Workshop 2025: Capturing instantons within the Functional Renormalization Group
Session Information
Location: Lecture room Schay
Day: Friday, 16 May
Time: 15:00 - 15:30
Chairperson: Michele Ruggeri
Presentation Details
Presentation Type: Oral presentation
Title: Capturing instantons within the Functional Renormalization Group
Abstract: We investigate localized excitations (instantons) in the scalar $\phi^4$ theory (Ising universality class) using Renormalization Group, RG. We do so in the regime of high interest for phase transitions, the vicinity of the lower critical dimension ($d_{lc}$) at which finite-temperature ordering becomes impossible due to such excitaions. Instantons are inherently nonperturbative, so we use Non-Perturbative Functional RG (NPFRG) allowing for nonperturbative Ansätze.
The 1+\epsilon regime of the Ising model has been studied in depth by Bruce and Wallace [1-3]. They consider "droplets", i.e., closed domain walls, to be the extension of the instantons to d=1+\epsilon (instantons being the excitations which prohibit ordering in $d_{lc}$=1). However, this is a very specialized method, and they had to introduce these excitations "by hand".
We do not aim to "re-solve" te Ising model. In reality, we investigate whether FRG, a tried and versatile method when it comes to phase transitions [4], can quantitatively capture the long-wave effects of localized excitations. This is challenging, as any RG method uses coarse-graining to build an effective action describing long-wave physics. The relevance comes from the potential for generalization to problems where the nature of the excitations is unknown. This is why we chose FRG, as its Ansätze are built from minimal information, such as the symmetry of the system and the nature of the order parameter. For instance, we calculate the value of $d_{lc}$ - it does not have to be known a priori.
Even in the lowest approximations of the NPFRG we find strong evidence that it captures instantons in the vicinity of ${d_lc}$. The hallmark of the approach to the lower critical dimension is nonuniform (in field) convergence of the fixed point potential. This is due to a a boundary layer forming near the minimum of the effective potential, the width of which shrinks to 0 as $d_{lc}$ is approached. The emergence and shape of this shrinking boundary layer allows us to recover the $T_c$ and gives strong evidence that the critical exponent $1/\nu$ goes to 0 in the limit of $d_{lc}$ regardless of approximation [5, 6].
Some systems dominated by localized, nonuniform excitations, that we hope to generalize our findings to, are hysteresis in the Random Field Ising model [7, 8] or even to tunneling problems that occur at the $d {lc}$ , like Josephson junctions [9].
[1] A. D. Bruce, D. J. Wallace, Droplet theory of low-dimensional Ising models, Phys. Rev. Lett. 47 (1981) 1743–1746
[2] A D Bruce ,D J Wallace, Droplet theory in low dimensions: Ising systems in zero field, Journal of Physics A: Mathematical and General, 16(8) (1983) 1721
[3] D. J. Wallace. Perturbative approach to surface fluctuations. In Les Houches Summer School in Theoretical Physics: Recent Advances in Field Theory and Statistical Mechanics (1982)
[4] N. Dupuis et al., The nonperturbative functional renormalization group and its applications, Physics Reports 910 (2021)
[5] L. N. Farkaš, G. Tarjus, I. Balog, Approach to the lower critical dimension of the ϕ4 theory in the derivative expansion of the functional renormalization group, Phys. Rev. E 108 (2023) 054107
[6] G. Tarjus, M. Marohnić, L. N. Farkaš, I. Balog, Capturing instantons within the Functional Renormalization Group, to appear
[7] L. X. Hayden, A. Raju, J. P. Sethna, Unusual scaling for two-dimensional avalanches: Curing the faceting and scaling in the lower critical dimension, Phys. Rev. Res. 1 (2019) 033060
[8] D. Spasojević, S. Janićević, M. Knežević, Numerical Evidence for Critical Behavior of the Two-Dimensional Nonequilibrium Zero-Temperature Random Field Ising Model, Phys. Rev. Lett. 106 (2011) 175701
[9] R. Daviet, N. Dupuis, Nature of the Schmid transition in a resistively shunted Josephson junction, Phys. Rev. B 108 (2023) 184514
Presenter
Dr Lucija Nora Farkaš
Institute of Physics | Croatia
Authors
1. Tarjus, Gilles | LPTMC, CNRS-UMR 7600, Sorbonne Université, 4 Place Jussieu, 75252 Paris cedex 05, France
2. Farkaš, Lucija Nora | Institute of Physics, Bijenička cesta 46, HR-10001 Zagreb, Croatia
3. Balog, Ivan | Institute of Physics, Bijenička cesta 46, HR-10001 Zagreb, Croatia