Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians

We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular $\mathcal{T}$ transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.

Figure 1

Single particle states $\phi (x,y)$ in the lowest Landau level of an infinite cylinder. $L$ is the circumference of the cylinder, and ${\ell }_B$ is the magnetic length.

Figure 2

(a) Entanglement entropy $S$ of the $\nu =2/5$ state as a function of the circumference $L$, for a bipartition into two half-infinite cylinders. The inset shows the estimated TEE $\gamma =L(dS/dL)-S(L)$ converging to $\gamma =(1/2)\log 5$ (green line) for large $L$. (b) Estimate of the chiral central charge $c/24$, the topological spin $h$ of the $e/5$ QP, and the Hall viscosity as expressed through the “shift,” $𝒮={\eta }_{H}8\pi {\ell }_B^2/\hslash \nu$ [46]. Dashed lines show expected hierarchy values of $2/24$, $1/5$, $4/16$, respectively. That $h$ is identically correct is peculiar to Abelian QPs.

Figure 3

(a) The relative energy difference between states in the 0110 and 0101 sectors at $\nu =1/2$ filling and $L=16{\ell }_B$, plotted against the ratio of pseudopotential strengths $V_3/V_1$. At small $V_3/V_1$, the energies $E_{0110}>E_{0101}$, and hence there is only a twofold GS degeneracy (CFL phase), whereas at large $V_3/V_1$, the energies are roughly equal, which gives rise to a sixfold GS degeneracy (MR phase). Inset shows the real-space entanglement spectrum of the 0101 state plotted versus $\overline{K}$. (b) Fixing a point $V_3/V_1=0.4$ in the MR phase, we increase $L$ and measure the difference in entanglement entropies $S_{0101}-S_{0110}$. The result is consistent with $d_{\sigma }=\sqrt{2}$ for the $e/4$ QP.

Figure 4

(a) The MPS used to represent a QP $a$. (b) The charge density $\rho (y)$ of the numerically optimized QP with charge $+e/3$ (solid blue) and $-e/3$ (dashed green) of the $\nu =1/3$ state. The cylinder has circumference $L=16{\ell }_B$, and the potential approximates a dipolar $r^{-3}$ interaction.

Figure 5

(a) A torus with dimensions $L_x\times L_y$. Taking the modular parameter $\tau \to \tau +1$ generates a modular transform $\mathcal{T}$ of the torus and acts on the set of GSs as a matrix $U_T$. (b) The wave function on the torus is represented by a periodic MPS. The sites are labeled by $n\in \mathbb{Z}+{\Phi }_x/2\pi$ due to the flux threading in the $x$ direction.