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, Solid state

Exact renormalization group exhibiting tricritical fixed point for a spin-one Ising model in one dimension

S. Krinsky, D. Furman

Published Apr 1, 1975• 67 citations
Publisher

Abstract

Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.

DOI
10.1103/physrevb.11.2602

Related papers

Exact Renormalization Group Exhibiting a Tricritical Fixed Point for a Spin-1 Ising Model in One Dimension

S. Krinsky, D. Furman

Exact renormalization-group recursion relations are studied for a spin-1 Ising model in one dimension. In this model, critical lines are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the tricritical point by a triple degeneracy. This is directly related to the tricritical fixed point being more unstable than the critical fixed points.

Physical Review Letters•Apr 1, 1974•25 citations

Three-component model and tricritical points: A renormalization-group study. Two dimensions

Miron Kaufman, Robert B. Griffiths, Julia M. Yeomans +1 more

The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary ($n=1$) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.

Physical review. B, Condensed matter•Apr 1, 1981•104 citations

Exact Renormalization Groups for One-Dimensional Spin Systems

David R. Nelson, Michael Ellis Fisher

Renormalization groups are constructed in exact closed form (as finite‐dimensional recursion relations) for a range of one‐dimensional Ising models including staggered and parallel magnetic fields, alternating and second neighbor pair coupling, four‐spin interactions, etc. Linearization about appropriate fixed points displays relevant, irrelevant, and marginal operators and yields the exact scaling behavior about the T=0 critical point (where η=1, δ=∞, and γ=ν). Both a fixed spin “dedecoration” group (based on the bond decoration process) and a spin‐rescaling group give identical critical behavior; but they lead to distinct realizations of the nonlinear scaling fields appearing in Wegner's analysis of corrections to scaling (which has been extended to discrete groups). The spin dedecoration and star‐triangle transformations are used to analyze a planar, multiconnected, “truncated tetrahedron” model.

Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•0 citations

Renormalization-group analysis of Lifshitz tricritical behavior

Amnon Aharony A, Eytan Domany, Richard M. Hornreich

A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in ${\mathrm{RbCaF}}_{3}$ and in some liquid-crystal systems.

Physical review. B, Condensed matter•Aug 1, 1987•21 citations

Exact differential renormalization group for the square Ising model

Wojciech Jeżewski

We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, ${y}_{T}=1$, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.

Physical review. B, Condensed matter•Oct 1, 1981•2 citations

An Exact Renormalization Group Transformation and Properties of Positive Symmetry Matrices

Chin‐Kun Hu

A positive symmetric m×m matrix P can be interpreted as the transfer matrix for an one-dimensional array of particles which have m-discrete energy levels and the most general nearest neighbor interactions. It is shown that P^1=P^2 can be considered as exact renormalization group (RG) transformation applied on such one dimensional system which does not have any phase transitions for T＞0. The exact RG transformation can be iterated to reach the high temperature fixed point. From the known asymptotic behaviour of RG transformations near the high temperature fixed point, we can derive and have a physical picture of properties of P^(2n) as n→∞, which can also be derived directly from the matrix theory. We also show that the fixed point of the RG transformations can be obtained directly from the eigenvector corresponding to the largest eigenvalue of P. The formulation is applied to the spin-j-Ising model as an illustration.

Chinese Journal of Physics•Dec 1, 1983•1 citation
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
Renormalization groups are constructed in exact closed form (as finite‐dimensional recursion relations) for a range of one‐dimensional Ising models including staggered and parallel magnetic fields, alternating and second neighbor pair coupling, four‐spin interactions, etc. Linearization about appropriate fixed points displays relevant, irrelevant, and marginal operators and yields the exact scaling behavior about the T=0 critical point (where η=1, δ=∞, and γ=ν). Both a fixed spin “dedecoration” group (based on the bond decoration process) and a spin‐rescaling group give identical critical behavior; but they lead to distinct realizations of the nonlinear scaling fields appearing in Wegner's analysis of corrections to scaling (which has been extended to discrete groups). The spin dedecoration and star‐triangle transformations are used to analyze a planar, multiconnected, “truncated tetrahedron” model.
A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in RbCaF3{\mathrm{RbCaF}}_{3}RbCaF3​ and in some liquid-crystal systems.
We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, yT=1{y}_{T}=1yT​=1, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.
A positive symmetric m×m matrix P can be interpreted as the transfer matrix for an one-dimensional array of particles which have m-discrete energy levels and the most general nearest neighbor interactions. It is shown that P^1=P^2 can be considered as exact renormalization group (RG) transformation applied on such one dimensional system which does not have any phase transitions for T＞0. The exact RG transformation can be iterated to reach the high temperature fixed point. From the known asymptotic behaviour of RG transformations near the high temperature fixed point, we can derive and have a physical picture of properties of P^(2n) as n→∞, which can also be derived directly from the matrix theory. We also show that the fixed point of the RG transformations can be obtained directly from the eigenvector corresponding to the largest eigenvalue of P. The formulation is applied to the spin-j-Ising model as an illustration.