Inferring effective interactions from the local density of states: application to STM data from Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$

TL;DR

This paper presents a method to infer local effective Hamiltonian structures from STM data by analyzing LDOS moments, revealing small electro-chemical potential variations but large gap fluctuations in a high-temperature superconductor.

Contribution

It introduces a novel approach to deduce local Hamiltonian features from STM data using LDOS moments, applied to Bi$_2$Sr$_2$CaCu$_2$O$_{8+eta}$.

Findings

Electro-chemical potential variations are minimal, indicating weak disorder.

Significant variations in the local d-wave gap parameter are observed.

The method effectively links STM data to underlying Hamiltonian properties.

Abstract

While the influence of impurities on the local density of states (LDOS) in a metal is notoriously non-local due to interference effects, low order moments of the LDOS in general can be shown to depend only on the local structure of the Hamiltonian. Specifically, we show that an analysis of the spatial variations of these moments permits one to ``work backwards'' from scanning tunneling microscopy (STM) data to infer the local structure of the underlying effective Hamiltonian. Applying this analysis to STM data from the high temperature superconductor, Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, we find that the variations of the electro-chemical potential are remarkably small (i.e., the disorder is, in a sense, weak) but that there are large variations in the local magnitude of the d-wave gap parameter.