2017 2024

24/4/2024

It's been a long time since I was not putting updates here about my research. Let me try to fill in the gap and talk about a few papers which I put out in the last couple of years.

• In a paper with an IHES postdoc Junchen Rong, I came back to the subject of multiscalar fixed points within the epsilon-expansion. It's kind of mind-boggling that such a simple problem resists a classification. The problem was reviewed in my previous work with Andy Stergiou where we in particular pointed out a puzzle that all known fixed points feature symmetry enhancement. We offered a bottle of Dom Pérignon champagne for resolving this puzzle (still unclaimed). For those interested to know more about this area of research, I also recommend the talk by Hugh Osborn, who offers a bottle of Trinity college port for solving another related conjecture. (I thought the talk was recorded but I currently only see the slides.) 
• In a paper with Bingxin Lao, a master student from ENS, I developed effective field theory approach for approximating spectra of quantum Hamiltonians on the two-sphere geometry. This is motivated by the recent "fuzzy sphere"revolution started in this paper. However the model we considered was even simpler - it was Transverse Field Ising Model on the icosahedron. The spectrum of this model was approximated by an effective Hamiltonian, taking the CFT Hamiltonian and perturbing it by integrals of local CFT operators on the two-sphere. I am fond of this paper which amply demonstrated the power of this technique. Congratulations to Bingxin Lao for making a crucial discovery about the importance of the spin-4 perturbation and for having been admitted to the PhD program at the Princeton Physics Department!
• In a paper with an IHES postdoc Aleix Gimenez-Grau and my esteemed colleague and friend Yu Nakayama (YITP, Kyoto), we clarified the role of shift symmetry in the phenomenon of scale without conformal invariance. The mechanism is general, although we mostly focused on a particular model of experimental relevance - ferromagnets with strong dipolar interactions. This paper has had an insanely long gestation period, going back to my 2018 talk. When I wrote to Yu Nakayama in April 2023 that I was finally ready to conclude, he sounded incredulous ("April Fool's was 10 days ago.") But we did conclude, and congratulations to Aleix Gimenez-Grau for crucial contributions along the way. The paper is currently under consideration in a condensed matter journal. 
• With Ning Su, we reviewed the status of the numerical conformal bootstrap, providing a necessary update to the 2018 review with David Poland and Alessandro Vichi. Ning Su having been on the forefront of so many recent advances in the numerical bootstrap, it was a pleasure to help him put together this review article.

17/12/2022

Rigorous approach to the lightcone bootstrap.

The lightcone bootstrap is a set of analytic arguments which allow to make predictions about the spectrum of Conformal Field Theories in the limit of large spin and fixed twist. These arguments go back to two seminal papers from 2012, Fitzpatrick, Kaplan, Poland Simmons-Duffin and Komargodski and Zhiboedov. The simplest prediction is that any CFT with a twist gap in the spectrum should contain infinite families of operators whose spin goes to infinity while whose twist tends to a limit. This was argued at an intuitive level in the original papers. The argument was compelling but nonrigorous.

Now, CFT is a subject which is mathematically well defined. Everything reduces to convergent expansions in conformal blocks. In this context one expects that any argument, which is actually valid, can be relatively easily upgraded to a rigorous one. I tried to find such an "upgrade"for the lightcone bootstrap since 2017. However my attempts did not work and they made me worried that perhaps the lightcone bootstrap predictions are not universally valid. Finally, I am happy to report good news. In a joint paper with Sridip Pal and Jiaxin Qiao, we found a rigorous proof of the simplest lightcone bootstrap prediction stated above.

The argument is actually quite simple and robust (see the paper), and it should have further applications.  One of the applications is already there: we adapted the argument to prove a result about the states in 2d CFTs with twist accumulating to (c-1)/24, from modular invariance.

14/10/2022

Since January 2021, I am collaborating with Tom Kennedy, mathematical physicist at the University of Arizona. Our long-term goal is to find a rigorous realization of the nontrivial renormalization group fixed point for the Ising model and other related lattice models. Our bet is that this can be achieved using Tensor Renormalization Group, a real-space renormalization framework which has various advantages with respect to the Kadanoff-Wilson block-spin transformations. So far we have developed rigorous analytic theory of Tensor RG near the high-temperature fixed point and the low-temperature fixed point in 2D. We are hopeful that a computer-assisted version of our approach will work for the nontrivial fixed point.

12/09/2022

See a non-technical article about my recent work on the Random Field Ising Model, joint with Apratim Kaviraj and Emilio Trevisani. I will soon be giving an IHES course about the theory we developed. 

From left to right: Apratim Kaviraj, Emilio Trevisani and myself at IHES in September 2019

29/6/2021

Navigator functions in the conformal bootstrap

With Marten Reehorst, David Simmons-Duffin, Benoit Sirois, Ning Su, and Balt van Rees, we recently proposed a new tool in the numerical conformal bootstrap which we named the "navigator function". The name comes from the fact that with the new tool one can "sail" in the sea of disallowed points to the bootstrap islands, following the gradient of the navigator function as a compass (while previously to find the islands one had to perform expensive scans). I believe this will allow extending the range of conformal bootstrap applications in quite amazing ways. To know more please see our paper and watch talks by Ning Su or Balt van Rees.

12/4/2021

Distributions in CFT II. Minkowski Space

Since a couple of years, my PhD student Jiaxin Qiao and myself were collaborating with Petr Kravchuk on Lorentzian CFT. Our long paper has finally appeared last week. In it we show that Euclidean CFT correlation functions, constructed using the usual Euclidean rules such as convergent OPE, can be analytically continued to Minkowski space, obtaining tempered distributions satisfying Wightman axioms. This result is new: in the past people were happy to accept Wightman axioms as an extra assumption. 

In our paper we also review a lot of old literature about the relation of Lorentzian and Euclidean quantum field theory, such as the Osterwalder-Schrader theorem. We also review modern Lorentzian CFT literature. Much of this literature is not on yet on the same rigorous footing as the Euclidean conformal bootstrap, and we point out the most pressing open problems. 

11/10/2020

Random Field Ising Model and Parisi-Sourlas Supersymmetry II. RenormalizationGroup

Another project started in Rome 3 years ago came to an end. In January 2018 I went to a conference on Disordered Systems organized at La Sapienza University (Rome) by Giorgio Parisi and the Cracking the Glass Simons Collaboration. I was an outsider invited to give a review talk about the conformal bootstrap, but I did listen to the other talks. Many of them were about the Parisi-Sourlas supersymmetry and dimensional reduction, or rather why they are absent, in the Random Field Ising and other related models. I previously heard about this problem from my ENS colleague Édouard Brézin, so I got curious and decided to study the literature. Thanks are due to my other ENS colleague Nicolas Sourlas for mentioning a poorly-known paper by John Cardy. After a quick look, it became clear that this is interesting both from CFT and Renormalization Group perspectives, and that something could be done complementary to the existing treatments. I came back to IHES, talked to my postdocs Apratim Kaviraj and Emilio Trevisani, and we set to work.

The first paper devoted to supersymmetric CFT aspects appeared in December 2019, and the second paper, about the RG aspects, is finally out in September 2020. I already discussed the SUSY CFT paper below. The RG paper is much longer and more subtle. It develops a systematic theory of RG stability of the SUSY fixed point. Complexity stems from the fact that two out of three potentially destablizing perturbation classes (susy-writable, non-susy-writable, and susy-null) cannot even be written in the SUSY variables. Yet we managed to prove that two such perturbations, one non-susy-writable and one susy-null, may destablize SUSY below a critical dimension d ≈ 4.2. See the paper for full details.

Emilio Trevisani, myself and my then-PhD student Bernardo Zan discuss the problem of dimensional reduction and Parisi-Sourlas supersymmetry (January 2018, IHES)

26/9/2020

Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Conformal bootstrap and Renormalization Group are two complementary, and rival, approaches in the theory of critical phenomena. Now that the conformal bootstrap is well and running, I feel a bit sorry for the non-perturbative RG. Majority of high-energy physics researchers view its most popular current incarnation (Functional Renormalization Group) with suspect, as a bunch of unjustified approximations and truncations. My point of view is that Wilson was right, the strongly coupled fixed points exist in a true mathematical sense, and the problem is that we simply haven't found yet a numerically stable implementation of Wilson's vision. Where to start? Mathematicians proved many theorems using the renormalization group, so they must know how to control it rigorously. It would seem natural to turn to them for inspiration, but the mathematical RG papers look so daunting... See my talk at a Princeton workshop about the dream to understand better the rigorous RG and how to make it useful for practical calculations.

This dream started to come true in this paper, written in collaboration with two remarkable mathematical physicists, Alessandro Giuliani and Vieri Mastropietro. Alessandro and I met in 2017 at a conference at the Accademia dei Lincei (Rome). Once Alessandro told me that he can control rigorously Wilsonian RG for fermions, I got excited about the possibility to use his techniques to construct the simplest non-gaussian fermionic fixed point, in a model of symplectic fermions with a long-range kinetic term and a quartic interaction (a fermionic analogue of the bosonic long-range model I studied earlier with Connor Behan, Leonardo Rastelli and Bernardo Zan). 

It was clear from the start to Alessandro and Vieri that this construction should be possible. I am the main culprit that it took 3 years to complete: I had to master all their tricks to the extent as to be able to review them for my hep-th colleagues (pedagogy was one of our goals) and I could not initially fathom the "Gallavotti-Niccolò" tree expansion used by the Italian school in the theory of fermionic fixed point. The final version of our paper implements instead a Banach contraction argument, arguably more "Wilsonian" in spirit (while the tree construction is in an appendix and in retrospect it does look simpler).

I am quite proud of this paper. We have a nice result, written in a self-contained way. It will certainly stimulate further work, e.g. on the general theory of fermionic fixed points, and on checking various common lore properties of RG fixed points (such as conformal invariance). It would be nice to have a similar pedagogical exposition for bosonic fixed points. Finally, talking about dreams, it would be nice to have a (computer-assisted) construction of the fixed point of the 3d Ising model. This would be similar to how my late IHES colleague Oscar Lanford constructed the fixed point for the Feigenbaum period doubling back in 1982. I have some ideas in this direction.

My bootstrap Zoom seminar about this paper

02/04/2020

Open Problems in Conformal Bootstrap part of this website started. Please send your problems you want to be posted on this site.

01/02/2020

Conformal bootstrap and liquid Helium. I first learned about an 8-standard deviations discrepancy between the best experimental measurement of the critical exponent α in the superfluid transition of Helium-4, and its best theoretical determination, from this paper by Ettore Vicari. My dream since 2014 was that one day conformal bootstrap will tell who's right and who's wrong? Once Kos, Poland and Simmons-Duffin discovered the O(N) archipelago, the problem appeared within reach: just shrink the O(2) island until it excludes the Monte Carlo or the experiment. But somehow the O(2) island was not shrinking as fast as the Ising island. Finally, last December a 7-strong collaboration (Chester, Landry, Liu, Poland, Simmons-Duffin, Su, Vichi) cracked this problem. In the plot below, the vertical axis is directly related to the critical exponent α, the other axes being two other relevant operators of the O(2) model CFT. Experiment gives the brown region, earlier theory (Monte Carlo) the green parallelepiped. The tiny blue region is the conformal bootstrap, which confirms Monte Carlo and excludes the experiment. (At least) three key ingredients went into this work:

• Shrinking the island from all three directions, i.e. with 3 relevant operators (2 operators gave weaker results).
• A new clever scanning algorithm over the OPE coefficient ratios
• Major improvements in the SDPB code, increasing speed and parallelization, and reducing memory usage.

See their paper for the full details, and my commentary for another introduction to this beautiful result. Congratulations!

Last but not least, I am impressed by the work of Campostrini,  Hasenbusch,  Pelissetto, and Vicari which first pointed out this anomaly back in 2006 using Monte Carlo and High Temperature expansion, and stood by their result for a decade. I heard they had to face some serious criticisms, while some other theorists were happy to fiddle with their calculations until they perfectly agreed with the experiment. 

27/1/2020

People and funding.

CFT and bootstrap have been rapidly developing in recent years, with applications to such disparate subjects as statistical physics, quantum gravity, or supersymmetric quantum field theory. I'm very happy that Paris now hosts many permanent researchers working in this direction. Two more senior people are joining Paris CFT community in 2020. Balt van Rees moved from Durham University to become a professor of mathematical physics at the Ecole Polytechnique, the most prestigious Grande Ecole university in France.  Eric Perlmutter is moving from Caltech to a permanent position at the famous Saclay Institute of Theoretical Physics (IPHT). Both these institutions are located in the south of Paris, a 10-minute drive from my own Institut des Hautes Etudes Scientfiiques (IHES), and I often go there for seminars.

This years postdoc hiring season also brought some good news. Yifei He is working on the applications of conformal bootstrap to the Potts model and percolation in 2d, as well as on S-matrix bootstrap, and she will stay in Paris on a postdoctoral fellowship from the Philippe Meyer Theoretical Physics institute at the Ecole Normale Superieure. Denis Karateev, a seasoned CFT warrior, will also join the Ecole Normale Superieure group. Marten Reehorst, graduating PhD student of Alessandro Vichi, will start a postdoc position at the IHES. With such a field of youngsters, the coming years will bring more Parisian CFT blasts! 

In the meantime, the Simons bootstrap collaboration for the nonperturbative bootstrap has been extended for 3 more years, until Sep 2023. 

27/01/2020

Parisi-Sourlas Supersymmetry Since a couple of years, I have been collaborating with my two postdocs Apratim Kaviraj and Emilio Trevisani on the puzzle of the Random Field Ising model and Parisi-Sourlas supersymmetry. (Nicolas Sourlas is my colleague at ENS and we shop at the same Parisian market). This 40-year old story concerns the phase transition in the Ising model with magnetic impurities. By one kind of lore, this leads to a non-unitary supersymmetric CFT with scalar supercharges (Parisi-Sourlas supersymmetry). By further lore, Parisi- Sourlas SCFT in D dimensions is equivalent to a non-SUSY CFT in D-2 dimensions (dimensional reduction). It is known that this story is not fully correct, and in our project we are trying to understand why. Last December we put out a paper which puts on firmer ground the dimensional reduction part of the story, using non-perturbative CFT methods.  We are now working on the second paper which aims to clarify the existence of the SUSY fixed point.

27/1/2020

Distributions in CFT

Petr Kravchuk, my PhD student Jiaxin Qiao, and myself have been working on deriving Lorentzian physics from Euclidean CFT axioms (see the 12/1/2020 post below). For example, we would like to show that CFT 4pt functions in Lorentzian signature satisfy Wightman axioms (in particular that they are tempered distributions). We have accumulated a lot of material and it's time to start publishing. Our first paper considers the simplest result: the distributional properties of CFT correlation functions in cross-ratio space on the boundary of the region of convergence. The next joint papers will consider CFT 4pt functions in flat Lorentzian space and on the Lorentzian cylinder. Jiaxin Qiao has also worked out an interesting classification of convergent OPE channels for CFT four-point functions, which he should publish soon.

12/01/2020

IS CATEGORY THEORY USEFUL FOR PHYSICS?

"Of course!" - will say the selected wise. However, until recently I belonged  to the great majority of people who believed that this subject was just "abstract nonsense". Things changed in a recent paper which I completed in collaboration with a Princeton PhD student Damon Binder. In this project we tried to clarify the pesky notion of O(N) symmetry when N is not an integer, and thus O(N) group does not exist. In theoretical physics, we love to perform analytic continuations in parameters which nominally have to be integers, and O(N) is one such example. Other example is the analytic continuation in the number of dimensions D. Even more excitingly, this continuation is not just a formal trick: models corresponding to the intermediate values of N do exist and are well-defined non-perturbatively (they are called loop models). Damon and mine's explanation for the non-integer O(N) symmetry is that it is a categorical symmetry, controlled not by a group or an algebra but by a `Deligne category' -  a symmetric tensor category defined by a famous Belgian mathematician Pierre Deligne (formerly at IHES, now at IAS) in 2004.

12/01/2020

"I remember telling someone that I wanted to learn about elementary particles by studying boiling water." (Alexander Polyakov)

In perturbative QFT, we can go back and forth between Euclidean and Lorentzian Feynman integrals - this is called Wick rotation. Under certain conditions, this is also true non-perturbatively: correlation functions can be analytically continued from Euclidean to Lorentzian and back. Even if one is eventually interested in Euclidean physics, the existence of this continuation allows to explore typically "Lorentzian" constraints such as e.g. causality. Recently I gave four lecturesabout such analytic continuations in Conformal Field Theories. This is an ongoing project with IAS postdoc Petr Kravchuk and my PhD student Jiaxin Qiao, and the papers should appear soon. 

05/02/2019

About a year ago I gave the 27^th Occam lecture at Merton college, Oxford, entitled: "Reductionism vs bootstrap: are things big always made of things elementary". The video recording is here.

9/11/2018

In my recorded talk at the annual Bootstrap Collaboration meeting (the Simons Foundation, New York), I talked about long-range interactions and structural phase transitions as possible bootstrap targets.

9/11/2018

I'd like to say a few words about a recent paper "General Properties of Multiscalar RG Flows in d=4-ε" by Andy Stergiou and myself. Epsilon-expansion is a classic subject on which there was a lot in the 70s and 80s, and it's even now in use. Of course now there are alternative conformal bootstrap techniques, which are nonperturbative and sometimes give much more powerful results. But the role of the epsilon-expansion is still significant - it provides us with conjectural theories to shoot for with the bootstrap. Andy and I took a fresh look at the epsilon-expansion, trying to get results valid in full generality (without any symmetry assumptions). Only a handful of such results are known. We got a new such result, which limits any fixed point to lie in a known compact region of coupling space. We prove \((\lambda_{ijkl})^2 \le (N/8)\epsilon^2\), where \(N\) is the number of scalar fields, and \((\lambda_{ijkl})^2\) is the square of the quartic coupling tensor at the fixed point. Writing this paper also gave us a chance to review a lot of old beautiful literature about the epsilon expansion.

31/10/2018

Long read: Walking, First-Order Phase Transitions, and Complex CFTs part I & part II

It's high time to report on this recent project I completed a couple of months ago with Victor Gorbenko and my PhD student Bernardo Zan. We linked two hitherto disconnected phenomena  - walking in 4D gauge theories and weakly first-order phase transitions in lattice models such as the 2D Potts model. Kaplan, Lee, Son and Stephanov proposed in 2009 that QCD conformal window terminates because the Banks-Zaks fixed point collides with an as yet unidentified fixed point they called `QCD*', after which the fixed points `go to the complex plane'. I was not convinced about their reasoning and examples, either exotic (Efimov physics), or inappropriate (BKT phase transition), or holographic (raising doubts that everything is just a large N artifact). However, in the summer of 2016 I learned by chance that the 2D Potts model realizes the fixed point annihilation scenario (the example missed by Kaplan et al). So I got converted, and then joined forces with Victor and Bernardo to convert the rest of the world. 

In part I we set up general theory of walking RG flows, and clarify numerous related confusions lingering in high energy physics due to the absence of reliable lattice data for many-flavor QCD. One such confusion is the light pseudodilaton, urban legend based on dodgy theoretical arguments and overly optimistic misinterpretations of the meager lattice data. We also introduced a novel concept of `complex CFT', to clarify another confusion related to the RG fixed points at complex couplings. Some authors seem to think that these fixed points are unphysical e.g. referring to them as `unstable'. We argued instead that these fixed points are bona fide nonperturbative CFTs with operator product expansion and other axioms, except that they are non-unitary in a rather radical way, with complex scaling dimensions (and the central charge). Thinking in terms of these CFTs we discovered a non-perturbative criterion for the existence of a walking RG flow - imaginary part of the scaling dimension of a certain operator has to be small.

In part II we treated in detail the 2D Potts model - a computable example where analytic continuation can be done thanks to the exact solution. While Q-state Potts models with Q=2,3 are well known (these are just unitary minimal models), here we needed to understand statistical physics literature on the Q-state Potts model with non-integer Q. That this model can be non-perturbatively defined may be a surprise in itself to a high-energy physicist, but it can, in terms of clusters and loops. It took us two years to decipher the stat-phys jargon, but the fruits we harvested were worth it. We were able to exhibit complex CFTs, compute characteristics of walking RG flow perturbing around them, and make predictions for lattice simulations of `drifting critical exponents'. Weak first order phase transitions seem to appear in the theory of deconfined quantum criticality (quantum cond-mat), so our work was timely and will lead to further results.

Our second paper will appear in SciPost. Do you know about their peer-witnessed refereeing system?

13/7/2018

About 4 years ago, Filip Kos, David Poland and David Simmons-Duffin discovered the 3d Ising island by means of the multi-correlator conformal bootstrap analysis of 3d CFTs having Z2 symmetry. This result opened a new era in the conformal bootstrap, and in particular has been extended to O(N) models, producing the O(N) archipelago. I am happy to announce that a new bootstrap island has just been discovered by Junchen Rong and Ning Su, which corresponds to the 3d CFT describing a supersymmetric version of the Ising model. I heard that the same result has been obtained by David Poland and collaborators, whose paper should appear soon.

28/5/2018

With conformal bootstrap revival turning 10 years old, David Poland, Alessandro Vichi and myself wrote a long review article "The Conformal Bootstrap: Numerical Techniques and Applications" which will hopefully appear in the Reviews of Modern Physics but is already available online: https://arxiv.org/abs/1805.04405. Our main goal was to review the euclidean CFT, numerical techniques, and applications to the main CFT universality classes discussed in statistical, condensed matter and high energy physics. We could not do justice to analytical techniques and applications to SUSY theories which are under active development. This will have to wait for their own reviewers.

23/1/2018

I started teaching my master course at the ENS on "Topics in strongly coupled QFT". The course will be devoted to the renormalization group in the Long Range Ising model with \(1/r^2\) interaction à la Anderson-Yuval, and the Polchinski equation.

31/12/2017

One of the subjects I've been focussing on recently is the Hamiltonian Truncation - a Hamiltonian approach to solving strongly coupled dynamics of quantum field theories, which takes inspiration of the Rayleigh-Ritz method in quantum mechanics. At present, most applications of this method have been in 1+1 dimensional field theories, but I am convinced that this method has a lot of unexplored potential and may one day lead to a way of solving even four-dimensional quantum field theories, such as QCD, which is radically different from the currently used lattice Monte Carlo techniques.

Links:

• My papers  1409.1581, 1412.3460, 1512.00493, 1706.06121, 1706.09929
• My talk at Strings 2016 (slides), at Accademia dei Lincei 2017, and especially the review at the Simons Bootstrap Collaboration 2017 meeting (slides). 

• Recent review article by  A. J. A. James, R. M. Konik, P. Lecheminant, N. J. Robinson and A. M. Tsvelik

• Workshop "Hamiltonian Methods for Strongly Coupled Quantum Field Theory" (IHES, 8-12/01/2018) (talks recorded)

26/10/2017

I have been recently appointed professor at the Institut des Hautes Études Scientifiques, a paradisiac home for theoretical physicists and mathematicians in Bures-sur-Yvette, a short metro ride to the hills south of Paris. I am also going to continue part-time affiliation with the Ecole Normale Supérieure in the center Paris.

If you are interested in working with me, please apply to these postdoc position openings at the IHES and the ENS:

https://academicjobsonline.org/ajo/jobs/10314 (Deadline Nov 26, 2017)

https://academicjobsonline.org/ajo/jobs/7800 (Deadline Nov 6, 2017)

There are also postdoc opening at other nodes of the Simons bootstrap collaboration:

https://academicjobsonline.org/ajo/jobs/10480 (Deadline Dec 8, 2017)

27/2/2017

Bootstrap in the news

Almost simultaneously, two popular articles about the conformal bootstrap appeared in New Scientist and Quanta. I like better the Quanta article, by Natalie Wolchover, which dives deeper into the ideas behind the bootstrap.

The connection with the early S-matrix bootstrap by Geoffrey Chew came out very well, and quotes by Sasha Polyakov, who has been a great inspiration all along, are great to see. Polyakov admits that he did not believe the first results coming out of the post-2008 bootstrap revival: "I thought originally that there is some mistake there,"This is very true - I remember his criticisms and incredulity when I came to Princeton to give a seminar at the IAS in 2008.  But this was also a positive influence of sorts - to work harder and to convince my former PhD advisor was a unique challenge :)

Unfortunately, both articles do have some inaccuracies in the history of the bootstrap. In particular the contribution of the Italian group (Ferrara, Gatto, Grillo) is completely absent, although I did stress it in my interviews. The New Scientist piece, by Gabriel Popkin, did not mention my collaborator Riccardo Rattazzi by name; it is also not quite correct in describing the sequence of events leading to the application of bootstrap methods to the 3d Ising model. 

Next time I meet Subir Sachdev, I have to ask him what he meant when he said that bootstrap "is like going from a Mercedes to a Roll-Royce". Is this a positive or a negative comment?