SciPost Submission Page
The random free field scalar theory
by Alessandro Piazza, Marco Serone, Emilio Trevisani
Submission summary
| Authors (as registered SciPost users): | Alessandro Piazza · Marco Serone |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00029v1 (pdf) |
| Date submitted: | 2024-09-24 09:03 |
| Submitted by: | Serone, Marco |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
Quantum field theories with quenched disorder are so hard to study that even exactly solvable free theories present puzzling aspects. We consider a free scalar field $\phi$ in $d$ dimensions coupled to a random source $h$ with quenched disorder. Despite the presence of a mass scale governing the disorder distribution, we derive a new description of the theory that allows us to show that the theory is gapless and invariant under conformal symmetry, which acts in a non-trivial way on $\phi$ and $h$. This manifest CFT description reveals the presence of exotic continuous symmetries, such as nilpotent bosonic ones, in the quenched theory. We also reconsider Cardy's CFT description defined through the replica trick. In this description, the nilpotent symmetries reveal a striking resemblance with Parisi-Sourlas supersymmetries. We provide explicit maps of correlation functions between such CFTs and the original quenched theory. The maps are non-trivial and show that conformal behaviour is manifest only when considering suitable linear combinations of averages of products of correlators. We also briefly discuss how familiar notions like normal ordering of composite operators and OPE can be generalized in the presence of the more complicated local observables in the quenched theory.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1. a simple exactly solvable model
2. a pedagogical paper.
Weaknesses
1. difficult to reconcile with other methods
2. the model is well defined only in d>4
Report
There has been growing interest in reconsidering random field spin models, particularly the Random Field Ising Model (RFIM), using a representation that dates back to the linear transformations of fields proposed by J. Cardy in 1985. This is a complex problem that has remained unresolved for many years. Significant progress was achieved about a decade ago through the use of functional renormalization group methods, especially in their non-perturbative form. Recent advances in conformal field theory methods have provided an alternative approach to studying this problem; however, to my knowledge, the perspectives offered by the two approaches have yet to be reconciled. This is difficult to realize because, despite the fact that both methods use the renormalization group language, the ways in which they attack this difficult problem are very different.
In the present manuscript the authors apply the conformal field theory (CFT) approach to a much simpler system, the random free field theory (RFFT). This is an exactly solvable model without interactions but with quenched random field. They show that the RFFT after changing variables can be viewed as a simple non-interacting CFT whose primary operators are related to the original elementary fields in a peculiar way.
In contrast to ordinary field theories, conformal behavior is observed only when considering specific linear combinations of correlators of local operators.
The authors provide two different effective conformal field theories with and without replicas and derive the explicit maps of correlation functions between these theories and the original theory with quenched random field.
The random field system with local interactions can then be studied using the RG flow considering the above effective conformal theories as a starting UV fixed point.
The problem under consideration is interesting, and the presentation is clear and pedagogically sound, making the manuscript suitable for publication. However, I am uncertain whether it meets the high standards of novelty and breakthrough required for general acceptance in SciPost Physics.
I encourage the authors to address several points that could enhance the manuscript and add significant value.
As mentioned by the authors, the RFFT model does not satisfy cluster decomposition for $d<4$. A natural question that arises for the reader, therefore, is: what are the consequences of this for the scaling behavior in physically relevant dimensions below $d=4$?
Is the RFFT model in the same universality class as the critical RFIM above $d=6$?
Since the the RFFT model is exactly solvable, could one compute the exact distribution of the order parameter at criticality?
The authors consider an extension of their model to the case where the action $S_0$ is replaced by a generalized free theory. Could the calculations be further extended to the case of correlated random fields, such as power-law correlated random fields?
In the seminal RG framework for the critical behavior of spin systems with random fields, one typically refers to a zero-temperature or infinite-disorder fixed point that governs the criticality of these disordered systems. It would be useful to discuss the connection between this picture and the approach developed by the authors.
The authors identify the $O(-2)$ symmetry in their CFT. Another known example of disordered systems that exhibit this hidden symmetry is Charge Density Waves (CDW) pinned by disorder, which is also described by a zero-temperature fixed point and can be mapped onto a pure system with $O(-2)$ symmetry (see K.J. Wiese and A.A. Fedorenko, Phys. Rev. Lett. 123, 197601 (2019)). Is this a common property of systems controlled by a zero-temperature fixed point?
Recommendation
Ask for major revision