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/9612139 · Physical review. B, Condensed matter

Plaquette basis for the study of Heisenberg ladders

J. Piekarewicz, James R. Shepard

Published Sep 1, 1997• 23 citations
arXivHTMLPDF

Abstract

We employ a plaquette basis---generated by coupling the four spins in a $2\ifmmode\times\else\texttimes\fi{}2$ lattice to a well-defined total angular momentum---for the study of Heisenberg ladders with antiferromagnetic coupling. Matrix elements of the Hamiltonian in this basis are evaluated using standard techniques in angular-momentum (Racah) algebra. We show by exact diagonalization of small $(2\ifmmode\times\else\texttimes\fi{}4$ and $2\ifmmode\times\else\texttimes\fi{}6)$ systems that in excess of 90% of the ground-state probability is contained in a very small number of basis states. These few basis states can be used to define a severely truncated basis which we use to approximate low-lying exact eigenstates. We show how, in this low-energy basis, the isotropic spin-$1/2$ Heisenberg ladder can be mapped onto an anisotropic spin-$1$ ladder for which the coupling along the rungs is much stronger than the coupling between the rungs. The mapping thereby generates two distinct energy scales which greatly facilitates understanding the dynamics of the original spin-$1/2$ ladder. Moreover, we use these insights to define an effective low-energy Hamiltonian in accordance to the contractor renormalization group (CORE) method. We show how a simple range-2 CORE approximation to the effective Hamiltonian to be used with our truncated basis reproduces the low-energy spectrum of the exact $2\ifmmode\times\else\texttimes\fi{}6$ theory at the $\ensuremath{\lesssim}1%$ level.

DOI
10.1103/physrevb.56.5366

Related papers

Dynamic spin response for Heisenberg ladders

J. Piekarewicz, James R. Shepard

We employ the recently proposed plaquette basis to investigate static and dynamic properties of isotropic two-leg Heisenberg spin ladders. Simple noninteracting multiplaquette states provide a remarkably accurate picture of the energy/site and dynamic spin response of these systems. Insights afforded by this simple picture suggest a very efficient truncation scheme for more precise calculations. When the small truncation errors are accounted for using recently developed contractor renormalization techniques, very accurate results requiring a small fraction of the computational effort of exact calculations are obtained. These methods allow us to determine the energy/site, gap, and spin response of $2\ifmmode\times\else\texttimes\fi{}16$ ladders. The former two values are in good agreement with density-matrix renormalization-group results. The spin-response calculations show that nearly all the strength is concentrated in the lowest triplet level and that coherent many-body effects enhance the response/site by nearly a factor of 1.6 over that found for $2\ifmmode\times\else\texttimes\fi{}2$ systems.

arXiv:cond-mat/9707207•Physical review. B, Condensed matter•May 1, 1998•24 citations

Perturbation theory for spin ladders using angular-momentum coupled bases

J. Piekarewicz, James R. Shepard

We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schr\"odinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\ifmmode\times\else\texttimes\fi{}12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}0.8,$ with $\ensuremath{\lambda}$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}1.25.$ Thus, we conclude that the plaquette basis offers the best currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(\ensuremath{\lambda}=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.

arXiv:cond-mat/9804261•Physical review. B, Condensed matter•Oct 1, 1998•9 citations

Low-energy properties of spin-1/2 two-leg antiferromagnetic Heisenberg ladders with ferromagnetic diagonal coupling

Ningsheng Zhu, Xiaoqun Wang, Changfeng Chen

Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.

Physical review. B, Condensed matter•Dec 11, 2000•14 citations

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

Mean-field theory for spin ladders using angular-momentum coupled bases

J. Piekarewicz, James R. Shepard

We study properties of two-leg Heisenberg spin ladders in a mean-field approximation using a variety of angular-momentum-coupled bases. The mean-field theory proposed by Gopalan, Rice, and Sigrist, which uses a rung basis, assumes that the mean-field ground state consists of a condensate of spin singlets along the rungs of the ladder. We generalize this approach to larger angular-momentum-coupled bases that incorporate---by their mere definition---a substantial fraction of the important short-range structure of these materials. In these bases the mean-field ground state remains a condensate of spin singlet---but now with each involving a larger fraction of the spins in the ladder. As expected, the ``purity'' of the ground state, as judged by the condensate fraction, increases with the size of the elementary block defining the basis. Moreover, the coupling to quasiparticle excitations becomes weaker as the size of the elementary block increases. Thus, the weak-coupling limit of the theory becomes an accurate representation of the underlying mean-field dynamics. We illustrate the method by computing static and dynamic properties of two-leg ladders in the various angular-momentum-coupled bases.

arXiv:cond-mat/9808310•Physical review. B, Condensed matter•Oct 1, 1999•8 citations

Numerical contractor renormalization method for quantum spin models

Sylvain Capponi, Andreas Martin Läuchli, Matthieu Mambrini

We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagom\'e lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagom\'e lattice we find that the previously proposed twofold degenerate $S=1∕2$ basis can account for a large number of phenomena of the spin $1∕2$ kagom\'e system. For spin $3∕2$, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice for the plaquette lattice or a 48-site kagom\'e lattice, which are beyond the possibilities of a standard exact diagonalization approach.

arXiv:cond-mat/0404712•Physical Review B•Sep 29, 2004•60 citations
We employ a plaquette basis---generated by coupling the four spins in a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 lattice to a well-defined total angular momentum---for the study of Heisenberg ladders with antiferromagnetic coupling. Matrix elements of the Hamiltonian in this basis are evaluated using standard techniques in angular-momentum (Racah) algebra. We show by exact diagonalization of small (2\ifmmode×\else\texttimes\fi4(2\ifmmode\times\else\texttimes\fi{}4(2\ifmmode×\else\texttimes\fi4 and 2\ifmmode×\else\texttimes\fi6)2\ifmmode\times\else\texttimes\fi{}6)2\ifmmode×\else\texttimes\fi6) systems that in excess of 90% of the ground-state probability is contained in a very small number of basis states. These few basis states can be used to define a severely truncated basis which we use to approximate low-lying exact eigenstates. We show how, in this low-energy basis, the isotropic spin-1/21/21/2 Heisenberg ladder can be mapped onto an anisotropic spin-111 ladder for which the coupling along the rungs is much stronger than the coupling between the rungs. The mapping thereby generates two distinct energy scales which greatly facilitates understanding the dynamics of the original spin-1/21/21/2 ladder. Moreover, we use these insights to define an effective low-energy Hamiltonian in accordance to the contractor renormalization group (CORE) method. We show how a simple range-2 CORE approximation to the effective Hamiltonian to be used with our truncated basis reproduces the low-energy spectrum of the exact 2\ifmmode×\else\texttimes\fi62\ifmmode\times\else\texttimes\fi{}62\ifmmode×\else\texttimes\fi6 theory at the \ensuremath≲1\ensuremath{\lesssim}1%\ensuremath≲1 level.
We employ the recently proposed plaquette basis to investigate static and dynamic properties of isotropic two-leg Heisenberg spin ladders. Simple noninteracting multiplaquette states provide a remarkably accurate picture of the energy/site and dynamic spin response of these systems. Insights afforded by this simple picture suggest a very efficient truncation scheme for more precise calculations. When the small truncation errors are accounted for using recently developed contractor renormalization techniques, very accurate results requiring a small fraction of the computational effort of exact calculations are obtained. These methods allow us to determine the energy/site, gap, and spin response of $2\ifmmode\times\else\texttimes\fi{}16$ ladders. The former two values are in good agreement with density-matrix renormalization-group results. The spin-response calculations show that nearly all the strength is concentrated in the lowest triplet level and that coherent many-body effects enhance the response/site by nearly a factor of 1.6 over that found for $2\ifmmode\times\else\texttimes\fi{}2$ systems.
We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schr\"odinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\ifmmode\times\else\texttimes\fi{}12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}0.8,$ with $\ensuremath{\lambda}$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}1.25.$ Thus, we conclude that the plaquette basis offers the best currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(\ensuremath{\lambda}=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
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 study properties of two-leg Heisenberg spin ladders in a mean-field approximation using a variety of angular-momentum-coupled bases. The mean-field theory proposed by Gopalan, Rice, and Sigrist, which uses a rung basis, assumes that the mean-field ground state consists of a condensate of spin singlets along the rungs of the ladder. We generalize this approach to larger angular-momentum-coupled bases that incorporate---by their mere definition---a substantial fraction of the important short-range structure of these materials. In these bases the mean-field ground state remains a condensate of spin singlet---but now with each involving a larger fraction of the spins in the ladder. As expected, the ``purity'' of the ground state, as judged by the condensate fraction, increases with the size of the elementary block defining the basis. Moreover, the coupling to quasiparticle excitations becomes weaker as the size of the elementary block increases. Thus, the weak-coupling limit of the theory becomes an accurate representation of the underlying mean-field dynamics. We illustrate the method by computing static and dynamic properties of two-leg ladders in the various angular-momentum-coupled bases.
We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagom\'e lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagom\'e lattice we find that the previously proposed twofold degenerate S=1∕2S=1∕2S=1∕2 basis can account for a large number of phenomena of the spin 1∕21∕21∕2 kagom\'e system. For spin 3∕23∕23∕2, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice for the plaquette lattice or a 48-site kagom\'e lattice, which are beyond the possibilities of a standard exact diagonalization approach.