Numerical and analytical analyses of a matrix model with non-pairwise contracted indices
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Regular Article - Theoretical Physics
Numerical and analytical analyses of a matrix model with non-pairwise contracted indices Naoki Sasakura1,a , Shingo Takeuchi2,b 1 2
Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan Phenikaa Institute for Advanced Study and Faculty of Basic Science, Phenikaa University, Hanoi 100000, Vietnam
Received: 25 July 2019 / Accepted: 23 December 2019 / Published online: 12 February 2020 © The Author(s) 2020
Abstract We study a matrix model that has φai (a = 1, 2, . . . , N , i = 1, 2, . . . , R) as its dynamical variable, whose lower indices are pairwise contracted, but upper ones are not always done so. This matrix model has a motivation from a tensor model for quantum gravity, and is also related to the physics of glasses, because it has the same form as what appears in the replica trick of the spherical p-spin model for spin glasses, though the parameter range of our interest is different. To study the dynamics, which in general depends on N and R, we perform Monte Carlo simulations and compare with some analytical computations in the leading and the next-leading orders. A transition region has been found around R ∼ N 2 /2, which matches a relation required by the consistency of the tensor model. The simulation and the analytical computations agree well outside the transition region, but not in this region, implying that some relevant configurations are not properly included by the analytical computations. With a motivation coming from the tensor model, we also study the persistent homology of the configurations generated in the simulations, and have observed its gradual change from S 1 to higher dimensional cycles with the increase of R around the transition region.
1 Introduction Quantization of gravity is one of the major fundamental problems in theoretical physics. The quantization of general relativity by the standard perturbative methods of quantum field theory fails due to non-renormalizable divergences. Various approaches have been proposed and being studied to solve the fundamental problem, depending on views of authors. In one approach, general relativity (with higher derivative terms) is directly quantized as quantum field thea e-mail:
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ory with the modern technique of the functional renormalization group [1]. In other approaches, fundamental discreteness is introduced to represent spacetimes, which include (causal) dynamical triangulations [2], loop quantum gravity [3], causal sets [4], quantum graphity [5], matrix models [6– 10], tensor models [11–14], and so on. In these discretized approaches, an important criterion for success is whether macroscopic spacetimes are generated, or in other words, whether there exist appropriate continuum limits that recover the usual continuum picture of spacetime with dynamics described by general relativity as low-energy effective theory. The criterion above can in principle be checked by studyin
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