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arXiv:0912.4589 · Physical Review B

Phase diagram of the frustrated spin ladder

Toshiya Hikihara, Oleg A. Starykh

Published Feb 25, 2010• 47 citations
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Abstract

We revisit the phase diagram of the frustrated spin-1/2 ladder with two competing interchain antiferromagnetic exchanges, rung coupling ${J}_{\ensuremath{\perp}}$ and diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$. We suggest, based on the accurate renormalization-group analysis of the low-energy Hamiltonian of the ladder, that marginal interchain current-current interaction plays central role in destabilizing previously predicted intermediate columnar dimer phase in the vicinity of classical degeneracy line ${J}_{\ensuremath{\perp}}=2{J}_{\ifmmode\times\else\texttimes\fi{}}$. Following this insight we then suggest that changing these competing interchain exchanges from the previously considered antiferromagnetic to the ferromagnetic ones eliminates the issue of the marginal interactions altogether and dramatically expands the region of stability of the columnar dimer phase. This analytical prediction is convincingly confirmed by the numerical density-matrix renormalization-group and exact-diagonalization calculations as well as by the perturbative calculation in the strong rung-coupling limit. The phase diagram for ferromagnetic ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$ is determined.

DOI
10.1103/physrevb.81.064432

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We revisit the phase diagram of the frustrated spin-1/2 ladder with two competing interchain antiferromagnetic exchanges, rung coupling J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ and diagonal coupling J\ifmmode×\else\texttimes\fi{J}_{\ifmmode\times\else\texttimes\fi{}}J\ifmmode×\else\texttimes\fi​. We suggest, based on the accurate renormalization-group analysis of the low-energy Hamiltonian of the ladder, that marginal interchain current-current interaction plays central role in destabilizing previously predicted intermediate columnar dimer phase in the vicinity of classical degeneracy line J\ensuremath⊥=2J\ifmmode×\else\texttimes\fi{J}_{\ensuremath{\perp}}=2{J}_{\ifmmode\times\else\texttimes\fi{}}J\ensuremath⊥​=2J\ifmmode×\else\texttimes\fi​. Following this insight we then suggest that changing these competing interchain exchanges from the previously considered antiferromagnetic to the ferromagnetic ones eliminates the issue of the marginal interactions altogether and dramatically expands the region of stability of the columnar dimer phase. This analytical prediction is convincingly confirmed by the numerical density-matrix renormalization-group and exact-diagonalization calculations as well as by the perturbative calculation in the strong rung-coupling limit. The phase diagram for ferromagnetic J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ and J\ifmmode×\else\texttimes\fi{J}_{\ifmmode\times\else\texttimes\fi{}}J\ifmmode×\else\texttimes\fi​ is determined.
We revisit the phase diagram of the frustrated s=1/2s=1/2s=1/2 spin ladder with antiferromagnetic rung and diagonal couplings. In particular, we reexamine the evidence for the columnar dimer phase, which has been predicted from analytic treatment of the model and has been claimed to be found in numerical calculations. By performing density-matrix renormalization-group (DMRG) calculations on long ladders, keeping a large number of states within the DMRG, and carrying out careful scaling in number of states kept and system size, we calculate the behavior of the dimer order parameter as a function of frustration strength. We find no positive numerical evidence for a finite dimer order parameter in the thermodynamic limit anywhere in the parameter regime in which the columnar dimer phase is expected to appear. Conservative error estimates in the scaling of our data place stringent limits on the maximum possible value of the dimer order parameter and the maximum possible extent of the parameter regime in which a sufficiently weak dimer phase could still exist.
We study the ground-state phase diagram of a spin-12\frac{1}{2}21​ frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.
In the present paper, by using the density-matrix renormalization-group technique, we investigate the existence of dimerized phases in a frustrated spin-1/2 ladder model. By calculating the two-rung localizable entanglement and, in particular, its first-order derivative in the ground state, we show that an intermediate columnar dimerized phase exists indeed in a narrow parameter region between the rung-singlet and the Haldane phases. Furthermore, by introducing an additional interchain next-nearest-neighbor exchange interaction, the stability of this phase can be greatly enhanced and a staggering dimerized phase also appears in the phase diagram. Our results confirm the previous conclusions derived by the bosonization methods.
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.
The phase diagram of an antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs is determined using numerical methods (exact diagonalization and density-matrix renormalization group) supplemented by strong-coupling and mean-field analysis. Interestingly, this model displays remarkable features, bridging the physics of the J1{J}_{1}J1​-J2{J}_{2}J2​ chain and of the unfrustrated ladder. The phase diagram as a function of the transverse coupling ${J}_{\ensuremath{\perp}}$ and the frustration J2{J}_{2}J2​ exhibits an Ising transition between a columnar phase of dimers and the usual rung-singlet phase of two-leg ladders. The transition is driven by resonating valence bond fluctuations in the singlet sector while the triplet spin gap remains finite across the transition. In addition, frustration brings incommensurability in the real-space spin-correlation functions, the onset of which evolves smoothly from the J1{J}_{1}J1​-J2{J}_{2}J2​ chain value to zero in the large-${J}_{\ensuremath{\perp}}$ limit. The onset of incommensurability in the spin structure factor and in the dispersion relation is also analyzed. The physics of the frustrated rung-singlet phase is well understood using perturbative expansions and mean-field theories in the large-${J}_{\ensuremath{\perp}}$ limit. Last, we discuss the effect of the nontrivial magnon dispersion relation on the thermodynamical properties of the system. The relation of this model and its physics to experimental observations on compounds that are currently investigated, such as BiCu2{}_{2}2​PO6{}_{6}6​, is eventually addressed.
The ground-state properties of spin-1/2 ladders are studied, emphasizing the role of frustration and ring exchange coupling. We present a unified field theory for ladders with general coupling constants and geometry. Rich phase diagrams can be deduced by using a renormalization group calculation for ladders with in-chain next-nearest-neighbor interactions and plaquette ring exchange coupling. In addition to established phases such as Haldane, rung singlet, and dimerized phases, we also observe a surprising instability towards an incommensurate phase for weak interchain couplings, which is characterized by an exotic coexistence of self-consistent ferromagnetic and antiferromagnetic order parameters.