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. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Excitations for lattice ferromagnetic classical spin systems at high temperature: Noneven single-spin distributions

Ricardo S. Schor, Michael O’Carroll

Published Jun 1, 2000• 5 citations
Publisher

Abstract

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region $(\ensuremath{\beta}\ensuremath{\ll}1).$ Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter $\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),$ where $\overline{s}=s\ensuremath{-}〈s〉,$ and $〈{s}^{k}〉$ denotes the $k\mathrm{th}$ moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold $2m.$ For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to $2m.$

DOI
10.1103/physreve.61.6156

Related papers

Bound states in the transfer matrix spectrum for general lattice ferromagnetic spin systems at high temperature

Ricardo S. Schor, Michael O’Carroll

We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 2000•18 citations

Transfer Matrix Spectrum for O(N) High Temperature Lattice Classical Ferromagnetic Spin Systems and Staggering Transformations

Petrus Henrique Ribeiro dos Anjos, Paulo A. Faria da Veiga, Michael O’Carroll

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d -dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

Physica Scripta•Jan 1, 2005•1 citation

Ising ferromagnets and antiferromagnets in an imaginary magnetic field

Roman Krčmár, Andrej Gendiar, Ladislav Šamaj

We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.

arXiv:2112.09536•Physical review. E•May 6, 2022•2 citations

Phase diagram of the mixed spin Ising model with four-spin interaction and nextnearest neighbor couplings

Majid Mouhib, Mouhcine Azhari, Noureddine Benayad

The phase diagrams of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’ are investigated by the use of the finite cluster approximation based on single-site cluster theory. The state equations are derived for the two-dimensional square lattice. The phase diagram is qualitatively and quantitatively different from that obtained when the interaction J’ is ignored. In fact, the system exhibits a variety of interesting features resulting from the introduction of the next-nearest neighbor interaction (NNN). It undergoes two kinds of behavior according to the negative and the positive value of J’. In particular, for J4/J2=-4, the system keeps the coexistence of the two ground state up to J’-dependent finite temperature.

PRSM•Aug 7, 2015•1 citation

Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series

P. Butera, M. Comi

We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling ${g}_{r}$ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ ${g}_{r}=1.404(3),$ and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.

arXiv:hep-lat/0112049•Physical review. B, Condensed matter•Apr 2, 2002•95 citations

Zero-temperature properties of quantum spin models on a triangular lattice I: the ferromagnet

Satoshi Fujiki, Donald D. Betts

The calculation of two- and four-spin correlations of the [Formula: see text] ferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse pair correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.5326 ± 0.003. We conjecture that the square of the magnetization per site vanishes very sharply as N−0.06. The nearest neighbour longitudinal correlation per bond [Formula: see text] for all two-dimensional lattices.

Canadian Journal of Physics•Aug 1, 1986•43 citations
We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (\ensuremathβ\ensuremath≪1).(\ensuremath{\beta}\ensuremath{\ll}1).(\ensuremathβ\ensuremath≪1). Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter \ensuremathα=(〈s‾4〉\ensuremath−3〈s‾2〉2\ensuremath−〈s‾3〉2〈s‾2〉\ensuremath−1)/(〈s‾4〉\ensuremath−〈s‾2〉2\ensuremath−〈s‾3〉2〈s‾2〉\ensuremath−1),\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),\ensuremathα=(〈s4〉\ensuremath−3〈s2〉2\ensuremath−〈s3〉2〈s2〉\ensuremath−1)/(〈s4〉\ensuremath−〈s2〉2\ensuremath−〈s3〉2〈s2〉\ensuremath−1), where s‾=s\ensuremath−〈s〉,\overline{s}=s\ensuremath{-}〈s〉,s=s\ensuremath−〈s〉, and 〈sk〉〈{s}^{k}〉〈sk〉 denotes the kthk\mathrm{th}kth moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass m\ensuremath∼ln\ensuremathβ.m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.m\ensuremath∼ln\ensuremathβ. Assuming 〈s3〉\ensuremath≠0〈{s}^{3}〉\ensuremath{\ne}0〈s3〉\ensuremath=0 we show that for \ensuremathα>0,\ensuremath{\alpha}>0,\ensuremathα>0, \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold 2m.2m.2m. For \ensuremathα<0\ensuremath{\alpha}<0\ensuremathα<0 bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to 2m.2m.2m.
We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.
We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d -dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.
The phase diagrams of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’ are investigated by the use of the finite cluster approximation based on single-site cluster theory. The state equations are derived for the two-dimensional square lattice. The phase diagram is qualitatively and quantitatively different from that obtained when the interaction J’ is ignored. In fact, the system exhibits a variety of interesting features resulting from the introduction of the next-nearest neighbor interaction (NNN). It undergoes two kinds of behavior according to the negative and the positive value of J’. In particular, for J4/J2=-4, the system keeps the coexistence of the two ground state up to J’-dependent finite temperature.
We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling gr{g}_{r}gr​ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ gr=1.404(3),{g}_{r}=1.404(3),gr​=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.
The calculation of two- and four-spin correlations of the [Formula: see text] ferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse pair correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.5326 ± 0.003. We conjecture that the square of the magnetization per site vanishes very sharply as N−0.06. The nearest neighbour longitudinal correlation per bond [Formula: see text] for all two-dimensional lattices.