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

Low-temperature behavior of the correlation length and the susceptibility of the ferromagnetic quantum Heisenberg chain

Peter Kopietz

Published Sep 1, 1989• 17 citations
Publisher

Abstract

The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/${\mathrm{JS}}^{3}$${)}^{1/2}$. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.

DOI
10.1103/physrevb.40.5194

Related papers

Low-temperature behavior of the correlation length and the susceptibility of a quantum Heisenberg ferromagnet in two dimensions

Peter Kopietz, Sudip Chakravarty

The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid $^{3}\mathrm{He}$ in two dimensions.

Physical review. B, Condensed matter•Sep 1, 1989•63 citations

Weak random bonds in the Heisenberg ferromagnet near two dimensions

Peter Kopietz

The quantum renormalization-group technique recently developed by Chakravarty and the present author is used to study weak random couplings in quantum-mechanical and classical nearest-neighbor Heisenberg ferromagnets in D=2+\ensuremath{\epsilon} dimensions and at low temperatures T. Momentum-shell recursion relations are obtained by considering the renormalization-group transformations of the cumulants of the probability distribution of the random exchange couplings. From a one-loop calculation it is found that, if the second cumulants ${\ensuremath{\sigma}}^{2}$ of the random couplings are smaller than a critical value ${\ensuremath{\sigma}}_{c}^{2}$, the only modification due to randomness in the low-temperature behavior of the order-parameter correlation length \ensuremath{\xi} in two dimensions is a renormalization of the spin stiffness ${\ensuremath{\rho}}_{s}$ according to ${\ensuremath{\rho}}_{s}$(X)=(1-X${)}^{\ensuremath{\varphi}}$${\ensuremath{\rho}}_{s}$(0), where X=${\ensuremath{\sigma}}^{2}$/${\ensuremath{\sigma}}_{c}^{2}$ and \ensuremath{\varphi}=(1/3 within the one-loop approximation. At zero temperature \ensuremath{\xi} diverges for with ${\ensuremath{\nu}}_{x}$=1/D. Precisely at X=1 the critical behavior of the system is characterized by new exponents; the dynamical critical exponent z is given by z=2+\ensuremath{\varphi}/${\ensuremath{\nu}}_{x}$. For D(1-\ensuremath{\varphi})>2 a new phase appears in the T-X phase diagram, and it is argued that this phase is characterized by spin-glass order.

Physical review. B, Condensed matter•Sep 1, 1989•5 citations

Monte Carlo renormalization-group analysis of the classical Heisenberg model in two dimensions

Stephen H. Shenker, Jan Tobochnik

Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.

Physical review. B, Condensed matter•Nov 1, 1980•259 citations

Thermodynamic properties of the two-dimensional quantum Heisenberg ferromagnet and the effects of bond dilution

E. F. Talbot

The energy, the specific heat, and the magnetic susceptibility of the bond-diluted S=(1/2) Heisenberg ferromagnet on a two-dimensional square lattice are studied using a Monte Carlo sampling of the terms contributing to the partition function. The full lattice is studied in detail for sizes up to 20\ifmmode\times\else\texttimes\fi{}20, and evidence is presented to suggest that the low-temperature correlation length behaves as \ensuremath{\xi}\ensuremath{\sim}${e}^{\ensuremath{\nu}\ensuremath{\beta}J}$ with \ensuremath{\nu}\ensuremath{\sim}1.91\ifmmode\pm\else\textpm\fi{}0.06. The temperature dependence of the susceptibility, however, is not quite a simple exponential in \ensuremath{\beta}J. Knowledge of the correlation length and finite-size scaling allows the determination of the prefactor. Our conclusion is that the low-temperature susceptibility behaves as ${T}^{\mathrm{\ensuremath{-}}1}$${e}^{2\ensuremath{\nu}\ensuremath{\beta}J}$. Square lattices of size ranging from 6\ifmmode\times\else\texttimes\fi{}6 to 12\ifmmode\times\else\texttimes\fi{}12 are studied for ${k}_{B}$T/J ranging from 0.1 to 3 and for several values of the bond probability p. Not surprisingly, the most drastic thermodynamic behavior near the critical bond-percolation probability is the sharp increase in the low-temperature susceptibility for p>${p}_{c}$. This corresponds to much larger correlation effects in the percolating lattice than below the percolation threshold. We also compare the thermodynamics at p=${p}_{c}$ to the scaling theories of Lubensky and Stanley et al. which suggest that the correlations at the percolation threshold are predominantly one dimensional.

Physical review. B, Condensed matter•Apr 1, 1986•5 citations

Renormalization-group approach to spin-wave theory of quantum Heisenberg ferromagnets

Naoum Karchev

The renormalization-group method is reformulated in a way that allows the analysis of two-dimensional Heisenberg ferromagnets with small spin. A set of recursion equations is derived in a one-loop approximation. The low-temperature asymptotics of the correlation length and the uniform susceptibility are obtained. The comparison of the results with the results obtained within the momentum shell recursion technique shows that even at one-loop approximation there is a sensitive renormalization due to small spin.

arXiv:cond-mat/9603123•Physical review. B, Condensed matter•Mar 1, 1997•6 citations

Random exchange effects in antiferromagnetic quantum spin chains: A Monte Carlo study

H.-B. Schüttler, Douglas J. Scalapino, Paul Michael Grant

We have carried out Monte Carlo studies of a random-exchange antiferromagnetic spin-(1/2) chain. For systems with XY-like (anisotropic) and with Heisenberg (isotropic) coupling, our results confirm the existence of a disorder-induced low-temperature (T) divergence in the long-wavelength ${S}^{z}$-${S}^{z}$ susceptibility \ensuremath{\chi} which was previously predicted by real-space renormalization-group (RSRG) treatments. Over the finite temperature range studied, these results are consistent with a 1/(T ${\mathrm{ln}}^{2}$T) behavior of \ensuremath{\chi}, and hence in qualitative agreement with the RSRG results. As in the XY-Heisenberg regime, we also find a disorder-induced enhancement of the low-T susceptibility for a system with Ising-like exchange coupling which, over the finite temperature range studied, is again consistent with RSRG results. However, there are inconsistencies between the RSRG predictions in the Ising-like regime at very low temperatures, and the exact results for the random-exchange Ising chain and the low-temperature behavior of \ensuremath{\chi} in the Ising-like regime may in fact be more complicated than predicted by RSRG. Finally, we also present results for the antiferromagnetic susceptibility and structure factor. For both Heisenberg and Ising-like systems, we find that disorder suppresses the long-range antiferromagnetic correlations at low T.

Physical review. B, Condensed matter•Mar 1, 1987•8 citations
The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=C\ensuremathχ(0){C}_{\ensuremath{\chi}}^{(0)}C\ensuremathχ(0)​(JS2{\mathrm{JS}}^{2}JS2/T) [1+C\ensuremathχ(1){C}_{\ensuremath{\chi}}^{(1)}C\ensuremathχ(1)​\ensuremath{\eta}+O(\ensuremathη2{\ensuremath{\eta}}^{2}\ensuremathη2)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=\ensuremathπ\ensuremath−1{\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}\ensuremathπ\ensuremath−1(T/{\mathrm{JS}}^{3}$${)}^{1/2}. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=C\ensuremathξ(0){C}_{\ensuremath{\xi}}^{(0)}C\ensuremathξ(0)​(JS2{\mathrm{JS}}^{2}JS2/T) [1+C\ensuremathξ(1){C}_{\ensuremath{\xi}}^{(1)}C\ensuremathξ(1)​\ensuremath{\eta}+O(\ensuremathη2{\ensuremath{\eta}}^{2}\ensuremathη2)], where a is the lattice spacing. The values of the coefficients C\ensuremathχ(j){C}_{\ensuremath{\chi}}^{(j)}C\ensuremathχ(j)​ and C\ensuremathξ(j){C}_{\ensuremath{\xi}}^{(j)}C\ensuremathξ(j)​ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.
The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.
The quantum renormalization-group technique recently developed by Chakravarty and the present author is used to study weak random couplings in quantum-mechanical and classical nearest-neighbor Heisenberg ferromagnets in D=2+\ensuremath{\epsilon} dimensions and at low temperatures T. Momentum-shell recursion relations are obtained by considering the renormalization-group transformations of the cumulants of the probability distribution of the random exchange couplings. From a one-loop calculation it is found that, if the second cumulants ${\ensuremath{\sigma}}^{2}$ of the random couplings are smaller than a critical value ${\ensuremath{\sigma}}_{c}^{2}$, the only modification due to randomness in the low-temperature behavior of the order-parameter correlation length \ensuremath{\xi} in two dimensions is a renormalization of the spin stiffness ${\ensuremath{\rho}}_{s}$ according to ${\ensuremath{\rho}}_{s}$(X)=(1-X${)}^{\ensuremath{\varphi}}$${\ensuremath{\rho}}_{s}$(0), where X=${\ensuremath{\sigma}}^{2}$/${\ensuremath{\sigma}}_{c}^{2}$ and \ensuremath{\varphi}=(1/3 within the one-loop approximation. At zero temperature \ensuremath{\xi} diverges for with ${\ensuremath{\nu}}_{x}$=1/D. Precisely at X=1 the critical behavior of the system is characterized by new exponents; the dynamical critical exponent z is given by z=2+\ensuremath{\varphi}/${\ensuremath{\nu}}_{x}$. For D(1-\ensuremath{\varphi})>2 a new phase appears in the T-X phase diagram, and it is argued that this phase is characterized by spin-glass order.
Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.
The energy, the specific heat, and the magnetic susceptibility of the bond-diluted S=(1/2) Heisenberg ferromagnet on a two-dimensional square lattice are studied using a Monte Carlo sampling of the terms contributing to the partition function. The full lattice is studied in detail for sizes up to 20\ifmmode\times\else\texttimes\fi{}20, and evidence is presented to suggest that the low-temperature correlation length behaves as \ensuremath{\xi}\ensuremath{\sim}${e}^{\ensuremath{\nu}\ensuremath{\beta}J}$ with \ensuremath{\nu}\ensuremath{\sim}1.91\ifmmode\pm\else\textpm\fi{}0.06. The temperature dependence of the susceptibility, however, is not quite a simple exponential in \ensuremath{\beta}J. Knowledge of the correlation length and finite-size scaling allows the determination of the prefactor. Our conclusion is that the low-temperature susceptibility behaves as ${T}^{\mathrm{\ensuremath{-}}1}$${e}^{2\ensuremath{\nu}\ensuremath{\beta}J}$. Square lattices of size ranging from 6\ifmmode\times\else\texttimes\fi{}6 to 12\ifmmode\times\else\texttimes\fi{}12 are studied for kB{k}_{B}kB​T/J ranging from 0.1 to 3 and for several values of the bond probability p. Not surprisingly, the most drastic thermodynamic behavior near the critical bond-percolation probability is the sharp increase in the low-temperature susceptibility for p>pc{p}_{c}pc​. This corresponds to much larger correlation effects in the percolating lattice than below the percolation threshold. We also compare the thermodynamics at p=pc{p}_{c}pc​ to the scaling theories of Lubensky and Stanley et al. which suggest that the correlations at the percolation threshold are predominantly one dimensional.
The renormalization-group method is reformulated in a way that allows the analysis of two-dimensional Heisenberg ferromagnets with small spin. A set of recursion equations is derived in a one-loop approximation. The low-temperature asymptotics of the correlation length and the uniform susceptibility are obtained. The comparison of the results with the results obtained within the momentum shell recursion technique shows that even at one-loop approximation there is a sensitive renormalization due to small spin.
We have carried out Monte Carlo studies of a random-exchange antiferromagnetic spin-(1/2) chain. For systems with XY-like (anisotropic) and with Heisenberg (isotropic) coupling, our results confirm the existence of a disorder-induced low-temperature (T) divergence in the long-wavelength Sz{S}^{z}Sz-Sz{S}^{z}Sz susceptibility \ensuremath{\chi} which was previously predicted by real-space renormalization-group (RSRG) treatments. Over the finite temperature range studied, these results are consistent with a 1/(T ln2{\mathrm{ln}}^{2}ln2T) behavior of \ensuremath{\chi}, and hence in qualitative agreement with the RSRG results. As in the XY-Heisenberg regime, we also find a disorder-induced enhancement of the low-T susceptibility for a system with Ising-like exchange coupling which, over the finite temperature range studied, is again consistent with RSRG results. However, there are inconsistencies between the RSRG predictions in the Ising-like regime at very low temperatures, and the exact results for the random-exchange Ising chain and the low-temperature behavior of \ensuremath{\chi} in the Ising-like regime may in fact be more complicated than predicted by RSRG. Finally, we also present results for the antiferromagnetic susceptibility and structure factor. For both Heisenberg and Ising-like systems, we find that disorder suppresses the long-range antiferromagnetic correlations at low T.