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The phase diagram of the anisotropic Spin-1 Heisenberg Chain

We applied the Density Matrix Renormalization Group to the XXZ spin-1 quantum chain. In studing this model we aim to clarify controversials about the point where the massive Haldane phase appears.

In 1983, Haldane conjectured that the isotropic antiferromagnetic spin-S Heisenberg chain has a gap or not depending if the spin is an integer or halfinteger [1]. This conjecture is today well established [2,3]. In the case where an exchange z-anisotropy is present the situation is less clear and the location of the point where the massive Haldane phase appears is still controversial even for the spin 1 case. The Hamiltonian in this last case is the spin-1 XXZ chain where S x j , S y j and S x j are the spin 1 operators. From finite-size studies ( for lattices sizes up to L = 14 ) and by exploring the results of conformal invariance [4] Alcaraz and Moreo [5] conjectured that for λ < λ c = 0 the Hamiltonian (1) is in a massless X-Y phase with critical fluctuations governed by a conformal field theory with central charge c = 1. Moreover in the critical phase (−1 ≤ λ ≤ 0) the dimensions of operators n,m , x (s) n governing the correlation functions in the bulk and in the surface are given by [5] x where , n, m = 0, ±1, ±2, ....
More recently Yajima and Takahashi [7] by calculating the spectra of (1) for lattice sizes up to L = 16 concluded that the Haldane phase should starts at λ = λ c = 0.068 ± 0.003. Since this result destroy the conjecture (2) we decide to make a more precise calculation of λ c by using the density matrix renormalization group (DMRG) introduced by White [8], which enable us to make spectral calculations in much larger lattice sizes. The direct way to estimate λ c would be by a direct mass gap evaluation of (1) in a periodic chain. However our results shows that for λ ≈ 0 the finite-size gaps are very small ( the transition at λ c should be of Kosterlitz-Thouless type ) and it is very difficult to decide if the point λ = 0.065 ( the worse estimative against the conjecture (2) ) is in a massive phase or not. Another way to estimate λ c is by calculating the exponents η x , governing the correlation S x i S x i+r (r) ∼ r −ηx . The critical phase disappears when η x reachs the value [7,9] 2x in order to estimate λ c . This is done by using the finite-size relations coming from the conformal invariance of the infinite system [4]. The exponent x s (1) is obtained by extrapolating (L → ∞) the sequence where E n j (λ, L) is the j th energy in the sector where S z i = n of the Hamiltonian (1) with free ends.
In table 1 we show for the special cases λ = 0 and λ = 0.065 the estimated values for x (s) 1 (L) for lattices up to L = 48. In our DMRG calculations we keep k = 50 and 80 states in the truncated Hilbert space [8]. A good estimator for the errors is 1 − P k , where P k is the trace of the truncated density matrix. The values in table 1 were obtained by taking P k → 1.
At λ = λ c , the operator x (b) 0,1 , which corrects the finite size scaling [5], should be marginal and logarithmic corrections in the sequence (4) are expected. In fig. 1 we show the data of table 1, at λ = 0, together with a mean square fit in the form showing a good numerical fit. A similar polynomial fit, which would be the case if λ c > 0, although possible shows larger deviations. On the other hand the data of table 1 at λ = 0.065 are not well fitted by (5). As we can see from table 1 the data at λ = 0.065 clear indicate a limiting value x (s) 1 (∞) bigger than 1/4. In order to see this more clearly we calculated at λ = 0.065 this estimator for L = 60 sites and obtain x (s) 1 (60) = 0.25023, which indicates that λ = 0.065 is already a point in the Haldane phase.
In conclusion, our results indicate that, in agreement with the conjecture (2), the Haldane phase appears for λ > λ c = 0, and the logarithmic corrections appearing at this point explains the numerical controversial in earlier numerical calculations.