In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for $d=2$ dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general $d$ is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with $d<~4$ . A dilution field is introduced which can be chosen to yield tricritical exponents for $d=2$ , in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for $d=2$ , and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for $d=2$ dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general $d$ is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with $d<~4$. A dilution field is introduced which can be chosen to yield tricritical exponents for $d=2$, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for $d=2$, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.

Physical review. B, Condensed matterSep 1, 198143 citations

Renormalization Group Approaches to Higher Order Critical Points.

G. F. Tuthill

Jan 1, 19750 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.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 matterApr 1, 1981104 citations

Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O( $N$ ) model in $d=3$ and $N=\infty$ Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O($N$) model in $d=3$ and $N=\infty$

Shunsuke Yabunaka, Bertrand Delamotte

Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical $O(N)$ model in three dimensions in the limit $N\to\infty$ . Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent $ν$ which takes the value $ν= 1/2$ along the regular part of the BMB line, that is, for $0 \leq λ< λ_{\rm BMB}$ , jumps to $ν= 1/3$ on the singular part of this line with the BMB FP, corresponding to $λ= λ_{\rm BMB}$ , being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical $O(N)$ model in three dimensions in the limit $N\to\infty$. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent $ν$ which takes the value $ν= 1/2$ along the regular part of the BMB line, that is, for $0 \leq λ< λ_{\rm BMB}$, jumps to $ν= 1/3$ on the singular part of this line with the BMB FP, corresponding to $λ= λ_{\rm BMB}$, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.

arXiv:2606.12269Jun 10, 20260 citations

In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for

d=2

dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general

d

is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with

d<~4

. A dilution field is introduced which can be chosen to yield tricritical exponents for

d=2

, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for

d=2

, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.

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.

Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O(

N

) model in

d=3

and

N=\infty

Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical

O(N)

model in three dimensions in the limit

N\to\infty

. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent

ν

which takes the value

ν= 1/2

along the regular part of the BMB line, that is, for

0 \leq λ< λ_{\rm BMB}

, jumps to

ν= 1/3

on the singular part of this line with the BMB FP, corresponding to

λ= λ_{\rm BMB}

, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.