Microscopic study of the Halperin–Laughlin interface through matrix product states

Abstract

Interfaces between topologically distinct phases of matter reveal a remarkably rich phenomenology. We study the experimentally relevant interface between a Laughlin phase at filling factor ν = 1/3 and a Halperin 332 phase at filling factor ν = 2/5. Based on our recent construction of chiral topological interfaces ( Nat. Commun . 10.1038/s41467-019-09168-z; 2019), we study a family of model wavefunctions that captures both the bulk and interface properties. These model wavefunctions are built within the matrix product state framework. The validity of our approach is substantiated through extensive comparisons with exact diagonalization studies. We probe previously unreachable features of the low energy physics of the transition. We provide, amongst other things, the characterization of the interface gapless mode and the identification of the spin and charge excitations in the many-body spectrum. The methods and tools presented are applicable to a broad range of topological interfaces.

Model states for a class of chiral topological order interfaces

“…The technical challenges inherent to such a computation breaking spatial symmetries are presented in Ref. [26]. We isolate the contribution of the interface edge mode from the area laws and corner contributions with a Levin-Wen addition subtraction scheme [52] depicted in Fig.…”

“…To capture the low energy features above the ground state, we can change the interface gapless mode momentum by choosing the correct level descendant P ⊥ of the ϕ ⊥ boson on the Laughlin side. We are able to reproduce the first few low lying excitations of the system above the finite size ground state with great accuracy [26]. Using matrix product operators and our ansatz, we can actually focus on and evaluate the dispersion relation of the gapless interface mode [26].…”

“…We are able to reproduce the first few low lying excitations of the system above the finite size ground state with great accuracy [26]. Using matrix product operators and our ansatz, we can actually focus on and evaluate the dispersion relation of the gapless interface mode [26].…”

“…By choosing the same m for the two model WFs, it is possible to embed the auxiliary space of the Laughlin state in that of the Halperin state. More precisely, up to compactification conditions [26,45] we have Aux H = Aux L ⊗Aux ⊥ with Aux ⊥ is the CFT Hilbert space for a free boson ϕ ⊥ . Representing the vertex operators in the same product basis for Aux H leads to the Halperin and B (n ↓ ,n ↑ ) H and Laughlin (B (n ↓ ) L ⊗I) iMPS matrices [45] (here the physical indices n ↑ and n ↓ denote the orbital occupation for particles with a spin ↑ and ↓).…”

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

“…We also checked this idea to remarkable numerical precision in the original Laughlin wave function. This was done using Monte Carlo techniques, complementing the MPS numerical results of [12][13][14][15] in cylinder geometries. We have also found a simple generalization to the multi-interval case in the non-interacting case.…”