© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

Real-space renormalization-group studies of low-dimensional quantum antiferromagnets

Marie-Bernadette Lepetit, Efstratios Manousakis

Published Jul 1, 1993• 16 citations

Abstract

We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.

DOI: 10.1103/physrevb.48.1028

Related papers

Ground-state energy of spin-1/2 and spin-1 antiferromagnetic chains: X Y model

C. Y. Pan, H. Q. Lin

The ground-state energy of the XY antiferromagnetic chains with spin-1/2 and spin-1 are obtained by a real space renormalization group method. For S=1/2 a large spin cluster is used in the mapping to improve the accuracy and a boundary theory is applied to extrapolate to the result which is in excellent agreement with the exact Bethe Ansatz result. For S=1 the ground-state energy of the model is obtained by using the renormalization group method and a finite chain calculation for lattices up to 16 sites. The two results are in very good agreement.

Journal of Applied Physics•Nov 15, 1988•1 citation

SPIN-1/2 TWO-DIMENSIONAL ANISOTROPIC HEISENBERG ANTIFERROMAGNETS

C. Y. Pan

A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.

International Journal of Modern Physics B•Sep 1, 1989•4 citations

GROUND-STATE ENERGY OF THE SPIN-1/2 TWO-DIMENSIONAL ANTIFERROMAGNETIC HEISENBERG MODEL

C. Y. Pan

The ground-state energy of the spin-1/2 two-dimensional antiferromagnetic Heisenberg model is obtained by a real space renormalization group method. A relative larger cluster (5×5) is used to improve the accuracy and a boundary theory is applied to extrapolate to the result which is in good agreement with the finite-size calculation and in fairly good agreement with the other available numerical estimates. How to further improve the calculation is discussed.

International Journal of Modern Physics B•Sep 1, 1989•1 citation

The Real-Space Renormalization-Group Study of a Heisenberg Antiferromagnetic Chain of Spin S

Yasushi Ishii, Yukito Tanabe

The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.

Journal of the Physical Society of Japan•Jul 15, 1981•0 citations

Quantum Spins and Quasiperiodicity: A Real Space Renormalization Group Approach

Anuradha Jagannathan

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.

arXiv:cond-mat/0312487•Physical Review Letters•Jan 28, 2004•54 citations

Renormalization-group analysis of long-range order in the two-dimensional antiferromagnetic Heisenberg model

Hua Lin, David K Campbell, Yingchun Cheng +1 more

We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J ${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}$ S(j)\ensuremath{\cdot}S(j+1)+ ${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}$(-1${)}^{\mathit{j}}$\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.

Physical review. B, Condensed matter•Nov 1, 1994•2 citations

We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.

The ground-state energy of the XY antiferromagnetic chains with spin-1/2 and spin-1 are obtained by a real space renormalization group method. For S=1/2 a large spin cluster is used in the mapping to improve the accuracy and a boundary theory is applied to extrapolate to the result which is in excellent agreement with the exact Bethe Ansatz result. For S=1 the ground-state energy of the model is obtained by using the renormalization group method and a finite chain calculation for lattices up to 16 sites. The two results are in very good agreement.

A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.

The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.

We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J

{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}

S(j)\ensuremath{\cdot}S(j+1)+

{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}

(-1

{)}^{\mathit{j}}

\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.