Abstract
We study the renormalization group flow of $\phi^4$ theory in two dimensions. Regularizing space into a fine-grained lattice and discretizing the scalar field in a controlled way, we rewrite the partition function of the theory as a tensor network. Combining local truncations and a standard coarse-graining scheme, we obtain the renormalization group flow of the theory as a map in a space of tensors. Aside from qualitative insights, we verify the scaling dimensions at criticality and extrapolate the critical coupling constant $f_{\rm c} = \lambda / \mu ^2$ to the continuum to find $f^{\rm cont.}_{\rm c} = 11.0861(90)$, which favorably compares with alternative methods.
Highlights
Solving interacting quantum field theories (QFTs) out of the perturbative regime is difficult, and is arguably one of the most pressing computational problems in theoretical physics nowadays
Of φ24, Monte Carlo methods [14,15] were quickly improved and seem back on a par [16]. As part of this effort, we propose in this paper a computation of the renormalization group (RG) flow of φ24 using tensor network renormalization techniques
It is a simple and fairly efficient algorithm, it has been known for several years that, despite its name, tensor renormalization group (TRG) does not yield a proper renormalization group flow
Summary
Solving interacting quantum field theories (QFTs) out of the perturbative regime is difficult, and is arguably one of the most pressing computational problems in theoretical physics nowadays. Of φ24, Monte Carlo methods [14,15] were quickly improved and seem back on a par [16] As part of this effort, we propose in this paper a computation of the renormalization group (RG) flow of φ24 using tensor network renormalization techniques. It was applied to φ24 in [39], and more recently in [10], where the authors report an estimate of the critical coupling of the theory in the continuum limit It is a simple and fairly efficient algorithm, it has been known for several years that, despite its name, TRG does not yield a proper renormalization group flow. We define φ24 theory on the lattice and in the continuum, and recall some of its basic properties
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