Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder

Given a microscopic lattice Hamiltonian for a topologically ordered phase, we propose a numerical approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. Then a quasiorthogonal basis on the torus is produced by chopping off and reconnecting the tensor network representation on the cylinder. From these two bases, and by using a number of previous results, most notably the recent proposal of Y. Zhang et al. [Phys. Rev. B 85, 235151 (2012)] to extract the modular $U$ and $S$ matrices, we obtain (i) a complete list of anyon types $i$, together with (ii) their quantum dimensions $d_i$ and total quantum dimension $D$, (iii) their fusion rules $N_{ij}{}^k$, (iv) their mutual statistics, as encoded in the off-diagonal entries $S_{ij}$ of $S$, (v) their self-statistics or topological spins ${\theta }_i$, (vi) the topological central charge $c$ of the anyon model, and, in a chiral phase (vii) the low energy spectrum of each sector of the boundary conformal field theory. As a concrete application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of its universal bulk and edge properties unambiguously shows that it realizes a $\nu =1/2$ bosonic fractional quantum Hall state.

Figure 1

(a) Honeycomb lattice on a cylinder with $L_x=\infty$ and $L_y=4$ (as measured in number of unit cells). A blue, dashed line indicates the entanglement cut used in dividing the cylinder into two semi-infinite cylinders. (b) Ground states $|{\Psi }_1^{\mathrm{cyl}}⟩$ and $|{\Psi }_s^{\mathrm{cyl}}⟩$ of $H_{\mathrm{Hal}.}$ in Eq. (1) with flux $i=1$, $s$. (c) Corresponding density matrices ${\rho }_1$ and ${\rho }_s$ for the left half of the infinite cylinder.

Figure 2

(a) MPS representation of a state $|{\Psi }_i^{\mathrm{cyl}}⟩$ of an infinite cylinder with $L_y=4$; see Fig. 1a. The bond indices that connect tensors form a snake. The MPS unit cell consists of eight tensors that are repeated indefinitely. (b) Periodic MPS for a state $|{\Psi }_i^{\mathrm{tor}}⟩$ of a torus with $4\times 4\times 2=32$ sites, obtained by reconnecting four MPS unit cells of $|{\Psi }_i^{\mathrm{cyl}}⟩$.

Figure 3

Entanglement spectrum of the reduced density matrix ${\rho }_i$ for half an infinite cylinder for ground state $|{\Psi }_i^{\mathrm{cyl}}⟩$, for $i=1$ (left) and $s$ (right). The vertical axis shows $E_{i,\alpha }\equiv -\log (p_{i,\alpha })$ (up to a global shift and rescaling), where $\{p_{i,\alpha }\}$ are the eigenvalues of ${\rho }_i$. The horizontal axis shows the momentum of the corresponding eigenvector of ${\rho }_i$, artificially extended beyond its $2\pi$ periodicity. All eigenvalues of ${\rho }_i$ with the same particle number (also indicated) are connected with a tilted line. They obey the degeneracy pattern $\{1,1,2,3,5,\cdots \}$, characteristic of a bosonic Gaussian theory.