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arXiv:cond-mat/9808342 · Journal of Physics A Mathematical and General

Optimum ground states for spin- ladders with two legs

H Niggemann, J Zittartz

Published Dec 11, 1998• 12 citations
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Abstract

We construct the exact ground state for an antiferromagnetic spin- model on the two-leg ladder as an optimum ground state . The ground state contains a discrete parameter and a continuous parameter a which controls z -axis anisotropy. For most values of a the global ground state is unique. It has vanishing sublattice magnetization and exponentially decaying correlation functions. By using the transfer matrix technique, we calculate exactly the fluctuations of the magnetization, the nearest-neighbour correlation, and the longitudinal correlation length as functions of the parameters.

DOI
10.1088/0305-4470/31/49/006

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We construct the exact ground state for an antiferromagnetic spin- model on the two-leg ladder as an optimum ground state . The ground state contains a discrete parameter and a continuous parameter a which controls z -axis anisotropy. For most values of a the global ground state is unique. It has vanishing sublattice magnetization and exponentially decaying correlation functions. By using the transfer matrix technique, we calculate exactly the fluctuations of the magnetization, the nearest-neighbour correlation, and the longitudinal correlation length as functions of the parameters.
We use the matrix product approach to construct all optimum ground states of general anisotropic spin-2 chains with nearest-neighbour interactions and common symmetries. These states are exact ground states of the model and their properties depend on up to three parameters. We find three different antiferromagnetic Haldane phases, one weak antiferromagnetic and one weak ferromagnetic phase. The antiferromagnetic phases can be described as spin liquids with exponentially decaying correlation functions. The variety of phases found with the matrix product ansatz also gives insight into the behaviour of spin chains with arbitrary higher spins.
We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to two-legged spin ladders with spins , 1 and and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder. We compute ground-state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the three different nonferromagnetic spin ladders. We show that the long-range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally, for ladders with spin equal to or greater than we define a class of AKLT states whose matrix product coefficients are given by 9- j symbols.
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.
We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either SSS or σσσ-type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both SSS and σσσ-type of spins are anti-ferromagnetically aligned with each other, two nearest SSS spins along the rung form an anisotropic singlet bond whereas two nearest σσσ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like Cv(T)Cv(T)Cv(T), χ(T)χ(T)χ(T) and S(T)S(T)S(T) are also calculated using the transfer matrix method for different phase. The magnetic gap in the SRFM and the SLFM can be notice from χ(T)χ(T)χ(T) and Cv(T)Cv(T)Cv(T) curves.
We study the ground-state spin correlations in the gapless incommensurate regime of a S=1/2S=1/2S=1/2 XXZ\mathrm{XXZ}XXZ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open XXZ\mathrm{XXZ}XXZ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter K and the amplitudes of the correlation functions. The value of K estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of K on the magnetization M. The $K\ensuremath{-}M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate 1/T1{1/T}_{1}1/T1​ and dynamical spin structure factors.
The ground-state phase diagram of a Heisenberg $\text{spin}\ensuremath{-}\frac{1}{2}$ system on a two-leg ladder with rung alternation is studied by combining analytical approaches with numerical simulations. For the case of ferromagnetic leg exchanges a unique ferrimagnetic ground state emerges, whereas for the case of antiferromagnetic leg exchanges several different ground states are stabilized depending on the ratio between exchanges along legs and rungs. For the more general case of a honeycomb-ladder model for the case of ferromagnetic leg exchanges besides the usual rung-singlet and saturated ferromagnetic states we obtain a ferrimagnetic Luttinger liquid phase with both linear and quadratic low-energy dispersions and ground-state magnetization continuously changing with system parameters. For the case of antiferromagnetic exchanges along legs, different dimerized states including states with additional topological order are suggested to be realized.