Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0206092 · Physical review. B, Condensed matter

Ground state of the random-bond spin-1 Heisenberg chain

Sara Bergkvist, Patrik Henelius, Anders Rosengren

Published Oct 9, 2002• 25 citations
arXivHTMLPDF

Abstract

Stochastic series expansion quantum Monte Carlo is used to study the ground state of the antiferromagnetic spin-1 Heisenberg chain with bond disorder. Typical spin- and string-correlation functions behave in accordance with real-space renormalization group predictions for the random-singlet phase. The average string-correlation function decays algebraically with an exponent of $\ensuremath{-}0.378(6),$ in very good agreement with the prediction of $\ensuremath{-}(3\ensuremath{-}\sqrt{5})/2\ensuremath{\simeq}\ensuremath{-}0.382,$ while the average spin-correlation function is found to decay with an exponent of about $\ensuremath{-}1,$ quite different from the expected value of $\ensuremath{-}2.$ By implementing the concept of directed loops for the spin-1 chain we show that autocorrelation times can be reduced by up to two orders of magnitude.

DOI
10.1103/physrevb.66.134407

Related papers

Crossover effects in the random-exchange spin- 12 antiferromagnetic chain

Nicolas Laflorencie, Heiko Rieger, Anders W. Sandvik +1 more

The random antiferromagnetic spin-$1∕2$ 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 $\mathcal{D}$ 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.

arXiv:cond-mat/0312572•Physical Review B•Aug 31, 2004•36 citations

Numerical renormalization-group study of spin correlations in one-dimensional random spin chains

Takashi Hikihara, Akira Furusaki, Manfred Sigrist

We calculate the ground-state two-spin correlation functions of spin-$\frac{1}{2}$ quantum Heisenberg chains with random exchange couplings using the real-space renormalization-group scheme. We extend the conventional scheme to take account of the contribution of local higher multiplet excitations in each decimation step. This extended scheme can provide highly accurate numerical data for large systems. The random average of staggered spin correlations of the chains with random antiferromagnetic (AF) couplings shows algebraic decay like ${1/r}^{2},$ which verifies the Fisher's analytic results. For chains with random ferromagnetic (FM) and AF couplings, the random average of generalized staggered correlations is found to decay more slowly than a power law, in the form close to $1/\mathrm{ln}(r).$ The difference between the distribution functions of the spin correlations of the random AF chains and of the random FM-AF chains is also discussed.

arXiv:cond-mat/9905352•Physical review. B, Condensed matter•Nov 1, 1999•40 citations

Adaptive density matrix renormalization group study of the disordered antiferromagnetic spin- 12 Heisenberg chain

Alexander H. O. Wada, José A. Hoyos

Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.

arXiv:2112.10420•Physical review. B./Physical review. B•Mar 24, 2022•7 citations

Ground State of the S = 1 / 2 Heisenberg Spin Chain with Random Ferromagnetic and Antiferromagnetic Couplings

Sibei Li, Hui Shao, Anders W. Sandvik

We study the Heisenberg S=1/2 chain with random ferro- and antiferromagnetic couplings using quantum Monte Carlo simulations at ultra-low temperatures, converging to the ground state. Finite-size scaling of correlation functions and excitation gaps demonstrate an exotic critical state in qualitative agreement with previous strong-disorder renormalization group calculations but with scaling exponents depending on the coupling distribution. We find dual scaling regimes of the transverse correlations versus the distance, with an L independent form C(r)=r^{-μ} for r≪L and C(r,L)=L^{-η}f(r/L) for r/L>0, where μ>η and the scaling function is delivered by our analysis. These results are at variance with previous spin-wave and density-matrix renormalization group calculations, thus highlighting the power of unbiased quantum Monte Carlo simulations.

arXiv:2408.12911•Physical Review Letters•Feb 24, 2025•2 citations

Properties of the random-singlet phase: From the disordered Heisenberg chain to an amorphous valence-bond solid

Yu-Rong Shu, Dao‐Xin Yao, Chih-Wei Ke +2 more

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study $S=1/2$ spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

arXiv:1603.04362•Physical review. B./Physical review. B•Nov 28, 2016•28 citations

Randomness-driven Quantum Phase Transition in Bond-alternating Haldane Chain

Takayuki Arakawa, Synge Todo, Hajime Takayama

The effect of bond randomness on the spin-gapped ground state of the spin-1 bond-alternating antiferromagnetic Heisenberg chain is discussed. By using the loop cluster quantum Monte Carlo method, we investigate the stability of topological order in terms of the recently proposed twist order parameter [M. Nakamura and S. Todo: Phys. Rev. Lett. 89 (2002) 077204]. It is observed that the dimer phases as well as the Haldane phase of the spin-1 Heisenberg chain are robust against a weak randomness, though the valence-bond-solid-like topological order in the latter phase is destroyed by introducing a disorder stronger than the critical value.

arXiv:cond-mat/0410755•Journal of the Physical Society of Japan•Apr 1, 2005•6 citations
Stochastic series expansion quantum Monte Carlo is used to study the ground state of the antiferromagnetic spin-1 Heisenberg chain with bond disorder. Typical spin- and string-correlation functions behave in accordance with real-space renormalization group predictions for the random-singlet phase. The average string-correlation function decays algebraically with an exponent of \ensuremath−0.378(6),\ensuremath{-}0.378(6),\ensuremath−0.378(6), in very good agreement with the prediction of \ensuremath−(3\ensuremath−5)/2\ensuremath≃\ensuremath−0.382,\ensuremath{-}(3\ensuremath{-}\sqrt{5})/2\ensuremath{\simeq}\ensuremath{-}0.382,\ensuremath−(3\ensuremath−5​)/2\ensuremath≃\ensuremath−0.382, while the average spin-correlation function is found to decay with an exponent of about \ensuremath−1,\ensuremath{-}1,\ensuremath−1, quite different from the expected value of \ensuremath−2.\ensuremath{-}2.\ensuremath−2. By implementing the concept of directed loops for the spin-1 chain we show that autocorrelation times can be reduced by up to two orders of magnitude.
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.
We calculate the ground-state two-spin correlation functions of spin-12\frac{1}{2}21​ quantum Heisenberg chains with random exchange couplings using the real-space renormalization-group scheme. We extend the conventional scheme to take account of the contribution of local higher multiplet excitations in each decimation step. This extended scheme can provide highly accurate numerical data for large systems. The random average of staggered spin correlations of the chains with random antiferromagnetic (AF) couplings shows algebraic decay like 1/r2,{1/r}^{2},1/r2, which verifies the Fisher's analytic results. For chains with random ferromagnetic (FM) and AF couplings, the random average of generalized staggered correlations is found to decay more slowly than a power law, in the form close to 1/ln(r).1/\mathrm{ln}(r).1/ln(r). The difference between the distribution functions of the spin correlations of the random AF chains and of the random FM-AF chains is also discussed.
Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.
We study the Heisenberg S=1/2 chain with random ferro- and antiferromagnetic couplings using quantum Monte Carlo simulations at ultra-low temperatures, converging to the ground state. Finite-size scaling of correlation functions and excitation gaps demonstrate an exotic critical state in qualitative agreement with previous strong-disorder renormalization group calculations but with scaling exponents depending on the coupling distribution. We find dual scaling regimes of the transverse correlations versus the distance, with an L independent form C(r)=r^{-μ} for r≪L and C(r,L)=L^{-η}f(r/L) for r/L>0, where μ>η and the scaling function is delivered by our analysis. These results are at variance with previous spin-wave and density-matrix renormalization group calculations, thus highlighting the power of unbiased quantum Monte Carlo simulations.
We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2S=1/2S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.
The effect of bond randomness on the spin-gapped ground state of the spin-1 bond-alternating antiferromagnetic Heisenberg chain is discussed. By using the loop cluster quantum Monte Carlo method, we investigate the stability of topological order in terms of the recently proposed twist order parameter [M. Nakamura and S. Todo: Phys. Rev. Lett. 89 (2002) 077204]. It is observed that the dimer phases as well as the Haldane phase of the spin-1 Heisenberg chain are robust against a weak randomness, though the valence-bond-solid-like topological order in the latter phase is destroyed by introducing a disorder stronger than the critical value.