Diamagnetic response and phase stiffness for interacting isolated narrow bands

Abstract

A platform that serves as an ideal playground for realizing ``high'' temperature superconductors are materials where the electrons' kinetic energy is completely quenched, and interactions provide the only energy scale in the problem for $T_c$. However, when the non-interacting bandwidth for a set of isolated bands is small compared to the scale of the interactions, the problem is inherently non-perturbative and requires going beyond the traditional mean-field theory of superconductivity. In two spatial dimensions, $T_c$ is controlled by the superconducting phase stiffness. Here we present a general theoretical framework for computing the electromagnetic response for generic model Hamiltonians, which controls the maximum possible superconducting phase stiffness and thereby $T_c$, without resorting to any mean-field approximation. Importantly, our explicit computations demonstrate that the contribution to the phase stiffness arises from (i) ``integrating out'' the remote bands that couple to the microscopic current operator, and (ii) the density-density interactions projected onto the isolated narrow bands. Our framework can be used to obtain an upper bound on the phase stiffness, and relatedly the superconducting transition temperature, for a range of physically inspired models involving both topological and non-topological narrow bands with arbitrary density-density interactions. We discuss a number of salient aspects of this formalism by applying it to a specific model of interacting flat bands and compare it against the known $T_c$ from independent numerically exact computations.

7 + 4 pages, 3 figures, new results added

References in corpus (14)

• Tunable Phase Boundaries and Ultra-Strong Coupling Superconductivity in Mirror Symmetric Magic-Angle Trilayer Graphene
• Electric field tunable unconventional superconductivity in alternating twist magic-angle trilayer graphene
• Superfluidity and Quantum Geometry in Twisted Multilayer Systems
• Exact Landau Level Description of Geometry and Interaction in a Flatband
• Dichroic f-sum rule and the orbital magnetization of crystals
• Fragile topology and flat-band superconductivity in the strong-coupling regime
• Effective theory and emergent $SU(2)$ symmetry in the flat bands of attractive Hubbard models
• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Kähler geometry and Chern insulators: Relations between topology and the quantum metric
• Optical Spectral Weight, Phase Stiffness and Tc Bounds for Trivial and Topological Flat Band Superconductors
• Superconductivity, charge density wave, and supersolidity in flat bands with tunable quantum metric
• Quantum Metric Induced Phases in Moiré Materials
• Heuristic bounds on superconductivity and how to exceed them
• Superconductivity and bosonic fluid emerging from Moiré flat bands

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