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Physical review. B, Condensed matter

Universality class of integer quantum spin chains: S=2 case study

Shaojin Qin, Xiaoqun Wang, Lu Yu

Published Dec 1, 1997• 18 citations
Publisher

Abstract

The density-matrix renormalization-group technique is employed to show explicitly that $S=2$ quantum spin chains belong to the same universality class as $S=1$ spin chains. The dispersion of the single-magnon band for periodic chains is shown to be $\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{q}^{2}}$, where $\ensuremath{\Delta}$ is the Haldane gap and ${\ensuremath{\omega}}_{q}=v|\mathrm{sin}(\ensuremath{\pi}\ensuremath{-}q)|$ with $v=\ensuremath{\Delta}\ensuremath{\xi}$, $\ensuremath{\xi}$ as the correlation length, $\ensuremath{\pi}\ensuremath{-}q=2\ensuremath{\pi}n/L$, $L$ as the chain length, and $n$ as an integer. Moreover, the existence of spin-$1/2$ topological excitations due to the breaking of one singlet bond is demonstrated by finite-size scaling of correlation functions decaying as $\mathrm{exp}(\ensuremath{-}r/\ensuremath{\xi})/\sqrt{L}$. These stringent numerical tests, as well as the repulsive interactions between magnons, provide strong evidence in favor of the existence of a universality class for integer quantum spin chains conjectured by Haldane.

DOI
10.1103/physrevb.56.r14251

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The density-matrix renormalization-group technique is employed to show explicitly that S=2S=2S=2 quantum spin chains belong to the same universality class as S=1S=1S=1 spin chains. The dispersion of the single-magnon band for periodic chains is shown to be \ensuremathΔ2+\ensuremathωq2\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{q}^{2}}\ensuremathΔ2+\ensuremathωq2​​, where \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the Haldane gap and \ensuremathωq=v∣sin(\ensuremathπ\ensuremath−q)∣{\ensuremath{\omega}}_{q}=v|\mathrm{sin}(\ensuremath{\pi}\ensuremath{-}q)|\ensuremathωq​=v∣sin(\ensuremathπ\ensuremath−q)∣ with v=\ensuremathΔ\ensuremathξv=\ensuremath{\Delta}\ensuremath{\xi}v=\ensuremathΔ\ensuremathξ, \ensuremathξ\ensuremath{\xi}\ensuremathξ as the correlation length, \ensuremathπ\ensuremath−q=2\ensuremathπn/L\ensuremath{\pi}\ensuremath{-}q=2\ensuremath{\pi}n/L\ensuremathπ\ensuremath−q=2\ensuremathπn/L, LLL as the chain length, and nnn as an integer. Moreover, the existence of spin-1/21/21/2 topological excitations due to the breaking of one singlet bond is demonstrated by finite-size scaling of correlation functions decaying as exp(\ensuremath−r/\ensuremathξ)/L\mathrm{exp}(\ensuremath{-}r/\ensuremath{\xi})/\sqrt{L}exp(\ensuremath−r/\ensuremathξ)/L​. These stringent numerical tests, as well as the repulsive interactions between magnons, provide strong evidence in favor of the existence of a universality class for integer quantum spin chains conjectured by Haldane.
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.
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We explore the physics of the quantum Hall effect using the Haldane mapping of dimerized SU(N+M)\mathrm{SU}(N+M)SU(N+M) spin chains, the large NNN expansion, and the density matrix renormalization group technique. We show that while the transition is first order for N+M>2N+M>2N+M>2, the system at zero temperature nevertheless displays a continuously diverging length scale $\ensuremath{\xi}$ (correlation length). The numerical results for (N,M)=(3,1),(2,2),(5,1)(N,M)=(3,1),\phantom{\rule{4pt}{0ex}}(2,2),\phantom{\rule{4pt}{0ex}}(5,1)(N,M)=(3,1),(2,2),(5,1), and (7,1)(7,1)(7,1) indicate that $\ensuremath{\xi}$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\ensuremath{\vartheta}$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of superuniversality which says that the dimerized spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of NNN and MMM. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on NNN and MMM. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\ensuremath{\xi}$ with varying values of $\ensuremath{\vartheta}\ensuremath{\sim}\ensuremath{\pi}$.
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