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arXiv:2105.00010 · Quantum

Tensor Renormalization Group for interacting quantum fields

Manuel Campos, Germán Sierra, Esperanza López

Published Nov 23, 2021• 3 citations
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Abstract

We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.

DOI
10.22331/q-2021-11-23-586

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We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.
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Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.
We propose a method to construct a tensor network representation of partition functions without singular value decompositions nor series expansions. The approach is demonstrated for one- and two-dimensional Ising models and we study the dependence of the tensor renormalization group (TRG) on the form of the initial tensors and their symmetries. We further introduce variants of several tensor renormalization algorithms. Our benchmarks reveal a significant dependence of various TRG algorithms on the choice of initial tensors and their symmetries. However, we show that the boundary TRG technique can eliminate the initial tensor dependence for all TRG methods. The numerical results of TRG calculations can thus be made significantly more robust with only a few changes in the code. Furthermore, we study a three-dimensional Z 2 gauge theory without gauge fixing and confirm the applicability of the initial tensor construction. Our method can straightforwardly be applied to systems with longer range and multisite interactions, such as the next-nearest neighbor Ising model. Published by the American Physical Society 2024
We propose a real-space renormalization group algorithm for accurately coarse-graining two-dimensional tensor networks. The central innovation of our method lies in utilizing variational boundary tensors as a globally optimized environment for the entire system. Based on this optimized environment, we construct renormalization projectors that significantly enhance accuracy. By leveraging the canonical form of tensors, our algorithm maintains the same computational complexity as the original tensor renormalization group (TRG) method, yet achieves higher accuracy than existing approaches that do not incorporate entanglement filtering. Our work offers a practical pathway for extending TRG methods to higher dimensions while keeping computational costs manageable.
We propose a method to construct the initial tensor representation of partition functions and observables for the tensor renormalization group (TRG). The TRG is a numerical calculation technique that utilizes a tensor network representations of physical quantities to investigate physical properties without encountering the sign problem. To apply the TRG, it is essential to construct a locally connected tensor network suitable for recursive coarse-graining. We present a systematic approach for translating a general tensor representation of the partition function to this form. Furthermore, we show the dependence of TRG algorithms on the choice of the initial tensor network representation and propose an improvement of TRG algorithms in this respect
We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)ddd) complex ϕ4\phi^4ϕ4 theory at finite density, (1+1)ddd pure U(1) lattice gauge theory with a θ\thetaθ term, (3+1)ddd Nambu--Jona-Lasinio model at finite density and (1+1)ddd and (2+1)ddd Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.