Lattice studies of quantum chromodynamics-like theories with many fermionic degrees of freedom

DeGrand, Thomas

doi:10.1098/rsta.2010.0380

Cited by 22 publications

(26 citation statements)

References 81 publications

(117 reference statements)

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“…We are hoping to be able to extend the TRG method for lattice gauge theories with fermions. Being able to block spin accurately would provide an efficient tool to study the continuum limits of asymptotically free models and the conformal window of models that could provide alternatives to the fundamental Higgs mechanism [24].…”

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Accurate exponents from approximate tensor renormalizations

Meurice

2013

Phys. Rev. B

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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...

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Accurate exponents from approximate tensor renormalizations

Meurice

2013

Phys. Rev. B

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“…Different numerical and analytical techniques have been applied to QCD-like models with a large number of fermion flavors [1,2,3,4,5,6,7] or with fermions in higher representations [8,9,10,11]. See also [12,13,14,15] for recent reviews of results and expectations. It is important to understand the critical behavior of lattice models from various points of view.…”

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confidence: 99%

Fisher zeros and conformality in lattice models

Meurice¹

2012

Proceedings of the 30th International Symposium on Lattice Field Theory — PoS(Lattice 2012)

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Fisher zeros are the zeros of the partition function in the complex β = 2N c /g 2 plane. When they pinch the real axis, finite size scaling allows one to distinguish between first and second order transition and to estimate exponents. On the other hand, a gap signals confinement and the method can be used to explore the boundary of the conformal window. We present recent numerical results for 2D O(N) sigma models, 4D U(1) and SU(2) pure gauge and SU(3) gauge theory with N f = 4 and 12 flavors. We discuss attempts to understand some of these results using analytical methods. We discuss the 2-lattice matching and qualitative aspects of the renormalization group (RG) flows in the Migdal-Kadanoff approximation, in particular how RG flows starting at large β seem to move around regions where bulk transitions occur. We consider the effects of the boundary conditions on the nonperturbative part of the average energy and on the Fisher zeros for the 1D O(2) model.

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“…Therefore lattice calculations are essential. Despite the intrinsic theoretical interest of the realization of conformal symmetry near an IRFP, it seems that phenomenologically, the preferred option is a "walking" coupling constant for extended technicolor interactions [47,48]. Given this situation, it is necessary to develop new methods that would lead to an unambiguous determination of the existence of IRFP or of a non-perturbative "walking" coupling constant.…”

Section: Introductionmentioning

confidence: 99%

Renormalization Group and Phase Transitions in Spin, Gauge, and QCD Like Theories

Liu¹

2013

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Lattice studies of quantum chromodynamics-like theories with many fermionic degrees of freedom

Cited by 22 publications

References 81 publications

Accurate exponents from approximate tensor renormalizations

Accurate exponents from approximate tensor renormalizations

Fisher zeros and conformality in lattice models

Renormalization Group and Phase Transitions in Spin, Gauge, and QCD Like Theories

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