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arXiv:2410.09485 · Journal of High Energy Physics

Grassmann tensor renormalization group approach to (1+1)-dimensional two-color lattice QCD at finite density

Kwok Ho Pai, Shinichiro Akiyama, Synge Todo

Published Mar 5, 2025• 5 citations
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Abstract

A bstract We construct a Grassmann tensor network representing the partition function of (1+1)-dimensional two-color QCD with staggered fermions. The Grassmann path integral is rewritten as the trace of a Grassmann tensor network by introducing two-component auxiliary Grassmann fields on every edge of the lattice. We introduce an efficient initial tensor compression scheme to reduce the size of initial tensors. The Grassmann bond-weighted tensor renormalization group approach is adopted to evaluate the quark number density, fermion condensate, and diquark condensate at different gauge couplings as a function of the chemical potential. Different transition behavior is observed as the quark mass is varied. We discuss the efficiency of our initial tensor compression scheme and the future application toward the corresponding higher-dimensional models.

DOI
10.1007/jhep03(2025)027

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A bstract We construct a Grassmann tensor network representing the partition function of (1+1)-dimensional two-color QCD with staggered fermions. The Grassmann path integral is rewritten as the trace of a Grassmann tensor network by introducing two-component auxiliary Grassmann fields on every edge of the lattice. We introduce an efficient initial tensor compression scheme to reduce the size of initial tensors. The Grassmann bond-weighted tensor renormalization group approach is adopted to evaluate the quark number density, fermion condensate, and diquark condensate at different gauge couplings as a function of the chemical potential. Different transition behavior is observed as the quark mass is varied. We discuss the efficiency of our initial tensor compression scheme and the future application toward the corresponding higher-dimensional models.
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Two-color lattice QCD in (1+1)(1+1)(1+1) dimensions with Grassmann tensor renormalization group
The (1+1)(1+1)(1+1)-dimensional two-color lattice QCD is studied with the Grassmann tensor renormalization group. We construct tensor network representations of theories with the staggered fermion and the Wilson fermion and show that Grassmann tensor networks can describe both cases with the same bond dimension. We also propose an efficient initial tensor compression scheme to gauge degrees of freedom. We compute the number density, chiral condensate, and diquark condensate at finite density, employing the staggered fermions. For the theory with Wilson fermion, a critical point in the negative mass region is identified by inspecting the pseudoscalar condensate and the conformal field theory data.
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.
We present a tensor-network approach for the strong-coupling expansion of two-dimensional QCD with staggered quarks at non-zero chemical potential. After expanding the Boltzmann factor in the gauge and fermion actions, all gauge fields can be integrated out exactly and the partition function can be evaluated using the Grassmann higher-order tensor renormalization group approach. The method is modified to compute the μ\muμ dependence of the quark number density and the chiral condensate up to order β3\beta^3β3 with complete absence of higher-order terms infiltrating the result. Although the expansion itself is only a good approximation to the full theory at small β<0.1\beta<0.1β<0.1, the range can be extended, by using judiciously chosen fits. Moreover, these fits also yield a valuable expansion in β\betaβ for the critical chemical potential.
We present a tensor-network method for 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. We apply the method to the two-dimensional case and validate it by comparing results for the partition function, the chiral condensate and the baryon density with exact analytical expressions on small lattices up to volumes of 4×44\times44×4. For larger two-dimensional volumes, we present 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. Furthermore, our results for the number density as a function of the chemical potential hint at a first-order phase transition. Finally, we present some preliminary tensor results for three-dimensional strong-coupling QCD.
We present a tensor-network formulation for the strong-coupling expansion of QCD with staggered quarks at nonzero chemical potential, for arbitrary number of dimensions, colors, and flavors. We integrate out the gauge and quark degrees of freedom and rewrite the partition function as the complete trace of a tensor network. This network consists of local tensors that contain a numerical and a Grassmann part. We truncate the initial tensor at a fixed order in the inverse coupling β and compute analytical results for the partition function, the free energy, and the chiral condensate on a 2 × 2 lattice up to order β 4 . In a follow-up paper we will introduce an enhanced tensor-network method, order-separated GHOTRG, to explicitly compute the expansion coefficients of the partition function for larger lattices. To demonstrate its potential, first results obtained with this new method are already presented here.