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

Self-Consistent Spin-Wave Approach to Quantum Spin Systems

Tosizumu Aoki

Published Aug 15, 1992• 12 citations
Publisher

Abstract

On the basis of the method of equation of motion, the ground states of the quantum spin systems with a long-range order are studied. The equations of motion for the spin operators are linearized by introduction of the effective field, which is the order parameter itself. The linearized equations of motion and the use of approximate commutation relations for spin operators lead to the boson representation of the systems. It is emphasized that in the ground state with the long-range order along the x -axis, the relationship ( S j y ) 2 +( S j z ) 2 =1/2 should be retained in the average. By use of the above constraint for spin fluctuations, the values of the long-range order and ground-state energy are calculated within the boson approximation. The results agree well with ones obtained by different methods.

DOI
10.1143/jpsj.61.2909

Related papers

Limitations of spin-wave theory in T=0 spin dynamics

James H. Taylor, Gerhard H. Muller

The $T=0$ dynamics of a family of one-dimensional quantum spin systems with fully ordered (ferromagnetic or spin-flop) ground states is investigated. By rigorous calculations, it is demonstrated that the $T=0$ dynamic structure factor is, in general, not a $\ensuremath{\delta}$ function as predicted by linear spin-wave theory but characterized by more complicated structures with nonzero linewidths. The exact result approaches the spin-wave result in the classical limit $s\ensuremath{\rightarrow}\ensuremath{\infty}$. The results of this study demonstrate how quantum effects may invalidate spin-wave theory even in the presence of saturated magnetic long-range order. Hence the failure of spin-wave theory is not necessarily linked to the absence or strong reduction of long-range order typical for one-dimensional quantum spin systems. The unreliability of spin-wave theory has therefore to be suspected also for the dynamics of three-dimensional quantum spin systems where the long-range order is always strong at $T=0$.

Physical review. B, Condensed matter•Aug 1, 1983•29 citations

Variational study on the ground state of the spin XY magnet

Masuo Suzuki, Seiji Miyashita

An approximate wave function of the ground state of the spin [Formula: see text] XY magnet is derived using a variational method. This wave function yields estimates of the ground state energy and long-range order which agree very well with the results obtained by Betts and Oitmaa by a finite lattice method.

Canadian Journal of Physics•Jul 1, 1978•57 citations

Schwinger-boson mean-field theory of the Heisenberg ferrimagnetic spin chain

Congjun Wu, Bin Chen, Xi Dai +2 more

The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at $T=0 \mathrm{K}.$ These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at $T>0 \mathrm{K},$ are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.

arXiv:cond-mat/9904291•Physical review. B, Condensed matter•Jul 1, 1999•65 citations

Spin-Wave Theory on Finite Lattices: Application to the J 1 - J 2 Heisenberg Model

Q. F. Zhong, Sandro Sorella

We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.

Europhysics Letters (EPL)•Feb 10, 1993•48 citations

Quantum field theory of phase transitions in Fermi systems

R. D. Mattuck, Börje Johansson

The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.

Advances In Physics•Jul 1, 1968•62 citations

Ground-State Energy of a System of Interacting Bose Particles

Masahiko HIROOKA, Shuichiro Yamasaki

The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.

Progress of Theoretical Physics•Oct 1, 1982•1 citation
On the basis of the method of equation of motion, the ground states of the quantum spin systems with a long-range order are studied. The equations of motion for the spin operators are linearized by introduction of the effective field, which is the order parameter itself. The linearized equations of motion and the use of approximate commutation relations for spin operators lead to the boson representation of the systems. It is emphasized that in the ground state with the long-range order along the x -axis, the relationship ( S j y ) 2 +( S j z ) 2 =1/2 should be retained in the average. By use of the above constraint for spin fluctuations, the values of the long-range order and ground-state energy are calculated within the boson approximation. The results agree well with ones obtained by different methods.
The T=0T=0T=0 dynamics of a family of one-dimensional quantum spin systems with fully ordered (ferromagnetic or spin-flop) ground states is investigated. By rigorous calculations, it is demonstrated that the T=0T=0T=0 dynamic structure factor is, in general, not a $\ensuremath{\delta}$ function as predicted by linear spin-wave theory but characterized by more complicated structures with nonzero linewidths. The exact result approaches the spin-wave result in the classical limit $s\ensuremath{\rightarrow}\ensuremath{\infty}$. The results of this study demonstrate how quantum effects may invalidate spin-wave theory even in the presence of saturated magnetic long-range order. Hence the failure of spin-wave theory is not necessarily linked to the absence or strong reduction of long-range order typical for one-dimensional quantum spin systems. The unreliability of spin-wave theory has therefore to be suspected also for the dynamics of three-dimensional quantum spin systems where the long-range order is always strong at T=0T=0T=0.
The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0K.T=0 \mathrm{K}.T=0K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0K,T>0 \mathrm{K},T>0K, are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.
We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.
The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.
The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.