arXiv:cond-mat/9910512 · Physical review. B, Condensed matter

Nonperturbative real-space renormalization group scheme for the spin- 12 XXXHeisenberg model

Andreas Degenhard

Published Oct 3, 2001• 4 citations

Abstract

In this paper we apply an analytical real-space renormalization group formulation which is based on numerical concepts of the density-matrix renormalization group. Within a rigorous mathematical framework we construct nonperturbative renormalization group transformations for the spin-$\frac{1}{2}$ $\mathrm{XXX}$ Heisenberg model in the finite-temperature regime. The developed renormalization group scheme allows for calculating the renormalization group flow behavior in the temperature-dependent coupling constant. The constructed renormalization group transformations are applied within the ferromagnetic and the antiferromagnetic regime of the Heisenberg chain. The ferromagnetic fixed point is computed and compared to results derived by other techniques.

DOI: 10.1103/physrevb.64.174408

Related papers

Finite-temperature quantum renormalization-group theory for the one-dimensional anisotropic Heisenberg model

A. J. Berlinsky, Catherine Kallin

Physical review. B, Condensed matterMar 1, 19811 citation

The density matrix renormalization group and critical phenomena

Robert J. Bursill, Fatma Göde

Journal of Physics Condensed MatterDec 11, 19954 citations

Block-spin renormalization group in the large- n limit

J. E. Hirsch, Stephen H. Shenker

Physical review. B, Condensed matterFeb 1, 198336 citations

Real-Space Renormalization Group Approach for the Corner Hamiltonian

Kouichi Okunishi

arXiv:cond-mat/0507195Journal of the Physical Society of JapanDec 1, 200510 citations

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

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

Physical review. B, Condensed matterNov 1, 19942 citations

Real-space renormalization group and critical phenomena in the two-dimensional spin- 12 X−Y model

Attilio L. Stella, Flavio Toigo

Physical review. B, Condensed matterMar 1, 197832 citations

arXiv:cond-mat/9910512 · Physical review. B, Condensed matter

Nonperturbative real-space renormalization group scheme for the spin- 12 XXXHeisenberg model

Andreas Degenhard

Published Oct 3, 2001• 4 citations

arXiv HTML PDF

Abstract

\frac{1}{2}

\mathrm{XXX}

Heisenberg model in the finite-temperature regime. The developed renormalization group scheme allows for calculating the renormalization group flow behavior in the temperature-dependent coupling constant. The constructed renormalization group transformations are applied within the ferromagnetic and the antiferromagnetic regime of the Heisenberg chain. The ferromagnetic fixed point is computed and compared to results derived by other techniques.

DOI: 10.1103/physrevb.64.174408

Related papers

Finite-temperature quantum renormalization-group theory for the one-dimensional anisotropic Heisenberg model

A. J. Berlinsky, Catherine Kallin

A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.

Physical review. B, Condensed matterMar 1, 19811 citation

The density matrix renormalization group and critical phenomena

Robert J. Bursill, Fatma Göde

We adapt White`s (1992, 1993) density matrix renormalization group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalized blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2 Heisenberg chain. In the simplest case where the two most probable states in odd-sized blocks are used to construct approximate renormalization group transformations, we find qualitative improvement upon the standard real-space renormalization group method for the thermal exponent nu .We adapt White`s (1992, 1993) density matrix renormalization group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalized blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2 Heisenberg chain. In the simplest case where the two most probable states in odd-sized blocks are used to construct approximate renormalization group transformations, we find qualitative improvement upon the standard real-space renormalization group method for the thermal exponent nu .

Journal of Physics Condensed MatterDec 11, 19954 citations

Block-spin renormalization group in the large- n limit

J. E. Hirsch, Stephen H. Shenker

We present a real-space renormalization-group (RG) study of the $n$ -component classical Heisenberg model in the large- $n$ limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.We present a real-space renormalization-group (RG) study of the $n$-component classical Heisenberg model in the large-$n$ limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.

Physical review. B, Condensed matterFeb 1, 198336 citations

Real-Space Renormalization Group Approach for the Corner Hamiltonian

Kouichi Okunishi

We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.

arXiv:cond-mat/0507195Journal of the Physical Society of JapanDec 1, 200510 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.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 matterNov 1, 19942 citations

Real-space renormalization group and critical phenomena in the two-dimensional spin- 12 X−Y model

Attilio L. Stella, Flavio Toigo

A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin- $\frac{1}{2}$ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-$\frac{1}{2}$ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.

Physical review. B, Condensed matterMar 1, 197832 citations

\frac{1}{2}

\mathrm{XXX}

We present a real-space renormalization-group (RG) study of the

n

-component classical Heisenberg model in the large-

n

limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.

S=1/2

XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.

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.

A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-

\frac{1}{2}

$X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.