Ground-state properties of spin-1/2 Heisenberg antiferromagnets with frustration on the diamond-like-decorated square and triangular lattices

The exploration of the frustration and quantum effects in the spin system has been one of the most interesting issues in condensed matter physics. In particular, significant interest has been focused on the spin-1/2 frustrated Heisenberg antiferromagnets on a diamond-like-decorated square lattice^1–4 because the Rokhsar-Kivelson quantum dimer model (QDM),⁵ which is a phenomenological model of the resonating valence bond (RVB)⁶ theory, can be realized as a low-energy effective Hamiltonian.^2,3 Our recipe for constructing the QDM can be adopted not only for the square lattice but also for other lattices. In this study, we investigate the ground-state phase diagrams and properties of spin-1/2 Heisenberg models on the diamond-like-decorated square and triangular lattices.

A diamond-like-decorated square (triangular) lattice is a lattice in which the bonds of a square (triangular) lattice are replaced with diamond units, as shown in Fig. 1(a) (Fig. 1(b)). For this lattice, if we define the interaction strength of the four sides of a diamond unit as J and that of the diagonal bond as J′ = λJ, the ratio λ determines the ground-state properties.¹ As shown in Fig. 1(c), we denote the four S = 1/2 operators in a diamond unit as s_i, s_j, s_k,a, and s_k,b. Therefore, the Hamiltonian can be written as

where ⟨i, j⟩ represents a nearest-neighbor pair of the square and triangular lattices. Here, we call s_i and s_j the edge spins (closed circles in Fig. 1(c)) and the pair (s_k,a, s_k,b) as the bond spin-pair (open circles). In Eq. (1), note that the energy of the bond spin-pair is measured from that of the singlet dimer.

For both the diamond-like-decorated square and triangular lattices, we obtain three types of ground-state phases: the dimer-monomer (DM) states for λ > 2, the macroscopically degenerated tetramer-dimer (MDTD) states for λ_c < λ < 2, and the ferrimagnetic states for λ < λ_c, where the value of λ_c depends on the lattices. In particular, the MDTD states are very fascinating because they are equivalent to the dimer covering states of the original lattice,^1,7 and we can derive a QDM as the second-order effective Hamiltonian by introducing the further neighbor couplings.^2,3 Recently, we have found that one of our obtained QDMs has a region of λ with the possibility of realizing the RVB state. One of the purposes of this study is to determine the phase boundaries λ_c between the MDTD and ferrimagnetic states. Therefore, using the modified spin wave (MSW) method⁸ and the quantum Monte Carlo (QMC) method,⁹ we estimate the ferrimagnetic ground-state energies, for which we can consider the mixed spin-1 and spin-1/2 Lieb-lattice and triangular Lieb-lattice Heisenberg antiferromagnets instead, and obtain the phase boundaries $λc(square)=0.974$ and $λc(triangular)=0.988$⁠. We also calculate the long-range order (LRO) parameters using the MSW and QMC methods and find the scaling relations where the spin reductions of each sublattice are inversely proportional to the number of sublattice sites. We prove these scaling relations by applying an infinitesimal uniform magnetic field to cause a spontaneous symmetry breaking. Furthermore, using the Marshall-Lieb-Mattis theorem and the MSW theory, we focus on the total spin of the ground state and examine the mathematical structure behind the scaling relations for the sublattice LRO parameters.¹⁰ 

This paper is organized as follows. In Section II, we show the ground-state properties and energies of each state on the diamond-like-decorated square and triangular lattices and determine the phase boundaries $λc(square)$ and $λc(triangular)$⁠. In Section III, we show the scaling relations and examine the mathematical structure behind the scaling relations for the sublattice LRO parameters. In Section IV, we summarize the results obtained in this study.

In this section, we show the ground-state properties and energies of the DM states, MDTD states, and ferrimagnetic states, and determine the phase boundaries $λc(square)$ and $λc(triangular)$ on the diamond-like-decorated square and triangular lattices, respectively.

In the case of λ > 2, we obtain the DM states, which have singlet dimers on all of the bond spin-pairs and monomers (free spins) on all of the edge spins, as shown in Figs. 2(a) and 2(b). It is worth noting that the singlet dimers on the bond spin-pairs cause the four interactions J in the diamond unit to vanish effectively, i.e., J(s_i + s_j) ⋅ (s_k,a + s_k,b)P_k,a;k,b = 0, where P_k,a;k,b = 1/4 − s_k,a ⋅ s_k,b is the projection operator on to the singlet sector.¹ Thus, from Eq. (1), both the ground-state energies of the DM states on the diamond-like-decorated square and triangular lattices are zero:

In the case of λ_c < λ < 2, we obtain the MDTD states, which have a nontrivial macroscopic degeneracy. In the MDTD states, as shown in Figs. 3(a) and 3(b), diamonds with triplet dimers (shaded blue ovals) and with singlet dimers (unshaded red ovals) are arranged where the number of diamonds with triplet dimers is the largest and two or more diamonds with triplet dimers are not placed continuously, which can be explained by the variational method and the Lieb-Mattis theorem.¹ Furthermore, the eigenstate of one diamond with a triplet dimer becomes a nonmagnetic tetramer–singlet state, which is given by $|ϕgi,j,k=(|t+i,j|t−k+|t−i,j|t+k−|t0i,j|t0k)/3$ where {|t⁺⟩_k, |t⁰⟩_k, |t⁻⟩_k} and {|t⁺⟩_i,j, |t⁰⟩_i,j, |t⁻⟩_i,j} represent the triplet states of the bond spin-pair and of the edge spins, respectively. The tetramer–singlet state is the same as the plaquette RVB state. Because the MDTD states consist only of the tetramer–singlet states and the singlet dimers, the MDTD states are of the nonmagnetic ground state. The ground-state energy of the tetramer–singlet state is obtained as J(λ − 2) and the number of the tetramer–singlet states in the whole system is N/2, where N is the total number of lattice points, which is the same in both the diamond-like-decorated square and triangular lattices. Therefore, both the ground-state energies of the MDTD states on both the diamond-like-decorated square and triangular lattices are obtained by

If we regard a tetramer singlet as a “dimer” in the square (triangular)-lattice QDM, the MDTD states are identical to the square (triangular)-lattice dimer-covering states,^1,7 because the tetramer–singlet state does not contain an orientation and our dimer coverings have orthogonality. It is noteworthy that the transition of triplet (singlet) dimers between diamonds does not occur because the $(Sk,a+Sk,b)2$ is a conserved quantity. When we introduce the further neighbor couplings, the transition of triplet (singlet) dimers becomes possible and the QDM can be derived as the second-order effective Hamiltonian.^2,3

As mentioned above, the MDTD states are stabilized in the region of λ_c < λ < 2; however, in the region of λ < λ_c, the systems can be in the ferrimagnetic states, which contain triplet dimers on all of the bond spin-pairs, as shown in Figs. 4(a) and 4(b). It is noteworthy that the ferrimagnetic states are identical to the mixed spin systems on the Lieb lattice and triangular Lieb-lattice, as shown in Figs. 4(c) and 4(d), when we regard the triplet dimers as the spin-1 sites. Therefore, we define the mixed-spin Hamiltonian on the Lieb lattice and triangular Lieb-lattice as $H̃=∑i∈A∑j∈BJi,jSi⋅Sj$⁠, where J_i,j = J is for the nearest-neighbor pairs, otherwise J_i,j = 0, $Si2=SA(SA+1)$ for the A-sublattice sites i (closed circles in Figs. 4(c) and 4(d)), and $Sj2=SB(SB+1)$ for the B-sublattice sites j (closed triangles). The above-mentioned ferrimagnetic states correspond to the case of (S_A, S_B) = (1/2, 1). Using the MSW and QMC methods, the ferrimagnetic ground-state energies on the diamond-like-decorated square and triangular lattices are obtained by

and

In our QMC method, we use three points L = 16, 24, 32 (⁠$L̃=32,48,64$⁠) for the definition of L² ≡ N (⁠$L̃2≡4N$⁠) in the case of the Lieb lattice (triangular Lieb-lattice) to create a linear extrapolation with respect to L⁻³ (⁠$L̃−3$⁠) and set the temperature as T = 0.001J, which can be regarded as the absolute zero temperature in the system sizes used.

From Eqs. (3), (4), and (5), we obtain the phase boundaries between the MDTD and the ferrimagnetic states on the diamond-like-decorated square- and triangular lattices:

and

The equations above show a good agreement within three digits after the decimal point. Therefore, by comparing the results of the two methods above, we finally determine that $λc(square)=0.974$ and $λc(triangular)=0.988$⁠.

Note that we assume the direct transitions from the MDTD states to the ferrimagnetic states at λ = λ_c. However, there is a possibility that the new states exist between these states, but it has not been found yet.¹ 

In this section, we consider the LRO parameters $mA(SA)$ for the A-sublattice and $mB(SB)$ for the B-sublattice on the Lieb lattice and triangular Lieb-lattice using the MSW and QMC methods. In our QMC method, we used three points L = 16, 24, 32 (⁠$L̃=32,48,64$⁠) in the case of the Lieb lattice (triangular Lieb lattice) to create a linear extrapolation with respect to L⁻¹ (⁠$L̃−1$⁠) for $mA(SA)=SA/NA$ and $mB(SB)=SB/NB$⁠, where $SA=1/NA∑i,j∈A⟨Si⋅Sj⟩$ and $SB=1/NB∑i,j∈B⟨Si⋅Sj⟩$ are the structure factors for A- and B-sublattices; N_A and N_B are numbers of the A- and B-sublattice sites. It is noteworthy that we have (N_A, N_B) = (N, 2N) and (N_A, N_B) = (N, 3N) on the Lieb lattice and triangular Lieb-lattice, respectively.

In Table I, we present the calculated results for the (S_A, S_B) = (1/2, 1) systems and obtain

in both the MSW and QMC methods, where $Δmα(Sα)≡Sα−mα(Sα)(α=A,B)$ denotes the spin reduction. It is noteworthy that, in the MSW, Eq. (8) holds exactly in the analytical expressions; and in the QMC, it holds in the range of the statistical error. Equation (8) indicates that the scaling relations exist where the spin reductions of each sublattice are inversely proportional to the number of sublattice sites. We can prove these scaling relations by applying an infinitesimal uniform magnetic field and causing the spontaneous symmetry breaking. However, we find that, even in the singlet state where the uniform fields do not give a spontaneous symmetry breaking, the scaling relations

hold. Note that the cases of (S_A, S_B) = (1, 1/2) on the Lieb lattice and of (S_A, S_B) = (3/2, 1/2) on the triangular Lieb-lattice give S_tot = N_BS_B − N_AS_A = 0, which is singlet state. Therefore, using the Marshall-Lieb-Mattis theorem and the MSW theory, we focus on the total spin S_tot of the ground state and examine the mathematical structure behind the scaling relations for sublattice LRO parameters. As a results, we find that the $Ntot2$ term of $⟨Stot2⟩$ is related to the scaling relations and this term is not affected by introducing an infinitesimal symmetry-breaking field, where N_tot = N_A + N_B and $Stot2=Stot(Stot+1)$⁠. On the other hand, introducing a symmetry-breaking field, the N_tot term of $⟨Stot2⟩$ has a positive value, which causes the generation of spontaneous staggered magnetization.¹⁰ 

We studied the ground-state phase diagram of spin-1/2 Heisenberg models on the diamond-like-decorated square and triangular lattices that contained three types of ground-state phases: the DM states, MDTD states, which are equivalent to the dimer covering states of the original lattices, and the ferrimagnetic states. We obtained the phase boundaries $λc(square)=0.974$ and $λc(triangular)=0.988$ between the MDTD and ferrimagnetic states employing the MSW and QMC methods.

We also calculated the LRO parameters of the mixed spin-1 and spin-1/2 Lieb-lattice and triangular Lieb-lattice Heisenberg antiferromagnets using the MSW and QMC methods. We showed the scaling relations for the sublattice spin-reductions, which can be proven applying a uniform field to cause a spontaneous symmetry breaking. Furthermore, using the Marshall-Lieb-Mattis theorem and examining the calculation process in the MSW, we clarified the mathematical structure behind the scaling relations for sublattice LROs.

We acknowledge Professor A. Oguchi for the helpful discussions. The authors thank the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo for the use of the facilities. This work was supported by JSPS KAKENHI Grant Numbers JP17J05190 and JP17K05519.