Ground state with nonzero spontaneous magnetization of the two-dimensional spin-1/2 Heisenberg antiferromagnet with frustration

The S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice is studied by the numerical-diagonalization method. This lattice is obtained by the combination of the Lieb lattice and the square lattice. It is known that when interaction on the square lattice is increased from the ferrimagnetic limit of strong interaction on the Lieb lattice, this system shows gradual decrease and disappearance of spontaneous magnetization in the ground state. The present study treats the region near the case of the square-lattice antiferromagnet accompanied by isolated spins by numerical-diagonalization calculations of finite-size clusters with the maximum size of 39 sites. Our numerical results suggest the existence of a new phase with small but nonzero spontaneous magnetization between two zero-spontaneous-magnetization phases.


I. INTRODUCTION
Ferrimagnetism is a fundamental magnetic phenomenon which shows ferromagnetic nature and antiferromagnetic nature at the same time. It is widely known that the ferrimagnetism is mathematically understood by Marshall-Lieb-Mattis (MLM) theorem 1,2) . The MLM theorem holds only when systems do not include magnetic frustration. It is a nontrivial problem how the ferrimagnetism is suppressed and disappears quantum-mechanically by such a frustrating situation. From such a viewpoint, there are several reports [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] which studied several models on lattices of various types. Some of them show intermediate states with spontaneous magnetization whose magnitude is smaller than that determined by the MLM theorem.
Among systems showing spontaneously magnetized intermediate states, the triangular-lattice Heisenberg antiferromagnet with distortion of the √3×√3 type shows a fascinating phenomenon [17][18][19][20] . The spontaneous magnetization decreases from the magnitude for the dice-lattice case 21) through smaller spontaneous magnetization to zero spontaneous magnetization for the undistorted triangular-lattice case. However, a recent report based on a numerical-diagonalization study 20) clarified the a Electronic mail: sakai@spring8.or.jp. b Electronic mail: hnakano@sci.u-hyogo.ac.jp existence of another spontaneous-magnetization phase in the region of the interaction controlling the distortion is further varied.
The new phase with spontaneous magnetization is different from the ferrimagnetic phase near the dice-lattice case and the previously reported spontaneous-magnetization phase neighboring the ferrimagnetic phase. One of the characteristic behaviors of the new phase is the discontinuity in the spontaneous magnetization at one boundary of the phase whereas the continuity at the other boundary. The nontrivial behavior of the distorted triangular-lattice system is confirmed by a density-matrixrenormalization-group study 22) . However, to understand the nontrivial behavior of the new spontaneous-magnetization state deeply, it is necessary to investigate another system showing the same behavior of such a spontaneous-magnetization phase.
Under these circumstances, the purpose of this study is to present a new system that shows such a spontaneous-magnetization phase. The lattice of the system is composed of the Lieb-lattice and square-lattice networks of antiferromagnetic interactions 23) . This paper is organized as follows. In the next section, the target system is introduced. The method used in my calculations is also explained. The third section is devoted to the presentation and the discussion of numerical results. In the final section, a summary of this study is given.

II. MODEL HAMILTONIAN AND METHOD
The Hamiltonian treated in this study is given by where S i denotes the S = 1/2 spin operator at site i. The case of isotropic interaction in spin space is considered. The site i is assumed to be the vertices of the Lieb lattice. The Lieb lattice is composed by interaction bonds J 2 when J 1 = 0. The vertices are divided into three sublattices A, B, and B'. Each site i in the A sublattice is linked by four bonds of J 2 ; each site i in the B or B' sublattice is linked by two bonds of J 2 and four bonds of J 1 . The local structure in each unit cell of the lattice looks like a pyramid cone. The square bottom face of each cone shares its vertices with the pyramid cone in a neighboring unit cell. Each site of B and B' sublattice is a vertex of the two-dimensional square lattice composed of J 1 bonds. Therefore, we will call the present lattice the two-dimensional pyramid (2DP) lattice hereafter. Note also that the 2DP lattice seems similar to the checkerboard lattice [24][25][26] ; the existence of A-site vertices in the present lattice is a significant difference from the checkerboard lattice. The number of spin sites is denoted by N; N/3 should be an integer because a unit cell includes three spins of A, B, and B' sublattices. We denote the ratio of J 2 /J 1 by r. Energies are measured in units of J 1 ; hereafter, we set J 1 = 1. We consider that all interactions are antiferromagnetic, namely, J 1 > 0 and J 2 > 0. Note that for J 2 → ∞, namely, r → ∞, J 1 gets relatively smaller; the 2DP lattice structure then forms the so-called Lieb lattice. Numerical-diagonalization study 23) clarified that the decreasing r makes the spontaneous magnetization get smaller gradually and disappear until r ～ 2. In the present study, on the other hand, we mainly investigate the region of r ≤ 2.
The finite-size clusters that we treat in the present study are depicted in Fig. 1, namely, N = 24, 27, 30, and 39. The periodic boundary condition is imposed for all N. Note here that all the clusters form a regular square although the clusters for N = 24, 30, and 39 are tilted. Since the cluster shape is a square, it is expected to capture the two-dimensionality of the system well. which was originally developed in the study of Haldane gaps 27) . The usefulness of our program was confirmed in large-scale parallelized calculations [28][29][30][31][32][33] .

III. RESULTS AND DISCUSSIONS
First, let us observe our numerical results for the M-dependence of the lowest-energy level; the observation makes us understand how to determine the spontaneous magnetization for a given N. Next, we examine the change of the spontaneous magnetization when r = J 2 /J 1 is varied around the region below r = 2; results are depicted in Fig. 3(a). For N = 24, we do not find the region of nonzero spontaneous magnetization; results for N = 24. For larger N, on the other hand, we actually find the region of nonzero spontaneous magnetization. It is noticeable that two steps indicating nonzero spontaneous magnetization appear for N = 30 and 39 whereas only a step appears for N = 27. This behavior for larger systems suggests that the appearance of this region is not a finite-size effect.
Then, let us examine the boundary of the region of nonzero spontaneous magnetization. For this examination, with decreasing r, we define r c2 as the value of r where M spo changes from 0 or 1/2 to larger values and r c1 as the value of r where M spo decreases again to 0 or 1/2 from larger values. The system size dependences of r c1 and r c2 are depicted in Fig. 3(b). One finds that both the dependences are small between N = 30 and 39 although the dependences between N = 27 and 30 are relatively larger. The small dependences for larger N suggest that the width of this region survives when N is increased. Therefore, these two panels of Fig. 3 gives a consequence that the region of nonzero spontaneous magnetization certainly exists as a macroscopic phenomenon in the thermodynamic limit.
Note here that the discontinuous behavior of M spo at r c1 appears whereas the continuous behavior at r c2 . To capture well the discontinuous behavior at r c1 , let us observe the M-dependence of the ground-state energy; our numerical results for N = 30 and 39 are depicted in Fig. 4.

IV. SUMMARY
We have studied the S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice by the numericaldiagonalization method. We find a region of nontrivial states showing spontaneous magnetization in the region near the case of the square-lattice system accompanied by isolated spins. When the strength of the frustrating interaction is increased from the square-lattice system accompanied by isolated spins, the spontaneous magnetization discontinuously appears at r = r c1 , but the spontaneous magnetization gradually decreases and continuously disappears at r = r c2 . The behavior of this system is similar to the one of the triangular-lattice system with distortion. Further comparison between these two cases contributes much for our understanding of frustration effects in magnetic materials.