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

Phase diagram of spin-1 quantum Ising models: Applications to systems of weakly coupled classical Ising chains

Yuval Gefen, Y. Imry, David Mukamel

Published Jun 1, 1981• 27 citations
Publisher

Abstract

The global phase diagram of the $S=1$ Ising model in a transverse field is studied using a mean-field approximation. The critical behavior has also been considered by real-space renormalization-group techniques. This model is used to analyze the phase diagram of weakly coupled ($\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}<<1$) classical Blume-Emery-Griffith chains. It is found that such systems display a first-order transition only in a very limited range of their parameters.

DOI
10.1103/physrevb.23.6099

Related papers

Renormalization-group approach to weakly interacting spin chains

J. Szṅajd

The thermodynamic properties of the weakly interacting spin-1/2 and spin-1 chains with the intrachain coupling J and interchain coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ are studied by using the linear renormalization-group method. The magnetic susceptibility of the weakly interacting $s=1/2$ quantum anisotropic Heisenberg chains in the longitudinal field (h) is investigated. The scaling shifts of the parallel susceptibility maxima in the external magnetic field are considered. The same method is used to study the specific heat and susceptibility of the weakly interacting $S=1$ Ising models. In the special case (Blume-Capel model) the phase diagram with critical and first-order transition lines is found. The location of the tricritical point for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}/J=0.2$ is estimated.

Physical review. B, Condensed matter•Sep 19, 2002•8 citations

Critical behavior of random transverse-field Ising spin chains

Daniel S. Fisher

A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.

Physical review. B, Condensed matter•Mar 1, 1995•828 citations

Phase Diagram of a Spin-1 Quantum Ising Model with Dipole and Quadrupole Interactions

Carla Buzano, Mario Rasetti, P. T. Valabrega

A quantum Ising-like spin-1 system with both dipole and quadrupole interactions is analyzed. Such model is a generalized quantum version of the Blume-Emery-Griffiths model in an external anisotropic field. Models of this type are interesting candidates to describe phase transitions in liquid crystals. The phase diagram as well as the thermodynamic potentials of the model are constructed by a recently formulated improved version of mean field theory. The latter, which has the usual mean field theory as its zero-th order approximation, takes into account fluctuations up to the fourth-order. The phase diagram is found to exhibit phase transition manifolds of both the first and the second order, as well as multicritical lines. Four phases can be recognized: one ferromagnetic and three paramagnetic. The phases are identified by the requirements of minimum free energy constrained by the condition of self consistency of the order parameters p and q

Molecular crystals and liquid crystals•Sep 1, 1983•4 citations

Phase diagram of spin-1 quantum Ising-Heisenberg models

Carla Buzano

The phase diagram at T = 0 of a family of spin-1 quantum Ising-Heisenberg models is studied using a mean-field approximation. The model Hamiltonian is characterized by single ion anisotropy terms D μ , (μ = x, y, z ) and by two positive parameters η x , η y measuring the degree of anisotropy of the interaction. They allow to describe a variety of models: among them the spin-1 Ising-Blume-Capel model (η x = η y = 0), and the isotropic Heisenberg model (η x = η y = 1). Both dipolar and quadrupolar order parameters are used to derive the phase diagram in the parameter space (η x , η y , D μ ). The resulting phase diagram is rather complex as it exhibits first and second order critical transitions as well as multicritical points.

Physica Scripta•Apr 1, 1988•4 citations

Exact renormalization of the random transverse-field Ising spin chain in the strongly ordered and strongly disordered Griffiths phases

Ferenc Iglói

Fisher's [Phys. Rev. B $51,$ 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.

arXiv:cond-mat/0108350•Physical review. B, Condensed matter•Jan 16, 2002•60 citations

Ising model with a transverse field in two dimensions: Phase diagram and critical properties from a real-space renormalization group

Zvi Friedman

Critical indices, the critical line, and the crossover scaling function for the free energy are calculated for the two-dimensional Ising model with a transverse field using a quantum-mechanical version of the real-space renormalization group. The critical behavior as function of the transverse field at a small finite temperature changes from that of a three-dimensional to that of a two-dimensional Ising behavior, as the critical line is approached.

Physical review. B, Condensed matter•Feb 1, 1978•24 citations
The global phase diagram of the S=1S=1S=1 Ising model in a transverse field is studied using a mean-field approximation. The critical behavior has also been considered by real-space renormalization-group techniques. This model is used to analyze the phase diagram of weakly coupled (J\ensuremath⊥J\ensuremath∥<<1\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}<<1J\ensuremath∥​J\ensuremath⊥​​<<1) classical Blume-Emery-Griffith chains. It is found that such systems display a first-order transition only in a very limited range of their parameters.
The thermodynamic properties of the weakly interacting spin-1/2 and spin-1 chains with the intrachain coupling J and interchain coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ are studied by using the linear renormalization-group method. The magnetic susceptibility of the weakly interacting s=1/2s=1/2s=1/2 quantum anisotropic Heisenberg chains in the longitudinal field (h) is investigated. The scaling shifts of the parallel susceptibility maxima in the external magnetic field are considered. The same method is used to study the specific heat and susceptibility of the weakly interacting S=1S=1S=1 Ising models. In the special case (Blume-Capel model) the phase diagram with critical and first-order transition lines is found. The location of the tricritical point for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}/J=0.2$ is estimated.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
A quantum Ising-like spin-1 system with both dipole and quadrupole interactions is analyzed. Such model is a generalized quantum version of the Blume-Emery-Griffiths model in an external anisotropic field. Models of this type are interesting candidates to describe phase transitions in liquid crystals. The phase diagram as well as the thermodynamic potentials of the model are constructed by a recently formulated improved version of mean field theory. The latter, which has the usual mean field theory as its zero-th order approximation, takes into account fluctuations up to the fourth-order. The phase diagram is found to exhibit phase transition manifolds of both the first and the second order, as well as multicritical lines. Four phases can be recognized: one ferromagnetic and three paramagnetic. The phases are identified by the requirements of minimum free energy constrained by the condition of self consistency of the order parameters p and q
The phase diagram at T = 0 of a family of spin-1 quantum Ising-Heisenberg models is studied using a mean-field approximation. The model Hamiltonian is characterized by single ion anisotropy terms D μ , (μ = x, y, z ) and by two positive parameters η x , η y measuring the degree of anisotropy of the interaction. They allow to describe a variety of models: among them the spin-1 Ising-Blume-Capel model (η x = η y = 0), and the isotropic Heisenberg model (η x = η y = 1). Both dipolar and quadrupolar order parameters are used to derive the phase diagram in the parameter space (η x , η y , D μ ). The resulting phase diagram is rather complex as it exhibits first and second order critical transitions as well as multicritical points.
Fisher's [Phys. Rev. B 51,51,51, 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.