Bootstrapping Flat-band Superconductors: Rigorous Lower Bounds on Superfluid Stiffness

TL;DR

This paper demonstrates how the quantum many-body bootstrap framework can be used to rigorously bound superfluid stiffness in flat-band superconductors, revealing new insights and potential for broader applications.

Contribution

It introduces a novel application of the RDM bootstrap to derive lower bounds on superfluid stiffness in frustration-free models, including effects of additional interactions.

Findings

Established a relation between superfluid stiffness and pair mass in flat-band models.

Showed magnetic interactions can enhance superfluid stiffness.

Revealed trion correlations are crucial for bounding stiffness.

Abstract

The superfluid stiffness fundamentally constrains the transition temperature of superconductors, especially in the strongly coupled regime. However, accurately determining this inherently quantum many-body property in microscopic models remains a significant challenge. In this work, we show how the \textit{quantum many-body bootstrap} framework, specifically the reduced density matrix (RDM) bootstrap, can be leveraged to obtain rigorous lower bounds on the superfluid stiffness in frustration-free interacting models with superconducting ground state. We numerically apply the method to a special class of frustration free models, which are known as quantum geometric nesting models, for flat-band superconductivity, where we uncover a general relation between the stiffness and the pair mass. Going beyond the familiar Hubbard case within this class, we find how additional interactions, notably simple magnetic couplings, can enhance the superfluid stiffness. Furthermore, we find that the RDM bootstrap unexpectedly reveals that the trion-type correlations are essential for bounding the stiffness, offering new insights on the structure of these models. A straightforward generalization of the method can lead to bounds on susceptibilities complementary to variational approaches. Our findings underscore the immense potential of the quantum many-body bootstrap as a powerful tool to derive rigorous bounds on physical quantities beyond energy.