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 Physics C Solid State Physics

The decoupling point of the axial next-nearest-neighbour Ising model and marginal crossover

David A. Huse, Michael Ellis Fisher

Published Jul 10, 1982• 21 citations
Publisher

Abstract

The axial next-nearest-neighbour Ising (or ANNNI) model and the four-state asymmetric clock model both exhibit a decoupling multicritical point where they decompose into two uncoupled Ising models at criticality. The corresponding scaling exponents for various operators are given generally, the leading modulation or chiral crossover exponent being phi = gamma - nu , where gamma and nu are the pure Ising susceptibility and correlation exponents. Implications for the ANNNI model phase diagrams are discussed. Crossover scaling forms appropriate to logarithmic and marginal behaviour, with non-universal exponents, are presented.

DOI
10.1088/0022-3719/15/19/002

Related papers

Three-dimensional axial next-nearest-neighbor Ising model: A series investigation

Z. Mo, Martin V. Ferer

We have derived a twelfth-order, high-temperature series for the axial next-nearest-neighbor Ising (ANNNI) models on the fcc and the sc lattices, and we present detailed results for one of the fcc models. We have determined the phase boundaries of these models and located the Lifshitz point from the intersection of the phase boundary with the ``disorder'' line, where the maximum in the susceptibility \ensuremath{\chi}(q) begins to occur at nonzero q signaling the onset of modulated correlations. As expected, the Lifshitz point appears to be an inflection point of the phase boundary. The effective susceptibility index shows the predicted universal values for definite regions of the parameter space, e.g., XY-like in the midrange modulated regime, with the expected smooth crossover near the multicritical points. Throughout the modulated regime from the Lifshitz point to the decoupling point, the variation in the critical wave vector shows only small deviations from mean-field predictions. Our best estimates for the Lifshitz-point indices, ${\ensuremath{\gamma}}_{\mathit{l}}$=1.62\ifmmode\pm\else\textpm\fi{}0.12, 4${\ensuremath{\nu}}_{\mathit{l}4}$=1.63\ifmmode\pm\else\textpm\fi{}0.10, and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{l}}$=0.20\ifmmode\pm\else\textpm\fi{}0.15, differ from previous estimates. We are confident that our estimates are more reliable not only because we have longer series for a close-packed lattice but also primarily because we have analyzed our series along the appropriate scaling direction.

Physical review. B, Condensed matter•May 1, 1991•17 citations

Mean-field renormalisation group approach for the axial next-nearest-neighbour Ising model

E. C. Valadares, J. A. Plascak

The critical properties of the axial next-nearest-neighbour Ising (ANNNI) model are investigated by using the mean-field renormalisation group approach. Linear and double chain clusters lying along the axial direction are considered. Phase diagrams for the model in two and three dimensions and estimates of thermal exponents are obtained.

Journal of Physics A Mathematical and General•Oct 1, 1987•5 citations

Dimensional crossover in quantal and finite-size Ising models

Ian D. Lawrie

The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.

Journal of Physics C Solid State Physics•Sep 28, 1978•27 citations

Quantum crossover in the transverse Ising model

Ian D. Lawrie

The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.

Journal of Physics C Solid State Physics•Mar 28, 1978•38 citations

High-Temperature-Series Study of Models Displaying Tricritical Behavior. II. A Nearest-Neighbor Ising Antiferromagnet with Next-Nearest-Neighbor Ferromagnetic Interactions

Fredric Harbus, Eugene Stanley

The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (${J}_{1}<0$) but also next-nearest-neighbor ferromagnetic exchange (${J}_{2}>0$). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power ${\overline{a}}_{2}$ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.

Physical review. B, Solid state•Aug 1, 1973•41 citations

The Classical Ground State of the Axial Next Nearest NeighbourIsing (ANNNI) Model in External Magnetic Fields

Kunle Adegoke

We obtain the classical critical antiferromagnetic---paramagnetic and antiphase---paramagnetic phase boundaries of the Axial Next Nearest Neighbour Ising Model in external magnetic fields

Archives of physics research•Jan 1, 2011•0 citations
The axial next-nearest-neighbour Ising (or ANNNI) model and the four-state asymmetric clock model both exhibit a decoupling multicritical point where they decompose into two uncoupled Ising models at criticality. The corresponding scaling exponents for various operators are given generally, the leading modulation or chiral crossover exponent being phi = gamma - nu , where gamma and nu are the pure Ising susceptibility and correlation exponents. Implications for the ANNNI model phase diagrams are discussed. Crossover scaling forms appropriate to logarithmic and marginal behaviour, with non-universal exponents, are presented.
We have derived a twelfth-order, high-temperature series for the axial next-nearest-neighbor Ising (ANNNI) models on the fcc and the sc lattices, and we present detailed results for one of the fcc models. We have determined the phase boundaries of these models and located the Lifshitz point from the intersection of the phase boundary with the ``disorder'' line, where the maximum in the susceptibility \ensuremath{\chi}(q) begins to occur at nonzero q signaling the onset of modulated correlations. As expected, the Lifshitz point appears to be an inflection point of the phase boundary. The effective susceptibility index shows the predicted universal values for definite regions of the parameter space, e.g., XY-like in the midrange modulated regime, with the expected smooth crossover near the multicritical points. Throughout the modulated regime from the Lifshitz point to the decoupling point, the variation in the critical wave vector shows only small deviations from mean-field predictions. Our best estimates for the Lifshitz-point indices, ${\ensuremath{\gamma}}_{\mathit{l}}$=1.62\ifmmode\pm\else\textpm\fi{}0.12, 4${\ensuremath{\nu}}_{\mathit{l}4}$=1.63\ifmmode\pm\else\textpm\fi{}0.10, and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{l}}$=0.20\ifmmode\pm\else\textpm\fi{}0.15, differ from previous estimates. We are confident that our estimates are more reliable not only because we have longer series for a close-packed lattice but also primarily because we have analyzed our series along the appropriate scaling direction.
The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.
The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.
The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (J1<0{J}_{1}<0J1​<0) but also next-nearest-neighbor ferromagnetic exchange (J2>0{J}_{2}>0J2​>0). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power a‾2{\overline{a}}_{2}a2​ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.