[Paper Review] Quantum geometry in superfluidity and superconductivity

Sebastiano Peotta, Kukka-Emilia Huhtinen+1 more|arXiv (Cornell University)|Aug 16, 2023

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper establishes that in multiband superfluid and superconducting systems—particularly flat bands—quantum geometry, specifically the quantum metric, governs supercurrents when conventional band curvature vanishes. It derives the superfluid weight as proportional to the minimal quantum metric, resolving a long-standing paradox in flat-band superconductivity by showing geometric contributions sustain supercurrents even when effective mass is infinite.

ABSTRACT

We review the theoretical description of the role of quantum geometry in superfluidity and superconductivity of multiband systems, with focus on flat bands where quantum geometry is wholly responsible for supercurrents. This review differs from previous ones in that it is based on the most recent understanding of the theory: the dependence of the self-consistent order parameter on the supercurrent is properly taken into account, and the superfluid weight in a flat band becomes proportional to the minimal quantum metric. We provide a recap of basic quantum geometric quantities and the concept of superfluid density. The geometric contribution of superconductivity is introduced via considering the two-body problem. The superfluid weight of a multiband system is derived within mean-field theory, leading to a topological bound of flat band superconductivity. The physical interpretation of the flat band supercurrent in terms of Wannier function overlaps is discussed.

Motivation & Objective

• To resolve the paradox of how supercurrents can exist in flat bands where conventional superfluid weight vanishes due to zero band curvature.
• To establish the role of quantum geometry in multiband superconductivity and superfluidity, particularly in systems with high symmetry and isolated flat bands.
• To derive the superfluid weight in multiband systems using mean-field BCS theory, distinguishing conventional and geometric contributions.
• To show that the superfluid weight in a flat band is determined by the minimal quantum metric, a single-particle quantity independent of the many-body ground state.
• To provide a framework for predicting the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature based on quantum geometric properties.

Proposed method

• Derives the superfluid weight in multiband systems using mean-field BCS theory, incorporating both conventional (band curvature) and geometric (quantum metric) contributions.
• Uses linear response theory to decompose the superfluid weight into conventional and geometric components, with the latter arising from the non-Abelian Berry connection.
• Introduces a gauge-invariant formulation of the order parameter that accounts for orbital positions within the unit cell, enabling consistent computation of the superfluid weight.
• Applies the principle of minimal quantum metric by optimizing orbital positions to saturate the inequality between the Hessian of the grand potential and the quantum metric.
• Demonstrates that in time-reversal symmetric systems, the superfluid weight is directly proportional to the quantum metric when the Hessian is positive semidefinite and the inequality is saturated.
• Uses the BKT theory to link the superfluid weight to the critical temperature for superfluidity in two-dimensional systems, showing Tc scales with the quantum metric.

Experimental results

Research questions

• RQ1 How can supercurrents persist in flat bands where the conventional superfluid weight vanishes due to zero effective mass?
• RQ2 What is the role of quantum geometry in determining the superfluid weight in multiband superconductors?
• RQ3 Can the superfluid weight in a flat band be fully attributed to the quantum metric, and if so, under what conditions?
• RQ4 How does the choice of orbital positions within the unit cell affect the relation between the superfluid weight and the quantum metric?
• RQ5 What is the connection between the minimal quantum metric and the Berezinskii-Kosterlitz-Thouless transition temperature in two-dimensional flat-band superfluids?

Key findings

• In isolated flat bands with time-reversal symmetry, the superfluid weight is proportional to the minimal quantum metric, which is a single-particle quantity computable without solving the full many-body BCS ground state.
• The geometric contribution to the superfluid weight arises from the non-Abelian Berry connection and is non-zero even when the band curvature is zero, resolving the paradox of supercurrents in flat bands.
• The superfluid weight is maximized when the orbital positions are chosen such that the derivatives of the order parameter phases vanish, which corresponds to minimizing the integrated quantum metric.
• The inequality between the Hessian of the grand potential and the quantum metric is saturated when the system is time-reversal symmetric and the chemical potential derivative vanishes at zero momentum.
• The Berezinskii-Kosterlitz-Thouless transition temperature in two-dimensional flat-band systems is directly proportional to the quantum metric, providing a route to high-Tc superconductivity.
• In systems without time-reversal symmetry, the geometric contribution cannot be fully eliminated by gauge choice, and the real part of the order parameter derivative affects the superfluid weight, breaking the direct proportionality to the quantum metric.