Phase Diagram of an S = 1/2 J1-J2 Anisotropic Heisenberg Antiferromagnet on a Triangular Lattice

We study the ground state of an 1 2 S = anisotropic α ( ) z xy J J ≡ Heisenberg antiferromagnet with nearest (J1) and next-nearest (J2) 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 36 N ≤ sites. We estimate the threshold ( ) ( ) 2 J α between the three-sublattice Néel state and the spin liquid (SL) state, and ( ) ( ) 2 J α between the stripe state and the SL state. The SL state exists over a wide range in the α-J2 plane. For 1 α > , the xy component of the magnetization is destroyed by quantum fluctuations, and the classical distorted 120 ̊ structure is replaced by the collinear state.


Introduction
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 ( ) where i S ν is the ν component ( , , x y z ν = ) of the quantum spin 1 2 S = at lattice site i, ( ) 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 1 1 J = as the unit of energy scaling.The ground state (GS) of an isotropic QAFH model ( 1 α = ) with 2 0 J = 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 [1].Since then, many authors have used various methods to study the model [2]- [16].The GS of the model is now widely believed to have the classical long-range-order (LRO) in the form of the 3-sublattice structure (120˚ Néel state).
However recent experiments on model compounds such as κ-(ET) 2 Cu 2 (CN) 3 [17], EtMe 3 Sb[Pd(dmit) 2 ] 2 [18], and Ba 3 IrTi 2 O 9 [19] have observed no LRO down to very low temperatures.Motivated by this discrepancy, the present authors [14] (hereafter referred to as SMFS) reexamined the GS of an anisotropic QAFH model with 0 α ≤ < ∞ on finite lattices having used an exact diagonalization technique, and found that the classical LRO is absent for 0.55 1.67 , it has no LRO and is the SL state.
In the present paper, we consider the effect of the NNN interaction 2 J on the anisotropic QAFH model.The isotropic QAFH model with 2 J was studied recently using approximations such as a variational Monte Carlo (VMC) method [20], a many-variable VMC (mVMC) method [21], a coupled cluster method (CCM) [22], and a density matrix renormalization group method [23] [24].
These approaches showed that the 120˚ Néel state occurs for ( ) In Section 2, we present our method with the finite lattices.In Section 3, we estimate the threshold ( ) ( ) J α between the 120˚ Néel state and the SL state.In Section 4, we consider the stripe Néel state and its threshold ( ) ( ) 2 s J α with the SL state.In Section 5, we propose a phase diagram of the model.

Method
It is known that the GS of the classical model is a 120˚ Néel state when .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 18 -30 N = (and partly 36 N = ) sites with periodic boundary conditions suitable for the three-sublattice structure (Figure 2(a)).
For the 4SLS, we consider the lattices with N = 24, 28, and 32 sites with periodic boundary conditions suitable for the stripe structure (Figure 2   In either case, we obtain the GS eigenfunction sG N ψ 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 l Ω sublattice is defined as We calculate the ζ component the squared sublattice magnetization, where z ζ = , xy, or xyz.
We study the Binder ratios [25] that are used by SMFS [14] to estimate the threshold α of the model with 2 0 J = .At the critical point, the Binder ratio is size invariant.If there is a LRO, the Binder ratio is expected to increase with the system size.In contrast, in the paramagnetic or SL state, the Binder ratio decreases with the system size.This means that the size dependence of the Binder ratio is different from each other with and without a LRO.The z, xy, and xyz components of the Binder ratio can be defined as .
Before estimating ( ) J , we should examine that no phase transi- tion will take place at ( ) . Figure 3 shows tri E and str E together with  E ) has its maximum value.
In the 3SLS, a bending of tri E accompanied by a discontinuous drop of  The phase boundary should be estimated by a different method.In contrast, we expect ( )

Three-Sublattice Néel State
In this section, we estimate the threshold ( ) in the GS of the 3SLS.Special attention should be paid to the ( ) subspace in which the GS belongs.For 1 α ≤ , the GS is in the minimum z M subspace.For 1 α > , however, the GS may not be restricted to the minimum z M sub- space depending on 2 J .We then consider the cases 1 α ≤ and 1 α > sepa- rately.

Ising-Like Case (α > 1)
For 1 α > , we are interested in  A typical result of these is shown in Figure 6 for 1.25 α = . For 2 0 J  , both , 0

J
and discontinuously jumps at ( ) ( ) ( ) . It exhibits its own N dependence in different ranges of 2 J .For 1) is almost independent of N. We believe that the classical ferrimagnetic state arises in this range, because However, for 3) ( )   B N to confirm the speculation given above and found that, in fact,

( )
xy tri B N de- creases increasing N for the whole range of 2 J .
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 J < , the LRO of the z component of the spin occurs.A question remains as to what the value of ( ) LRO of the z component of the spin occurs for : 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 2 0 J = .Further studies are necessary to answer this question.

Stripe State
In this section, we consider the stripe state.We obtain the GS as the eigenfunc-  We have also examined ( )

Summary
We have studied the 1 2 S = anisotropic antiferromagnetic model ( ) J J α ≡ 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 ( ) ( ) We have suggested that the SL state exists over a wide range in theα-J 2 plane in contrast with recent approximation studies [20] (b)).

Figure 1 .
Figure 1.(a) Sublattices A, B, and C in the three-sublattice state; (b) Sublattices A, B, C, and D in the four-sublattice state.
the 4SLS.The operators of the z, xy, and xyz components of the squared sublattice magnetization are defined as

Figure 3 .
Figure 3.The GS energies s E and the squared sublattice magnetizations

,
various α .For 1 α < , because the spins lie in the xy plane, we consider the xy the finite-size effect (FSE) for 2 0.05 J  is rather weak implying the occurrence of the 120˚ Néel state.As α is increased, the FSE becomes stronger.In the isotropic case of 1.0 α = , we can see a strong FSE even for 2 0.1.J < − N. Suzuki et al.DOI: 10.4236/jmp.2019.10100213 Journal of Modern Physics

Figure 5 .N
Figure 5. Binder ratios (a)-(c) occurs in the classical model.We obtain the eigenfunction which gives the lowest value among the occurrence of the LRO of the z component of the spin.

2 sJ
str E because the stripe state belongs to the 0 z M = subspace.In Figure8(a) and Figure8(b), we show these quantities as functions of 2 We readily see that the results for the 24 N = system are quantitatively different from those of the 28 N = and 32 systems, which lead us to consider mainly data for 28 N ≥ in order to evaluate ( ) .We see similar properties in the cases of 1 α < and 1 α > .The magnetization 2,str N m ζ rapidly increases around the peak posion 2 peak J of str E that implies the occurrence of the phase transition.When Figure 8.The GS energies str E and four-sublattice magnetizations

2 tJ 2 sJ
α between the three-sublattice Néel state and the disordered state, i.e., the spin liquid (SL) state, and the threshold ( ) ( ) α between the stripe state and the SL state.The results are summarized in the phase diagram shown in Figure10.For 1 α < , the classical 120˚ phase or the stripe phase oc- curs in the xy plane.For 1 α > , the xy component of the sublattice magnetiza- tion vanishes, i.e., the distorted 120˚ state is replaced by the collinear (up up down) antiferromagnetic state because of quantum fluctuations.
Figure 10.The α-J 2 phase diagram of the J 1 -J 2 anisotropic Heisenberg model on a triangular lattice.Cross symbols are those estimated in SMFS [14].