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

Some difficulties associated with parameter-dependent renormalization-group transformations

Michel Droz, Marcel Ausloos, J. D. Gunton

Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.

Physical review. B, Condensed matterJul 1, 19785 citations

Critical phenomena and renormalization-group theory

Andrea Pelissetto, Ettore Vicari

arXiv:cond-mat/0012164Physics ReportsOct 1, 20021,631 citations

Critical phenomena and the renormalization group

Franz Wegner

Springer eBooksApr 6, 20086 citations

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.