Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
We explore the ground-state phase diagram of the S = 1/2 two-leg ladder. The isotropic leg interactions J 1, a and J 1, b between nearest neighbor spins in the legs a and b , respectively, are different from each other. The xy and z components of the uniform rung interactions are denoted by J r and Δ J r , respectively, where Δ is the XXZ anisotropy parameter. This system has a frustration when J 1, a J 1, b < 0 irrespective of the sign of J r . The phase diagrams on the Δ (0≤Δ<1) versus J 1, b plane in the cases of J 1, a = − 0.2 and J 1, a = 0.2 with J r = −1 are determined numerically. We employ the physical consideration, the level spectroscopy analysis of the results obtained by the exact diagonalization method and also the density-matrix renormalization-group method. It is found that the non-collinear ferrimagnetic (NCFR) state appears as the ground state in the frustrated region of the parameters. Furthermore, the direct-product triplet-dimer (TD) state in which all rungs form the TD pair is the exact ground state , when J 1, a + J 1, b = 0 and 0≤ Δ ≲ 0.83. The obtained phase diagrams consist of the TD, XY and Haldane phases as well as the NCFR phase.
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.
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.
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.