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arXiv:1309.6623 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Controlling sign problems in spin models using tensor renormalization

A. Denbleyker, Yuzhi Liu A, Yannick Meurice, Mingpu Qin, Tao Xiang, Z. Y. Xie, Ji-Feng Yu, Haiyuan Zou

Published Jan 9, 2014• 60 citations
arXivHTMLPDF

Abstract

We consider the sign problem for classical spin models at complex $\ensuremath{\beta}=1/{g}_{0}^{2}$ on $L\ifmmode\times\else\texttimes\fi{}L$ lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im$\ensuremath{\beta}$ than the reweighting Monte Carlo method. For the Ising model with complex $\ensuremath{\beta}$ we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the $O(2)$ model on $L\ifmmode\times\else\texttimes\fi{}L$ lattices when the number of states ${D}_{s}$ increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the $O(2)$ model with a chemical potential is briefly discussed.

DOI
10.1103/physrevd.89.016008

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We consider the sign problem for classical spin models at complex \ensuremathβ=1/g02\ensuremath{\beta}=1/{g}_{0}^{2}\ensuremathβ=1/g02​ on L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im\ensuremathβ\ensuremath{\beta}\ensuremathβ than the reweighting Monte Carlo method. For the Ising model with complex \ensuremathβ\ensuremath{\beta}\ensuremathβ we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the O(2)O(2)O(2) model on L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattices when the number of states Ds{D}_{s}Ds​ increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the O(2)O(2)O(2) model with a chemical potential is briefly discussed.
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 (
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 study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential.So far, there is no universal method to overcome this problem.However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent.In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model.We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics.In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential.We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.
The study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential. So far, there is no universal method to overcome this problem. However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent. In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model. We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics. In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential. We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.
We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied tensor renormalization group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically, yielding a value of 0.964 for the critical exponent $\ensuremath{\nu}$ much closer to the exact value 1 than the 1.338 value obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to $\ensuremath{\nu}=0.987$ and 0.993, respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable for accurate calculations of critical exponents, taking continuum limits, and the study of near-conformal systems in arbitrarily large volumes.
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.