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arXiv:cond-mat/9809037 · Physical review. B, Condensed matter

Stability of the Haldane phase in anisotropic magnetic ladders

Örs Legeza, Jenő Sólyom

Published Dec 1, 1997• 19 citations
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Abstract

We have considered the properties of anisotropic two-leg ladder models with $S=1/2$ or $S=1$ spins on the rungs, using White's density-matrix renormalization-group method. We have generalized the method by taking into account the symmetries of the model in order to reduce the dimensions of the matrix to be diagonalized, thereby making it possible to consider more states. The boundaries in the parameter space of the extended region, where the Haldane phase exists, are estimated.

DOI
10.1103/physrevb.56.14449

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We have considered the properties of anisotropic two-leg ladder models with S=1/2S=1/2S=1/2 or S=1S=1S=1 spins on the rungs, using White's density-matrix renormalization-group method. We have generalized the method by taking into account the symmetries of the model in order to reduce the dimensions of the matrix to be diagonalized, thereby making it possible to consider more states. The boundaries in the parameter space of the extended region, where the Haldane phase exists, are estimated.
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 study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy Jz/Jxy.{J}^{z}{/J}^{\mathrm{xy}}.Jz/Jxy. It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of Jz/Jxy,{J}^{z}{/J}^{\mathrm{xy}},Jz/Jxy, and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as Jz/Jxy,{J}^{z}{/J}^{\mathrm{xy}},Jz/Jxy, and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.
We explore the ground-state phase diagram of the S=1/2S=1/2S=1/2 two-leg ladder with different leg interactions. The xyxyxy and zzz components of the leg interactions between nearest-neighbor spins in the aaa (bbb) leg are respectively denoted by Jl,aJ_{{\rm l},a}Jl,a​ and ΔlJl,aΔ_{\rm l} J_{{\rm l},a}Δl​Jl,a​ (Jl,bJ_{{\rm l},b}Jl,b​ and ΔlJl,bΔ_{\rm l} J_{{\rm l},b}Δl​Jl,b​). On the other hand, the xyxyxy and zzz components of the uniform rung interactions are respectively denoted by ΓrJrΓ_{\rm r} J_{\rm r}Γr​Jr​ and JrJ_{\rm r}Jr​. In the above, ΔlΔ_{\rm l}Δl​ and ΓrΓ_{\rm r}Γr​ are the XXZXXZXXZ-type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when Jl,aJl,b<0J_{{\rm l},a} J_{{\rm l},b}<0Jl,a​Jl,b​<0 irrespective of the sign of JrJ_{\rm r}Jr​. The phase diagram on the ΔlΔ_{\rm l}Δl​ (∣Δl∣≤1.0|Δ_{\rm l}| \leq 1.0∣Δl​∣≤1.0) versus Jl,bJ_{{\rm l},b}Jl,b​ (−2.0≤Jl,b≤3.0-2.0\leq J_{{\rm l},b}\leq 3.0−2.0≤Jl,b​≤3.0) plane in the case where Jl,a=0.2J_{{\rm l},a}=0.2Jl,a​=0.2, Jr=−1.0J_{\rm r}=-1.0Jr​=−1.0, and Γr=0.5Γ_{\rm r} = 0.5Γr​=0.5 is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{é}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and XY1XY1XY1 phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.
Using density-matrix renormalization group calculations, we investigate the general phase diagram of the frustrated two-leg ladder with Heisenberg interactions along legs, rungs, and diagonals. We confirm that all antiferromagnetic gapped states belong to the same universality class as the Haldane phase. In a three-dimensional phase diagram, we determine a continuous surface with singularities in the string-order parameter or its first derivative, corresponding to a transition between two Haldane phases with different topological order. Some parts of this transition surface are critical with zero gap and vanishing string-order parameter. In the complementary parts, the transition is first order with finite gap and string order. The boundary of this surface with the ferromagnetic region is a critical end line, when the surface is critical, and a triple line anywhere else. Part of this boundary coincides with the exactly soluble model proposed by Dmitriev, Krivnov, and Ovchinnikov [Phys. Rev. B 56, 5985 (1997)].
The ground state and low lying excitation of the spin 1/2 Heisenberg ladder with antiferromagnetic leg (JJJ) and ferromagnetic rung (−λJ,λ>0-λJ, λ>0−λJ,λ>0) interaction is studied by means of the density matrix renormalization group method. It is found that the state remains in the Haldane phase even for small λ∼0.02λ\sim 0.02λ∼0.02 suggesting the continuous transition to the gapless phase at λ=0λ= 0λ=0. The critical behavior for small λλλ is studied by the finite size scaling analysis. The result is consistent with the recent field theoretical prediction.
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.