Quantum spin model with frustration on the union jack lattice

Frustrated lattice spin models in two dimensions have attracted much discussion in recent years. They exhibit new and interesting phase structures and phase transitions; in particular, they may develop “spin-liquid” states, without longrange order.1 It is also believed that frustrating interactions may play a role in the high-temperature superconducting cuprate materials. Primary examples are the anisotropic triangular lattice Heisenberg antiferromagnet Fig. 1 a , the square lattice J1-J2 model Fig. 1 b , and the ShastrySutherland model Fig. 1 c . In this paper we discuss another member of this group, the Heisenberg antiferromagnet on the union jack lattice Fig. 1 d , which is another frustrated lattice model and might be expected to display some interesting properties. The spin-1 /2 J1-J2 model on the square lattice 2 involves antiferromagnetic Heisenberg spin interactions with coupling J1 between nearest neighbors and coupling J2 between diagonal next-nearest neighbors, as illustrated in Fig. 1 b . The union jack lattice model has J2 interactions on only half the diagonal bonds, in the pattern shown in Fig. 1 d , so that the lattice consists of two different site types A and B and the unit cell is 2 2 sites. Both models will exhibit quantum phase transitions as the coupling ratio =J2 /J1 is varied. In the J1-J2 model at small , the J1 coupling is dominant and produces antiferromagnetic Néel ordering of the spins. At large , the J2 interaction is dominant and produces antiferromagnetic ordering on the two diagonal sublattices, and then the effect of the J1 interaction is to align the two sublattices to form a columnar ordered state as illustrated in Fig. 2, an example of the “order-by-disorder” phenomenon. Numerical investigations3–5,7–10,13 have shown that the boundaries of these two phases lie at 0.38 and 0.60, respectively. The nature of the intermediate phase or phases remains controversial. Monte Carlo simulations are hampered by the “minus sign” problem, exact diagonalizations are limited to small lattices, and series expansions are based on some particular ordered reference state and are only valid within a single phase. It is generally believed that the intermediate phase is gapped and shows no long-range magnetic order. Field-theory approaches5,6 and dimer series expansions5,7–9 seem to indicate a columnar dimerized state in the intermediate region, with spontaneous breaking of translational symmetry, as illustrated in Fig. 2. Capriotti and co-workers, on the other hand, have suggested a homogeneous spin-liquid plaquette resonant valence bond RVB state10,11 and have found that exact diagonalization up to 6 6 sites shows no strong evidence of dimerization.12 Another Monte Carlo study has suggested a columnar dimer state with plaquettetype modulation.13 Sushkov et al.9 have even suggested that there may be three different phases in the intermediate region: reading from left to right, a Néel state with columnar dimerization, a columnar dimerized spin liquid, and a columnar dimerized spin liquid with plaquette-type modulation. Several discussions have centered on the Lieb-SchulzMattis theorem in higher dimensions,14–16 which shows that for a spin system with half-integer spin per unit cell, there is an excitation energy gap behaving like 1/L, where L is the linear size of the system. Takano et al.17 have argued that a uniform RVB state without gapless singlet excitations is excluded by the theorem and that the true ground state is a plaquette state with spontaneously broken translation invariance and fourfold degeneracy. Later arguments11,16,18 have refuted this, however, and shown that the theorem may be satisfied if the translation symmetry remains unbroken, but the ground state has a fourfold “topological” degeneracy instead, as in a simple dimer model.