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Physical review. B, Condensed matter

Spin-(1/2 antiferromagneticXXZchain: New results and insights

Rajiv R. P. Singh, Michael Ellis Fisher, R. Shankar

Published Feb 1, 1989• 89 citations
Publisher

Abstract

We study the criticality at the isotropic limit of the S=(1/2, antiferromagnetic, XXZ chain on approach from the Ising side. Long series are developed for the longitudinal and transverse correlation sums or structure factors ${S}_{\mathrm{zz}}$ and ${S}_{+\mathrm{\ensuremath{-}}}$ around the Ising limit. Their extrapolation indicates a divergence as the critical point is approached, with two different powers, a result quite unexpected if one employs a naive scaling argument since the critical correlations are isotropic and there is only one correlation length, \ensuremath{\xi}. An analytical calculation is presented, in which this paradox is resolved and the numerical results are confirmed.

DOI
10.1103/physrevb.39.2562

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We study the criticality at the isotropic limit of the S=(1/2, antiferromagnetic, XXZ chain on approach from the Ising side. Long series are developed for the longitudinal and transverse correlation sums or structure factors Szz{S}_{\mathrm{zz}}Szz​ and S+\ensuremath−{S}_{+\mathrm{\ensuremath{-}}}S+\ensuremath−​ around the Ising limit. Their extrapolation indicates a divergence as the critical point is approached, with two different powers, a result quite unexpected if one employs a naive scaling argument since the critical correlations are isotropic and there is only one correlation length, \ensuremath{\xi}. An analytical calculation is presented, in which this paradox is resolved and the numerical results are confirmed.
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
We have calculated the spin-correlation functions (in the x, y, and z directions) in the linear spin-(1/2 XY system for both isotropic and anisotropic couplings. The study of zero-temperature correlation functions shows that a long-range order develops in the direction in which the coupling is stronger, but no such ordering exists for isotropic systems. We have studied the temperature variation of inverse correlation lengths in these directions and it has been found that at T=0 all the correlation lengths (${\ensuremath{\xi}}_{x}$, ${\ensuremath{\xi}}_{y}$, ${\ensuremath{\xi}}_{z}$) diverge in the isotropic case. On the other hand, in the anisotropic case, the correlation length diverges in the direction (either x or y) in which the coupling becomes stronger. The results are compared with the experimental data on Cs2{\mathrm{Cs}}_{2}Cs2​CoCl4{\mathrm{CoCl}}_{4}CoCl4​. The critical exponents (\ensuremath{\eta}',\ensuremath{\nu}') of the correlation function and correlation length are also calculated near the critical temperature (T=0) for different anisotropies, and it is found that the system behaves like an Ising model when a little anisotropy is introduced.
The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field (hhh) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}\ensuremath{\infty}, \mathrm{\ensuremath{\Gamma}}$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\ensuremath{\Gamma}}>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\ensuremath{\Gamma}}$ the transition is of mixed order.
We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime ∆> 1, extending our previous results to the full parameter plane of the antiferromagnetic chain ( ∆> −1 and arbitrary field h). The factorized formulae are numerically efficient and allow for taking the isotropic limit ( ∆ = 1) and the Ising limit ( ∆ = ∞). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite ∆ inducing unexpected crossover phenomena in the correlations. 1
Using the quantum transfer-matrix method we have calculated the spin correlation functions of the S =3/2 XXZ chain and examined the temperature dependence of the mass (≡ the inverse of the correlation length), susceptibility χ x ( Q ) and the magnetic structure factor S z ( Q ). The result shows that the mass vanishes at T =0 for the planar and isotropic cases (-1≦ Δ ≦1) in agreement with the theoretical prediction. At a low temperature the peak of S z ( Q ) at Q =π has been observed to persist even in the ferromagnetic region (0< Δ <0.8). The susceptibility χ z ( Q =0) at T =0 has been obtained with use of the extrapolation. The critical indices have been also examined.
The random antiferromagnetic spin-1∕21∕21∕2 XX and XXZ chain is studied numerically for varying strength of the disorder, using exact diagonalization and stochastic series expansion methods. The spin-spin correlation function as well as the stiffness display a clear crossover from the pure behavior (no disorder) to the infinite randomness fixed point or random singlet behavior predicted by the real space renormalization group. The crossover length scale is shown to diverge as $\ensuremath{\xi}\ensuremath{\sim}{\mathcal{D}}^{\ensuremath{-}\ensuremath{\gamma}}$, where D\mathcal{D}D is the variance of the random bonds. Our estimates for the exponent $\ensuremath{\gamma}$ agrees well within the error bars with the one for the localization length exponent emerging within an analytical bosonization calculation. Exact diagonalization and stochastic series expansion results for the string correlation function are also presented.