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arXiv:2005.07570 · Journal of High Energy Physics

More about the Grassmann tensor renormalization group

Shinichiro Akiyama, Daisuke Kadoh

Published Oct 1, 2021• 30 citations
arXivHTMLPDF

Abstract

Abstract We derive a general formula of the tensor network representation for d-dimensional lattice fermions with ultra-local interactions, including Wilson fermions, staggered fermions, and domain-wall fermions. The Grassmann tensor is concretely defined with auxiliary Grassmann variables that play a role in bond degrees of freedom. Compared to previous works, our formula does not refer to the details of lattice fermions and is derived by using the singular value decomposition for the given Dirac matrix without any ad-hoc treatment for each fermion. We numerically test our formula for free Wilson and staggered fermions and find that it properly works for them. We also find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method.

DOI
10.1007/jhep10(2021)188

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Two-color lattice QCD in (1+1)(1+1)(1+1) dimensions with Grassmann tensor renormalization group
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We present a tensor-network approach for two-dimensional strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. During the coarsening procedure, the blocking of adjacent Grassmann tensors is performed analytically, and the total number of Grassmann variables in the tensor network is reduced by a factor of two at each coarsening step. The coarse-site numeric tensors are truncated using higher-order singular value decompositions. The method is validated by comparing the partition function, the chiral condensate and the baryon density computed with the tensor method with exact analytical results on small lattices up to volumes of 4×4. For larger volumes, we present first tensor results for the chiral condensate as a function of the mass and volume, and observe that the chiral symmetry is not broken dynamically in two dimensions. We also present tensor results for the number density as a function of the chemical potential, which hint at a first-order phase transition.
Abstract The strongly correlated fermions play a vital role in modern physics. For a given fermionic Hamiltonian system, the most widely used approach to exploring the underlying physics is to study the wave function that incorporates Fermi–Dirac statistics, which can be obtained variationally by energy minimization or by imaginary-time evolution. In this work, we develop an accurate tensor network method for one-dimensional interacting fermionic models based on the coherent-state path-integral representation of the fermionic partition function. Employing the coherent-state representation, the partition function is effectively represented as a (1+1)-dimensional anisotropic Grassmann-valued tensor network, and the Grassmann version of the corner transfer-matrix renormalization group algorithm is developed to contract the tensor network and evaluate physical quantities. We validate our method on the one-dimensional fermionic Hubbard model with a magnetic field, where the essential features of the phase diagram in the ( μ , B ) plane are quantitatively captured. Our work offers a promising approach to interacting fermionic models within the framework of tensor networks.