We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the higher order tensor renormalization group approach. Regarding the ${Z}_{N}$ symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a nonzero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the higher order tensor renormalization group approach. Regarding the ${Z}_{N}$ symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a nonzero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.

arXiv:2205.03548Physical review. D/Physical review. D.Oct 17, 20236 citations

The Tensor Track V: Holographic Tensors

Vincent Rivasseau, Nicolas Delporte

We review the fast developing subject of tensor models for the NAdS 2 /NCFT 1 holographic correspondence.We include a brief review of the Sachdev-Ye-Kitaev (SYK) model and then focus on the associated quantum mechanical tensor models (GW and CTKT).We examine their main features and how they compare with SYK.To end, we discuss different extensions: the large D limit of matrix-tensor models, the large N expansion of symmetric/antisymmetric tensors, the use of probes, the construction of a bilocal action for tensors, some attempts to extend the above models to higher dimensions and a proposal to break the tensor symmetry.We review the fast developing subject of tensor models for the NAdS 2 /NCFT 1 holographic correspondence.We include a brief review of the Sachdev-Ye-Kitaev (SYK) model and then focus on the associated quantum mechanical tensor models (GW and CTKT).We examine their main features and how they compare with SYK.To end, we discuss different extensions: the large D limit of matrix-tensor models, the large N expansion of symmetric/antisymmetric tensors, the use of probes, the construction of a bilocal action for tensors, some attempts to extend the above models to higher dimensions and a proposal to break the tensor symmetry.

Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)Aug 24, 201822 citations

Equivalent Representations in Symmetrized Tensors

Donald R. Tompkins

Physical Review LettersSep 26, 19661 citation

Symmetric Tensor Networks

Simone Montangero

Springer eBooksJan 1, 20180 citations

Symmetric cluster expansions with tensor networks

Bram Vanhecke, Laurens Vanderstraeten, Frank Verstraete

Cluster expansions for the exponential of local operators are constructed using tensor networks. In contrast to other approaches, the cluster expansion does not break any spatial or internal symmetries and exhibits a very favorable prefactor to the error scaling versus bond dimension. This is illustrated by time evolving a matrix product state using very large time steps, and by constructing a robust algorithm for finding ground states of two-dimensional Hamiltonians using projected entangled pair states as fixed points of two-dimensional transfer matrices.Cluster expansions for the exponential of local operators are constructed using tensor networks. In contrast to other approaches, the cluster expansion does not break any spatial or internal symmetries and exhibits a very favorable prefactor to the error scaling versus bond dimension. This is illustrated by time evolving a matrix product state using very large time steps, and by constructing a robust algorithm for finding ground states of two-dimensional Hamiltonians using projected entangled pair states as fixed points of two-dimensional transfer matrices.

arXiv:1912.10512Physical Review AFeb 22, 202122 citations

We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the higher order tensor renormalization group approach. Regarding the

{Z}_{N}

symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a nonzero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.