Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling ${\mathit{J}}_{1}$ and frustrating second-neighbor coupling ${\mathit{J}}_{2}$, particularly in the limit ${\mathit{J}}_{2}$\ensuremath{\gg}${\mathit{J}}_{1}$. This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.
Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling J1 and frustrating second-neighbor coupling J2, particularly in the limit J2\ensuremath{\gg}J1. This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.
Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange J2. For \ensuremath{\delta}=0, the system is gapless for J2J2c and has a gap for J2>J2c where J2c is about 0.241. For J2=J2c, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the J2-\ensuremath{\delta} plane, there is a disorder line 2J2+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at qmax=\ensuremath{\pi} (N\'eel phase), while to the right of the line, qmax decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as J2 is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.
The interplay between the Kondo effect, indirect magnetic interaction, and geometrical frustration is studied in the Kondo lattice on the one-dimensional zigzag ladder. Using the density-matrix renormalization group, the ground-state and various short- and long-range spin- and density-correlation functions are calculated for the model at half filling as a function of the antiferromagnetic Kondo interaction down to J=0.3t, where t is the nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown to lead to at least two critical points: Starting from the strong-J limit, where almost local Kondo screening dominates and where the system is a nonmagnetic Kondo insulator, antiferromagnetic correlations between nearest-neighbor and next-nearest-neighbor local spins become stronger and stronger, until at ${J}_{\mathrm{c}}^{\mathrm{dim}}\ensuremath{\approx}0.89t$ frustration is alleviated by a spontaneous breaking of translational symmetry and a corresponding transition to a dimerized state. This is characterized by antiferromagnetic correlations along the legs and by alternating antiferro- and ferromagnetic correlations on the rungs of the ladder. A mechanism of partial Kondo screening that has been suggested for the Kondo lattice on the two-dimensional triangular lattice is not realized in the one-dimensional case. Furthermore, within the symmetry-broken dimerized state, there is a magnetic transition to a ${90}^{\ensuremath{\circ}}$ quantum spin spiral with quasi-long-range order at ${J}_{\mathrm{c}}^{\mathrm{mag}}\ensuremath{\approx}0.84t$. The quantum-critical point is characterized by a closure of the spin gap (with decreasing J) and a divergence of the spin-correlation length and of the spin-structure factor S(q) at wave vector $q=\ensuremath{\pi}/2$. This is opposed to the model on the one-dimensional bipartite chain, which is known to have a finite spin gap for all J>0 at half filling.
A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-21 chain with frustrated antiferromagnetic exchange J1 and J2 between first and second neighbors. The modified algorithm yields accurate results up to ${J}_{2}/{J}_{1}\ensuremath{\approx}4$ for the magnetic gap $\ensuremath{\Delta}$ to the lowest triplet state, the amplitude B of the bond order wave phase, the wavelength $\ensuremath{\lambda}$ of the spiral phase, and the spin correlation length $\ensuremath{\xi}$. The J2/J1 dependences of $\ensuremath{\Delta}$, B, $\ensuremath{\lambda}$, and $\ensuremath{\xi}$ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.
Motivated by the magnetic properties of the spin-chain compounds ${\mathrm{LiCuSbO}}_{4}\ensuremath{\equiv}{\mathrm{LiSbCuO}}_{4}$ and Rb2Cu2Mo3O12, we study the ground state of the Heisenberg chain with dimerized nearest-neighbor ferromagnetic (FM) (${J}_{1},{J}_{1}^{\ensuremath{'}}<0$) and next-nearest-neighbor antiferromagnetic (J2>0) couplings. Using the density-matrix renormalization group technique and spin-wave theory, we find a first-order transition between a fully polarized FM and an incommensurate spiral state at $2\ensuremath{\alpha}=\ensuremath{\beta}/(1+\ensuremath{\beta})$, where $\ensuremath{\alpha}$ is the frustration ratio J2/∣J1∣ and $\ensuremath{\beta}$ the degree of dimerization ${J}_{1}^{\ensuremath{'}}/{J}_{1}$. In the singlet spiral state the spin-gap is vanishingly small in the vicinity of the FM transition, corresponding to a situation of LiCuSbO4. For larger $\ensuremath{\alpha}$, corresponding to Rb2Cu2Mo3O12, and smaller $\ensuremath{\beta}$ there is a crossover from this frustration induced incommensurate state to an Affleck-Lieb-Kennedy-Tasaki-type valence-bond solid state with substantial spin gaps.
The effect of next-nearest-neighbor hopping t2 on the ground-state phase diagram of the one-dimensional Kondo lattice is studied with density-matrix renormalization-group techniques and by comparing with the phase diagram of the classical-spin variant of the same model. For a finite t2, i.e., for a zigzag-ladder geometry, indirect antiferromagnetic interactions between the localized spins are geometrically frustrated. We demonstrate that t2 at the same time triggers several magnetic phases which are absent in the model with nearest-neighbor hopping only. For strong J, we find a transition from antiferromagnetic to incommensurate magnetic short-range order, which can be understood entirely in the classical-spin picture. For weaker J, a spin-dimerized phase emerges, which spontaneously breaks the discrete translation symmetry. The phase is not accessible to perturbative means but is explained, on a qualitative level, by the classical-spin model as well. Spin dimerization alleviates magnetic frustration and is interpreted as a key to understand the emergence of quasi-long-range spiral magnetic order, which is found at weaker couplings. The phase diagram at weak J, with gapless quasi-long-range order on top of the twofold degenerate spin-dimerized ground state, competing with a nondegenerate phase with gapped spin (and charge) excitations, is unconventional and eludes an effective low-energy spin-only theory.
A frustrated spin-1/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling J2>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of J1/J2 and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D+) phase with the same sign of the x,y- and z-component dimer order parameters, two VC dimer (VCD+/VCD${}_{\ensuremath{-}}$) phases with the sign of the z-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D+ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb2Cu2Mo3O12 is discussed.