Periodic Reporting for period 4 - RTFT (Random Tensors and Field Theory)
Período documentado: 2023-12-01 hasta 2024-08-31
Quantum field theory (QFT) is omnipresent in the physical world. It is a powerful formalism which describes both the interactions of the elementary constituents of matter and the effective behavior of large systems, like magnets or superconductors. More recently, quantum field theory concepts, like decimation and renromalization have found new applications to the study of biological systems or machine learning. A most surprising feature of QFT is universality: water at the critical point and a magnet at the Curie temperature are described by surprisingly similar equations. Deepening our understanding of QFT will not only provide us with a new understanding of physical phenomena, but also it is today our best bet to understand why machine learning actually works.
We have a good grasp of weakly coupled QFT, but in many instances one needs to deal with strongly coupled systems. Understanding such models has been an ongoing research endeavor for the past 50 years. A particularly fruitful approach to strongly interacting QFTs are the so called large N theories in which the fundamental building block, the field, is not a scalar but a vector, a matrix or a higher order tensor. The number of components of the field is counted by a parameter, N, which can be varied. The vector and matrix case have been studied since the '70s and led to numerous advances in conformal field theory (CFT), string theory etc. . Their success relies on the "1/N expansion": the models significantly simplify when the number of components N is take to be large.
The tensor case has been less studied and took off only about 10 years ago when the PI and collaborators discovered the 1/N expansion for random tensors and the corresponding melonic large N limit. Interest in the field increased significantly after E. Witten noted that such models are similar to the SYK model but do not require quenching and I. Klebanov and collaborators pioneered the study of tensor field theories. This research project studied the implications of the tensor 1/N expansion both for random tensors and tensor field theories. Objective 1 explored and extended the melonic universality class and lead to unexpected connections to quantum information theory and free probability. Objective 2 explored tensor field theories and melonic conformal field theories. Objective 3 applied such theories and lead to the introduction of an asymptotically free tensor field theory ideal for the study of strongly coupled systems and pioneered the study of quantum mechanical models with long range interactions.
Among the achievements of this project we count the discovery of new universality classes of critical phenomena, the identification of an asymptotically free tensor field theory suited for the study of strongly interacting physics, the formulation of a new criterion for entanglement detection in D-parttite quantum systems and the identification of the correct generalization of free cumulants for random tensors.
We have a good grasp of weakly coupled QFT, but in many instances one needs to deal with strongly coupled systems. Understanding such models has been an ongoing research endeavor for the past 50 years. A particularly fruitful approach to strongly interacting QFTs are the so called large N theories in which the fundamental building block, the field, is not a scalar but a vector, a matrix or a higher order tensor. The number of components of the field is counted by a parameter, N, which can be varied. The vector and matrix case have been studied since the '70s and led to numerous advances in conformal field theory (CFT), string theory etc. . Their success relies on the "1/N expansion": the models significantly simplify when the number of components N is take to be large.
The tensor case has been less studied and took off only about 10 years ago when the PI and collaborators discovered the 1/N expansion for random tensors and the corresponding melonic large N limit. Interest in the field increased significantly after E. Witten noted that such models are similar to the SYK model but do not require quenching and I. Klebanov and collaborators pioneered the study of tensor field theories. This research project studied the implications of the tensor 1/N expansion both for random tensors and tensor field theories. Objective 1 explored and extended the melonic universality class and lead to unexpected connections to quantum information theory and free probability. Objective 2 explored tensor field theories and melonic conformal field theories. Objective 3 applied such theories and lead to the introduction of an asymptotically free tensor field theory ideal for the study of strongly coupled systems and pioneered the study of quantum mechanical models with long range interactions.
Among the achievements of this project we count the discovery of new universality classes of critical phenomena, the identification of an asymptotically free tensor field theory suited for the study of strongly interacting physics, the formulation of a new criterion for entanglement detection in D-parttite quantum systems and the identification of the correct generalization of free cumulants for random tensors.
Objective 1. Member S. Harribey et al. proved that the melonic universality class includes rank 5 irreducible tensors.The PI et al. proved that the 1/N expansion of vectors is Borel summable; studied the small N limit and instanton contributions in the vector case and proved that the N to -N duality extends to tensors models. The PI introduced a resolvent operator for real symmetric tensors and found a generalization of the Wigner semicircle law. The PI et al. generalized the Itzykson-Zuber integral to the tensor case; formulated a new criterion for entanglement detection in multipartite quantum systems and found the appropriate notion of free cumulants for tensors.
Objective 2. The PI et al. investigated the long range O(N)^3 tensor field theory. We proved that the model exhibits a line of large N melonic fixed points; by studying the OPE coefficients we showed that (although the model has a complex coupling) the large N CFT appears to be unitary; we proved that (although the model does not have a local energy momentum tensor) conformal invariance emerges at the melonic fixed point for correlations of the fundamental field and checked conformal invarince for correlations with composite operators at low orders in perturbation theory; we studied the long-range multi-scalar models at three loops and proved that the 1/N corrections to the melonic fixed point explicitly break unitarity. Crucially, we proved that the F-theorem applies to the long range O(N)^3 model using conformal partial waves: this is surprising as the current proof of the F-theorem relies crucially on unitarity.
Objective 3. The PI et al. proved that the O(N)^3 model in exactly four dimensions is asymptotically free and stable yielding an ideal toy model for the study of strongly coupled phenomena like confinement. The PI et al. have introduced fractional Lifshitz theories describing quantum systems with long range interactions and uncovered new universality classes of critical phenomena.
Our results resulted in over 40 publications in first tier journals and in order to showcase our results the team members participated in numerous international conferences and workshops. Additionally, we organized in Heidelberg two scientific conferences, one on Random Geometry and one on Quantum Field Theory, with a large international participation.
Objective 2. The PI et al. investigated the long range O(N)^3 tensor field theory. We proved that the model exhibits a line of large N melonic fixed points; by studying the OPE coefficients we showed that (although the model has a complex coupling) the large N CFT appears to be unitary; we proved that (although the model does not have a local energy momentum tensor) conformal invariance emerges at the melonic fixed point for correlations of the fundamental field and checked conformal invarince for correlations with composite operators at low orders in perturbation theory; we studied the long-range multi-scalar models at three loops and proved that the 1/N corrections to the melonic fixed point explicitly break unitarity. Crucially, we proved that the F-theorem applies to the long range O(N)^3 model using conformal partial waves: this is surprising as the current proof of the F-theorem relies crucially on unitarity.
Objective 3. The PI et al. proved that the O(N)^3 model in exactly four dimensions is asymptotically free and stable yielding an ideal toy model for the study of strongly coupled phenomena like confinement. The PI et al. have introduced fractional Lifshitz theories describing quantum systems with long range interactions and uncovered new universality classes of critical phenomena.
Our results resulted in over 40 publications in first tier journals and in order to showcase our results the team members participated in numerous international conferences and workshops. Additionally, we organized in Heidelberg two scientific conferences, one on Random Geometry and one on Quantum Field Theory, with a large international participation.
During this project we obtained numerous results which go beyond the state of the art.
Work on Objective 1 brought numerous milestone results and lead to new research directions. The tensor resolvent and the generalized Wigner law introduced by the PI have been used for for "signal detection" in tensor PCA, that is detecting whether a random tensor consists in a fixed "signal" tensor plus a random noise. An achievement of the program is the formulation of an entanglement detection criterion for multipartite quantum systems. A milestone result of the program is the identification of the correct generalization of free cumulats to tensors, which yields the first step in the formulation of a "free probability theory" adapted to tensors, a major research objective for the future.
Our results on Objective 2 give a complete and consistent picture of the O(N)^3 tensor model in dimension less that 4. We proved that the long range melonic fixed points of the model correspond to conformally invariant field theories which are unitary at large N, a milestone result of the project. The rigorously study of the emergence of conformal invariance in this model is a very promising future research direction. Another milestone result is the proof that the F-Theorem is respected by the long range O(N)^3 model. This shows that unitarity is a sufficient but not necessary condition for the F-theorem and studying how unitarity can be relaxed to a less restrictive condition should lead to future breakthroughs.
Objective 3 lead to another two milestone results. We proved that the O(N)^3 model in dimension four is asymptotically free hence, seen as a toy model for strongly interacting infrared physics, this model is ideally suited for the study of the formation of bound states and confinement. We have also pioneered the study of fractional Lifshitz theories describing quantum mechanical systems with long range interactions and identified a new universality class of critical phenomena.
Work on Objective 1 brought numerous milestone results and lead to new research directions. The tensor resolvent and the generalized Wigner law introduced by the PI have been used for for "signal detection" in tensor PCA, that is detecting whether a random tensor consists in a fixed "signal" tensor plus a random noise. An achievement of the program is the formulation of an entanglement detection criterion for multipartite quantum systems. A milestone result of the program is the identification of the correct generalization of free cumulats to tensors, which yields the first step in the formulation of a "free probability theory" adapted to tensors, a major research objective for the future.
Our results on Objective 2 give a complete and consistent picture of the O(N)^3 tensor model in dimension less that 4. We proved that the long range melonic fixed points of the model correspond to conformally invariant field theories which are unitary at large N, a milestone result of the project. The rigorously study of the emergence of conformal invariance in this model is a very promising future research direction. Another milestone result is the proof that the F-Theorem is respected by the long range O(N)^3 model. This shows that unitarity is a sufficient but not necessary condition for the F-theorem and studying how unitarity can be relaxed to a less restrictive condition should lead to future breakthroughs.
Objective 3 lead to another two milestone results. We proved that the O(N)^3 model in dimension four is asymptotically free hence, seen as a toy model for strongly interacting infrared physics, this model is ideally suited for the study of the formation of bound states and confinement. We have also pioneered the study of fractional Lifshitz theories describing quantum mechanical systems with long range interactions and identified a new universality class of critical phenomena.