Fig. 2
From: Microscopic study of the Halperin–Laughlin interface through matrix product states
Construction of an interface model wavefunction. a Schematic representation of the MPS ansatz \(|\Psi _{{\mathrm{H}} - {\mathrm{L}}}\rangle\) for the Laughlin 1/3—Halperin 332 interface on a cylinder of perimeter L. The Halperin iMPS matrices BH (red) are glued to the Laughlin iMPS matrices \(B_{\mathrm{L}} \otimes {\Bbb I}\) (blue) in the Landau orbital space. Due to the embedding of one auxiliary space into the other, the quantum numbers of \(|\mu _ \bot \rangle\) (see Eq. (14)) are left unchanged by the Laughlin matrices all the way to the interface. It constitutes a direct access controlling the states of the interface chiral gapless mode, graphically sketched here with a double arrow. b Spin-resolved densities of the MPS ansatz state along the cylinder axis obtained at Pmax = 11 for \(L = 25\ell _B\). They smoothly interpolate between the Laughlin (2πρ↓ = 1/3 and ρ↑ = 0) and the Halperin (\(2\pi \rho _ \downarrow = 2\pi \rho _ \uparrow = 1/5\)) theoretical values. The density is a robust quantities for which it is safe to consider large perimeter with our truncation level. c The EE \(S_{\cal{A}}(L,x)\) follows an area law (Eq. 15) for the rotationally invariant bipartition \({\cal{A}} - {\cal{B}}\) depicted on top of the graph. The constant correction is numerically extracted by finite differences \(S_{\cal{A}}(L,x) - L\partial _LS_{\cal{A}}(L,x)\) and plotted for \(L = 13\ell _B\) as a function of the position along the cylinder axis x. It smoothly interpolates between its respective Laughlin and Halperin bulk values and we see no universal signature of the critical mode at the interface. Away from the interface, i.e., \(x \,< - 7\ell _B\) on the Halperin side and \(x \,> \,3\ell _B\) on the Laughlin side, the extracted γ(x) agree with the theoretical expectation within 3% accuracy. Thus, our MPS model WF describes the interface between two distinct topological orders. Note that the extraction and convergence of the subleading quantity γ(x) (see Eq. 15) requires large truncation parameters. The spikes appearing on both sides of the transitions are artifacts of the computations of the RSES (see ref. 34 for details). They corresponds to the points where a patch of three (resp. five) orbitals are added on the Laughlin (resp. Halperin) side of the finite size region which translate the bipartition \({\cal{A}} - {\cal{B}}\) to orbital space. They disappear with increasing Pmax, as shown by the points computed at Pmax = 14 (black dots)
