25] to determine the phases and thermodynamic of the antiferromagnetic XY model in the checkerboard lattice. Unlike the results of previous section, the SCHA provides an ordered phase at zero temperature for 0 ≤ r ≤ 1. There is no disordered phase above a critical point rc. The main idea of the SCHA is to replace the Hamiltonian (1) by a quadratic one with temperature dependent renormalized parameters that implement higher order contributions. We have used the Villain spin representation:

Si

A−

=

=

⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎜S + ⎟ − ⎜SizA + ⎟ ⎝ ⎝ 2⎠ 2⎠

⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎜S + ⎟ − ⎜SizA + ⎟ e−iϕiA ⎝ ⎝ 2⎠ 2⎠

(6a)

Si+ B = − eiϕiB

Si− B = −

⎞2

⎛ ⎛ 1 1 ⎜S + ⎟ − ⎜SizB + ⎟ e−iϕiB ⎝ ⎝ 2⎠ 2⎠

i

j

⎧ ⎪

˜2 ⎡

⎪ ⎩

4

∑ ⎨− J ′ S ρ′ ⎢ (ϕi A − ϕ jA ) 〈i, j〉

+

⎛ ⎞ ⎜S zA S zA + S zB S zB ⎟ i i ⎟ ⎜ i i ⎝ ⎠ 2

⎢⎣

⎤ 2 + ϕi B − ϕ jB ⎥ ⎥⎦

(

)

⎫ 2 2⎪ JS˜ ρ ϕi A − ϕ jB ⎬ , ⎪ 2 ⎭

(

)

(8)

where S˜ = S (S + 1) and the sums are over unit cells with two spins. The spin stiffness parameters ρ and ρ′ are obtained by the Bogoliubov variational principle. If we define the Helmholtz free energy F˜ = F0 + 〈H − H0 〉0 , the variational principle imposes F ≤ F˜ , where the mean-values are calculated using the quadratic Hamiltonian. Both parameters ρ and ρ′ take into account the contributions of anharmonic terms in the harmonic approximation. As the fields ϕi and Siz obey the fundamental Poisson bracket {ϕi , Siz } = δ ij for planar representation of a spin vector, the operators ϕi and Siz are canonically conjugate, i.e. [ϕi , S jz ] = iδ ij (for spins of the same sublattice). For spins in different sublattices all operators commutate. After a Fourier transform, we obtain

H0 =

⎡ T →

→z T

→

∑ ⎢⎢ϕq / qϕϕq + Sq q

⎣

→z ⎤ / qz Sq ⎥, ⎥⎦

(9) → ϕq = (ϕqA , ϕqB )T

where we have defined the vectors →z S q = (SqzA, SqzB )T . Already the matrices are given by

⎡A C ⎤ q q ⎥ / qϕ = ⎢ ⎢ Cq B q ⎥ ⎣ ⎦

and

⎡ 2J − J ′ 0 / qz = ⎢ ⎢ 2J − ⎣ 0

2 2 Aq = 2JS˜ ρ − J ′ S˜ ρ′(1 − γqA ),

with

⎤ ⎥, ⎥ J ′⎦

and

(10)

2 2 Bq = 2JS˜ ρ − J ′ S˜ ρ′(1 − γqB ),

˜2

Cq = − 2JS ργq . Since [/ ϕ, / z] = 0, we can diagonalize simultaneously both matrices defining new operators φq(A,B) and : (qA,B) through the Bogoliubov transform:

→ ϕq = Mq → φq,

→ → Sq = Mq η :q

(11)

where

⎡ ξq − ζq ⎤ ⎥ Mq = ⎢ ⎢⎣ iζq − iξq ⎥⎦

and

⎡ ⎤ 1 0 ⎥ η=⎢ . ⎢⎣0 − 1⎥⎦

(12)

The constraint |Mq | = i is imposed in order to keep the commutation relation [φq , : qz ] = i . After a straightforward work, we found

(6b)

for sublattice A and

⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎜S + ⎟ − ⎜SizB + ⎟ ⎝ ⎠ ⎝ 2 2⎠

⎛ ⎞ H0 = ⎜⎜2J − J ′ ⎟⎟ ∑ ⎝ ⎠ i

(5)

3. Self-consistent harmonic approximation

ei ϕ iA

i

quadratic Hamiltonian then is given by

+

when written in the continuous limit. We define the critical point rc as the point where the magnetization vanishes driven by quantum fluctuations. For spin-1/2 we have found rc ¼ 0.76 while for S¼ 1 the system does not show magnetic ordering for values bigger than rc ¼ 0.94. Above the critical point (up to r ¼1) the magnons still have a dispersive spectrum and the lack of an ordered state is due to emergence of spin–waves, according to the first order approximation. The situation is different when r ¼1. In this case, the lowest energy level is a flat region over the whole Brillouin zone. The absence of a minimal point in the momentum space is a typical characteristic of a system with zero spin stiffness. Therefore, there is no long-range order and the system is disordered by geometric frustration, as it is the case for the kagomé lattice. Obviously, the inclusion of interacting spin–waves could raise this flat region and revert the scenario to dispersive magnons again. However, instead of considering quartic order in Holstein– Primakoff representation, we have adopted a Self-consistent Harmonic Approximation in order to study beyond the critical point rc. The results of the XY model are comparable with those of the Heisenberg model obtained by Canals using the Dyson–Maleev bosons.

Si+ A

for sublattice B. A minus signal appears in sublattice B operators due to the rotation of the angle ϕB by π. The mean-field 〈ϕ〉 (for any sublattice) is not well defined to angles measured relative to a fixed axis. Therefore, in order to avoid divergences the quantized direction is choosen relative to the instantaneous magnetization. Considering a smooth spin field, we assume (ϕi − ϕ j )⪡1 and expand the Hamiltonian (1) in powers of (S z /S˜ )2 and (ϕ − ϕ )2. The

(7a)

⎞2

(7b)

ξq2 =

⎤ 1 ⎡ A q − Bq ⎢ + 1⎥ ⎥⎦ 2 ⎢⎣ ωq

(13a)

ζq2 =

⎤ 1 ⎡ A q − Bq ⎢ − 1⎥ , ⎢ ⎥⎦ 2 ⎣ ωq

(13b)

where ωq = (Aq − Bq )2 + (2Cq )2 . It is appropriated to introduce new bosonic operators αq and βq given by

A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

⎡ ⎤1/4 1 ⎢ 2 (2J − J ′ ) ⎥ (αq† + α−q ) 2 ⎢⎣ A q + Bq + ωq ⎥⎦

φ−αq =

(14a)

⎡ ⎤1/4 i ⎢ A q + Bq + ω q ⎥ (αq† − α−q ) 2 ⎢⎣ 2 (2J − J ′ ) ⎥⎦

: −zαq =

1 = 2

(14b)

(14c)

⎡ A q + Bq − ωq ⎤1/4 ⎢ ⎥ (β † − β−q ). q ⎢⎣ 2 (2J − J ′ ) ⎥⎦

∑ q

(ϕ =

A i

(17)

0

⎡

2J − J ′ (2π )2

⎛

∫BZ ⎢⎢ ω1α ⎜⎜1 − ⎣

q

⎝

⎛ βω α ⎞ 2γq Cq ⎞ ⎟ coth ⎜ q ⎟ ⎟ ⎜ 2 ⎟ ωq ⎠ ⎝ ⎠

⎛ ⎛ βω β ⎞ ⎤ 2γq Cq ⎞ 1 ⎜ ⎟ coth ⎜ q ⎟ ⎥ d2q 1 + ⎜ ⎟ ⎜ 2 ⎟⎥ ωq ⎠ ωqβ ⎝ ⎝ ⎠⎦

(18)

and

(ϕ (15) =

with ωqα = 2 (2J − J′ )(Aq + Bq + ωq ) and ωqβ = 2 (2J − J′ )(Aq + Bq − ωq ) being the spin–wave energies for the two modes α and β, showed 2 in Fig. 2. The α spin–wave energy has a gap Δ α = 4S˜ Jρ (2J − J ′)

A j

− ϕi A

2J − J ′ (2π )2

thereby ρ and ρ′ are written as the self-consistent equations:

⎛ Siz ⎞2 ⎜ ˜⎟ ⎝S⎠

⎤ ⎥ −(1/2)〈(ϕ jA − ϕiB )2〉0 ⎥e 0⎦

(16a)

⎛ Siz ⎞2 ⎜ ˜⎟ ⎝S⎠

⎤ ⎥ −(1/2)〈(ϕ jA − ϕiA )2〉0 , ⎥e 0⎦

(16b)

and

where the mean-values 〈⋯〉0 are determined using the Hamiltonian (15). In the continuous limit, the mean-value equations are given by

2

)

0

⎡ (1 − γ A ) ⎛ ⎛ βω α ⎞ A q − Bq ⎞⎟ q ⎜ q⎟ 1+ coth ⎜⎜ α ⎜ ⎟ ⎟ 2 ω ω q q ⎝ ⎠ ⎝ ⎠ ⎣

∫BZ ⎢⎢

⎞ ⎛ β ⎞⎤ (1 − γqA ) ⎛ ⎜1 − A q − Bq ⎟ coth ⎜ βωq ⎟ ⎥ d2q. ⎜ 2 ⎟⎥ ωq ⎟⎠ ωqβ ⎜⎝ ⎝ ⎠⎦

+

q=0

(between 3.736J /K B when r¼ 1 and 5.474J /K B for r ¼0, considering S ¼1) while the β mode is gapless which corresponds to the Goldstone modes. Consequently, at finite low temperatures only the β mode is accessible for the lower energy spin excitations. Using the SCHA we do not have found any flat region, as occurs in the linear spin–wave approximation described in previous section, and the magnons are always dispersive. Since ϕi and Siz are uncoupled operators in the harmonic approximation, the mean-value fields specified in the spin stiffness can easily approximated by the use of the harmonic Hamiltonian. Moreover, the angle field ϕ has a Gaussian distribution and

⎡ ⎢ ρ′ ≈ ⎢1 − ⎣

∫BZ

2

)

− ϕ jB

(14d)

⎤ ⎡ ⎛ ⎢ωqα ⎜αq† αq + 1 ⎞⎟ + ωqβ ⎛⎜β † β + 1 ⎞⎟ ⎥, q q ⎝ ⎢⎣ ⎝ 2⎠ 2 ⎠ ⎥⎦

⎡ ⎢ ρ ≈ ⎢1 − ⎣

8 (2π )2

0

+

Therefore, we obtain the harmonic Hamiltonian:

H0 =

( )

⎤ ⎡ ⎛ βω α ⎞ ⎛ βω β ⎞ ⎥ ⎢ q q α β ⎜ ⎟ ⎜ ⎟ ⎢ωq coth ⎜ ⎟ + ωq coth ⎜ 2 ⎟ ⎥ ⎢ ⎝ 2 ⎠ ⎝ ⎠⎥ ⎦ ⎣

d2q,

⎡ ⎤1/4 i ⎢ 2 (2J − J ′ ) ⎥ (βq† + β−q ) 2 ⎢⎣ A q + Bq − ωq ⎥⎦

φ−βq =

: −zβq

−1

(2J − J′) =

2 Siz

63

(19)

Initially we have determined the behavior at zero temperature on which we consider coth (βωq /2) = 1. In Fig. 3 we plot the square magnetization 〈(Siz )2〉0 as a function of r = J ′ /J . Unlike the linear spin–wave results from previous section, the antiferromagnetic XY model presents a finite magnetization even at r ¼1. Therefore, higher corrections provided by the SCHA raise the degeneracy and the system has an ordered state for all physical spins. This transition, from a semi-classical disordered state to a quantum ordered state, is a typical feature for the order-by-quantum disorder phenomenon. Already in Fig. 4 we can note the finite spin stiffness in whole range 0 ≤ r ≤ 1. This implies a finite energy to slowly twist the spins. Consequently, there is no disordered state caused by geometric frustration as it occurs when only linear spin–wave approximation is considered. Naturally, there must be a critical value Sc from which there is no more magnetization. In Fig. 5 we show the critical Sc as a function of r. According to SCHA, for all physical values of spin there is an ordered state at zero temperature, even for S = 1/2. Sz

2

0

0.4

0.3

0.2 S 1 0.1 S 1 2 Fig. 2. The two modes of the spin–wave energies. The ωqβ is gapless while ωqα has a gap Δqα=0 = 4S˜

2

Jρ (2J − J′ ) . Here we have consider J′ = 0.5J but the behavior is

similar for all values of J′.

0.2

0.4

0.6

0.8

1.0

r

Fig. 3. The square magnetization 〈(Siz )2〉0 as a function of ratio r = J′ /J . The ground state is ordered even in the limit r ¼1.

64

A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66 2

2TBKT /πJS˜ (when r ¼0). If we hold this assumption for 0 ≤ r ≤ 1, we can find the BKT temperature by the intersection of the curve ρ (T ) 2 with the line η = 2T /πJS˜ . The same result is obtained when ρ′(T ) is used. Eqs. (16a) can be simplified in the semi-classical limit k B T ⪢ωq(α,β ). Using the approximation coth (ωq(α,β )/2T ) ≈ 2T /ωq(α,β ) (adopting kB ¼1), the spin stiffness ρ is written as

⎡ ⎤ T ⎥ e−RT , ρ = ⎢1 − ⎢⎣ 2 (2J − J ′ ) ⎥⎦

(20)

where

R=

Fig. 4. The spin stiffness ρ and ρ′ as a function of r. Finite values indicate the lack of a geometric frustration as observed in the linear spin–wave approximation. Sc

1 2π 2

∫BZ

A q + Bq + 2γq Cq (A q + Bq )2 − ωq2

d2q. (21)

Although the above equation provides a simpler calculation, to include the full quantum effects we have to consider the original Eq. (16a). Both results, semi-classical and quantum, are shown in Fig. 6. The results for r > 0 are new in the literature but, for 2 comparison, when r ¼ 0 we have found TBKT = 0.84JS˜ , close to the 2

known value TBKT = 0.83JS˜ [20] (considering S¼1). As we can see,

0.37

the diagonal interaction J′ destabilizes the topological structures. The vortex coupling decreases with the increasing of r, resulting in lower TBKT temperatures. Presumably, in the limit of uncoupled

0.36

0.35

0.34

0.33 0.2

0.4

0.6

0.8

1.0

r

Fig. 5. The critical spin Sc as a function of r. For values below Sc there is no magnetic ordering at zero temperature.

4. Thermodynamic 4.1. Semi-classical limit Since we have a two-dimensional short-range interaction model, a disordered phase at finite temperatures is expected, as dictated by the Mermin–Wagner theorem. In this case, although the O(3) symmetry cannot be spontaneously broken at finite temperatures, there is a phase transition not associated with a spontaneous symmetry-breaking. As is well known, the XY model presents the famous Berezinskii–Kosterlitz–Thouless (BKT) transition which is involved with the emergence of topological configurations at finite temperatures, the magnetic vortices. Below TBKT, there is a state with quasi-long-range order (algebraic decay for the spin order-parameter correlation) due to the bounding of the vortex–antivortex pairs. Above the TBKT, the free vortices guide the system to a more disordered phase where the order-parameter decays exponentially (without quasi-long-range order). We apply Eqs. (16a) at finite temperature to determine the critical temperature Tc, defined as the temperature for which the spin stiffness vanishes discontinuously. In a first approximation, Tc can be taken as the BKT temperature. However, we have adopted a more precise treatment. In the limit r ⪡1, the angle field ϕ (r) (for both A and B sublattice) should be split in two parts ϕ (r) = ϕ0 (r) + ψ (r), where ϕ0 (r) describes the small phase fluctuations of the order parameter while ψ (r) is the vortex field. The spin stiffness described heretofore just consider the spin–wave contribution, ignoring entirely the vortices. It is a known fact that the spin stiffness should present a universal jump given by

chains ( J ′⪢J ) the vortices are free even at zero temperature providing a phase without quasi-long-range order, which is primordial to a spin liquid phase. Unfortunately, the limit r⪢1 is out of scope in the current work since an ordered state is one of the assumptions for the formulation of the SCHA. In addition to the BKT transition, it is possible to induce a quantum phase transition by the inclusion of extras interactions in the Hamiltonian (1). The single-ion anisotropy, for instance, defined by the Hamiltonian HD = D ∑i (Siz )2 (with D > 0) is one traditional term that creates a competition with the exchange energy [26–29]. For small values of D, the quantum fluctuations are negligible and the spins tend to align with the z-axis, providing 〈S z〉 ≠ 0 (quasi-long-range order). However, for values bigger than a critical anisotropy Dc, the system goes in the so-called large-D phase. In this phase the energetic cost to keep a non-null magnetization is too high and the system keeps a phase with 〈Siz 〉 = 0 . The energy spectrum is gapped and the spin order-parameter correlation decays exponentially. In our model, the inclusion of the single-ion anisotropy changes the Hamiltonian / qz to

/ qz = (2J − J ′ + D) 2x2. The necessary modifications are straightforward. Fig. 7 shows the main result obtained. Again, as expected, we have obtained a lower critical anisotropy Dc for larger r. S 1 2, Sem i classical T

JS

S 1, Sem i classical

0.8

S 1 2, Qu an tu m S 1, Qu an tu m

0.6

0.4

0.2

0.2

0.4

0.6

0.8

Fig. 6. The TBKT temperature as a function of r.

1.0

r

A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

Dc

65

temperature proportionality. As we can see, when quantum perturbation is considered we have a null entropy at zero temperature unlike of the classical result in the isotropic limit. A similar behavior (no residual entropy at zero temperature) is also obtained in the semi-classical limit.

6.5

6.0

5. Discussion and conclusions 5.5

In this paper we have studied the antiferromagnetic XY model in the checkerboard lattice for arbitrary spin. In special, we have determined the behavior for spin-1 and 1/2. However, the model is applicable to any value of spin. When the crossing and square

5.0

0.2

0.4

0.6

0.8

1.0

r

Fig. 7. The critical single-ion anisotropy (given in units of J) as function of r.

4.2. Quantum limit Let us consider now the low temperature limit for small S and

ωq(α,β ) ⪡k B T . Writing the coth function as coth x = 1 +

2 e2x − 1

, the

spin stiffness equations are given by

⎞ 0 ⎛ I0 Iz T 3 ⎟ − Iϕ + Iϕ T 3 ρ = ⎜⎜1 − z − 2 ⎟e 2 2 S˜ S˜ ⎠ ⎝

(22)

⎛ ⎞ 0 I0 Iz T 3 ⎟ − Iϕ′ + 2Iϕ T 3 ρ′ = ⎜⎜1 − z − e , 2 ⎟ 2 2 S˜ S˜ ⎠ ⎝ where

Iz0 , Iϕ0 A

and

Iϕ′0

are the mean-values of

(23) 〈(Siz )2〉0 ,

A

〈(ϕi −

ϕ jB )2〉0

ϕ jA )2〉0 ,

and 〈(ϕi − respectively, calculated at zero temperature while the finite temperature coefficients are defined by

Iz =

ζ (3) 2 4π (2J − J ′ )2 (Jρ − J ′ ρ′) S˜

(24)

and

Iϕ =

ζ (3) 4π (2J − J ′ )(Jρ − J ′ ρ′)2S˜

4

, (25)

−3 ≈ 1.202 is the Riemann zeta function. In the where ζ (3) = ∑∞ n=1 n above equations we have consider only excitations in the β spin– wave mode provided that the α mode is gapped with Δqα=0 ⪢ωqβ=0 . Therefore, the gapless mode at low temperatures is ωqβ (T ) ≈ [c0 − c (T )] q, where the time independent c0 and time dependent c(T) component of the spin–wave velocity in the limit of long wavelength are

c0 =

⎛ 2 ⎞⎛ ⎞ 0 0 2J − J ′ ⎜S˜ − Iz0 ⎟⎜Je−(Iϕ /2) − J ′ e−(Iϕ′ /2) ⎟ ⎝ ⎠⎝ ⎠

c (T ) =

⎤ 3 ⎡ ⎛ Iϕ0 Iϕ′0 ⎞ 1 2 T 2J − J ′ ⎢ (S˜ − I0z )(J − 2J ′ ) Iϕ+⎜⎜Je− 2 − J ′ e− 2 ⎟⎟ Iz ⎥ , ⎥ ⎢2 ⎠ ⎦ 2 ⎝ ⎣

1 S = N (2π )2

∫BZ

⎛ ⎞2 T d2q ∝ ⎜⎜ ⎟⎟ , 2 ⎝ JS˜ ⎠ −1

References

(26b)

ωqβ /T β

e ωq / T

entropy at zero temperature. For small ratio r = J ′ /J , the model has similarities with the traditional XY antiferromagnet. We have demonstrated distinct behaviors whether the model is treated considering a linear spin–wave approximation or a self-consistent method which includes higher order perturbation. Using the Holstein–Primakoff representation and ignoring spin–wave interactions, the model shows two different phases. There is a magnetic ordering only below a critical ratio rc. For rc ≤ r < 1 the magnetization is null due to the first order quantum perturbations. The r = rc is a special point where the disordered phase is a consequence of the structure lattice. The Brillouin zone presents a flat region associated with the degenerate ground state and a null spin stiffness. Already when we adopted the SCHA, the model has a magnetic phase in whole range 0 ≤ r ≤ 1 at zero temperature. The high order perturbation, included as renormalized parameters in the method, restores the magnetic ordering. This effect is a typical feature of an order-by-quantum disorder. However, more studies are necessary to prove this assumption. We determined the BKT transition at finite temperatures (the thermal phase transition associated with the vortices uncoupling) as a function of r. Since there is a decrease of TBKT with the increase of r, the vortices coupling becomes weaker and we expect free vortices in the limit r⪢1 even at zero temperature. The spin liquid phase of the uncoupled chains could be associated with the unbound vortices. Unfortunately the limit r⪢1 is out of the scope of the SCHA and a different approach is required. We have also considered the inclusion of a single-ion anisotropy D which induces a Quantum Phase Transition. Below the critical value Dc the model presents gapless energy spectrum and the spin order-parameter correlation decays algebraically. Already D > Dc the decay is exponential and the phase is gapped. The critical Dc is determined as a function of r and there is a decreasing behavior with respect to r. Finally, the special limit of small spin is studied in semi-classical and quantum limit and the obtained results are in agreement with the literature.

(26a)

in agreement with the antiferromagnetic XY model when J ′ = 0 [30]. Yet in the small S limit, we can calculate the specific entropy:

s=

interactions ( J′ and J, respectively) are equal, the checkerboard lattice is the planar projection of the pyrochlore lattice and keeps similar properties with the three-dimensional case. For example, the classical ground state is disordered and there is a residual

(27)

where we have discarded again lower spin–wave excitation in the α mode and used the long wavelength limit to get the square

[1] [2] [3] [4] [5] [6] [7]

F. Haldane, Phys. Lett. A 93 (1983) 464. F. Haldane, Phys. Rev. Lett. 50 (1983) 1153. F. Haldane, J. Appl. Phys. 57 (1985) 3359. S. Bramwell, M. Gingras, Science 294 (2001) 1495. M. Gingras, Science 326 (2009) 375. S. Petit, et al., Phys. Rev. B 90 (2014) 060410. M. Zhitomirsky, M. Gvozdikova, P. Holdsworth, R. Moessner, Phys. Rev. Lett. 109 (2012) 077204. [8] L. Savary, K. Ross, B.D. Gaulin, J. Ruff, L. Balents, Phys. Rev. Lett. 109 (2012) 167201. [9] K. Ross, Y. Qiu, J.R.D. Copley, H. Dabkowska, B. Gaulin, Phys. Rev. Lett. 112 (2014) 057201. [10] B. Canals, T. Zhitomirsky, J. Phys.: Condens. Matter. 16 (2004) S759.

66

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

A.R. Moura / Journal of Magnetism and Magnetic Materials 394 (2015) 60–66

R. Singh, O. Starykh, P. Freitas, J. Appl. Phys. 83 (1998) 7387. O. Starykh, R. Singh, G. Levine, Phys. Rev. Lett. 88 (2002) 167203. B. Canals, Phys. Rev. B 65 (2002) 184408. E.H. Lieb, P. Schupp, Phys. Rev. Lett. 83 (1999) 5362. T. Holstein, H. Primakoff, Phys. Rev. 58 (1940) 1908. G. Gomez-Santos, J. Joannopoulos, Phys. Rev. B 36 (1987) 8707. J. Hirsch, S. Tang, Phys. Rev. B 40 (1989) 4769. J. Villain, J. Phys. (France) 35 (1974) 27. S. Menezes, M.E. Gouvêa, A.S.T. Pires, Phys. Lett. A 166 (1992) 330. A.S.T. Pires, M.E. Gouvêa, Phys. Rev. B 48 (1993) 12698. A.S.T. Pires, Phys. Rev. B 53 (1996) 235.

[22] [23] [24] [25] [26] [27] [28] [29] [30]

A.S.T. Pires, Solid State Commun. 104 (1997) 771. M.E. Gouvêa, A.S.T. Pires, Physica Status Solidi 242 (2005) 2138. A.S.T. Pires, L.S. Lima, M.E. Gouvêa, J. Phys.: Condens. Matter 20 (2008) 015208. A.S.T. Pires, M.E. Gouvea, Physica A 388 (2009) 21. H. Xingand, G. Su, S. Gao, J. Chu, Phys. Rev. B 66 (2002) 054419. H. Wang, Y. Wang, Phys. Rev. B 71 (2005) 104429. A.R. Moura, A.S.T. Pires, A.R. Pereira, J. Magn. Magn. Mater. 357 (2014) 45. A.R. Moura, J. Magn. Magn. Mater. 369 (2014) 62. J. Villain, R. Bidaux, J. Carton, R. Conte, J. Phys. France 41 (1980) 1263.