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/9803159 · Physical review. B, Condensed matter

Correlation amplitude for the S=12 XXZ spin chain in the critical region: Numerical renormalization-group study of an open chain

Takashi Hikihara, Akira Furusaki

Published Jul 1, 1998• 70 citations
arXivHTMLPDF

Abstract

The density-matrix renormalization-group technique is used to calculate the spin-correlation functions $〈{S}_{j}^{x}{S}_{k}^{x}〉$ and $〈{S}_{j}^{z}{S}_{k}^{z}〉$ of the one-dimensional $S=\frac{1}{2}$ $\mathrm{XXZ}$ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerically.

DOI
10.1103/physrevb.58.r583

Related papers

Correlation amplitudes for the spin- 12 XXZ chain in a magnetic field

Takashi Hikihara, Akira Furusaki

We present accurate numerical estimates for the correlation amplitudes of leading and main subleading terms of the two- and four-spin correlation functions in the one-dimensional spin-$1/2$ $\mathrm{XXZ}$ model under a magnetic field. These data are obtained by fitting the correlation functions, computed numerically with the density-matrix renormalization-group method, to the corresponding correlation functions in the low-energy effective theory. For this purpose we have developed the Abelian bosonization approach to the spin chain under the open-boundary conditions. We use the numerical data of the correlation amplitudes to quantitatively estimate spin gaps induced by a transverse staggered field and by exchange anisotropy.

arXiv:cond-mat/0310391•Physical Review B•Feb 24, 2004•102 citations

Dimer correlation amplitudes and dimer excitation gap in spin- 12 XXZ and Heisenberg chains

Toshiya Hikihara, Akira Furusaki, Sergei L. Lukyanov

Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-$\frac{1}{2}$ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.

arXiv:1706.02538•Physical review. B./Physical review. B•Oct 26, 2017•21 citations

Finite-size corrections of the entanglement entropy of critical quantum chains

J. C. Xavier, Francisco Castilho Alcaraz

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with $U(1)$ symmetry such as the spin-1/2 $XXZ$ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.

arXiv:1111.6577•Physical Review B•Jan 24, 2012•43 citations

Kondo effect in XXZ spin chains

Akira Furusaki, Takashi Hikihara

The Kondo effect in a one-dimensional spin-$\frac{1}{2}$ $\mathrm{XXZ}$ model in the gapless $\mathrm{XY}$ regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin $(S=1/2)$ is coupled to a single spin in the $\mathrm{XXZ}$ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host $\mathrm{XXZ}$ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host $\mathrm{XXZ}$ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.

arXiv:cond-mat/9802258•Physical review. B, Condensed matter•Sep 1, 1998•25 citations

Numerical renormalization-group study of the correlation functions of the antiferromagnetic spin-1/2 Heisenberg chain

K. Hallberg, Peter Horsch, Gerardo Martı́nez

We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈${\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$〉=(-1${)}^{\mathit{l}}$\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 ${\mathit{g}}_{\mathrm{eff}}$(${\mathit{l}}_{0}$)=1/ln(${\mathit{cl}}_{0}$), as predicted by field theoretical treatments.

arXiv:cond-mat/9505132•Physical review. B, Condensed matter•Jul 1, 1995•68 citations

Density-matrix renormalization-group analysis of the spin- 12 XXZ chain in anXYsymmetric random magnetic field

Laura Urba, Anders Rosengren

The spin-$\frac{1}{2} \mathrm{XXZ}$ chain in a random magnetic field pointing in the Z direction is numerically studied using the density-matrix renormalization-group method. The phase diagram as a function of the anisotropy of the $\mathrm{XXZ}$ Hamiltonian and the strength of the random field is analyzed by computing the spin correlations and the superfluid density. To obtain the superfluid density we consider a superblock configuration representing a closed system with an arbitrary twist at the boundary. This allows us to estimate the size of the critical region where the quasi-long-range order persists, that is, where the spin correlation length is infinite and the superfluid density is nonzero.

arXiv:cond-mat/0204539•Physical review. B, Condensed matter•Mar 14, 2003•11 citations
The density-matrix renormalization-group technique is used to calculate the spin-correlation functions 〈SjxSkx〉〈{S}_{j}^{x}{S}_{k}^{x}〉〈Sjx​Skx​〉 and 〈SjzSkz〉〈{S}_{j}^{z}{S}_{k}^{z}〉〈Sjz​Skz​〉 of the one-dimensional S=12S=\frac{1}{2}S=21​ XXZ\mathrm{XXZ}XXZ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerically.
We present accurate numerical estimates for the correlation amplitudes of leading and main subleading terms of the two- and four-spin correlation functions in the one-dimensional spin-1/21/21/2 XXZ\mathrm{XXZ}XXZ model under a magnetic field. These data are obtained by fitting the correlation functions, computed numerically with the density-matrix renormalization-group method, to the corresponding correlation functions in the low-energy effective theory. For this purpose we have developed the Abelian bosonization approach to the spin chain under the open-boundary conditions. We use the numerical data of the correlation amplitudes to quantitatively estimate spin gaps induced by a transverse staggered field and by exchange anisotropy.
Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-12\frac{1}{2}21​ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.
Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with U(1)U(1)U(1) symmetry such as the spin-1/2 XXZXXZXXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.
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.
The spin-12XXZ\frac{1}{2} \mathrm{XXZ}21​XXZ chain in a random magnetic field pointing in the Z direction is numerically studied using the density-matrix renormalization-group method. The phase diagram as a function of the anisotropy of the XXZ\mathrm{XXZ}XXZ Hamiltonian and the strength of the random field is analyzed by computing the spin correlations and the superfluid density. To obtain the superfluid density we consider a superblock configuration representing a closed system with an arbitrary twist at the boundary. This allows us to estimate the size of the critical region where the quasi-long-range order persists, that is, where the spin correlation length is infinite and the superfluid density is nonzero.