Tensor Journal Club

The eigenstate thermalization hypothesis (ETH) was developed to explain the mechanism by which "chaotic" systems reach thermal equilibrium from a generic state. ETH is an ansatz for the matrix elements of physical operators in the basis of the Hamiltonian, and since its postulation, numerous studies have characterized these quantities in increasingly fine detail, providing a solid framework for understanding the (thermo)dynamics of quantum many-body systems. ETH can be viewed as a generalisation of random matrix theory and, in fact, within this ansatz matrix elements are modeled as random variables. In our work, we have generalized the ETH ansatz in order to take into account correlations between matrix elements which are essential to describe high-order correlation functions. By analogy with the theory of random matrices, one can assume a certain hierarchy between these correlations and show how this generalized ansatz underlies a relationship between ETH and free probability, a branch of mathematics that studies non-commutative random variables. This relationship allowed us to unveil a particular structure of the time-dependent correlation functions in thermal equilibrium.

One aspect of the black hole information puzzle concerns the issue that the radiation entropy seems ever increasing. Assuming unitarity, Page showed in a toy model that the entropy must follow the Page curve that stops increasing after the black hole "half-life". In semiclassical gravity, the entropy of Hawking radiation is recently shown to be given by the "island formula", which was derived using the replica trick with replica wormhole contributions. It successfully produces the qualitative behavior of the Page curve. However, the island formula doesn't capture the radiation entropy accurately when the effect of replica symmetry breaking is significant, which could occur in presence of non-trivial entanglement spectra between the radiation and the black hole. In such scenarios, the quantum extremal surfaces and islands cannot be identified. After reviewing the path integral method for the gravitational replica trick, we show how free probability can help us compute the fine-grained radiation entropy for an arbitrary bulk radiation state, in the context of the Penington-Shenker-Stanford-Yang (PSSY) model of two-dimensional black holes. We observe that the relevant gravitational path integral is combinatorially matching the free multiplicative convolution between the spectra of the gravitational sector and the matter sector respectively. We then work out how the free convolution formula can be evaluated using free harmonic analysis. The convolution formula computes the radiation entropy accurately even in cases when the island formula fails to apply. Using the close tie between free probability and random matrix theory, we show that the PSSY model can be described as a generalization of Page's model.

In this talk, we will present an asymptotic analysis of large asymmetric spiked tensor models, relying on tools of random matrix theory. In the first part of the talk, we will provide the analysis of a rank-one model along with some algorithmic implications in terms of possible signal recovery with a polynomial time algorithm. The second part of the talk will present a generalization of a low-rank spiked model with non-independent components, by studying two types of deflation procedures and proposing an improved algorithm.

Asymptotic freedom is a hallmark of the strong interaction and in four space-time dimensions conventionally linked to local space-time symmetries of non-Abelian gauge fields. I will discuss asymptotic freedom in a tensor quantum field theory with global \(O(N)^3\) symmetry and imaginary tetrahedral coupling. A perturbative analysis uncovers that the renormalization group flow of the quartic couplings connects a Gaussian ultraviolet fixed point to a strongly interacting theory in the infrared. The full quantum effective action is real and the path integral is bounded from below in the melonic large-N limit. The analytical tractability of the non-trivial melonic large-N limit raises the possibility of studying the renormalization group flow towards the strongly interacting infrared.

Eigenvalue distributions play important roles in understanding the dynamics of matrix models. Then it would be a natural question what roles tensor eigenvalue/vector distributions play in tensor models. We consider here the most basic case of Gaussian tensor models, and compute the eigenvalue/vector distributions. In fact essentially the same problem has already been solved by matrix model techniques in the context of the p-spin spherical model for spin glasses. But in this talk I use more familiar field theoretical methods. A picked-up result is that, in the large-N limit, the eigenvalue distribution is Gaussian, which can be contrasted with Wigner’s semicircle law of the matrix model.

In the past 20 years classical concepts of algebraic geometry such as the notion of dual varieties have been introduced in the quantum information literature to distinguish different classes of entanglement, a quantum property recognized as a resource in quantum information processing. In this talk, after introducing the connection between the basics of quantum information and the geometry of tensors, I will explain how new equations of the duals of homogeneous varieties can be obtained from graded simple Lie algebras. I will in particular focus on the dual of the spinor varieties \(\mathbb{S}_{16}\), the projectivization of the highest weight orbit of the 128 dimensional spin module and its connection with what is known in physics as Fermionic Fock spaces. This is joint work with Luke Oeding.

We will briefly review Wilson-Kadanoff type renormalization group (RG) maps for Ising spin systems and the lack of progress in proving that there is a non-trivial fixed point for these maps. The Ising model can be written as a tensor network, and RG maps can be defined in the tensor network formalism. Numerical studies of these tensor network RG maps by many groups have been remarkably successful in two dimensions. In joint work with Slava Rychkov we have begun a rigorous study of such tensor network RG maps. In particular we proved that in two dimensions there is a high temperature fixed point tensor which is locally stable. We have also proved results in the low temperature phase. Here there are two stable fixed points and one unstable fixed point which is related to behavior near the phase coexistence curve. Our long range goal is to prove the existence of a non-trivial fixed point which describes the second order phase transition in the Ising model.

I will discuss two counting problems in the context of random Gaussian tensors. The first considers the notion of typical rank of a tensors. I will show a result of Bergqvist on the probabilities of such ranks. The second is on the expected number of zeros of a system of polynomial equations.

In recent years, the import of quantum information techniques to quantum gravity has opened up new perspectives in the study of the microscopic structure of spacetime. I will present recent results in this direction, based on the possibility to regard (superposition of) spin networks - graphs decorated by volume and area quantum numbers, modelling spatial geometries in several approaches to quantum gravity - as graphs built up from geometric entanglement, and put them in correspondence with tensor networks. In particular, I will show that spin network states can be interpreted as maps from the bulk to the boundary of the dual spatial geometry, whose holographic behaviour increases with the inhomogeneity of the geometric data. I will also illustrate how exceeding a certain volume-entanglement threshold leads to the emergence of a horizon-like region, revealing intriguing perspectives for quantum cosmology. Based on joint work with Daniele Oriti, Goffredo Chirco and Simon Langenscheidt, available at arXiv:2012.12622, arXiv:2105.06454, arXiv:2110.15166 and arXiv:2207.07625.

Holographic models of quantum gravity conjecture a duality between the geometry of a bulk gravitational theory and the entanglement properties of a boundary conformal field theory. Tensor networks provide discrete toy-models for such correspondence, which appear to successfully reproduce several of the conjectured formulas. In particular, boundary states constructed from a bulk tensor network satisfy by construction an area law of entanglement, which is the primary expected feature of holographic states. In this talk, I will explain how picking the tensors composing the network at random provides a mathematically tractable model. Indeed, when doing so, the asymptotic spectral distribution of the corresponding random boundary states can be precisely characterized, and hence their asymptotic entanglement entropy as well. The latter turns out to be, as wanted, proportional to the area of the minimal surface inside the bulk enclosing the boundary subregion (rather than to its volume), with a correction term that depends on the number of such minimal surfaces. I will show how results of this kind can be obtained using tools from random matrix theory and free probability. Based on joint work with Newton Cheng, Geoff Penington, Michael Walter and Freek Witteveen, available at arXiv:2206.10482.

An important ingredient in the asymptotic safety scenario for quantum gravity is the existence of a suitable point in theory space modeling quantum geometry with exact scaling symmetry. It is natural to look for candidates in scaling limits of random discrete geometries, like the random triangulated spheres featuring in Euclidean Dynamical Triangulations (EDT). However, in dimensions higher than two, there are serious mathematical challenges in studying these models analytically, while numerical simulations are yet to uncover promising critical phenomena in these systems. I will discuss recent joint work (arXiv:2203.16105) with Luca Lionni, in which we considered a restricted family of triangulations of the 3-sphere that can be encoded in certain trees. These triangulation are under better enumerative control (they are locally constructible and admit explicit exponential bounds) and exploratory numerical simulations point at qualitative differences with vanilla EDT.

The Barrett-Crane spin foam and GFT model is a state-sum model which provides a quantization of first order Lorentzian Palatini gravity. Its complete formulation has only recently been accomplished. It is conjectured that the collective dynamics of the quanta of this model, which correspond to discrete building blocks of spacetime with spacelike, timelike and lightlike components, gives rise to continuum spacetime at criticality via phase transition. In this talk, we discuss how phase transitions for this and related models can be studied using Landau-Ginzburg mean-field theory. To this aim, we restrict the building blocks of the complete model such that the Feynman diagrams are dual to spacelike triangulations. We also include local degrees of freedom which may be interpreted as discretized scalar fields typically employed in quantum gravity to furnish a matter reference frame. This setting lays the groundwork to study the critical behavior when arbitrary Lorentzian building blocks are incorporated and represents a crucial advance to understand how phase transitions to continuum spacetime can be achieved in this setting. It also paves the way for the analysis of the phase structure of such models via functional renormalization group techniques in the future. This work is based on arXiv:2112.00091, arXiv:2206.15442, arXiv:2209.04297.

The Klebanov-Tarnopolsky tensor model is a quantum field theory for rank-three tensor scalar fields with certain quartic potential. The theory possesses an unusual large \(N\) limit known as the melonic limit that is strongly coupled yet solvable, producing at large distance a rare example of non-perturbative non-supersymmetric conformal field theory that admits analytic solutions. We study the dynamics of defects in the tensor model defined by localized magnetic field couplings on a \(p\)-dimensional subspace in the \(d\)-dimensional spacetime. While we work with general \(p\) and \(d\), the physically interesting cases include line defects in \(d=2,3\) and surface defects in \(d=3\). By identifying a novel large \(N\) limit that generalizes the melonic limit in the presence of defects, we prove that the defect one-point function of the scalar field only receives contributions from a subset of the Feynman diagrams in the shape of melonic trees. These diagrams can be resummed using a closed Schwinger-Dyson equation which enables us to determine non-perturbatively this defect one-point function. At large distance, the solutions we find describe nontrivial conformal defects and we discuss their defect renormalization group (RG) flows. In particular, for line defects, we solve the exact RG flow between the trivial and the conformal lines in \(d=4−ϵ\). We also compute the exact line defect entropy and verify the g-theorem. Furthermore we analyze the defect two-point function of the scalar field and its decomposition via the operator-product-expansion, providing explicit formulae for one-point functions of bilinear operators and the stress-energy tensor.

The local-potential approximation is a powerful tool to determine the RG flow in local field theories, \(O(N)\) theories being the paradigmatic example where full analytic solutions can be obtained. For fields interacting non-locally such as in tensor-invariant theories, a local potential is strictly speaking not at hand. In this talk we show under what conditions the local-potential technique is still applicable and that it can lead again to Wilson-Fisher type fixed points but also beyond.

The large charge approach allows to describe special sectors of strongly-coupled systems. When another controlling parameter is present, it is still useful because it allows to unveil interesting mathematical structures and access previously unexplored parts of the spectrum. In this talk I will concentrate on the \(O(N)\) vector model in the large-\(N\) limit, give a geometrical interpretation of the asymptotic expansion of the dimension of the lowest operator of fixed charge, and use resurgence to resum this expansion. The result suggests an unexpected interplay between conformal symmetry and resurgence and will lead to an intuitive interpretation for the impressive agreement of the large-charge predictions with lattice data also in the \(O(2)\) model case. Based on 1610.04495, 2008.03308, 2102.12488.

Over the last few years, it has become clear that working in sectors of large global charge leads to significant simplifications when studying strongly coupled CFTs, theories which are otherwise often inaccessible to analytic methods. It allows us in particular to calculate the CFT data as an expansion in inverse powers of the large charge. In this overview talk, I will introduce the large-charge expansion via the simple example of the O(2) model and will then apply it to the O(N) vector model which has a non-Abelian global symmetry group. Using large-N methods in conjunction with large charge gives us even more control over the dynamics and lets us study the system away from the conformal point. For large charge literature, see the arxiv's 2008.03308 (review), Large N: 1909.02571, 2102.12488, 2110.07616, 2110.07617.

The reputation of the Harish-Chandra-Itzykson-Zuber (HCIZ) integral is now firmly established. Among its numerous applications, the identification of the class of planar graphs and the recovery of some of the key results of free probability can both be deduced from the study of its large N series expansions. To understand their possible generalizations for random tensor models, we introduced with Collins and Gurau a tensor version of the HCIZ integral. I will describe its large N limits, and if time allows, explain why it applies to the detection of entanglement in quantum systems in the context of randomized measurements.

The Hermitian 1-matrix model, or equivalently the generating series of all oriented maps, satisfies the equations of the KP hierarchy. This means that its expansion in a special basis, the Schur functions, has minor determinants as coefficients. I will recall how to prove it from the matrix integral itself and also from the celebrated Frobenius formula in the more general case of constellations (i.e. weighted Hurwitz numbers). A nice application of this formalism is recurrence formulas for a few families of maps, which are the most efficient formulas to compute the numbers of maps, and which completely bypass Tutte's resolution method of the usual loop equations. Then I will motivate the generalization to non-oriented maps through a beautiful conjecture of Goulden and Jackson, known as the b-conjecture. I will explain the difficulties in proving integrability for non-oriented maps, in particular for matrix models with an external matrix. In addition to recurrence formulas counting the numbers of non-oriented maps (like in the oriented case), we found another application, which is a Pfaffian expression for the \(O(N)\)-BGW matrix integral in the case of even multiplicities.

In a recent work, Halverson, Maiti and Stoner proposed a description of a statistical ensemble of neural networks in terms of a quantum field theory (dubbed NN-QFT correspondence). The infinite-width limit is mapped to a free field theory while finite N corrections are taken into account by interactions. In this talk, after reviewing the correspondence, I will describe how to use non-perturbative renormalization in this context. An important difference with the usual analysis is that the effective (IR) 2-point function is known, while the microscopic (UV) 2-point function is not, which requires setting the problem with care. Finally, I will discuss preliminary numerical results for translation-invariant kernels. A major result is that changing the standard deviation of the neural network weight distribution can be interpreted as a renormalization flow in the space of networks.

I discuss the general motivating ideas about how cosmology could emerge from full quantum gravity in a TGFT formalism. In particular, I emphasize the use of discrete quantum geometric data, at the technical level, and the focus on relational observables for defining time evolution and spatial localization, at the conceptual level. I then summarize how these ideas have a concrete implementation in TGFT models with mixed local and non-local directions. In this context, I overview recent results on emerging bouncing cosmology at early times, phantom-like dark energy at late times and cosmological perturbations.

In high energy physics, one achievement of noncommutative geometry (NCG) is the possibility to actually derive the observed particle spectrum (the Standard Model) from a simple input. This noncommutative geometrical description of matter works only at a classical level (and for the rest, the theory is 'patched' with ordinary quantum field theory methods). The main topic of this talk is a path-integral quantization approach that leads to the concept of 'random noncommutative geometry', i.e. ensembles of Dirac operators, the finite approximations of which can be restated as random multi-matrix ensembles with wildly non-factorizable measures and multi-trace interactions. After a friendly introduction to these topics, I will present a Dirac ensemble which can be identified with Yang-Mills-Higgs matrix theory. If time allows, I will discuss the functional renormalization of the type multimatrix models that NCG motivates.

We will present Selective Multiple Power Iterations (SMPI), a new gradient descent-based algorithm that we introduced recently to address the important Tensor PCA problem. This problem consists in recovering a spike \(\mathbf{v}_0^{\otimes k}\) corrupted by a Gaussian noise tensor \(\mathbf{Z} \in (\mathbb{R}^n)^{\otimes k}\) such that \(\mathbf{T}=\sqrt{n} \beta \mathbf{v}_0^{\otimes k} + \mathbf{Z}\) where \(\beta\) is the signal-to-noise ratio. Various numerical simulations show that the experimental performances of SMPI improve drastically upon existent algorithms and becomes comparable to the theoretical optimal recovery. We show that, these unexpected performances are due to a powerful mechanism appearing at low \(\beta\) and in which the noise plays a key role for the signal recovery. These remarkable results may have strong impact on both practical and theoretical applications of Tensor PCA. (i) We provide multiple variants of this algorithm to tackle low-rank CP tensor decomposition. These proposed algorithms also outperforms existent methods even on real data which shows a huge potential impact for practical applications. (ii) We present new theoretical insights on the behavior of SMPI and gradient descent methods for the optimization in high-dimensional non-convex landscapes that are present in spin glass models and in various machine learning problems. (iii) We expect that these results may help the discussion concerning the existence of the conjectured statistical-algorithmic gap.

Canonical quantization (CQ) has been successful in solving many problems, and is a tool used for nearly all cases. But CQ has not worked well for some problems that have resisted any acceptable solution. Happily, the harmonic oscillator, with \(-\infty < p,q < \infty\), is well solved. However, for the half-harmonic oscillator, where \(-\infty < p <\infty\) while \(0 < q < \infty\) CQ fails significantly. Likewise, quantum gravity has failed with CQ as well. A new quantization process called affine quantization (AQ) will be introduced. AQ leads to acceptable solutions for the half-harmonic oscillator and shows considerable success for quantum gravity. This lecture will demonstrate the solution of the half-harmonic oscillator and leads to a meaningful Schrödinger equation for quantum gravity, which is in position to seek appropriate solutions from complex differential equations.

Recent work by Kartsaklis, Ramgoolam, Sadrzadeh, Sword on random matrix theory discovered that Gaussian matrix distributions with permutation symmetry are good statistical models of words within the framework of compositional distributional semantics. Particular interest lies in the expectation values of permutation invariant polynomials in matrix variables. It was observed that permutation invariant matrix polynomials are in correspondence with directed graphs for sufficiently large matrices. In the talk, I will outline the proof of this statement and the generalization to the two-matrix case. Furthermore, we will find that the statement continues to hold for arbitrary sizes of matrices, with a small put precisely stated modification. We then move on to expectation values. I will describe the general structure of expectation values and give a procedure for computing them. We will see how the procedure can be understood as a graph algorithm.

In recent work Kartsaklis, Ramgoolam and Sadrzadeh developed a class of Gaussian matrix models for which the usual \(U(N)\) symmetry is relaxed to a less restrictive \(S_N\), the group of permutations. Utilising the combinatorics of Wick contractions from QFT and representation theory helps to uncover the rich mathematical structure of these models and permits the computation of expectation values of the most general permutation invariant Gaussian theories. An application of these models to the statistics of words in computational linguistics is described. This application was central in motivating the development of these models and, more recently, provided the stimulus for extending this programme to general permutation invariant Gaussian two-matrix models.

Weingarten calculus addresses the problem of computing the integral of polynomial functions against the Haar measure over compact groups. In the case of classical series, it gives a rational fraction depending on the dimension. In a natural sense, it extends Gaussian (Wick) calculus. We will explain how these computations work, and focus on the case of tensor matrix integrals, whose study we initiated in a series of work in collaboration with Luca Lionni and Razvan Gurau.

Powerful computers and acquisition devices have made it possible to capture and store large real-world multidimensional data. For practical applications, analyzing and organizing these high dimensional arrays (formally called tensors) lead to the well-known curse of dimensionality. Thus, dimensionality reduction is frequently employed to transform a high-dimensional data set by projecting it into a lower dimensional space while retaining most of the information and underlying structure. One of these techniques is Principal Component Analysis (PCA), which has made remarkable progress in a large number of areas thanks to its simplicity and adaptability. These last years, tools based on tensor contractions (trace invariants) have been developed by theoretical physicists where random tensors have emerged as a generalization of random matrices. In this work, we investigate the algorithmic threshold of tensor PCA and some of its variants using the theoretical physics approach and we show that it leads to new insights and knowledge in tensor PCA.

Observables in a tensor model can be enumerated using colored graphs, which have a description in terms of permutation triples subject to an equivalence relation generated by permutation products. A gauge-fixing of the equivalence relation relates the enumeration to bipartite ribbon graphs and Belyi maps between surfaces. Fourier transformation on permutation group algebras relates the counting of observables to a sum of squares of Kronecker coefficients. In the recent paper arxiv.org/abs/2010.04054, we develop these observations to give a combinatoric construction of the Kronecker coefficient for a fixed triple of Young diagrams. The Kronecker coefficient counts vectors in a lattice of ribbon graphs, determined as null vectors of integer matrices. The result motivates a discussion of quantum mechanical systems and algorithms to determine non-vanishing Kronecker coefficients. Using the link to Belyi maps, these quantum mechanical systems have an interpretation in terms of quantum membrane geometries interpolating between algebraic string worldsheets.

In this talk, I will explain how to build from a classical rational spectral curve (\(P(x,y)=0\)) a "quantum curve" \(\tilde{P}(x,\hbar \partial_x)\Psi=0\) using the Chekhov-Eynard-Orantin topological recursion. The strategy is to regroup the correlation functions generated by the topological recursion into a matrix wave function \(\Psi\) and show that it satisfies a rational linear differential system with the same pole structure as the initial curve. In the hyper-elliptic case, we explain why this procedure is naturally connected with isomonodromic deformations and recovers the standard Lax pair formulations (Painlevé). The talk is mostly based on arxiv:1911.07739 with N. Orantin.

Bicolored maps with boundaries can be seen as a special, simple case of the Ising model on maps. However, they are yet to be enumerated explicitly. I will present in this talk a work in progress in collaboration with Jérémie Bouttier, where we used analytic-combinatorial tools for the specific case of planar bicolored maps with alternating boundaries. This yields an explicit rational parametrization in the case of m-angulations and m-constellations. In the special case of Eulerian triangulations, this rational parametrization was a key ingredient in proving in arXiv:1912.13434 that this family of maps admits the Brownian map as a scaling limit.

The Hermitian 1-matrix model has been known for 30 years to satisfy the KP hierarchy. It is an infinite set of partial differential equations, here with respect to the coupling constants of the model. At least three proofs of this statement are known. Other systems in two-dimensional enumerative geometry such as, famously, intersection numbers of the moduli space of Riemann surfaces, satisfy the KP hierarchy. In this talk, I will introduce the KP hierarchy from the point of view of the Sato Grassmannian as an orbit of an action of \(GL(\infty)\). This will then give us the opportunity to present the proof of Kazarian and Zograf that the generating function of bipartite maps satisfies the KP hierarchy. I will then focus on constellations, which are generalizations of bipartite maps, and are also the dual to the jackets of the bipartite edge-colored graphs of unitary-invariant tensor models. Constellations are known to also satisfy the KP hierarchy but only a proof via algebraic combinatorics is known. I will report some progress towards understanding their integrable properties using Tutte/Schwinger-Dyson--like equations. Only this last part is an original contribution.

It is known that the Kontsevich model is equivalent to the hermitian 1-matrix model by choosing a special relation between the moments of the external matrix (Kontsevich times) and the parameters of the potential of the hermitian 1-matrix model. This equivalence can easily be proven by topological recursion. We will present the quartic analogue of the Kontsevich model with the action \(S=\text{Tr}(E \phi^2+\frac{\lambda}{4}\phi^4)\), where \(E\) is the external matrix and \(\lambda\) a real number. The Dyson-Schwinger equations (loop equations) will be shown, as well as some results for the first correlation functions. From these results the genus zero spectral curve of the model is identified, wich has a the symmetry \(x(z)=-y(-z)\). Furthermore, it turns out that the results and loop equations of the quartic model are structurally the same as from the hermitian 2-matrix model with a genus zero assumption for the spectral curve. The hermitian 2-matrix model satisfies a more general topological expansion.