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arXiv:1311.4826 · International Symposium on Lattice Field Theory (LATTICE)

Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models

Yannick Meurice

Published Apr 28, 2014• 5 citations
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Abstract

We show that the Tensor Renormalization Group (TRG) method can be applied to O(N) spin models, principal chiral models and pure gauge theories (Z2, U(1) and SU(2)) on (hyper) cubic lattices.We explain that contrarily to some common belief, it is very difficult to write compact formulas expressing the blockspinning of lattice models.We show that in contrast to other approaches, the TRG formulation allows us to write exact blocking formulas with numerically controllable truncations.The basic reason is that the TRG blocking separates neatly the degrees of freedom inside the block and which are integrated over, from those kept to communicate with the neighboring blocks.We argue that the TRG is a method that can handle large volumes, which is crucial to approach quasi-conformal systems.The method can also get rid of some sign problems.We discuss recent results regarding the critical properties of the 2D O(2) nonlinear sigma model with complex β and chemical potential.As some of these results appeared in a recently published paper (PRD 88, 056005) and two recent preprints (

DOI
10.22323/1.187.0329

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We show that the Tensor Renormalization Group (TRG) method can be applied to O(N) spin models, principal chiral models and pure gauge theories (Z2, U(1) and SU(2)) on (hyper) cubic lattices.We explain that contrarily to some common belief, it is very difficult to write compact formulas expressing the blockspinning of lattice models.We show that in contrast to other approaches, the TRG formulation allows us to write exact blocking formulas with numerically controllable truncations.The basic reason is that the TRG blocking separates neatly the degrees of freedom inside the block and which are integrated over, from those kept to communicate with the neighboring blocks.We argue that the TRG is a method that can handle large volumes, which is crucial to approach quasi-conformal systems.The method can also get rid of some sign problems.We discuss recent results regarding the critical properties of the 2D O(2) nonlinear sigma model with complex β and chemical potential.As some of these results appeared in a recently published paper (PRD 88, 056005) and two recent preprints (
Using the example of the two-dimensional (2D) Ising model, we show that in contrast to what can be done in configuration space, the tensor renormalization group formulation allows one to write exact, compact, and manifestly local blocking formulas and exact coarse-grained expressions for the partition function. We argue that similar results should hold for most models studied by lattice gauge theorists. We provide exact blocking formulas for several 2D spin models [the O(2)O(2)O(2) and O(3)O(3)O(3) sigma models and the SU(2) principal chiral model] and for the three-dimensional gauge theories with groups Z2{Z}_{2}Z2​, U(1) and SU(2). We briefly discuss generalizations to other groups, higher dimensions and practical implementations.
The idea of blocking in configuration space has played an important role in the development of the RG ideas. However, despite being half a century old and having had a huge intellectual impact, generic numerical methods to perform blocking for lattice models have progressed more slowly than sampling methods. Blocking may be essential to deal with near conformal situations. Typically, blocking methods have smaller statistical errors but larger systematic errors than sampling methods. This situation is evolving with recent developments based on the Tensor RG (TRG) method. We report recent results for spin and gauge lattice models obtained with this new method regarding searches for fixed points, calculations of critical exponents and resolutions of sign problems. An interesting model for comparison is the 2-dimensional O(2) model with a chemical potential which has a sign problem with conventional Monte Carlo but allows sampling with the worm algorithm and blocking with various TRG formulations. We compare the efficiency and accuracy of these two methods.
The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.
The idea of blocking in configuration space has played an important role in the development of the RG ideas.However, despite being half a century old and having had a huge intellectual impact, generic numerical methods to perform blocking for lattice models have progressed more slowly than sampling methods.Blocking may be essential to deal with near conformal situations.Typically, blocking methods have smaller statistical errors but larger systematic errors than sampling methods.This situation is evolving with recent developments based on the Tensor RG (TRG) method.We report recent results for spin and gauge lattice models obtained with this new method regarding searches for fixed points, calculations of critical exponents and resolutions of sign problems.An interesting model for comparison is the 2-dimensional O(2) model with a chemical potential which has a sign problem with conventional Monte Carlo but allows sampling with the worm algorithm and blocking with various TRG formulations.We compare the efficiency and accuracy of these two methods.
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.
This thesis describes the Tensor Renormalization Group (TRG) method used to solve the Z5 model in lattice gauge theory. The TRG method provides exact blocking for the models and is much less expansive in terms of computational costs. Here we consider the Z5 model in two dimensions, we discuss the formulation of the tensor and the process we used to do the TRG blocking, in order to find the phase transition and critical behavior of the model. We give some simple results and plots, and some expectation for further research in the model.