A conjecture on the lower bound of the length-scale critical exponent $ν$ at continuous phase transitions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912957069262848 |
|---|---|
| author | Pelissetto, Andrea Vicari, Ettore |
| author_facet | Pelissetto, Andrea Vicari, Ettore |
| contents | A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $ν$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $φ$ and a unique $φ\cdot φ$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $Δ_φ=(d-2+η)/2$ is the dimension of the order-parameter field $φ$, and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[φ\cdot φ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon \ge 2 Δ_φ$, which implies $ν\ge (2-η)^{-1}$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $ε$-expansion results for generic LGW $Φ^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $Φ^4$ theories. In particular, since unitarity requires $η\ge 0$, the above inequality implies $ν\ge 1/2$ for unitary theories. This lower bound is more restrictive than $ν> 1/d$, derived by noting that $ν=1/d$ characterizes the singular finite-size behavior at first-order transitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A conjecture on the lower bound of the length-scale critical exponent $ν$ at continuous phase transitions Pelissetto, Andrea Vicari, Ettore Statistical Mechanics High Energy Physics - Lattice High Energy Physics - Theory A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $ν$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $φ$ and a unique $φ\cdot φ$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $Δ_φ=(d-2+η)/2$ is the dimension of the order-parameter field $φ$, and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[φ\cdot φ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon \ge 2 Δ_φ$, which implies $ν\ge (2-η)^{-1}$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $ε$-expansion results for generic LGW $Φ^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $Φ^4$ theories. In particular, since unitarity requires $η\ge 0$, the above inequality implies $ν\ge 1/2$ for unitary theories. This lower bound is more restrictive than $ν> 1/d$, derived by noting that $ν=1/d$ characterizes the singular finite-size behavior at first-order transitions. |
| title | A conjecture on the lower bound of the length-scale critical exponent $ν$ at continuous phase transitions |
| topic | Statistical Mechanics High Energy Physics - Lattice High Energy Physics - Theory |
| url | https://arxiv.org/abs/2510.17637 |