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

Journal of the Physical Society of Japan

Green's Function Approach to the Modified Quantum Heisenberg Model

Tosizumi Aoki

Published Nov 1, 1998• 1 citation
Publisher

Abstract

The two-time Green's function method is applied to the modified quantum Heisenberg model with the following exchange interactions: The interactions decay as inverse square of the distance between sites and have alternate positive and negative signs with increasing distance between sites; moreover, they work well only along the coordinate axes. The critical temperature and zero-temperature spin reduction are calculated for the one-, two- and three-dimensional cases. The temperature dependence of spin reductions in two and three dimensions is estimated in the low-temperature region. The critical behavior of long-range order and susceptibility is also investigated. Lastly, it is attempted to estimate the long-distance behavior of static correlation function in one dimension.

DOI
10.1143/jpsj.67.3742

Related papers

Magnetic properties of quantum Heisenberg ferromagnets with long-range interactions

Hiroki Nakano, Masahiro Takahashi

Quantum Heisenberg ferromagnets with long-range interactions in low dimensions are inves- tigated by means of the Green's frunction method. The model Hamiltonian is given by scrH=-(${\mathit{J}}_{0}$/2)${\mathcal{J}}_{\mathit{i}\mathrm{\ensuremath{\ne}}\mathit{j}}$${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathit{p}}$${\mathbf{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{j}}$ -H${\mathcal{J}}_{\mathit{j}}$${\mathit{S}}_{\mathit{j}}^{\mathit{z}}$. It is shown that there exists a finite-temperature phase transition in the region dp2d for the d-dimensional case and that no transitions at any finite temperature exist for p\ensuremath{\ge}2d; the critical temperature is also estimated. We study the magnetic properties of this model. We calculate the critical exponents' dependence on p; these exponents also satisfy a scaling relation. Some of the results were also found using the modified spin-wave theory and are in remarkable agreement with each other.

arXiv:cond-mat/9505066•Physical review. B, Condensed matter•Sep 1, 1995•48 citations

Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

Jun Kondo, Yamaji K.

The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.

Progress of Theoretical Physics•Mar 1, 1972•246 citations

Green’s function theory of the spin-1 low-dimensional quantumXYferromagnet

M. E. Gouvêa, Antonio S. T. Pires

We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional $\mathrm{XY}$ models with $S=1,$ using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.

Physical review. B, Condensed matter•Mar 5, 2001•9 citations

Quantum Heisenberg Chain with Long-Range Ferromagnetic Interactions at Low Temperatures

Hiroki Nakano, Minoru Takahashi

A modified spin-wave theory is applied to the one-dimensional quantum Heisenberg model with long-range ferromagnetic interactions. ( H =- J ∑ i < j ( r i j ) - p S i · S j ). Low-temperature properties of this model are investigated. The susceptibility and the specific heat are calculated; the relation between their behaviors and strength of the long-range interactions is obtained. This model includes both the Haldane-Shastry model and the nearest-neighbor Heisenberg model; the corresponding results in this paper are in agreement with the solutions of both the models. It is shown that there exists an ordering transition at a finite temperature for 1< p <2 where the model has longer-range interactions than the HS model. The critical temperature is estimated.

Journal of the Physical Society of Japan•Mar 15, 1994•23 citations

Field-induced quantum critical point in planar Heisenberg ferromagnets with long-range interactions: Two-time Green’s function framework

L. S. Campana, L. De Cesare, Ugo Esposito +2 more

The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a $d$-dimensional spin-$S$ planar Heisenberg ferromagnet with long-range interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}>d$) with the distance $r$ between spins. We adopt the Callen scheme for spin $S$ and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the $(\ensuremath{\alpha},d)$ plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of $\ensuremath{\alpha}$, we find that also for dimensionalities $d\ensuremath{\leqslant}2$ a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of $(\ensuremath{\alpha},d)$-dependent scaling functions and effective critical exponents.

Physical Review B•Jul 13, 2010•20 citations

Quantum Monte Carlo study of a Heisenberg spin system in two dimensions with long-range interactions

Oleg N. Vassiliev, Michael G Cottam, I. V. Rojdestvenski

The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.

Journal of Applied Physics•Jun 1, 2001•5 citations
The two-time Green's function method is applied to the modified quantum Heisenberg model with the following exchange interactions: The interactions decay as inverse square of the distance between sites and have alternate positive and negative signs with increasing distance between sites; moreover, they work well only along the coordinate axes. The critical temperature and zero-temperature spin reduction are calculated for the one-, two- and three-dimensional cases. The temperature dependence of spin reductions in two and three dimensions is estimated in the low-temperature region. The critical behavior of long-range order and susceptibility is also investigated. Lastly, it is attempted to estimate the long-distance behavior of static correlation function in one dimension.
Quantum Heisenberg ferromagnets with long-range interactions in low dimensions are inves- tigated by means of the Green's frunction method. The model Hamiltonian is given by scrH=-(J0{\mathit{J}}_{0}J0​/2)${\mathcal{J}}_{\mathit{i}\mathrm{\ensuremath{\ne}}\mathit{j}}$${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathit{p}}$Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Sj{\mathbf{S}}_{\mathit{j}}Sj​ -HJj{\mathcal{J}}_{\mathit{j}}Jj​Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​. It is shown that there exists a finite-temperature phase transition in the region dp2d for the d-dimensional case and that no transitions at any finite temperature exist for p\ensuremath{\ge}2d; the critical temperature is also estimated. We study the magnetic properties of this model. We calculate the critical exponents' dependence on p; these exponents also satisfy a scaling relation. Some of the results were also found using the modified spin-wave theory and are in remarkable agreement with each other.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional XY\mathrm{XY}XY models with S=1,S=1,S=1, using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.
A modified spin-wave theory is applied to the one-dimensional quantum Heisenberg model with long-range ferromagnetic interactions. ( H =- J ∑ i < j ( r i j ) - p S i · S j ). Low-temperature properties of this model are investigated. The susceptibility and the specific heat are calculated; the relation between their behaviors and strength of the long-range interactions is obtained. This model includes both the Haldane-Shastry model and the nearest-neighbor Heisenberg model; the corresponding results in this paper are in agreement with the solutions of both the models. It is shown that there exists an ordering transition at a finite temperature for 1< p <2 where the model has longer-range interactions than the HS model. The critical temperature is estimated.
The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a ddd-dimensional spin-SSS planar Heisenberg ferromagnet with long-range interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}>d$) with the distance rrr between spins. We adopt the Callen scheme for spin SSS and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the $(\ensuremath{\alpha},d)$ plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of $\ensuremath{\alpha}$, we find that also for dimensionalities $d\ensuremath{\leqslant}2$ a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of $(\ensuremath{\alpha},d)$-dependent scaling functions and effective critical exponents.
The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.