arXiv:1811.12376 · Journal of High Energy Physics

Tensor network analysis of critical coupling in two dimensional ϕ4 theory

Daisuke Kadoh, Y. Kuramashi, Y. Nakamura, Ryo Sakai, Shinji Takeda, Yusuke Yoshimura

Published May 1, 2019• 57 citations

Abstract

Abstract We make a detailed analysis of the spontaneous Z 2-symmetry breaking in the two dimensional real ϕ 4 theory with the tensor renormalization group approach, which allows us to take the thermodynamic limit easily and determine the physical observables without statistical uncertainties. We determine the critical coupling in the continuum limit employing the tensor network formulation for scalar field theories proposed in our previous paper. We obtain [λ/μ c 2 ]cont. = 10.913(56) with the quartic coupling λ and the renormalized critical mass μ c . The result is compared with previous results obtained by different approaches.

DOI: 10.1007/jhep05(2019)184

Related papers

Tensor network study of two dimensional lattice $\phi^{4}$ theory

Ryo Sakai, Daisuke Kadoh, Y. Kuramashi +3 more

arXiv:1812.00166International Symposium on Lattice Field Theory (LATTICE)May 29, 20191 citation

Computing the renormalization group flow of two-dimensional ϕ4 theory with tensor networks

Clement Delcamp, Antoine Tilloy

arXiv:2003.12993Physical Review ResearchAug 20, 202029 citations

Phase transition of four-dimensional lattice ϕ4 theory with tensor renormalization group

Shinichiro Akiyama, Y. Kuramashi, Yusuke Yoshimura

arXiv:2101.06953Physical review. D/Physical review. D.Aug 30, 202120 citations

Critical analysis of two-dimensional classical XY model

Raghav G Jha

Abstract We consider the two-dimensional classical XY model on a square lattice in the thermodynamic limit using tensor renormalization group and precisely determine the critical temperature corresponding to the Berezinskii–Kosterlitz–Thouless (BKT) phase transition to be 0.89290(5) which is an improvement compared to earlier studies using tensor network methods.

arXiv:2004.06314Journal of Statistical Mechanics Theory and ExperimentAug 1, 202022 citations

Tensor network formulation of two-dimensional gravity

Muhammad Asaduzzaman, Simon M. Catterall, Judah Unmuth-Yockey

arXiv:1905.13061Physical review. D/Physical review. D.Sep 17, 202019 citations

Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a θ term

Yoshinobu Kuramashi, Yusuke Yoshimura

arXiv:1911.06480Journal of High Energy PhysicsApr 1, 202037 citations

arXiv:1811.12376 · Journal of High Energy Physics

Tensor network analysis of critical coupling in two dimensional ϕ4 theory

Daisuke Kadoh, Y. Kuramashi, Y. Nakamura, Ryo Sakai, Shinji Takeda, Yusuke Yoshimura

Published May 1, 2019• 57 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep05(2019)184

Related papers

Tensor network study of two dimensional lattice $\phi^{4}$ theoryTensor network study of two dimensional lattice $\phi^{4}$ theory

Ryo Sakai, Daisuke Kadoh, Y. Kuramashi +3 more

The tensor renormalization group attracts great attention as a new numerical method that is free of the sign problem. In addition to this striking feature, it also has an attractive aspect as a coarse-graining of space-time; the computational cost scales logarithmically with the space-time volume. This fact allows us to aggressively approach the thermodynamic limit. While taking this advantage, we study the critical coupling of the two dimensional $\phi^{4}$ theory on large and fine lattices. We present the numerical results along with the extrapolation procedure to the continuum limit and compare them with the previous ones by Monte Carlo simulations.The tensor renormalization group attracts great attention as a new numerical method that is free of the sign problem. In addition to this striking feature, it also has an attractive aspect as a coarse-graining of space-time; the computational cost scales logarithmically with the space-time volume. This fact allows us to aggressively approach the thermodynamic limit. While taking this advantage, we study the critical coupling of the two dimensional $\phi^{4}$ theory on large and fine lattices. We present the numerical results along with the extrapolation procedure to the continuum limit and compare them with the previous ones by Monte Carlo simulations.

arXiv:1812.00166International Symposium on Lattice Field Theory (LATTICE)May 29, 20191 citation

Computing the renormalization group flow of two-dimensional ϕ4 theory with tensor networks

Clement Delcamp, Antoine Tilloy

This paper presents a method to write the partition function of ${\ensuremath{\phi}}^{4}$ theory as a tensor network and obtain its renormalization group flow as a map in a space of tensors. The authors also compute the critical coupling constant ${f}_{\mathrm{c}}=\ensuremath{\lambda}/{\ensuremath{\mu}}^{2}$ in the continuum to find ${f}_{\mathrm{c}}^{\mathrm{cont}.}=11.0861(90)$ and compare with other methods.This paper presents a method to write the partition function of ${\ensuremath{\phi}}^{4}$ theory as a tensor network and obtain its renormalization group flow as a map in a space of tensors. The authors also compute the critical coupling constant ${f}_{\mathrm{c}}=\ensuremath{\lambda}/{\ensuremath{\mu}}^{2}$ in the continuum to find ${f}_{\mathrm{c}}^{\mathrm{cont}.}=11.0861(90)$ and compare with other methods.

arXiv:2003.12993Physical Review ResearchAug 20, 202029 citations

Phase transition of four-dimensional lattice ϕ4 theory with tensor renormalization group

Shinichiro Akiyama, Y. Kuramashi, Yusuke Yoshimura

We investigate the phase transition of the four-dimensional single-component ${\ensuremath{\phi}}^{4}$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $⟨\ensuremath{\phi}⟩$ at a finite quartic coupling $\ensuremath{\lambda}$ on large volumes up to $V=102{4}^{4}$ in order to detect the spontaneous breaking of the ${\mathbb{Z}}_{2}$ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.We investigate the phase transition of the four-dimensional single-component ${\ensuremath{\phi}}^{4}$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $⟨\ensuremath{\phi}⟩$ at a finite quartic coupling $\ensuremath{\lambda}$ on large volumes up to $V=102{4}^{4}$ in order to detect the spontaneous breaking of the ${\mathbb{Z}}_{2}$ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.

arXiv:2101.06953Physical review. D/Physical review. D.Aug 30, 202120 citations

Critical analysis of two-dimensional classical XY model

Raghav G Jha

arXiv:2004.06314Journal of Statistical Mechanics Theory and ExperimentAug 1, 202022 citations

Tensor network formulation of two-dimensional gravity

Muhammad Asaduzzaman, Simon M. Catterall, Judah Unmuth-Yockey

We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the tensor renormalization group to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the tensor renormalization group to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.

arXiv:1905.13061Physical review. D/Physical review. D.Sep 17, 202019 citations

Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a θ term

Yoshinobu Kuramashi, Yusuke Yoshimura

Abstract We make an analysis of the two-dimensional U(1) lattice gauge theory with a θ term by using the tensor renormalization group. Our numerical result for the free energy shows good consistency with the exact one at finite coupling constant. The topological charge density generates a finite gap at θ = π toward the thermodynamic limit. In addition finite size scaling analysis of the topological susceptibility up to V = L × L = 1024 × 1024 allows us to determine the phase transition at θ = π is the first order.Abstract We make an analysis of the two-dimensional U(1) lattice gauge theory with a θ term by using the tensor renormalization group. Our numerical result for the free energy shows good consistency with the exact one at finite coupling constant. The topological charge density generates a finite gap at θ = π toward the thermodynamic limit. In addition finite size scaling analysis of the topological susceptibility up to V = L × L = 1024 × 1024 allows us to determine the phase transition at θ = π is the first order.

arXiv:1911.06480Journal of High Energy PhysicsApr 1, 202037 citations

Tensor network study of two dimensional lattice

\phi^{4}

theory

\phi^{4}

theory on large and fine lattices. We present the numerical results along with the extrapolation procedure to the continuum limit and compare them with the previous ones by Monte Carlo simulations.

{f}_{\mathrm{c}}^{\mathrm{cont}.}=11.0861(90)

and compare with other methods.

V=102{4}^{4}

in order to detect the spontaneous breaking of the

{\mathbb{Z}}_{2}

symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.