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Physical review. B, Condensed matter

Correlation functions of the antiferromagnetic Heisenberg model using a modified Lanczos method

Eduardo R. Gagliano, Elbio R. Dagotto, Adriana Moreo, Francisco Castilho Alcaraz

Published Aug 1, 1986• 213 citations
Publisher

Abstract

Using a modified Lanczos algorithm, we study the correlation functions in the ground state of the one-dimensional antiferromagnetic Heisenberg model. We obtain numerical results for rings up to 24 sites. There are no indications of the anomalous behavior of these correlation functions recently observed in chains with 16 sites. We also present a pedagogical description of the hashing technique which is an efficient algorithm for searching and storage purposes.

DOI
10.1103/physrevb.34.1677

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Ground state of the one-dimensional antiferromagnetic Heisenberg model

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We have calculated the two-point correlation functions ${\ensuremath{\omega}}_{\mathrm{il}}$(N)=(4/3)〈S${\ensuremath{\rightarrow}}_{i}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{i+l}$ 〉 and their averages over i,${\ensuremath{\omega}}_{l}$ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4,6,8,...,16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in \ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(\ensuremath{\infty})\ensuremath{\Vert} for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert} for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) and in ${S}_{N}$, the structure factor at wave vector \ensuremath{\pi}, then finite-size contributions to ${\ensuremath{\omega}}_{l}$(N) will have to contribute to the (seemingly) logarithmic divergence of ${S}_{N}$ as N\ensuremath{\rightarrow}\ensuremath{\infty}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.

Physical review. B, Condensed matter•Feb 1, 1985•40 citations
Using a modified Lanczos algorithm, we study the correlation functions in the ground state of the one-dimensional antiferromagnetic Heisenberg model. We obtain numerical results for rings up to 24 sites. There are no indications of the anomalous behavior of these correlation functions recently observed in chains with 16 sites. We also present a pedagogical description of the hashing technique which is an efficient algorithm for searching and storage purposes.
We have calculated the two-point correlation functions ${\ensuremath{\omega}}_{\mathrm{il}}$(N)=(4/3)〈S${\ensuremath{\rightarrow}}_{i}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{i+l}$ 〉 and their averages over i,${\ensuremath{\omega}}_{l}$ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4,6,8,...,16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in \ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(\ensuremath{\infty})\ensuremath{\Vert} for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert} for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) and in SN{S}_{N}SN​, the structure factor at wave vector \ensuremath{\pi}, then finite-size contributions to ${\ensuremath{\omega}}_{l}$(N) will have to contribute to the (seemingly) logarithmic divergence of SN{S}_{N}SN​ as N\ensuremath{\rightarrow}\ensuremath{\infty}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.