Characteristic properties of excited states in anitiferromagnetic ladder systems with spin-1/2 are investigated in relation with the theoretical analysis of the quite asymmetric shape of the peak for twice the spin gap Δg in magnetic Raman spectra. Here, the structure of exchange-scattering spectra are reproduced by the numerical calculation for the finite Heisenberg ladder, in which legs with antiferromagnetic bond Jc is connected by rungs with antiferromagnetic bond Jr. The singlet ground state in this system can be expressed as the ordering of singlet dimers on rungs in the ladder for Jc/Jr →+0. In fact, the value of 0.1 is adopted to be the ratio of Jc /Jr in order to discuss the experimental work of the magnetic Raman spectra for the ladder of V4+ ions in the oxide compound CaV2O5 reported in Konstantinović et al., Phys. Rev. B61 (2000), 15185. As for this peak of 2Δg , quite an asymmetric observed shape is explained by the present calculation. According to the analysis of spectra, we discuss the expression of excited states by triplet dimers on rungs: The dominant contribution of the pair of triplet dimers on adjacent rungs to the corresponding state for 2Δg peak is pointed out.
The exchange-scattering spectra are reproduced by the numerical calculation for the finite Heisenberg ladder, in which two-legs with antiferromagnetic bond J c is connected by rungs with antiferromagnetic bond J r . The numerical calculation is made for the system of 12 ×2 sites by Lanczos method. The ratio of J c / J r is considered to be 0.6–1.0 in order to discuss the experimental work for ladders of Cu 2+ ions in oxide compounds Sr 14- x Ca x Cu 24 O 41 . In addition to the Raman edge about J r corresponding twice the singlet-triplet spin gap, the characteristic structure appears around 2 or 3 times of J r i.e. 3 or 5 times the spin-gap. We concentrate our attention on characteristic features of excited levels, transitions to which are Raman-active. In particular, the relationship between correlations for spins in those states and corresponding Raman intensities is discussed in the picture of triplet pairs excited from singlet dimers. By the perturbation theory, the essential dependence of those intensities on distances between triplet dimers is investigated.
The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange
JA to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of
2N spins-
21 with periodic boundary conditions. The ground state is a singlet
(S=0) and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin-
21 string correlation functions
g1(N) and
g2(N) are defined for an even number
N of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum
N∗ of
g2(N) is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and
g1(N) decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of
JF,JL, and
JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-
21 systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and
JA. Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at
J2/J1=1/2.
The frustrated ladder with alternate ferromagnetic(F) exchange
−JF and AF exchange
JA to first neighbors and F exchange
−JL to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of
2N spins-1/2 with periodic boundary conditions. The ground state is a singlet
(S=0) and the singlet-triplet gap
εT is finite for the exchanges considered. Spin-1/2 string correlation functions
g1(N) and
g2(N) are defined for an even number
N of consecutive spins in systems with two spins per unit cell; the ladder has string order
g2(∞)>0 and
g1(∞)=0. The minimum
N∗ of
g2(N) is related to the range of ground-state spin correlations. Convergence to
g2(∞) is from below, and
g1(N) decreases exponentially for
N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of
JF,
JL and
JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the
J1−J2 model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at
JL=JF/2≤JA, that is reversed on interchanging
−JF and
JA. Inversion symmetry is spontaneously broken in the dimer phase of the
J1−J2 model where the Kekulé diagrams are the doubly degenerate ground states at
J2/J1=1/2.
Abstract Ladders with 2 N spins-1/2 and singlet ( S = 0) ground states are studied numerically using exact diagonalization and density matrix renormalization group (DMRG) calculations. The curvatures E ″ ( s ) of states with the lowest energy E ( s ) at spin s = S / N ≤ 1 are obtained in the thermodynamic limit. All states have E ″ ( s ) > 0 in two phases, one with strong F exchange J F in rungs, the other with strong F exchange J L in legs. The Dimer phase with frustrated F exchanges has one or more states with E ″ ( s ) < 0. All three phases have finite gaps to the lowest triplet state. The ladder at J F → ∞ is adiabatically connected to the spin-1 Heisenberg AF chain, and at J L → ∞ to the J 1 − J 2 model.
Thermodynamic properties of spin=1/2 two-leg ladders with alternating chains including ferromagnetic and antiferromagnetic interactions are discussed on the basis of the numerical results obtained by quantum Monte Carlo simulations. As well as the role of exchange interaction of rungs, the essential role of the bond alternation in the chain on ladder systems is investigated in connection with the appearance of characteristic properties in magnetic susceptibilities and heat capacities. Furthermore, quantum effects on the ground-state energy are obtained. On the basis of calculated results, the relationship between singlet dimer phases and Haldane states is discussed.