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

Physical review. B, Condensed matter

Symmetric-group approach to the studies of spin-1/2 lattices

Norbert Flocke, Jacek Karwowski

Published Apr 1, 1997• 13 citations
Publisher

Abstract

A technique for studying spin-1/2 lattices, based on the properties of the symmetric group ${\mathrm{S}}_{\mathrm{N}}$, is presented. It is shown that the symmetric-group approach, applied to the Heisenberg Hamiltonian, leads to efficient diagonalization algorithms. Some examples of calculations for the lowest singlet and triplet states of the 2\ifmmode\times\else\texttimes\fi{}L isotropic antiferromagnetic Heisenberg ladders are compared with results in the literature. For the singlet-triplet gap we have obtained, by extrapolation to the bulk limit, the values 0.498 949 and 0.499 545 for a polynomial and an exponential fit, respectively.

DOI
10.1103/physrevb.55.8287

Related papers

Exact diagonalization study on spin- 12 ladders as a function of two coupling parameters using the symmetric group approach

Norbert Flocke

Using the symmetric-group-approach method [N. Flocke and J. Karwowski, Phys. Rev. B 55, 8287 (1997)], exact diagonalization studies on a set of spin-1/2 Heisenberg ladders containing three different couplings $J$, ${J}^{\ensuremath{'}}$ (chain), and ${J}_{\ensuremath{\perp}}$ (rungs) were performed and the behavior of the singlet-triplet energy gap was analyzed as a function of ${J}^{\ensuremath{'}}$ and ${J}_{\ensuremath{\perp}}$ in units of $J$. Exact diagonalizations for the ground and first-exited state were performed for up to $2\ifmmode\times\else\texttimes\fi{}L=30$ spins for the isotropic ladder ${(J=J}^{\ensuremath{'}}{=J}_{\ensuremath{\perp}})$ and for up to $2\ifmmode\times\else\texttimes\fi{}L=28$ spins for all the anisotropic ones. All calculations were done under periodic/M\"obius boundary conditions for $L$ even/odd, respectively. I found some evidence, that for certain ${J}_{\ensuremath{\perp}},$ ${J}^{\ensuremath{'}}$ pairs the gap may vanish in the bulk limit. For the isotropic ladder I obtained for the gap $\ensuremath{\Delta}(\ensuremath{\infty})=0.49996$, giving further support for the conjecture that for this system the exact value of $\ensuremath{\Delta}(\ensuremath{\infty})$ is equal to 1/2.

Physical review. B, Condensed matter•Dec 1, 1997•10 citations

Two-dimensional Heisenberg antiferromagnet: A numerical study

Ted Barnes, Eric S. Swanson

We describe a numerical simulation of the isotropic spin-\textonehalf{} Heisenberg antiferromagnet on square lattices of size 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. A Hamiltonian Monte Carlo method is used to measure the energies of the lowest-lying singlet and triplet states, and we extrapolate these results to give estimates of the properties of large antiferromagnets. Our result for the ground-state energy per spin is $\frac{{E}_{0}}{{N}^{2}}=\ensuremath{-}0.672\ifmmode\pm\else\textpm\fi{}0.001$ for an infinite system. Within our statistical accuracy, the singlet-triplet gap behaves as ${E}_{1}\ensuremath{-}{E}_{0}\ensuremath{\approx}\frac{2.1}{N}$, which implies that the infinite system is gapless.

Physical review. B, Condensed matter•Jun 1, 1988•63 citations

Numerical solution of larges=1/2 ands=1 Heisenberg antiferromagnetic spin chains using a truncated basis expansion

Mitchel D. Kovarik

A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.

Physical review. B, Condensed matter•Apr 1, 1990•15 citations

Strong-coupling expansions for antiferromagnetic Heisenberg spin-one-half ladders

Martin Reigrotzki, Hirokazu Tsunetsugu, Thomas Maurice Rice

The properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.

arXiv:cond-mat/9408023•Journal of Physics Condensed Matter•Oct 24, 1994•79 citations

Bond operator theory of isotropic spin-1 antiferromagnetic chains

Hanting Wang, Jue-Lian Shen, Zhao-Bin Su

The bond operator representation of quantum $S=\frac{1}{2}$ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.

Physical review. B, Condensed matter•Dec 1, 1997•29 citations

Numerical Solution of the spin-$\frac {1}{2}$ Heisenberg Antiferromagnetic Chains Using a Projector Method

Gang Sun

An efficient projector method is proposed to calculate the energy gap between the ground, state and the first excited state of the spin- Heisenberg antiferromagnetic chains. We exactly computed the two lowest-lying energies for the magnetization and the lowest-lying energy for on chains up to 16 spins. The lowest-lying energy stat6 for is a singlet ground state. The second lowest-lying energy state for and the lowest-lying energy states for are triplet degenerate, which are the first excited energy states. We also imposed the Monte Carlo technique to estimate the lowest-lying energies for both and 1 on longer chains up to 32 spins. We found that the singlet-triplet energy gap trends to zero as .

Communications in Theoretical Physics•Sep 1, 1991•1 citation
A technique for studying spin-1/2 lattices, based on the properties of the symmetric group SN{\mathrm{S}}_{\mathrm{N}}SN​, is presented. It is shown that the symmetric-group approach, applied to the Heisenberg Hamiltonian, leads to efficient diagonalization algorithms. Some examples of calculations for the lowest singlet and triplet states of the 2\ifmmode\times\else\texttimes\fi{}L isotropic antiferromagnetic Heisenberg ladders are compared with results in the literature. For the singlet-triplet gap we have obtained, by extrapolation to the bulk limit, the values 0.498 949 and 0.499 545 for a polynomial and an exponential fit, respectively.
Using the symmetric-group-approach method [N. Flocke and J. Karwowski, Phys. Rev. B 55, 8287 (1997)], exact diagonalization studies on a set of spin-1/2 Heisenberg ladders containing three different couplings JJJ, ${J}^{\ensuremath{'}}$ (chain), and ${J}_{\ensuremath{\perp}}$ (rungs) were performed and the behavior of the singlet-triplet energy gap was analyzed as a function of ${J}^{\ensuremath{'}}$ and ${J}_{\ensuremath{\perp}}$ in units of JJJ. Exact diagonalizations for the ground and first-exited state were performed for up to $2\ifmmode\times\else\texttimes\fi{}L=30$ spins for the isotropic ladder ${(J=J}^{\ensuremath{'}}{=J}_{\ensuremath{\perp}})$ and for up to $2\ifmmode\times\else\texttimes\fi{}L=28$ spins for all the anisotropic ones. All calculations were done under periodic/M\"obius boundary conditions for LLL even/odd, respectively. I found some evidence, that for certain ${J}_{\ensuremath{\perp}},$ ${J}^{\ensuremath{'}}$ pairs the gap may vanish in the bulk limit. For the isotropic ladder I obtained for the gap $\ensuremath{\Delta}(\ensuremath{\infty})=0.49996$, giving further support for the conjecture that for this system the exact value of $\ensuremath{\Delta}(\ensuremath{\infty})$ is equal to 1/2.
We describe a numerical simulation of the isotropic spin-\textonehalf{} Heisenberg antiferromagnet on square lattices of size 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. A Hamiltonian Monte Carlo method is used to measure the energies of the lowest-lying singlet and triplet states, and we extrapolate these results to give estimates of the properties of large antiferromagnets. Our result for the ground-state energy per spin is $\frac{{E}_{0}}{{N}^{2}}=\ensuremath{-}0.672\ifmmode\pm\else\textpm\fi{}0.001$ for an infinite system. Within our statistical accuracy, the singlet-triplet gap behaves as ${E}_{1}\ensuremath{-}{E}_{0}\ensuremath{\approx}\frac{2.1}{N}$, which implies that the infinite system is gapless.
A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.
The properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.
The bond operator representation of quantum S=12S=\frac{1}{2}S=21​ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.
Numerical Solution of the spin-12\frac {1}{2}21​ Heisenberg Antiferromagnetic Chains Using a Projector Method
An efficient projector method is proposed to calculate the energy gap between the ground, state and the first excited state of the spin- Heisenberg antiferromagnetic chains. We exactly computed the two lowest-lying energies for the magnetization and the lowest-lying energy for on chains up to 16 spins. The lowest-lying energy stat6 for is a singlet ground state. The second lowest-lying energy state for and the lowest-lying energy states for are triplet degenerate, which are the first excited energy states. We also imposed the Monte Carlo technique to estimate the lowest-lying energies for both and 1 on longer chains up to 32 spins. We found that the singlet-triplet energy gap trends to zero as .