Note on searching for critical lattice models as entropy critical points from strange correlator

Abstract

An entropy function is proposed in [Phys. Rev. Lett. 131, 251602] as a way to detect criticality even when the system size is small. In this note we apply this strategy in the search for criticality of lattice transfer matrices constructed based on the topological holographic principle. We find that the combination of strategy is indeed a cost-effective and efficient way of identifying critical boundary conditions, estimating central charges and moreover, plotting entire phase diagrams in a multi-dimensional phase space.

Figures (9)

• Figure 1: The entropy function Eq. \ref{['eq:entropy function']} can be defined on (a) an infinite line and (b) a closed circle. $\eta$ is the corresponding cross-ratio.
• Figure 2: (a) The square-octagon lattice (black dashed lines) and its dual triangulation (blue lines and red vertices) used in this work. Each blue-edged triangle represents the front face of a tetrahedron with the three red vertices representing three radial edges connecting to the same central vertex. (b) The topological ground state is given by the product of every tetrahedron, each carrying a weight equaling to the quantum 6$j$-symbol of its three surface edges $\{i,j,k\}$ and three radial edges $\{\alpha,\beta,\gamma\}$, scaled by their quantum dimensions. Equivalently, the topological ground state can be seen as a tensor network state with physical legs $\{i,j,k\}$ and auxiliary legs $\{\alpha,\beta,\gamma\}$.
• Figure 3:
• Figure 4: The entropy function values of the competition between 0 and $0\oplus 2$ with common module 1 in input categories $A_{k+1}$, for $k$ from 2 to 6. The estimated central charge is three times of the entropy function value. The entropy function shows clear peaks near the critical points, as summarized in Table \ref{['tab:Ising critical']}. Each curve contains 100 data points.
• Figure 5: The entropy function values of the competition between $0\oplus 2$ and $0\oplus 3$ with common module $1\oplus2$ in input categories $A_4$. The peak of the entropy function is located at $r = 1.1747$, and the theoretical critical point is $r^* = 1.1441$. The curve contains 100 data points.