Periodic Reporting for period 4 - CFT-MAP (Charting the space of Conformal Field Theories: a combined nuMerical and Analytical aPproach)
Okres sprawozdawczy: 2022-11-01 do 2024-02-29
Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials. A prototypical example is boiling water at the critical pressure and temperature.
In addition, conformal field theories sit at the centre of our modern understanding of quantum field theory, describing the asymptotic behaviour of quantum systems at infinitesimally small or large distances.The description of this theories in a quantitative way is often very difficult: due to the absence of typical scales, all the degrees of freedom of the theory interact together, giving rise to a strong dynamics which eludes any description as a perturbation of simpler and solvable system.
For decades it has been a dream to study these intricate strongly coupled theories non-perturbatively using only symmetries and other consistency conditions. This approach, which takes the name of Conformal Bootstrap, was extremely successful in two dimensions and has led to exact solutions, as in the case of the planar Ising model. On the other hand, only recently we understood how to formulate the conformal bootstrap in higher dimensions and concretely apply it to extract quantitative results.
This project explores new directions in this field and develops more efficient numerical techniques and complementary analytical tools.
By using these techniques CFT-MAP is able to scan the space of possible conformal field theories and identify where and how they can exist. CFT- MAP connected high energy theory and the study of phase transitions.
Besides the innovative methodologies, a fundamental outcome of CFT-MAP is a word record determination of critical exponents in second phase transition, together with additional information that allows an approximate reconstruction of a quantum field theory in the proximity of a conformal field theory.
In addition, conformal field theories sit at the centre of our modern understanding of quantum field theory, describing the asymptotic behaviour of quantum systems at infinitesimally small or large distances.The description of this theories in a quantitative way is often very difficult: due to the absence of typical scales, all the degrees of freedom of the theory interact together, giving rise to a strong dynamics which eludes any description as a perturbation of simpler and solvable system.
For decades it has been a dream to study these intricate strongly coupled theories non-perturbatively using only symmetries and other consistency conditions. This approach, which takes the name of Conformal Bootstrap, was extremely successful in two dimensions and has led to exact solutions, as in the case of the planar Ising model. On the other hand, only recently we understood how to formulate the conformal bootstrap in higher dimensions and concretely apply it to extract quantitative results.
This project explores new directions in this field and develops more efficient numerical techniques and complementary analytical tools.
By using these techniques CFT-MAP is able to scan the space of possible conformal field theories and identify where and how they can exist. CFT- MAP connected high energy theory and the study of phase transitions.
Besides the innovative methodologies, a fundamental outcome of CFT-MAP is a word record determination of critical exponents in second phase transition, together with additional information that allows an approximate reconstruction of a quantum field theory in the proximity of a conformal field theory.
The initial phase of the project focused on analyzing correlation functions of operators found in quantum field theory, such as globally conserved symmetry currents. This involved integrating these techniques with existing methods to evaluate the method's viability. Our primary aim was to compare the new approach with existing ones, considering both the precision of results and the computational resources required.
The outcomes were promising, indicating the potential for further development. Incorporating conserved currents alongside other operators not only enhanced the numerical bootstrap's ability to constrain but also enabled the extraction of data inaccessible via standard scalar-based approaches.
Throughout this phase, our team developed various algorithms to efficiently tackle large-scale bootstrap problems. These algorithms were instrumental in resolving a longstanding discrepancy between experimental and simulated results regarding a specific observable related to the transition from fluid to superfluid helium. Similarly, they facilitated the rigorous demonstration of isotropic magnet instability under cubic perturbations, addressing another enduring challenge. Notably, the project achieved a breakthrough by designing an algorithm that integrates numerical properties with analytical input. This novel algorithm enabled the consideration of universal behaviors inherent in CFT within specific limits for the first time.
Additionally, CFT-MAP achieved groundbreaking precision in determining numerous observables, including critical exponents for various theories like the supersymmetric Ising model and the O(2) and O(3) models. Some of these computations would have been exceedingly difficult, if not impossible, using alternative methods.
Furthermore, the new algorithm enabled studies beyond our initial objectives. For example, we could non-perturbatively track the state spectrum of the Ising model across dimensions from 4 to 2. CFT-MAP also initiated exploration into exotic phase transitions characterized by diverse symmetry-breaking patterns.
The project also accomplished significant milestones beyond its primary objectives. This included delving into the constraints imposed by supersymmetric conformal field theories across various degrees of supersymmetry. Additionally, the project undertook a formal exploration of conformal field theories (CFT), with a focused examination on 2D CFTs.
In the project's final stages, we discovered that the techniques employed in numerical conformal bootstrap could be adapted to study scattering amplitudes in weakly coupled effective field theories. This led to investigations into phenomena like photon and pion scattering in theories resembling quantum chromodynamics, yielding bounds applicable to the real world and accurately reproducing low-lying meson resonance spectra with minimal assumptions.
The results of CFT-MAP have been disseminated through 38 publications in prestigious international scientific journals, with several more currently under review. A review on conformal bootstrap methodologies published in Reviews of Modern Physics has garnered over 600 citations since its release, with a follow-up review on numerical methods recently completed. Our team regularly presented project findings at international conferences, workshops, and local seminars worldwide.
To enhance project visibility and foster collaboration, we organized two significant events: a five-week workshop at the Galileo Galilei Institute in Florence, Italy, in October 2022, and a one-week conference at the University of Pisa, Italy, in February 2024, featuring international guests from leading universities.
The outcomes were promising, indicating the potential for further development. Incorporating conserved currents alongside other operators not only enhanced the numerical bootstrap's ability to constrain but also enabled the extraction of data inaccessible via standard scalar-based approaches.
Throughout this phase, our team developed various algorithms to efficiently tackle large-scale bootstrap problems. These algorithms were instrumental in resolving a longstanding discrepancy between experimental and simulated results regarding a specific observable related to the transition from fluid to superfluid helium. Similarly, they facilitated the rigorous demonstration of isotropic magnet instability under cubic perturbations, addressing another enduring challenge. Notably, the project achieved a breakthrough by designing an algorithm that integrates numerical properties with analytical input. This novel algorithm enabled the consideration of universal behaviors inherent in CFT within specific limits for the first time.
Additionally, CFT-MAP achieved groundbreaking precision in determining numerous observables, including critical exponents for various theories like the supersymmetric Ising model and the O(2) and O(3) models. Some of these computations would have been exceedingly difficult, if not impossible, using alternative methods.
Furthermore, the new algorithm enabled studies beyond our initial objectives. For example, we could non-perturbatively track the state spectrum of the Ising model across dimensions from 4 to 2. CFT-MAP also initiated exploration into exotic phase transitions characterized by diverse symmetry-breaking patterns.
The project also accomplished significant milestones beyond its primary objectives. This included delving into the constraints imposed by supersymmetric conformal field theories across various degrees of supersymmetry. Additionally, the project undertook a formal exploration of conformal field theories (CFT), with a focused examination on 2D CFTs.
In the project's final stages, we discovered that the techniques employed in numerical conformal bootstrap could be adapted to study scattering amplitudes in weakly coupled effective field theories. This led to investigations into phenomena like photon and pion scattering in theories resembling quantum chromodynamics, yielding bounds applicable to the real world and accurately reproducing low-lying meson resonance spectra with minimal assumptions.
The results of CFT-MAP have been disseminated through 38 publications in prestigious international scientific journals, with several more currently under review. A review on conformal bootstrap methodologies published in Reviews of Modern Physics has garnered over 600 citations since its release, with a follow-up review on numerical methods recently completed. Our team regularly presented project findings at international conferences, workshops, and local seminars worldwide.
To enhance project visibility and foster collaboration, we organized two significant events: a five-week workshop at the Galileo Galilei Institute in Florence, Italy, in October 2022, and a one-week conference at the University of Pisa, Italy, in February 2024, featuring international guests from leading universities.
This project has significantly advanced the conformal bootstrap methodology, surpassing previous limitations and setting new standards in research methodologies. Through the introduction of practical tools, methodologies, and strategies, it has provided invaluable contributions to the field, with implications expected to unfold over the coming years.
One notable achievement lies in the development of numerical algorithms tailored for solving convex optimization problems with a large number of parameters. These algorithms hold promise for application beyond the immediate scope of this project, potentially influencing various other fields.
Moreover, the project's success in computing rigorous bounds on observables of physical interest, surpassing the precision achievable with existing techniques, has opened up new avenues for exploration. Additionally, the computation of previously inaccessible observables underscores the project's contribution to expanding the breadth of research in the field.
Overall, this project's advancements represent a significant step forward in scientific inquiry, offering both practical tools and theoretical insights that stand to benefit researchers across disciplines.
One notable achievement lies in the development of numerical algorithms tailored for solving convex optimization problems with a large number of parameters. These algorithms hold promise for application beyond the immediate scope of this project, potentially influencing various other fields.
Moreover, the project's success in computing rigorous bounds on observables of physical interest, surpassing the precision achievable with existing techniques, has opened up new avenues for exploration. Additionally, the computation of previously inaccessible observables underscores the project's contribution to expanding the breadth of research in the field.
Overall, this project's advancements represent a significant step forward in scientific inquiry, offering both practical tools and theoretical insights that stand to benefit researchers across disciplines.