Physical Review Letters

Phase Transitions in the Quantum Heisenberg Model

Freeman J. Dyson, Élliott H. Lieb, Barry Simon A

Published Jul 19, 1976• 67 citations

Abstract

We rigorously prove that in three or more dimensions, the nearest-neighbor, simple-cubic, ferromagnetic, quantum Heisenberg model of spin $S(=\frac{1}{2}, 1, \dots{})$ has a phase transition at nonzero temperature.

DOI: 10.1103/physrevlett.37.120

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Some Critical Properties of the Heisenberg Model

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Physical ReviewJan 10, 196948 citations

Computer simulation of the three-dimensional short-range Heisenberg spin glass

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The results of large-scale Monte Carlo simulations on a nearest-neighbor Heisenberg spin-glass model are presented. We find that the transition is at zero temperature and that 3 is below the lower critical dimension. Relaxation times follow conventional dynamical scaling and diverge with a power of the temperature T as T\ensuremath{\rightarrow}0.

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Existence of Phase Transitions for Anisotropic Heisenberg Models

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Physical Review Letters

Phase Transitions in the Quantum Heisenberg Model

Freeman J. Dyson, Élliott H. Lieb, Barry Simon A

Published Jul 19, 1976• 67 citations

Publisher

Abstract

We rigorously prove that in three or more dimensions, the nearest-neighbor, simple-cubic, ferromagnetic, quantum Heisenberg model of spin

S(=\frac{1}{2}, 1, \dots{})

has a phase transition at nonzero temperature.

DOI: 10.1103/physrevlett.37.120

Related papers

Some Critical Properties of the Heisenberg Model

Roger Bowers, Michael E. Woolf

A detailed series-extrapolation investigation of some critical properties ($T\ensuremath{\rightarrow}{T}_{c}+$) of the Heisenberg model is presented. Systematic use is made of ratio and Pad\'e-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\ensuremath{\alpha}=0, \ensuremath{-}\frac{1}{8}<~{\ensuremath{\alpha}}_{s}<~\ensuremath{-}\frac{1}{16}$, $\ensuremath{\gamma}=1.375\ifmmode\pm\else\textpm\fi{}0.002$, $0.6875<~\ensuremath{\nu}<~0.7125$, $0<~\ensuremath{\eta}<~0.07$. Estimates for $\ensuremath{\gamma}$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\ensuremath{\gamma}=1\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-\textonehalf{} Heisenberg ferromagnet, it is estimated that $\ensuremath{\gamma}=1.3744\ifmmode\pm\else\textpm\fi{}0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-\textonehalf{} nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\ensuremath{\gamma}$ has the same value $1\frac{3}{8}$ irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.A detailed series-extrapolation investigation of some critical properties ($T\ensuremath{\rightarrow}{T}_{c}+$) of the Heisenberg model is presented. Systematic use is made of ratio and Pad\'e-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\ensuremath{\alpha}=0, \ensuremath{-}\frac{1}{8}<~{\ensuremath{\alpha}}_{s}<~\ensuremath{-}\frac{1}{16}$, $\ensuremath{\gamma}=1.375\ifmmode\pm\else\textpm\fi{}0.002$, $0.6875<~\ensuremath{\nu}<~0.7125$, $0<~\ensuremath{\eta}<~0.07$. Estimates for $\ensuremath{\gamma}$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\ensuremath{\gamma}=1\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-\textonehalf{} Heisenberg ferromagnet, it is estimated that $\ensuremath{\gamma}=1.3744\ifmmode\pm\else\textpm\fi{}0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-\textonehalf{} nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\ensuremath{\gamma}$ has the same value $1\frac{3}{8}$ irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.

Physical ReviewJan 10, 196948 citations

Computer simulation of the three-dimensional short-range Heisenberg spin glass

J. A. Olive, A. P. Young, D. Sherrington

Physical review. B, Condensed matterNov 1, 1986166 citations

Existence of Phase Transitions for Anisotropic Heisenberg Models

Jürg Martin Fröhlich, Élliott H. Lieb

Physical Review LettersFeb 21, 197745 citations

Quantum phase transition in the three-dimensional anisotropic frustrated Heisenberg antiferromagnetic model

J. Roberto Viana, J. Ricardo de Sousa, M. A. Contínentino

The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest $({J}_{1})$ - and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state $(T=0)$ and the finite- $T$ phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest $({J}_{1})$- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state $(T=0)$ and the finite-$T$ phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.

Physical Review BMay 27, 200824 citations

Critical temperature for the 2D Heisenberg model

E. Praveczki

Journal of Physics C Solid State PhysicsApr 20, 19803 citations

Quantum Ferrimagnets

Ramin Abolfath, H. Hamidian, A. Langari

We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear $σ$ -model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h ( $h_{c1} < h < h_{c2}$ ) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear $σ$-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h ($h_{c1} < h < h_{c2}$) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.

arXiv:cond-mat/9901063Jan 8, 19991 citation

We rigorously prove that in three or more dimensions, the nearest-neighbor, simple-cubic, ferromagnetic, quantum Heisenberg model of spin

S(=\frac{1}{2}, 1, \dots{})

has a phase transition at nonzero temperature.

1\frac{3}{8}

irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.

The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest

({J}_{1})

- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state

(T=0)

and the finite-

T

phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.

σ

-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (

h_{c1} < h < h_{c2}

) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.