Renormalization in tensor Group Field Theory

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) $d$ ) complex $\phi^4$ theory at finite density, (1+1) $d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1) $d$ Nambu--Jona-Lasinio model at finite density and (1+1) $d$ and (2+1) $d$ 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.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)$d$) complex $\phi^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ 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.

arXiv:2111.04240International Symposium on Lattice Field Theory (LATTICE)May 16, 20224 citations

Toward tensor renormalization group study of three-dimensional non-Abelian gauge theory

Takaaki Kuwahara, Asato Tsuchiya

Abstract We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group (TRG) with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for the free energy agrees with the analytical results in the weak and strong coupling regimes.Abstract We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group (TRG) with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for the free energy agrees with the analytical results in the weak and strong coupling regimes.

arXiv:2205.08883Progress of Theoretical and Experimental PhysicsJul 25, 202221 citations

d

) complex

\phi^4

theory at finite density, (1+1)

d

pure U(1) lattice gauge theory with a

\theta

term, (3+1)

d

Nambu--Jona-Lasinio model at finite density and (1+1)

d

and (2+1)

d

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.