Abstract In Gurau and Keppler 2022 ( arxiv:2207.01993 ), a relation between orthogonal and symplectic tensor models with quartic interactions was proven. In this paper, we provide an alternative proof that extends to polynomial interactions of arbitrary order. We consider tensor models of order D with no symmetry under permutation of the indices that transform in the tensor product of D fundamental representations of O(N) and Sp(N). We explicitly show that the models obey the N to
−N - N duality graph by graph in perturbation theory.
The groups \mathrm{O}(N) and \mathrm{Sp}(N) are related by an analytic continuation to negative values of N , \mathrm{O}(-N)\simeq\mathrm{Sp}(N) . This duality has been studied for vector models, \mathrm{SO}(N) and \mathrm{Sp}(N) gauge theories, as well as some random matrix ensembles. We extend this duality to real random tensor models of arbitrary order D with no symmetry under permutation of the indices and with quartic interactions. The N to -N duality is shown to hold graph by graph to all orders in perturbation theory for the partition function, the free energy and the connected two-point function.
The Ising model is the simplest to describe many-body effects in classical statistical mechanics. Duality analysis leads to a critical point under several assumptions. The Ising model itself has
Z(2) symmetry. The basis of the duality analysis is a nontrivial relationship between low and high-temperature expansions. However, the discrete Fourier transformation finds the hidden relationship automatically. The duality analysis can be naturally generalized into the case with the degrees of freedom with
Z(q) symmetry and random spin systems. We further obtain the duality in a series of permutation models in the present study by considering the symmetric group
Sq and its Fourier transformation. The permutation model in the symmetric group is closely related to the random quantum circuits and random tensor network model, often discussed in the context of quantum computing and the holographic principle, a property of string theories and quantum gravity. We provide a systematic way by our duality analysis to analyze the phase transition in these models.
Abstract The Ising model is the simplest model for describing many-body effects in classical statistical mechanics. A duality analysis leads to its critical point under several assumptions. The Ising model has
Z2-symmetry. The basis of duality analysis is a nontrivial relationship between low- and high-temperature expansions. However, discrete Fourier transformation automatically determines the hidden relationship. The duality analysis can naturally extend to systems with various degrees of freedom,
Zq symmetry, and random spin systems. Furthermore, we obtained the duality relation in a series of permutation models in the present study by considering the symmetric group
Sq and its Fourier transformation. The permutation model in a symmetric group is closely related to random quantum circuits and random tensor network models, which are frequently discussed in quantum computing. It also relates to the holographic principle, a property of string theories and quantum gravity. We provide a systematic approach using duality analysis to examine the phase transition in these models.
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-
N melonic expansion of tensor models. For the first model, invariant under
U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless
U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into
U(N2)×U(N/2)×U(N/2). A matrix model invariant under
U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups
U(N)3 and
O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in
N drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-
N, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension
4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances,
O(N,R),
O(N,C), and
U(N,C), respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.