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arXiv:cond-mat/0212254 · Physical review. B, Condensed matter

Density matrix renormalization group study of critical behavior of the s p i n − 1 2 alternating Heisenberg chain

T. Papenbrock, Ted Barnes, David J. Dean, M. V. Stoitsov, Michael R. Strayer

Published Jul 28, 2003• 37 citations
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Abstract

We investigate the critical behavior of the $S=1/2$ alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin, ${e}_{0},$ and singlet-triplet energy gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}$ are determined for a range of alternations $\ensuremath{\delta}.$ Our results for the approach of ${e}_{0}$ to the uniform chain limit are well described by $c{\ensuremath{\delta}}^{p},$ with $p\ensuremath{\approx}1.45.$ The singlet-triplet gap is also well described by a power law, with $p\ensuremath{\approx}0.73,$ half of the ${e}_{0}$ power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter ${\ensuremath{\delta}}_{0}$ is present in the logarithms.

DOI
10.1103/physrevb.68.024416

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We investigate the critical behavior of the S=1/2S=1/2S=1/2 alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin, e0,{e}_{0},e0​, and singlet-triplet energy gap \ensuremathΔ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}\ensuremathΔ\ifmmode~\else˜\fi​ are determined for a range of alternations \ensuremathδ.\ensuremath{\delta}.\ensuremathδ. Our results for the approach of e0{e}_{0}e0​ to the uniform chain limit are well described by c\ensuremathδp,c{\ensuremath{\delta}}^{p},c\ensuremathδp, with p\ensuremath≈1.45.p\ensuremath{\approx}1.45.p\ensuremath≈1.45. The singlet-triplet gap is also well described by a power law, with p\ensuremath≈0.73,p\ensuremath{\approx}0.73,p\ensuremath≈0.73, half of the e0{e}_{0}e0​ power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter \ensuremathδ0{\ensuremath{\delta}}_{0}\ensuremathδ0​ is present in the logarithms.
We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-1/21/21/2 Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.
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.
By using the coupled cluster method and the numerical density matrix renormalization group method, we investigate the properties of the quantum plateau state in an alternating Heisenberg spin chain. In the absence of a magnetic field, the results obtained from the coupled cluster method and density matrix renormalization group method both show that the ground state of the alternating chain is a gapped dimerized state when the parameter α exceeds a critical point α c . The value of the critical points can be determined precisely by a detailed investigation of the behavior of the spin gap. The system therefore possesses an m = 0 plateau state in the presence of a magnetic field when α > α c . In addition to the m = 0 plateau state, the results of density matrix renormalization group indicate that there is an m = 1/4 plateau state that occurs between two critical fields in the alternating chain if α > 1. The mechanism for the m = 1/4 plateau state and the critical behavior of the magnetization as one approaches this plateau state are also discussed.
In conformal field theory, key properties of spin-12\frac{1}{2}21​ chains, such as the ground state energy per site and the excitation gap, scale with dimerization $\ensuremath{\delta}$ as ${\ensuremath{\delta}}^{\ensuremath{\alpha}}$ with known exponents $\ensuremath{\alpha}$ and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest (J=1)(J=1)(J=1) and next-nearest-neighbor interactions (J2)({J}_{2})(J2​), for J2c=0.2411{J}_{2\mathrm{c}}=0.2411J2c​=0.2411. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of $\ensuremath{\alpha}$, and the scaling relation is valid up to the physically realized range, $\ensuremath{\delta}\ensuremath{\sim}0.1$. However, chains with logarithmic corrections (J2<0.2411J)({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})(J2​<0.2411J) are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately 3∕43∕43∕4 to 2∕32∕32∕3 for the spin gap and from approximately 3∕23∕23∕2 to 4∕34∕34∕3 for the energy per site, and error bars in the exponent also decrease as J2{J}_{2}J2​ approaches to J2c{J}_{2\mathrm{c}}J2c​.
The ground-state phase diagram of S=2S=2S=2 antiferromagnetic Heisenberg chains with coexisting uniform and alternating single-site anisotropies is investigated by the numerical exact diagonalization and density matrix renormalization group methods. We find the Haldane, large-DDD, Néel, period-doubled Néel, gapless spin fluid, quantized and partial ferrimagnetic phases. The Haldane phase is limited to the close neighborhood of the isotropic point. Within numerical accuracy, the transition from the gapless spin-fluid phase to the period-doubled Néel phase is a direct transition. Nevertheless, the presence of a narrow spin-gap phase between these two phases is suggested on the basis of the low-energy effective theory. The ferrimagnetic ground state is present in a wide parameter range. This suggests the realization of magnetized single-chain magnets with a uniform spin magnitude by controlling the environment of each magnetic ion without introducing ferromagnetic interactions.
The ground-state energy per site, \ensuremath{\epsilon}(\ensuremath{\delta}), and the singlet-triplet gap, \ensuremath{\Delta}(\ensuremath{\delta}), of the linear spin-(1/2) alternating Heisenberg antiferromagnet are found by extrapolation of direct solutions to N\ensuremath{\le}26 sites for even rings and N\ensuremath{\le}21 sites for odd rings, whose orbital degeneracy leads to qualitatively different critical behavior. The \ensuremath{\epsilon}(\ensuremath{\delta}) accuracy of ${<10}^{\mathrm{\ensuremath{-}}4}$ (in units of J) decisively rules out power-law behavior for any alternation \ensuremath{\delta}>0.001 and is consistent with ${\ensuremath{\delta}}^{4/3}$/\ensuremath{\Vert}ln\ensuremath{\delta}\ensuremath{\Vert} for \ensuremath{\delta}<0.05. The gap \ensuremath{\Delta}(\ensuremath{\delta}) is slightly sublinear in \ensuremath{\delta}. Direct ground-state solutions are found via diagrammatic valence-bond methods and provide sensitive tests for theoretical approaches to linear spin systems.