Renormalization procedure for random tensor networks and the canonical tensor model

Field theoretic renormalization group is applied to measurement and entanglement phase transitions. The renormalization group flows for both random tensor networks and monitored systems show that, in high dimensions, there exist distinct weak- and strong-coupling universality classes for these phase transitions. In addition, one of the field theories exhibits a supersymmetry that leads to dimensional reduction by four.

arXiv:2303.07848Physical review. B./Physical review. BSep 12, 202315 citations

Scaling dimensions from linearized tensor renormalization group transformations

Xinliang Lyu, RuQing G. Xu, Naoki Kawashima

arXiv:2102.08136Physical Review ResearchApr 16, 202115 citations

Renormalization procedure for random tensor networks and the canonical tensor model

Naoki Sasakura, Yūki Sato

Published Apr 1, 2015• 10 citations

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Abstract

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Renormalization procedure for random tensor networks and the canonical tensor model

Naoki Sasakura, Yuki Sato

We discuss a renormalization procedure for random tensor networks, and show that the corresponding renormalization-group flow is given by the Hamiltonian vector flow of the canonical tensor model, which is a discretized model of quantum gravity. The result is a generalization of the previous one concerning the relation between the Ising model on random networks and the canonical tensor model with | $N=2$ |⁠. We also prove a general theorem that relates discontinuity of the renormalization-group flow and the phase transitions of random tensor networks.We discuss a renormalization procedure for random tensor networks, and show that the corresponding renormalization-group flow is given by the Hamiltonian vector flow of the canonical tensor model, which is a discretized model of quantum gravity. The result is a generalization of the previous one concerning the relation between the Ising model on random networks and the canonical tensor model with |$N=2$|⁠. We also prove a general theorem that relates discontinuity of the renormalization-group flow and the phase transitions of random tensor networks.

arXiv:1501.05078Progress of Theoretical and Experimental PhysicsApr 1, 20158 citations

Renormalization Group-Like Flows in Randomly Connected Tensor Networks

Naoki Sasakura

Abstract Randomly connected tensor networks (RCTN) are dynamical systems defined by summing over all the possible networks of tensors. Because of the absence of fixed lattice structure, RCTN is not expected to have renormalization procedures. In this paper, however, we consider RCTN with a real tensor, and it is proven that a Hamiltonian vector flow of a tensor model in the canonical formalism with a positive cosmological constant has the properties that a renormalization group (RG) flow of RCTN would have: The flow has fixed points on phase transition surfaces; every flow line is asymptotically terminated by fixed points at both ends, where an upstream fixed point has higher criticality than a downstream one; the flow goes along phase transition surfaces; there exists a function that monotonically decreases along the flow, analogously to the $a$ - and $c$ -functions of RG. A complete classification of fixed points is given. Although there are no cyclic flows in the strict sense, they exist, if infinitesimal jumps are allowed near fixed points.Abstract Randomly connected tensor networks (RCTN) are dynamical systems defined by summing over all the possible networks of tensors. Because of the absence of fixed lattice structure, RCTN is not expected to have renormalization procedures. In this paper, however, we consider RCTN with a real tensor, and it is proven that a Hamiltonian vector flow of a tensor model in the canonical formalism with a positive cosmological constant has the properties that a renormalization group (RG) flow of RCTN would have: The flow has fixed points on phase transition surfaces; every flow line is asymptotically terminated by fixed points at both ends, where an upstream fixed point has higher criticality than a downstream one; the flow goes along phase transition surfaces; there exists a function that monotonically decreases along the flow, analogously to the $a$- and $c$-functions of RG. A complete classification of fixed points is given. Although there are no cyclic flows in the strict sense, they exist, if infinitesimal jumps are allowed near fixed points.

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Renormalization Group Flows of Hamiltonians Using Tensor Networks

Matthias Bal, Michael Mariën, Jutho Haegeman +1 more

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.

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Tensor Network Renormalization

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We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

arXiv:1412.0732Physical Review LettersOct 29, 2015362 citations

Renormalization group for measurement and entanglement phase transitions

Adam Nahum, Kay Jörg Wiese

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Scaling dimensions from linearized tensor renormalization group transformations

Xinliang Lyu, RuQing G. Xu, Naoki Kawashima

We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions.The tensor RG methods have had great success in producing accurate free energy compared with the conventional real-space RG schemes.However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes.We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory.This approach is benchmarked in the context of the Ising models in one dimension and two dimensions.Based on a pure RG argument, the proposed method has potential applications to three-dimensional systems, where the existing bread-and-butter method is inapplicable.We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions.The tensor RG methods have had great success in producing accurate free energy compared with the conventional real-space RG schemes.However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes.We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory.This approach is benchmarked in the context of the Ising models in one dimension and two dimensions.Based on a pure RG argument, the proposed method has potential applications to three-dimensional systems, where the existing bread-and-butter method is inapplicable.

arXiv:2102.08136Physical Review ResearchApr 16, 202115 citations

N=2

|⁠. We also prove a general theorem that relates discontinuity of the renormalization-group flow and the phase transitions of random tensor networks.

a

- and

c

-functions of RG. A complete classification of fixed points is given. Although there are no cyclic flows in the strict sense, they exist, if infinitesimal jumps are allowed near fixed points.