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/0001275 · Physical review. B, Condensed matter

Density-matrix renormalization-group analysis of quantum critical points: Quantum spin chains

Shan-Wen Tsai, John Bradley Marston

Published Sep 1, 2000• 27 citations
arXivHTMLPDF

Abstract

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-$1/2$ 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.

DOI
10.1103/physrevb.62.5546

Related papers

Scaling exponents in spin- 1 2 Heisenberg chains with dimerization and frustration studied with the density-matrix renormalization group

Manoranjan Kumar, S. Ramasesha, Diptiman Sen +1 more

In conformal field theory, key properties of spin-$\frac{1}{2}$ 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 $(J=1)$ and next-nearest-neighbor interactions $({J}_{2})$, for ${J}_{2\mathrm{c}}=0.2411$. 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 $({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})$ are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately $3∕4$ to $2∕3$ for the spin gap and from approximately $3∕2$ to $4∕3$ for the energy per site, and error bars in the exponent also decrease as ${J}_{2}$ approaches to ${J}_{2\mathrm{c}}$.

arXiv:cond-mat/0608395•Physical Review B•Feb 8, 2007•27 citations

Study of critical behavior of the supersymmetric spin chain that models plateau transitions in the integer quantum Hall effect

Shan-Wen Tsai, John Bradley Marston

We use the density-matrix renormalization-group (DMRG) algorithm and finite-size scaling to study a supersymmetric (SUSY) spin chain that models plateau transitions in the integer quantum Hall effect. To illustrate the method, we first present results for the S=1 antiferromagnetic spin chain. Different scaling behavior of the dimerization operator is obtained for the Haldane phase, the dimerized phase, and at the critical point separating the two phases. We then investigate the supersymmetric spin chain. As the on-site Hilbert space has infinite dimension, it is non-compact. We present DMRG calculations for supersymmetric truncations of the on-site Hilbert space and also for non-supersymmetric truncations. These two cases scale differently, in accord with a Lieb-Schultz-Mattis type theorem.

arXiv:cond-mat/9908180•Aug 12, 1999•1 citation

Density Matrix Renormalization Group Study of Random Dimerized Antiferromagnetic Heisenberg Chains

Kazuo Hida

The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as u a with a > 2 where u denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as u 2β with β∼0.37 in agreement with the recent prediction of real space renormalization group theory (\(\beta =(3-\sqrt{5})/2 \simeq 0.382\)). The physical picture of this behavior in this model is also discussed.

Journal of the Physical Society of Japan•Oct 15, 1997•14 citations

Scaling of entanglement entropy at quantum critical points in random spin chains

Prashant Kumar, R. N. Bhatt

We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ${c}_{\mathrm{eff}}=1.17(4)$ at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.

arXiv:2307.00062•Physical review. B./Physical review. B•Dec 15, 2023•2 citations

Density-matrix renormalization-group algorithm for quantum lattice systems with a large number of states per site

Robert J. Bursill

A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain and a dimerized $\mathrm{XY}$ spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.

arXiv:cond-mat/9812349•Physical review. B, Condensed matter•Jul 15, 1999•31 citations

Boundary Effects in the Critical Scaling of Entanglement Entropy in 1D Systems

Nicolas Laflorencie, Erik Schwartz Sørensen, Ming-Shyang Chang +1 more

We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.

arXiv:cond-mat/0512475•Physical Review Letters•Mar 15, 2006•287 citations
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.
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​.
We use the density-matrix renormalization-group (DMRG) algorithm and finite-size scaling to study a supersymmetric (SUSY) spin chain that models plateau transitions in the integer quantum Hall effect. To illustrate the method, we first present results for the S=1 antiferromagnetic spin chain. Different scaling behavior of the dimerization operator is obtained for the Haldane phase, the dimerized phase, and at the critical point separating the two phases. We then investigate the supersymmetric spin chain. As the on-site Hilbert space has infinite dimension, it is non-compact. We present DMRG calculations for supersymmetric truncations of the on-site Hilbert space and also for non-supersymmetric truncations. These two cases scale differently, in accord with a Lieb-Schultz-Mattis type theorem.
The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as u a with a > 2 where u denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as u 2β with β∼0.37 in agreement with the recent prediction of real space renormalization group theory (β=(3−5)/2≃0.382\beta =(3-\sqrt{5})/2 \simeq 0.382β=(3−5​)/2≃0.382). The physical picture of this behavior in this model is also discussed.
We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ceff=1.17(4){c}_{\mathrm{eff}}=1.17(4)ceff​=1.17(4) at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.