Order-separated tensor-network method for QCD in the strong-coupling expansion

Abstract

We introduce the order-separated Grassmann higher-order tensor renormalization group (OS-GHOTRG) method for QCD with staggered quarks in the strong-coupling expansion. The method allows us to determine the expansion coefficients of the partition function, from which we can obtain the strong-coupling expansions of thermodynamical observables. We use the method in two dimensions to compute the free energy, the particle-number density, and the chiral condensate as a function of the chemical potential up to third order in the inverse coupling $β$. Although near the phase transition the expansion is only a good approximation to the full theory at small $β$, we show that the range of applicability can be greatly extended by fits to judiciously chosen transition functions.

Figures (20)

• Figure 1: Blue: Edge variables $n$ and $\bar{n}$ for the oriented plaquettes starting at site $x$ in the $\mu\nu$ and $\nu\mu$ directions, respectively. The subscript indicates the link, while the superscript indicates the direction (including sign) in which the plaquette extends perpendicular to the link. For a valid configuration the four edge variables around an oriented plaquette must be equal. Red: Edge occupations of the unoriented plaquette as defined in \ref{['eq:def_plaq_excitation']}, which must also be equal for a valid configuration.
• Figure 2: For valid configurations we require two adjacent edge occupations of the same unoriented plaquette to be equal and call this requirement the hook condition. The quantity $\Delta_x$ in \ref{['eq:pc']} is a product of hook conditions.
• Figure 3: Blocking procedure to reduce the tensor network on a two-dimensional $4\times 4$ lattice to a single tensor using alternating contractions in the horizontal and vertical directions. Each contraction step results in a new tensor network defined on a coarse grid with half the number of sites in the contraction direction and with fat links in the perpendicular directions, which are then truncated.
• Figure 4: Illustration of a contraction step. On the fine grid (left), the tensors at the sites $x$ and $y=x+\hat{\mu}$ are contracted, resulting in a new tensor at the site $X$ on the coarse grid (right). Every plaquette that has the contracted link as an edge is collapsed onto the corresponding fat link perpendicular to $\hat{\mu}$.
• Figure 5: Visualization of the construction of the occupation numbers on a coarse site. The occupations on the coarse site (right) are computed from those on the fine sites (left) according to \ref{['eq:coarse_occupations']}.