Blocking versus Sampling

Meurice, Yannick; Liu, Yuzhi; Unmuth-Yockey, Judah; Yang, Liping; Zou, Haiyuan

doi:10.22323/1.214.0319

Cited by 2 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The anisotropic TRG methods discussed above involve less contractions or external legs and the computational cost for N s states can be limited [11] to a N 7 s growth in two dimensions and a N 11 s growth in three dimensions. Numerical implementations of our method for up to 12 states are now in progress [20]. In all cases, we observe a sharp split between a few large eigenvalues, their number being characteristic of the phase, and the small ones as in the two states case.…”

mentioning

confidence: 82%

See 1 more Smart Citation

Accurate exponents from approximate tensor renormalizations

Meurice

2013

Phys. Rev. B

View full text Add to dashboard Cite

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 0.964 for the critical exponent ν much closer to the exact value 1 than 1.338 obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to ν = 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 to calculate accurately critical exponents, take continuum limits and study near-conformal systems in arbitrarily large volumes.PACS numbers: 05.10. Cc,05.50.+q,11.10.Hi,64.60.De,75.10.Hk The Renormalization Group (RG) ideas have triggered considerable conceptual and numerical progress in many branches of physics [1,2]. However, the basic method to thin down the number of degrees of freedom in configuration space [3], often called "block spinning", has remained a formidable computational challenge for most classical lattice models (e. g., O(N ) spin models and lattice gauge theories). A few years ago, inspired by the so-called tensor network states [4,5] introduced in the context of the density matrix RG method [6,7], a Tensor RG (TRG) approach of two-dimensional (2D) classical lattice models was proposed [8]. Successful approximations [8][9][10] were found for the Ising model on honeycomb and triangular lattices.Very recently, the TRG method was successfully extended to the Ising model on square and cubic lattices by Tao Xiang's group [11]. There are two important ingredients in their calculations. First, their formulation allows an exact block spinning procedure which separates neatly the degrees of freedom inside the block, which are integrated over, from those kept to communicate with the neighboring blocks. As explained below, this provides a more systematic way to implement ideas initiated by Migdal [12] and Kadanoff [13] (abbreviated as MK hereafter). The indices of the tensors run over some finite vector space of "states" associated with finite volume link configurations. Second, they used a new method, based on higher order singular value decomposition, which selects in a very economical way the most important states that insure the communication among the blocks. Calculations using of the order of 20 states can be carried on a laptop computer. The excellent agreement found with the Onsager solution in 2D for arbitrarily large volume suggests that TRG-based methods could become competitors for conventional Monte Carlo methods.In this Letter, we show that the truncation method of Ref. [11] for the 2D Ising model sharply singles out a twodimensional subspace of states. Keeping...

show abstract

mentioning

confidence: 82%

“…In all cases, we observe a sharp split between a few large eigenvalues, their number being characteristic of the phase, and the small ones as in the two states case. The interesting behavior of these new maps will be discussed elsewhere [20].…”

mentioning

confidence: 94%

Accurate exponents from approximate tensor renormalizations

Meurice

2013

Phys. Rev. B

View full text Add to dashboard Cite

show abstract

“…One perspective has been using Monte Carlo sampling to approximate tensor network contractions, either to contract or optimize a quantum state [37,38,39], or to approximate classical partition functions represented by a TN [24]. For the latter task, focusing on the square lattice O(2) model, a thorough comparison between TN-based and Markov Chin Monte Carlo techniques was presented in [40,41]. A different angle has been the construction of statistical mixtures of pure tensor network states to represent the thermal ensemble of a quantum system [42,43,44].…”

Section: Introductionmentioning

confidence: 99%

Collective Monte Carlo updates through tensor network renormalization

Pérez

Mariën²,

García

et al. 2023

SciPost Phys.

View full text Add to dashboard Cite

We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmark computations for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass instances, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff’s algorithm: equilibration times appear to be reduced by a factor that varies between 4040 and 20002000, depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro- and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.

show abstract

Blocking versus Sampling

Cited by 2 publications

References 12 publications

Accurate exponents from approximate tensor renormalizations

Accurate exponents from approximate tensor renormalizations

Collective Monte Carlo updates through tensor network renormalization

Contact Info

Product

Resources

About