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arXiv:cond-mat/9809038 · Physical review. B, Condensed matter

Response of finite spin- S Heisenberg chains to local perturbations

Örs Legeza, Jenő Sólyom

Published Feb 1, 1999• 3 citations
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Abstract

We consider the properties of finite isotropic antiferromagnetic Heisenberg chains with $S=1/2,1,3/2$ spins when a weak magnetic field is applied on a few sites, using White's density-matrix renormalization-group method. For the $S=1$ chain there exists only one length scale in the system which determines the behavior of the one- and two-point correlation functions both around the local perturbation and near the free boundary. For the critical half-odd-integer spin cases the exponent of the spin-spin correlation function was found to be $\ensuremath{\eta}=1$ and the exponent of the decay of the site magnetization around the perturbed site is ${x}_{m}=\ensuremath{\eta}/2.$ Close to a free boundary, however, the behavior is completely different for $S=1/2$ and $S>1/2.$

DOI
10.1103/physrevb.59.3606

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We consider the properties of finite isotropic antiferromagnetic Heisenberg chains with S=1/2,1,3/2S=1/2,1,3/2S=1/2,1,3/2 spins when a weak magnetic field is applied on a few sites, using White's density-matrix renormalization-group method. For the S=1S=1S=1 chain there exists only one length scale in the system which determines the behavior of the one- and two-point correlation functions both around the local perturbation and near the free boundary. For the critical half-odd-integer spin cases the exponent of the spin-spin correlation function was found to be \ensuremathη=1\ensuremath{\eta}=1\ensuremathη=1 and the exponent of the decay of the site magnetization around the perturbed site is xm=\ensuremathη/2.{x}_{m}=\ensuremath{\eta}/2.xm​=\ensuremathη/2. Close to a free boundary, however, the behavior is completely different for S=1/2S=1/2S=1/2 and S>1/2.S>1/2.S>1/2.
We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈Sn+lz{\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}Sn+lz​ Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉=(-1)l{)}^{\mathit{l}})l\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant geff{\mathit{g}}_{\mathrm{eff}}geff​(l0{\mathit{l}}_{0}l0​)=1/ln(cl0{\mathit{cl}}_{0}cl0​), as predicted by field theoretical treatments.
By means of the density matrix renormalization group (DMRG) method, the magnetic properties of the J-J-J′^{\prime}′ quantum Heisenberg chains with spin S=1/2S=1/2S=1/2, 1, 3/2 and 2 in the ground states are investigated in the presence of a magnetic field. Two different cases are considered: (a) when JJJ is antiferromagnetic and J′J^{\prime}J′ is ferromagnetic (i.e. the AF-AF-F chain), the system is a ferrimagnet. The plateaus of the magnetization are observed. It is found that the width of the plateaus decreases with increasing the ferromagnetic coupling, and disappears when % J^{\prime}/J passes over a critical value. The saturated field is observed to be independent of the ferromagnetic coupling; (b) when JJJ is ferromagnetic and J′J^{\prime}J′ is antiferromagnetic (i.e. the F-F-AF chain), the system becomes an antiferromagnet. The plateaus of the magnetization are also seen. The width of the plateaus decreases with decreasing the antiferromagnetic coupling, and disappears when $J^{\prime}/J $ passes over a critical value. Though the ground state properties are quite different, the magnetization plateaus in both cases tend to disappear when the ferromagnetic coupling becomes more dominant. Besides, no fundamental difference between the systems with spin half-integer and integer has been found.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
Extensive density-matrix renormalization-group calculations for spin S=1/2S=1/2S=1/2 and S=3/2S=3/2S=3/2 disordered antiferromagnetic Heisenberg chains show a rather distinct behavior in the two cases. While at sufficiently strong disorder both systems are in a random singlet phase, we show that weak disorder is an irrelevant perturbation for the S=3/2S=3/2S=3/2 chain, contrary to what expected from a naive application of the Harris criterion. The observed irrelevance is attributed to the presence of a new correlation length due to enhanced end-to-end correlations. This phenomenon is expected to occur for all half-integer S>1/2S>1/2S>1/2 chains. A possible phase diagram of the chain for generic S is also discussed.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λλλ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λλλ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.