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arXiv:1508.03941 · Journal of the Korean Physical Society

The antiferromagnetic cross-coupled spin ladder: Quantum fidelity and tensor networks approach

Xi-Hao Chen, Sam Young Cho, Huan-Qiang Zhou, Murray T. Batchelor

Published May 1, 2016• 11 citations
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Abstract

We investigate the phase diagram of the cross-coupled Heisenberg spin ladder with antiferromagnetic couplings. For this model, the results for the existence of the columnar dimer phase, which was predicted on the basis of weak coupling field theory renormalization group arguments, have been conflicting. The numerical work on this model has been based on various approaches, including exact diagonalization, series expansions and density-matrix renormalization group calculations. Using the recently-developed tensor network states and groundstate fidelity approach for quantum spin ladders, we find no evidence for the existence of the columnar dimer phase. We also provide an argument based on the symmetry of the Hamiltonian, which suggests that the phase diagram for antiferromagnetic couplings consists of a single line separating the rung-singlet and the Haldane phases.

DOI
10.3938/jkps.68.1114

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We investigate the phase diagram of the cross-coupled Heisenberg spin ladder with antiferromagnetic couplings. For this model, the results for the existence of the columnar dimer phase, which was predicted on the basis of weak coupling field theory renormalization group arguments, have been conflicting. The numerical work on this model has been based on various approaches, including exact diagonalization, series expansions and density-matrix renormalization group calculations. Using the recently-developed tensor network states and groundstate fidelity approach for quantum spin ladders, we find no evidence for the existence of the columnar dimer phase. We also provide an argument based on the symmetry of the Hamiltonian, which suggests that the phase diagram for antiferromagnetic couplings consists of a single line separating the rung-singlet and the Haldane phases.
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 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.
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.
Dimerized spin-1/2 ladders exhibit a variety of phase structures, which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder. Using the density matrix renormalization group (DMRG) algorithm, we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J ×. Two types of spin systems, staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder, are investigated. To clarify the phase transition behaviors, we simultaneously analyze the string order parameter (SOP), the twisted order parameter (TOP), as well as a measurement of the quantum information analysis. Based on measuring this different observables, we establish the phase diagram accurately and give the fitting functions of the phase boundaries. In addition, the phase transition of cross-coupled spin ladder (in the absence of intrinsic dimerization) is also discussed.
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 a frustrated S=1∕2S=1∕2S=1∕2 antiferromagnetic spin ladder with additional next-nearest neighbor exchanges, both diagonal and in-chain, is studied by a weak-coupling effective field theory approach combined with exact diagonalization for finite systems. In addition to two known phases with rung-singlet and Haldane-type ground states, we observe two new phases with dimerization along the chains. Furthermore, the transitions between the different phases are studied and shown to be either first order or to belong to the universality class of the two-dimensional Ising model. The nature of elementary excitations is discussed briefly.