Inner workings of fractional quantum Hall parent Hamiltonians: A matrix product state point of view

TL;DR

This paper explores the connection between matrix product state representations and frustration-free parent Hamiltonians in fractional quantum Hall systems, revealing insights into their structure and solvability beyond finite bond dimension models.

Contribution

It establishes a direct link between infinite-bond-dimension MPS structures of Laughlin states and their parent Hamiltonians, extending understanding beyond short-range models.

Findings

Link between MPS structure and parent Hamiltonians for FQH states

Extension of solvable lattice models to infinite bond dimension

Insights into the structure of Laughlin and other CFT-MPS states

Abstract

We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized "analytic clustering properties". In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration-free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short-range Hamiltonians and MPSs of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin-conformal field theory (CFT) states and their parent Hamiltonians. While focusing on the Laughlin state, generalizations to other CFT-MPSs will become transparent.