We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a two-leg ladder geometry using the density-matrix renormalization-group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=sin\ensuremath{\phi}$ and $J=cos\ensuremath{\phi}$ with an angle parameter $\ensuremath{\phi}$. Based on the results for several types of order parameters, excitation gaps, string order parameter, and entanglement spectra, the $\ensuremath{\phi}$-dependent ground state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range-ordered states, namely rung singlet (analog to N\'eel in three dimensions), zigzag, ferromagnetic, and stripy, and the presence of gapped spin liquids around the exactly solvable Kitaev points $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\ensuremath{\pi}$-flux state in order to establish how the gapped Kitaev spin liquid extends away from the $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.
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.
We describe a numerical simulation of the isotropic spin-\textonehalf{} Heisenberg antiferromagnet on square lattices of size 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. A Hamiltonian Monte Carlo method is used to measure the energies of the lowest-lying singlet and triplet states, and we extrapolate these results to give estimates of the properties of large antiferromagnets. Our result for the ground-state energy per spin is $\frac{{E}_{0}}{{N}^{2}}=\ensuremath{-}0.672\ifmmode\pm\else\textpm\fi{}0.001$ for an infinite system. Within our statistical accuracy, the singlet-triplet gap behaves as ${E}_{1}\ensuremath{-}{E}_{0}\ensuremath{\approx}\frac{2.1}{N}$, which implies that the infinite system is gapless.
The two-layer Heisenberg antiferromagnet exhibits a zero temperature quantum phase transition from a disordered dimer phase to a collinear Neel phase, with long range order in the ground state. The spin-wave gap vanishes as ? ? ( J ? – J ? c ) &ngr; approaching the transition point. To account for strong correlations, the S = 1 elementary excitations triplets are described as a dilute Bose gas with infinite on-site repulsion. We apply the Brueckner diagram approach which gives the critical index &ngr; ˜ 0 . 5. We demonstrate also that the linearised in density Brueckner equations give the mean field result &ngr; = 1. Finally, an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, gives &ngr; ˜ 0 . 67. This value agrees reasonably with that of the nonlinear O(3) s model. Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear s model. We discuss the conditions for this to occur.
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.