Magnetic properties of the spin-1 J 1 − J 3 Heisenberg model on a triangular lattice

We study the spin-1 J 1 − J 3 Heisenberg model with antiferromagnetic nearest neighbors J 1 > 0 and the ferromagnetic third nearest neighbor J 3 < 0 for a triangular lattice. The ground-state and thermodynamic properties are evaluated under the Popov-Fedotov functional integral method and the Luttinger-Tisza procedure. Perturbation expansion of the auxiliary field around mean-field theory was carried out to investigate the effect of thermal fluctuations on ground state energy and sublattice magnetization. The results are compared with those of the spin wave approximation and other methods.


Introduction
One of the simplest cases of geometrical magnetic frustration is the triangular lattice antiferromagnetic.Although it has been studied for the past five decades, triangular lattice antiferromagnetism still fascinates the scientific community because it can lead to the competition of some novel states when the system is broken magnetic symmetry and also because it is the fundamental lattice of many real magnetic materials [1].It is now widely accepted that the ground state of the spin-1/2 triangular lattice Heisenberg antiferromagnetism with the nearest-neighbor bond is the Néel state with a three-sublattice structure in which the spin of each sublattice creates an angle of 120 0 in combination with the remaining two sublattices [2][3][4].Theoretical researches mainly focus on two key issues, one is to study the properties of the ground state, the magnon excitation spectrum of ordered phases [5][6][7][8][9][10][11][12][13][14], the other is to determine the nature of the new phases without magnetic orders by adding to the prototype model [1] additional factors such as geometrical frustration [15][16][17][18], spatial anisotropy [19,20], biquadratic interactions [21][22][23][24], etc.However, the results of calculating the magnetization of the three-sublattice structure [5][6][7], or determining the quantum phase transition points as well as the range of appearance of new phases without magnetic order and the nature of the phase [15][16][17][18][19] depends on the methods.It is noteworthy that while most theoretical and experimental works have focused on the spin-1/2 Heisenberg antiferromagnetic model and have achieved much success, research works on the spin-1 Heisenberg antiferromagnetic model are very scarce [21][22][23][24][25][26][27], in particular there are no experimental examples of isotropic triangular lattice antiferromagnets with spin S=1 that possess regular long-range order.Therefore, studying the magnetic properties in an isotropic triangular lattice system with spin S=1, in which there are possible collective magnetic phases, is necessary and very useful.
In this paper we investigate the magnetic properties of the spin-1 J 1 − J 3 Heisenberg model on a triangular lattice using two basic techniques, namely: the Luttinger-Tisza process [28] and the Popov-Fedotov functional integral method [29].The Luttinger-Tisza procedure leads to the classical ground state energy represented by the magnetic ordering wave vector Q while the Popov-Fedotov functional integral method allows us to calculate the magnon spectrum, sublattice magnetization and their temperature dependence in a mean-field approach within the one-loop approximation.The results obtained are compared with those of linear spin wave approximation and other methods.
The rest of the article is organized as follows: Section 2, recalls some analytical results of the Popov-Fedotov functional integration method received from the report in the 41st Vietnam Theoretical Physics Conference [30].The results are presented in the section 3, first for the classical approximation and then for the one-loop approximation, and compared with the results obtained from other methods.Finally, conclusions and discussions are presented in Section 4.

Model and formalism
We study the Hamiltonian where < ij >, << kl >> denote the sum extends over all nearest-neighbor (NN) sites, and third-nearest-neighbor (TNN) sites, respectively.J 1 , J 3 are the magnitudes of the coupling exchange constants.J 1 > 0, J 3 < 0; J 3 = αJ 1 , − 1 9 ≤ α ≤ 0. The energies are given in the unit of J 1 .S i is a spin-1 operator whose components satisfy the relation Where i, j, k, l are indices of sites over the triangular lattice with the distance between two NN sites a ≡ 1.The triangular lattice and coupling exchange are illustrated in Figure 1.First, the Tisza-Luttinger procedure is applied to describe the ground state of the Hamiltonian (1).In the classical approximation, when the ground state of the Hamiltonian (1) has long-range magnetic order, its spin configuration can be described by the magnetic ordering vector Q such that the spin at site i is given by Where n 1 , n 2 are two orthogonal unit vectors in spin space, r i is the position vector of site i.
The classical energy is represented with respect to the magnetic ordering vector Q as follows where J(k) is the Fourier transform of J ij and has the form (5) Next, starting from the assumption of the existence of long-range magnetic order, we calculate the quantities characterizing the magnetic properties in the mean-field approximation and the principal fluctuations for those quantities on the level of the one-loop approximation.Applying the PF formalism to the S = 1 spin system in the Bravais lattice (for details, see reference [31]), we obtain analytical expressions for the sublattice magnetization taking into account quantum and thermal fluctuations, and magnon spectrum as follows: Where m 0 , δm zz , δm +− , and ω(k) are given by with In the above expressions, X(k), Y (k), W (k) are the components of the exchange interaction after the Fourier transform m 0 is the mean -field sublattice magnetization under the constraint of strict single site occupancy, δm zz and δm +− are the fluctuating contributions to the longitudinal and transerve parts of the sublattice magnetization respectively.ω(k) is the frequency of the spin excitations.
δ i denote the sum extends over all nearest-neighbor δ (1) and third nearest-neighbor δ (3) sites of sit i: ) is the inverse temperature, and N is number sites of lattice.
Finally, the quantities found above are numerically solved and graphed using Mathematica.The results are presented in the section below.

Results
This section presents the authors' research findings on the magnetic properties of the spin-1 model on an isotropic triangular lattice, where the Néel state with structure 120 0 has been used as the model state.First we determine the classical ground state, study the sublattice magnetization in the mean field approximation, and then calculate the influence of quantum fluctuations on the sublattice magnetization and ground state energy.Finally, we study the temperature effects on sublattice magnetization and magnon spectrum.

Ordering wave vector
The classical ground state can be found by minimizing the function ( 4) with weaker global conditions on the spin value It is supposed to make the partial derivative of the equation (5) with respect to k x , k y and then set those equations equal to zero.If this system of equations is satisfied at k = Q, then the positive definiteness condition of Hessian is verified here γ, β = x, y.
After this step we obtain the local minima of the function J(k) in the Brillouin zone (BZ) of the triangular lattice as a function of α = J 3 /J 1 .The condition (15) is necessary but not sufficient to have a total minimum value.Therefore, to find the exact ground state, it is necessary to compare different local minima and choose the one with the lowest energy.Specifically, the dependence on α of the classical ground state is described as follows: When −1/9 ≤ α ≤ 0, J(k) has a minimum value at six points located at the corners of hexagonal BZ, these points are equivalent, one of the points has coordinates Q 1 = (4π/3, 0) shown in figure 2. The minimum energy configuration of system now is the well known Néel state with 120 0 three-sublattice structure.When α < −1/9, the minimum value of J(k) appears at the midpoints of edges of the BZ, such as the point with coordinate Q 2 = (0, 2π/ √ 3).In this case, the system is in a collinear-state with four periodic sublattices so that the total spin on the four-sublattice is zero.In this paper, we only study the magnetic properties of the 120 0 three-sublattice phase, so the frustrated parameter α will be investigated in the range −1/9 ≤ α ≤ 0.

Sublattice magnetization
First we consider the system at the temperature T = 0. From the expression (8) the sublattice magnetization is found at the mean field approximation level when the constraint of a single occupancy condition is taken into account m 0 = 1.Under the influence of quantum fluctuations, the sublattice magnetization is decreased.Indeed, at low temperatures (T → 0) the sublattice magnetization taking into account fluctuations is given by the expression ( 8) is reduced to Solving numerically the equation ( 16) we obtain m = 0.738703 for α = 0.This result is quite consistent with those obtained from the spin wave method [5, 6], the large S expansion method [7], and the numerical diagonal method [27].The dependence of the sublattice magnetization m on the frustration parameter α is shown in Figure 3. From Figure 3 we see that m < m 0 , and when |α| increases inversely proportional to m decreases.So at temperature T = 0, quantum fluctuations decrease the sublattice magnetization and this decrease is enhanced by frustration, the ground state is still the Néel state with the 120 0 structure.
Dimensionless temperature t = k B T /J 1 and frustration parameter α dependence of sublattice magnetization at the mean field level (for three different frustration parameter values), and taking into account fluctuations are plot in Figure 4.
Figure 4a shows the difference of the mean-field sublattice magnetization when the single occupancy condition is taken into account in the PF method and when it is neglected.From these curves, we see that in the low temperature region (t < 0.5), the curves overlap and almost remain unchanged (m 0 = 1), they only differ significantly in the higher temperature region.In addition, the critical temperature when the single occupancy condition is taken into account is greater than that when this condition is disregarded.This feature is originated from the fact  that we impose the constraint of the single occupancy condition taken thermal fluctuations enter unphysical spinless states, reducing magnetic moments.
Figure 4b shows the dependence on temperature t and the frustration parameter α of the sublattice magnetization m, considering effects of fluctuations.When there is no geometric frustration α = 0, the magnetic moment at temperature t = 0 is maximum m max = 0.738703.When the temperature increases, the magnetic moment decreases rapidly and becomes zero at the critical temperature t max = 0.345.Under the influence of the frustration parameter at temperature t = 0, the sublattice magnetization also gradually decreases from the value m 1 = m max = 0.738703 for α 1 = 0 to the value m 2 = 0.489087 for α 2 = −1/9.At a fixed temperature below the critical temperature, the sublattice magnetization is also greatest for α = 0, and gradually decreases as α decreases to the value α = −1/9.
Therefore, quantum and thermal fluctuations have a great influence on the decrease in sublattice magnetization, this decrease is enhanced by geometric frustration.To see this, let's compare the critical temperature of the sublattice magnetization which included fluctuations with the one of the mean field sublattice magnetization.For example, for α = 0, the critical temperature when considering fluctuations is t c(f l) = 0.345 while the mean field critical temperature is t c(mf ) = 2.001; For α = −0.05, the critical temperature considering fluctuations is t c(f l) = 0.2634 while the mean field critical temperature is t c(mf ) = 1.905.

Ground state energy and magnon excitation spectrum
We now calculate the ground-state energy per spin then study the magnon excitation spectrum.In the zero temperature limit T → 0 (i.e.β → ∞), the free energy expression per site obtained from ref [30] is the energy of the ground state per site Solving numerically equation (17) we get the result E 0 /N = −1.82762J 1 .This result is in good agreement with previous calculation results using other methods (spin wave method [6], coupled cluster method [26], numerical diagonalization method [27]).
The energy spectrum of magnons at t = 0 and t > 0 is plotted in figures 5, 6. Figure 5a shows the spectrum of spin excitations in the case J 3 = 0.The spin excitation frequencies, ω(α, k) = ω(0, k) = ω(k) vanishes at k = 0 and k = ±Q 1 corresponds to the center and opposite corners of the BZ, where Q 1 = (4π/3, 0) , 2π/3, 2π/ √ 3 , −2π/3, 2π/ √ 3 .The zero modes are the three Goldstone modes that appear because the SO(3) rotational symmetry is broken.The evolution of the spin excitation spectrum at zero temperature with respect to α is shown in figure 5(b,c).In the range of −1/9 ≤ α < 0 frequencies of the spin excitations still vanish at the wave vector k = 0 and k = ±Q 1 .However, the dispersion relation shape is only close to the dispersion relation shown in Figure 5a with small values of |α| (small J 3 interaction), When |α| increases, in the low-energy region of the magnon spectrum a flat region appears over a wide momentum range (Fig. 5b).Furthermore, |α| approaching a value 1/9 in the spectrum occurs a roton-like minimum at the midpoints of the BZ edges (Figure 5c).These characteristics are unlike the prototype model [1] (J 3 = 0) whose appearance is due to the frustration effect.The dispersion curves at zero temperature for different values of α are plot in Figure 5d.The temperature dependence of the magnon excitation spectrum is shown in Figure 6.In the low temperature region t < 0.5, the energy spectrum remains virtually unchanged compared to it at zero temperature.In the temperature region near the critical temperature, although the magnon excitation spectrum changes significantly, its shape is remained unchanged.This is because the magnon spectrum given by the expression (7) at temperature zero and temperature t differs only by the multiplication factor m 0 (t).

Discussion and Conclusions
The above results demonstrate that the considered Heisenberg model has the classical longrange-order in the form of the three-sublattice structure of the Néel type.At zero temperature, quantum fluctuations combined with geometric frustration have a great influence on the Néel order.Although they do not break this structure, they reduce the sublattice magnetization from 26.13% compared to its classical value.When they are enhanced by thermal fluctuations, the sublattice magnetization decreases rapidly with increasing temperature.We have calculated the maximum critical temperature of the model corresponding to α = 0 (ie excluding the thirdnearest-neighbor interaction, J 3 = 0) is T N = 0.345J 1 /k B (K).When the temperature crosses the critical temperature T N the sublattice magnetization m = 0, which means that no long-range In this paper, the PF function integral method and the Luttinger-Tisza process were employed to find the expression of the magnon spectrum and sublattice magnetization.Starting from the mean field hypothesis, we take into account the first-order contribution in the loop expansion to study the effect of fluctuation corrections on the mean field in the Gaussian approximation.To fully describe the picture of the magnetic properties of new synthetic layered materials containing N i 2+ ions with spin S = 1, it may be necessary to include all three interactions J 1 − J 2 − J 3 , anisotropy of the space or include other additional interactions such as biquadratic interactions.This will be our future research.

Figure 1 .
Figure 1.The triangular lattice with coupling J 1 and J 3 .

Figure 3 .
Figure 3. Sublattice magnetization m as a function of α.

Figure 4 .
Figure 4.The left for temperature dependence of mean field magnetization m o : Dashed line: average-constraint, solid line: exact-constraint; The right for temperature and frustrated coupling parameter dependence of one-loop corrected magnetization m.

48thFigure 5 .
Figure 5.The dispersions are shown in the ONMP' quadrant of the BZ in the triangular lattice HAF and the dependencies of the spin-excitation frequency on the wave vector along the line OMPN for different values of the frustration parameter α at T = 0.

48thFigure 6 .
Figure 6.The dispersion of spin excitations E(k) for different values of the frustration parameter α at t = 0 and t = 1.5 (where t = k B T /J 1 is the dimensionless temperature).