Periodic Reporting for period 3 - RTFT (Random Tensors and Field Theory)
Okres sprawozdawczy: 2022-06-01 do 2023-11-30
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 studies the implications of the tensor 1/N expansion both for random tensors and tensor field theories. Objective 1 explores and extends the melonic universality class. Objective 2 explores tensor field theories and melonic conformal field theories. Objective 3 applies such theories to the AdS/CFT correspondence, condensed matter systems, models of random geometry or non equilibrium QFT.
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 studies the implications of the tensor 1/N expansion both for random tensors and tensor field theories. Objective 1 explores and extends the melonic universality class. Objective 2 explores tensor field theories and melonic conformal field theories. Objective 3 applies such theories to the AdS/CFT correspondence, condensed matter systems, models of random geometry or non equilibrium QFT.
Significant progress has been achieved on all the objectives and new directions of research have been uncovered.
Objective 1. Project member S. Harribey et al. proved that the melonic universality class includes rank 5 irreducible tensors; new techniques need to be developed in order to generalize further.The PI et al. proved that the 1/N expansion of vectors is Borel summable; we furthermore studied the small N limit and instanton contributions in the vector case.This studies will be pursued for matrices and tensors. We proved that the N to -N duality of vector models extends to the tensor case. Applications of random tensors to machine learning and quantum information have been discovered: the PI introduced a resolvent operator for real symmetric tensors and found a generalization of the Wigner semicircle law relevant for signal detection in random tensor noise, an active topic in machine learning; the PI et al. generalized the Itzykson-Zuber integral to the tensor case and formulated a new criterion for the detection of entanglement in multipartite quantum systems.
Objective 2, renormalziation in tensor field theory, has been completed. The PI et al. exhaustively 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 observed that, although the model has a complex coupling, the large N CFT appears to be unitary (real OPE coefficients, real operator dimensions obeying unitarity bounds). We furthermore proved that, although the model does not have a local energy momentum tensor, conformal invariance emerges at the melonic fixed point for correlations involving only 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. We proved that the F-theorem applies to the long range O(N)^3 model using a newly rediscovered approach to CFT via conformal partial waves; this is surprising as the current proof of the F-theorem relies crucially on unitarity.
Objective 3, applications of tensor field theories. The PI et al. proved that the O(N)^3 model in exactly four dimensions is asymptotically free and stable, which is quite unexpected as the models has only bosonic fields; this provides a simple toy model for the study of confinement. The PI et al. have introduced fractional Lifshitz theories describing quantum systems with long range interactions.
Objective 1. Project member S. Harribey et al. proved that the melonic universality class includes rank 5 irreducible tensors; new techniques need to be developed in order to generalize further.The PI et al. proved that the 1/N expansion of vectors is Borel summable; we furthermore studied the small N limit and instanton contributions in the vector case.This studies will be pursued for matrices and tensors. We proved that the N to -N duality of vector models extends to the tensor case. Applications of random tensors to machine learning and quantum information have been discovered: the PI introduced a resolvent operator for real symmetric tensors and found a generalization of the Wigner semicircle law relevant for signal detection in random tensor noise, an active topic in machine learning; the PI et al. generalized the Itzykson-Zuber integral to the tensor case and formulated a new criterion for the detection of entanglement in multipartite quantum systems.
Objective 2, renormalziation in tensor field theory, has been completed. The PI et al. exhaustively 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 observed that, although the model has a complex coupling, the large N CFT appears to be unitary (real OPE coefficients, real operator dimensions obeying unitarity bounds). We furthermore proved that, although the model does not have a local energy momentum tensor, conformal invariance emerges at the melonic fixed point for correlations involving only 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. We proved that the F-theorem applies to the long range O(N)^3 model using a newly rediscovered approach to CFT via conformal partial waves; this is surprising as the current proof of the F-theorem relies crucially on unitarity.
Objective 3, applications of tensor field theories. The PI et al. proved that the O(N)^3 model in exactly four dimensions is asymptotically free and stable, which is quite unexpected as the models has only bosonic fields; this provides a simple toy model for the study of confinement. The PI et al. have introduced fractional Lifshitz theories describing quantum systems with long range interactions.
The tensor resolvent and the generalized Wigner law will be studied further. Departures from the generalized Wigner law are associated with "signal detection", that is detecting that the tensor consists of a fixed "signal" tensor plus a random noise. The next task is the "signal reconstruction", that is a well defined procedure to recover the fixed signal tensor. Progress will be made by making contact with the tensor principal component analysis pioneered by Ben Arous et al.
A milestone of the program so far is the formulation of an entanglement detection criterion for multipartite quantum systems. We will study the complementary small N large D limit of the tensor HCIZ integral and adapt our entanglement detection criterion to the more common case of a system of many q-bits.
Our results on Objective 2 form a consistent picture: the long range melonic fixed points correspond to conformally invariant field theories which are unitary at large N. This is a milestone result of the project. We will study rigorously the emergence of full conformal invariance using flow equations along the lines of Delamotte et al. However, a proof of unitarity at large N seems for now our of reach: it is difficult to envisage how one can recover a unitary CFT as a limit of non unitary ones.
Another milestone result is our proof that the F-Theorem is respected by the long range O(N)^3 model. We will study further to what extent unitarity is required for the F-theorem: our results on the O(N)^3 model suggest that, at least in some cases, the unitarity requirement can be relaxed to a less restrictive condition. Identifying such a condition requires a novel approach to the F-theorem and would be a major breakthrough. This is especially relevant as it has been recently observed that CFTs in non integral dimensions are non unitary.
We will continue to explore Objective 3, i.e. the implications of the melonic large N limit in various fields. We will study the strongly interacting infrared limit of the O(N)^3 model in dimension four and search for bound states. Another project will be the analytic study of large N tensor theories out of equilibrium, which require to Wick rotated to real time. We will continue the study of fractional Lifshitz theories and their interplay with the large N limit in the vector, matrix and tensor case.
A milestone of the program so far is the formulation of an entanglement detection criterion for multipartite quantum systems. We will study the complementary small N large D limit of the tensor HCIZ integral and adapt our entanglement detection criterion to the more common case of a system of many q-bits.
Our results on Objective 2 form a consistent picture: the long range melonic fixed points correspond to conformally invariant field theories which are unitary at large N. This is a milestone result of the project. We will study rigorously the emergence of full conformal invariance using flow equations along the lines of Delamotte et al. However, a proof of unitarity at large N seems for now our of reach: it is difficult to envisage how one can recover a unitary CFT as a limit of non unitary ones.
Another milestone result is our proof that the F-Theorem is respected by the long range O(N)^3 model. We will study further to what extent unitarity is required for the F-theorem: our results on the O(N)^3 model suggest that, at least in some cases, the unitarity requirement can be relaxed to a less restrictive condition. Identifying such a condition requires a novel approach to the F-theorem and would be a major breakthrough. This is especially relevant as it has been recently observed that CFTs in non integral dimensions are non unitary.
We will continue to explore Objective 3, i.e. the implications of the melonic large N limit in various fields. We will study the strongly interacting infrared limit of the O(N)^3 model in dimension four and search for bound states. Another project will be the analytic study of large N tensor theories out of equilibrium, which require to Wick rotated to real time. We will continue the study of fractional Lifshitz theories and their interplay with the large N limit in the vector, matrix and tensor case.