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Abstract. We explore, at T = 0, the magnetic properties of the S = 1/2 antiferromagnetic distorted diamond chain described by the Hamiltonian H =, which well models A 3 Cu 3 (PO 4 ) 4 with A = Ca, Sr, Bi 4 Cu 3 V 2 O 14 and azurite Cu 3 (OH) 2 (CO 3 ) 2 . We employ the physical consideration, the degenerate perturbation theory, the level spectroscopy analysis of the numerical diagonalization data obtained by the Lanczos method and also the density matrix renormalization group (DMRG) method. We investigate the mechanisms of the magnetization plateaux at M = M s /3 and M = (2/3)M s , and also show the precise phase diagrams on the (J 2 /J 1 , J 3 /J 1 ) plane concerning with these magnetization plateaux, where M = N l=1 S z l and M s is the saturation magnetization. We also calculate the magnetization curves and the magnetization phase diagrams by means of the DMRG method.

We study the ground state of the model Hamiltonian of the trimerized S = 1/2 quantum Heisenberg chain Cu3Cl6(H2O)22H8C4SO2 in which a non-magnetic ground state was observed recently. This model consists of stacked trimers and has three kinds of coupling constant describing the couplings between spins: the intra-trimer coupling constant J1 and the inter-trimer coupling constants J2 and J3. All of these constants are assumed to be antiferromagnetic. By use of an analytical method and physical considerations, we show that there are three phases on the 2 - 3 plane (2J2 / J1, 3J3 / J1): the dimer phase, the spin-fluid phase and the ferrimagnetic phase. The dimer phase is caused by the frustration effect. In the dimer phase, there exists an excitation gap between the twofold-degenerate ground state and the first excited state, which explains the non-magnetic ground state observed in Cu3Cl6(H2O)2·2H8C4SO2. We also obtain the phase diagram on the 2 - 3 plane from the numerical diagonalization data for finite systems by use of the Lanczos algorithm.

The ground-state phase diagram of frustrated S = 1 XXZ spin chains with the competing nearest-and next-nearest-neighbor antiferromagnetic couplings is studied using the infinitesystem density-matrix renormalization-group method. We find six different phases, namely, the Haldane, gapped chiral, gapless chiral, double Haldane, Néel, and double Néel (uudd) phases. The gapped and gapless chiral phases are characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, κ l =S , whereas the spin correlation decays either exponentially or algebraically. These chiral ordered phases appear in a broad region in the phase diagram for ∆ < 0.95, where ∆ is an exchange-anisotropy parameter. The critical properties of phase transitions are also studied.KEYWORDS: frustration, spin-1 chain, ground-state phase diagram, density-matrix renormalization-group method, chiral ordering §1. Introduction Frustrated antiferromagnetic quantum spin systems have attracted considerable attention over the decades since they exhibit a wealth of fascinating phenomena in their ground states and lowlying excitations. In general, frustration supresses antiferromagnetic correlations and the tendency towards the Néel order. Classical systems, for example, often show a helical ordered state in their ground states in the presence of strong frustration. In quantum systems, the interplay of frustration and quantum fluctuations plays an important role which causes exotic phenomena, e.g., a spin-liquid state and a novel type of spontaneous symmetry breaking.In this paper, we study a one-dimensional anisotropic spin system with the antiferromagnetic nearest-neighbor coupling J 1 and the frustrating next-nearest-neighbor coupling J 2 . The model is * E-mail address: hikihara@phys03.phys.sci.kobe-u.ac. where S l is a spin-S spin operator at site l and ∆ ≥ 0 represents an exchange anisotropy. Hereafter, we put j ≡ J 2 /J 1 (j ≥ 0).In the classical limit, S → ∞, the system exhibits a magnetic long-range order (LRO) characterized by a wavenumber q. The order parameter is defined bywhere L is the total number of spins. While the magnetic LRO is of the Néel-type (q = π) when j is smaller than a critical value, j ≤ 1/4, it becomes of helical-type for j > 1/4 with q = cos −1 (−1/4j).It should be noticed that both the time-reversal and parity symmetries are broken in this helical ordered state. When the system has an XY -like anisotropy (∆ < 1), the helical ordered state possesses a two-fold discrete degeneracy according as the helix is either right-or left-handed, in addition to a continuous degeneracy associated with the original U (1) symmetry of the XY spin.This discrete degeneracy is characterized by mutually opposite signs of the total chirality definedNote that this chirality is distinct from the scalar chirality of the Heisenberg spin often discussed in the literature 2) defined by χ l = S l−1 · S l × S l+1 : The chirality O κ changes its sign under the parity operation but is invariant under the time-reversal op...

Ordering of frustrated S=1/2 and 1 XY and Heisenberg spin chains with the competing nearest-and next-nearest-neighbor antiferromagnetic couplings is studied by exact diagonalization and densitymatrix renormalization-group methods. It is found that the S=1 XY chain exhibits both gapless and gapped 'chiral' phases characterized by the spontaneous breaking of parity, in which the longrange order parameter is a chirality, κi=S, whereas the spin correlation decays either algebraically or exponentially. Such chiral phases are not realized in the S=1/2 XY chain nor in the Heisenberg chains.Ordering of frustrated quantum spin chains has attracted considerable interest since these systems exhibit a rich variety of magnetic phases due to the interplay between quantum effect and frustration. We consider here an anisotropic frustrated quantum spin chain described by the XXZ Hamiltonian,where S ℓ is the spin-S spin operator at the ℓth site, J ρ > 0 is the antiferromagnetic interaction between the nearest-neighbor (ρ=1) and the next-nearest-neighbor (ρ=2) spin pairs, and λ (0 ≤ λ ≤ 1) represents an exchange anisotropy. Note that λ = 0 and λ = 1 correspond to the XY and Heisenberg chains, respectively.The ground state phase diagram of the corresponding S=1/2 system has been extensively studied either numerically [1,2] or analytically [3,4]. These studies have revealed that when J 2 is smaller than a critical value, i.e., j≡J 2 /J 1 ≤ j c , the system is in the gapless spin-fluid phase in which the antiferromagnetic spin correlation decays algebraically. By contrast, for larger values of j > j c , the system is in the dimer phase with a finite energy gap above the doubly degenerate ground states. The dimer phase is characterized by the spontaneously breaking of both parity and translation symmetries with preserving time-reversal symmetry. The value of j c has been estimated to be j c ∼ =0.241 for the Heisenberg chain [2]. Although there is no magnetic long-range order (LRO), the nature of the magnetic short-range order (SRO) changes at the Lifshitz point j L , where j L ≃0.5 for the Heisenberg chain [1]. For j ≤ j L , the system has the standard Néel-type antiferromagnetic SRO and the structure factor S(q) has a maximum at q = π, while for j > j L , the system has a helical SRO with the maximum of S(q) at some q=Q < π.In the case of S=1, by contrast, no dimer phase occurs [5,6]. The Heisenberg chain is in the Haldane phase characterized by a singlet ground state and a finite energy gap above it. A first-order transition takes place at j=j T ≃ 0.744 between the 'single-chain' Haldane phase at j < j T and the 'double-chain' Haldane phase at j > j T [6]. In the XY case, on the other hand, the situation remains not entirely clear. Analytical studies based on the bosonization method suggested that the gapless phase at j = 0 (the so-called XY 1 phase) extended to finite j > 0 [7] whereas numerical studies suggested that the Haldane phase was stabilized for j > 0 [8]. In any case, the fate of such XY 1 or Haldane phase at larger j has...