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arXiv:2304.03625 · Letters in Mathematical Physics

Duality of orthogonal and symplectic random tensor models: general invariants

Hannes Keppler, Thomas Müller

Published Jul 12, 2023• 5 citations
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Abstract

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.

DOI
10.1007/s11005-023-01706-7

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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−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.
Abstract In a recent series of papers, a duality between orthogonal and symplectic random tensor models has been proven, first for quartic models and then for models with interactions of arbitrary order. However, the tensor models considered so far in the literature had no symmetry under permutation of the indices. In this paper, we generalize these results for tensors models with interactions of arbitrary order which further have non-trivial symmetry under the permutation of the indices. Totally symmetric and anti-symmetric tensors are thus treated as a particular case of our result.
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 Z2Z_2Z2​-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, ZqZ_qZq​ 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 SqS_qSq​ 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.
The representation theory of the Clifford group is playing an increasingly prominent role in quantum information theory, including in such diverse use cases as the construction of protocols for quantum system certification, quantum simulation, and quantum cryptography. In these applications, the tensor powers of the defining representation seem particularly important. The representation theory of these tensor powers is understood in two regimes. 1. For odd qudits in the case where the power t is not larger than the number of systems n: Here, a duality theory between the Clifford group and certain discrete orthogonal groups can be used to make fairly explicit statements about the occurring irreps (this theory is related to Howe duality and the η-correspondence). 2. For qubits: Tensor powers up to t = 4 have been analyzed on a case-by-case basis. In this paper, we provide a unified framework for the duality approach that also covers qubit systems. To this end, we translate the notion of rank of symplectic representations to representations of the qubit Clifford group, and generalize the η correspondence between symplectic and orthogonal groups to a correspondence between the Clifford and certain orthogonal-stochastic groups. As a sample application, we provide a protocol to efficiently implement the complex conjugate of a black-box Clifford unitary evolution.
Abstract We demonstrate that random tensors transforming under rank-5 irreducible representations of O(N)\mathrm {O}(N)O(N) O ( N ) can support melonic large N expansions. Our construction is based on models with sextic (5-simplex) interaction, which generalize previously studied rank-3 models with quartic (tetrahedral) interaction (Benedetti et al. in Commun Math Phys 371:55, 2019. arXiv:1712.00249 ; Carrozza in JHEP 06:039, 2018. arXiv:1803.02496 ). Beyond the irreducible character of the representations, our proof relies on recursive bounds derived from a detailed combinatorial analysis of the Feynman graphs. Our results provide further evidence that the melonic limit is a universal feature of irreducible tensor models in arbitrary rank.
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)Z(2)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)Z(q)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 SqS_qSq​ 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.