^{1}

^{*}

^{2}

^{1}

^{3}

We study the ground state of an
*S*=1/2 anisotropic
*a *(
≡J
_{z}/J
_{xy}) Heisenberg antiferromagnet with nearest (
*J*
_{1}) and next-nearest (
*J*
_{2}) neighbor exchange interactions on a triangular lattice using the exact diagonalization method. We obtain the energy, squared sublattice magnetizations, and their Binder ratios on finite lattices with
*N*
≤36 sites. We estimate the threshold
*<span style="white-space:nowrap;">J<sup>(t)</sup><sub style="margin-left:-6px;"> 2</sub></span> (a)* between the three-sublattice Néel state and the spin liquid (SL) state, and
* <span style="white-space:nowrap;">J<sup>(s)</sup><sub style="margin-left:-6px;"> 2</sub></span> (a) * between the stripe state and the SL state. The SL state exists over a wide range in the

*α*-

*J*

_{2}plane. For

*α>1*, the xy component of the magnetization is destroyed by quantum fluctuations, and the classical distorted 120 ° structure is replaced by the collinear state.

Over the past three decades, the low temperature properties of low-dimensional quantum systems have been studied because of the exotic spin states that can arise from quantum fluctuations. The quantum antiferromagnetic Heisenberg (QAFH) model on a triangular lattice is a typical quantum frustrated system. This involves a generalized model with an antiferromagnetic nearest-neighbor (NN) interaction J 1 ( > 0 ) and a next-nearest-neighbor (NNN) interaction J 2 , the model Hamiltonian of which given by

H = 2 J 1 ∑ 〈 i , j 〉 [ S i x S j x + S i y S j y + α S i z S j z ] + 2 J 2 ∑ 〈 〈 i , j 〉 〉 [ S i x S j x + S i y S j y + α S i z S j z ] , (1)

where S i ν is the ν component ( ν = x , y , z ) of the quantum spin S = 1 / 2 at lattice site i, α ( ≡ J z / J x y ) is an exchange anisotropy, and the sums 〈 i , j 〉 and 〈 〈 i , j 〉 〉 run over all NN and NNN pairs of sites, respectively. Hereafter we set J 1 = 1 as the unit of energy scaling. The ground state (GS) of an isotropic QAFH model ( α = 1 ) with J 2 = 0 is the central issue of the model. Anderson proposed a resonating-valence-bond state or a spin liquid (SL) state as the GS of the model [_{2}Cu_{2}(CN)_{3} [_{3}Sb[Pd(dmit)_{2}]_{2} [_{3}IrTi_{2}O_{9} [

In the present paper, we consider the effect of the NNN interaction J 2 on the anisotropic QAFH model. The isotropic QAFH model with J 2 was studied recently using approximations such as a variational Monte Carlo (VMC) method [_{2} plane.

In Section 2, we present our method with the finite lattices. In Section 3, we estimate the threshold J 2 ( t ) ( α ) between the 120˚ Néel state and the SL state. In Section 4, we consider the stripe Néel state and its threshold J 2 ( s ) ( α ) with the SL state. In Section 5, we propose a phase diagram of the model.

It is known that the GS of the classical model is a 120˚ Néel state when J 2 < J 2 ( c l a s ) , and the stripe Néel state when J 2 > J 2 ( c l a s ) , where J 2 ( c l a s ) = 1 / 8 for α ≤ 1 and tends to zero as α → ∞ . The unit cells of these states are shown in

We first consider this problem. Hereafter we refer to the spin space of a lattice with N = 3 n sites with three-sublattice symmetry as the three-sublattice space (3SLS), where n is a natural number. Similarly, we refer to the spin space of a lattice with N = 4 n sites with four-sublattice symmetry as the four-sublattice space (4SLS). The minimum energies per site in the 3SLS and 4SLS are labeled as E t r i and E s t r , respectively. The spin state is in the 3SLS when E t r i < E s t r and in the 4SLS when E s t r < E t r i . In the classical model, the threshold of J 2 ( c l a s ) is one at which the spin space changes from one to the other, and the phase transition at J 2 ( c l a s ) is of the first order. In the quantum model, although the spin space changes at some threshold J 2 ( q u a n ) , no phase transition will take place at J 2 ( q u a n ) , because there would be no LRO in those spin spaces at J 2 ∼ J 2 ( q u a n ) . We must then consider the thresholds and natures of the phase transitions in the 3SLS and in the 4SLS, separately.

For the 3SLS, we consider the lattices with N = 18 - 30 (and partly N = 36 ) sites with periodic boundary conditions suitable for the three-sublattice structure (

For the 4SLS, we consider the lattices with N = 24, 28, and 32 sites with periodic boundary conditions suitable for the stripe structure (

In either case, we obtain the GS eigenfunction | ψ s G 〉 N of the N sites using the Lanczos method, where s = tri or str for the 3SLS or 4SLS, respectively. The ν component of the magnetization on the Ω l sublattice is defined as

μ l ν = 2 N s u b N ∑ i ∈ Ω l S i ν , (2)

where N s u b = 3 and l = A , B , and C for the 3SLS, and N s u b = 4 and l = A , B , C , and D for the 4SLS. The operators of the z, xy, and xyz components of the squared sublattice magnetization are defined as

m 2 z = 1 N s u b ∑ l ( μ l z ) 2 , (3)

m 2 x y = 1 N s u b ∑ l ( ( μ l x ) 2 + ( μ l y ) 2 ) , (4)

m 2 x y z = 1 N s u b ∑ l ( ( μ l x ) 2 + ( μ l y ) 2 + ( μ l z ) 2 ) . (5)

We calculate the ζ component the squared sublattice magnetization, 〈 m 2, s ζ 〉 N , as

〈 m 2, s ζ 〉 N = N 〈 ψ s G | m 2 ζ | ψ s G 〉 N (6)

where ζ = z , xy, or xyz.

We study the Binder ratios [

B s z ( N ) = ( 3 − 〈 ( m 2, s z ) 2 〉 N / 〈 m 2, s z 〉 N 2 ) / 2 , (7)

B s x y ( N ) = 2 − 〈 ( m 2 , s x y ) 2 〉 N / 〈 m 2 , s x y 〉 N 2 , (8)

B s x y z ( N ) = ( 5 − 3 〈 ( m 2, s x y z ) 2 〉 N / 〈 m 2, s x y z 〉 N 2 ) / 2 . (9)

Before estimating J 2 ( t ) and J 2 ( s ) , we should examine that no phase transition will take place at J 2 ( q u a n ) .

phase transition between the 120˚ Néel state and the SL state because J 2 p e a k > J 2 ( q u a n ) . The phase boundary should be estimated by a different method. In contrast, we expect J 2 ( s ) to be near J 2 p e a k , because J 2 ( q u a n ) < J 2 p e a k . In Section 4, we consider J 2 p e a k together with the Binder ratio B s t r ζ in order to estimate J 2 ( s ) .

In this section, we estimate the threshold J 2 ( t ) . We consider 〈 m 2, t r i ζ 〉 N in the GS of the 3SLS. Special attention should be paid to the M z ( = ∑ i S i z ) subspace in which the GS belongs. For α ≤ 1 , the GS is in the minimum | M z | subspace. For α > 1 , however, the GS may not be restricted to the minimum | M z | subspace depending on J 2 . We then consider the cases α ≤ 1 and α > 1 separately.

For α ≤ 1 , the LRO has the 120˚ Néel state symmetry, and 〈 m 2 , t r i x y 〉 N and 〈 m 2 , t r i x y z 〉 N are calculated for various α . For α < 1 , because the spins lie in the xy plane, we consider the ζ = x y component, whereas the ζ = x y z component for α = 1 . In Figures 4(a)-(d), we present 〈 m 2, t r i ζ 〉 N for α = 0.0 , 0.4, 0.8, and 1.0 as functions of J 2 , respectively. As J 2 is decreased, 〈 m 2, t r i ζ 〉 N increases revealing the development of the 120˚ spin correlation. For small α ( = 0 , 0.4 ) , the finite-size effect (FSE) for J 2 ≲ 0.05 is rather weak implying the occurrence of the 120˚ Néel state. As α is increased, the FSE becomes stronger. In the isotropic case of α = 1.0 , we can see a strong FSE even for J 2 < − 0.1.

We consider the Binder ratio B t r i ζ ( N ) [

In the case of α = 1.0 , B t r i ζ ( N ) exhibits a somewhat different behavior from those for α < 1.0 . Although B t r i ζ ( N ) increases with decreasing J 2 , its increment

depends only very weakly on N, especially for J 2 < 0 . We could see no definite intersection point of B t r i ζ ( N ) for N ≥ 24 down to J 2 = − 0.4 , i.e., we could not evaluate J 2 l . Therefore we believe J 2 ( t ) ( 1 ) < − 0.1 , because J 2 u ∼ − 0.1 .

For α > 1 , we are interested in 〈 m 2, t r i z 〉 N and 〈 m 2 , t r i x y 〉 N because a distorted 120˚ Néel state occurs in the classical model. We obtain the eigenfunction | ψ ( M z ) 〉 N with the minimum energy E ( M z ) for each M z subspaces. Note that we consider only the subspaces of M z ≤ N / 6 because the GS is in the M z = N / 6 subspace for J 2 → − ∞ . The GS eigenfunction | ψ s G 〉 N of the system is one which gives the lowest value among E ( M z ) ’s. When J 2 ≥ 0 , the GS eigenfunction | ψ s G 〉 N is | ψ ( 0 ) 〉 N . As J 2 is decreased, | ψ s G 〉 N successively changes to | ψ ( 1 ) 〉 N , | ψ ( 2 ) 〉 N , ⋯ , | ψ ( N / 6 ) 〉 N at J 2 ( 1 ) , J 2 ( 2 ) , ⋯ , J 2 ( N / 6 ) , respectively. Using | ψ s G 〉 N , we obtain 〈 m 2, t r i z 〉 N and 〈 m 2 , t r i x y 〉 N for various α . A typical result of these is shown in

In

of J 2 for α = 1.25 and α = 2.5 , respectively. For (a) α = 1.25 , we evaluate the lower and upper bounds of J 2 ( t ) as J 2 l ∼ − 0.15 and J 2 u ∼ 0 , i.e., J 2 ( t ) ( 1.25 ) ∼ − 0.08 ± 0.08 . For (b) α = 2.5 , we estimate J 2 l ∼ − 0.08 and J 2 u ∼ 0.04 , i.e., J 2 ( t ) ( 2.5 ) ∼ − 0.02 ± 0.06 . Note that we also examined B t r i x y ( N ) to confirm the speculation given above and found that, in fact, B t r i x y ( N ) decreases with increasing N for the whole range of J 2 .

To close this subsection, we emphasize that the distorted 120˚ Néel state is absent in the QAFH model, in contrast to the classical model. We find that, when J 2 < J 2 ( 1 ) , the LRO of the z component of the spin occurs. A question remains as to what the value of J 2 ( 1 ) for N → ∞ , J 2, ∞ ( 1 ) . If J 2 , ∞ ( 1 ) = J 2 ( t ) ( α ) , the LRO of the z component of the spin occurs for J 2 ≤ J 2 ( t ) ( α ) . If not, two possibilities exist in the range J 2 , ∞ ( 1 ) < J 2 < J 2 ( t ) ( α ) : either there is still the LRO, or the system is in a critical state that is similar to the spin state of the Ising model with J 2 = 0 . Further studies are necessary to answer this question.

In this section, we consider the stripe state. We obtain the GS as the eigenfunction | ψ s t r G 〉 N = | ψ ( 0 ) 〉 N with energy E s t r because the stripe state belongs to the M z = 0 subspace. In

Binder ratio B s t r ζ ( N ) . In

We have also examined J 2 ( s ) for α = 1 . We may evaluate the lower bound of J 2 l = J 2 p e a k ∼ 0.20 . However, we could not evaluate J 2 u because B s t r x y z ( 32 ) < B s t r x y z ( 28 ) even up to J 2 = 0.3 [

We have studied the S = 1 / 2 anisotropic antiferromagnetic model ( α ≡ J z / J x y ) with nearest-neighbor (J_{1}) and next-nearest-neighbor (J_{2}) interactions on a triangular lattice using the exact diagonalization method. We have obtained the ground-state energy and the sublattice magnetizations for systems of different size N. We have examined Binder ratios to investigate the stability of the long-range order of the system. The N-dependences of Binder ratios suggest the threshold J 2 ( t ) ( α ) between the three-sublattice Néel state and the disordered state, i.e., the spin liquid (SL) state, and the threshold J 2 ( s ) ( α ) between the stripe state and the SL state. The results are summarized in the phase diagram shown in

We have suggested that the SL state exists over a wide range in theα-J_{2} plane in contrast with recent approximation studies [

establish the thresholds for α ∼ 1 .

Some of the results in this research were obtained using the supercomputing resources at the Cyberscience Center of Tohoku University.

The authors declare no conflicts of interest regarding the publication of this paper.

Suzuki, N., Matsubara, F., Fujiki, S. and Shirakura, T. (2019) Phase Diagram of an S = 1/2 J_{1}-J_{2} Anisotropic Heisenberg Antiferromagnet on a Triangular Lattice. Journal of Modern Physics, 10, 8-19. https://doi.org/10.4236/jmp.2019.101002