[Paper Review] Eigenprojectors, Bloch vectors and quantum geometry of $N$-band systems

Ansgar Graf, Frédéric Piéchon|arXiv (Cornell University)|Feb 19, 2021

Topological Materials and Phenomena70 references4 citations

TL;DR

This paper introduces a gauge-invariant framework for computing quantum geometric tensors—Berry curvature and quantum metric—in $N$-band systems using eigenprojectors and generalized Bloch vectors, bypassing problematic eigenstate constructions. It derives explicit, $N$-band generalizations of the two-band Berry curvature formula and reveals distinct geometric properties in systems with identical spectra but different symmetries.

ABSTRACT

The eigenvalues of a parameter-dependent $N imes N$ Hamiltonian matrix form a band structure in parameter space. Quantum geometric properties (Berry curvature, quantum metric, etc.) of such $N$-band systems are usually computed from parameter-dependent eigenstates. This approach faces several difficulties, including gauge ambiguities and singularities in the multicomponent eigenfunctions. In order to circumvent this problem, this work exposes an alternative approach based on eigenprojectors and (generalized) Bloch vectors. First, an expansion of each eigenprojector as a matrix polynomial in the Hamiltonian is deduced, and using SU($N$) Gell-Mann matrices an equivalent expansion of each Bloch vector is found. In a second step, expressions for the $N$-band Berry curvature and quantum metric in terms of Bloch vectors are obtained. This leads to new explicit Berry curvature formulas in terms of the Hamiltonian vector, generalizing the well-known two-band formula to arbitrary $N$. Moreover, a detailed treatment is given for the case of a particle-hole symmetric energy spectrum, which occurs in systems with a chiral or charge conjugation symmetry. For illustrating the formalism, several model Hamiltonians featuring a multifold linear band crossing are discussed; they have identical energy spectra but completely different geometric and topological properties. The methodology used in this work is more broadly applicable to compute any physical quantity, or to study the quantum dynamics of any observable without the explicit construction of energy eigenstates.

Motivation & Objective

• To overcome gauge ambiguities and singularities in eigenstate-based computation of quantum geometric tensors in $N$-band systems.
• To develop a formalism that computes Berry curvature and quantum metric without explicitly constructing multicomponent eigenstates.
• To generalize the two-band Berry curvature formula to arbitrary $N$-band systems using SU($N$) group theory.
• To analyze systems with particle-hole symmetry, such as those with chiral or charge conjugation symmetry, using the new formalism.
• To demonstrate that identical energy spectra can support vastly different geometric and topological properties through model Hamiltonians with multifold band crossings.

Proposed method

• Derive an expansion of each eigenprojector as a matrix polynomial in the Hamiltonian matrix.
• Express each eigenprojector using SU($N$) Gell-Mann matrices to define generalized Bloch vectors.
• Obtain explicit expressions for the $N$-band Berry curvature and quantum metric in terms of these Bloch vectors.
• Use the Hamiltonian vector to generalize the two-band Berry curvature formula to arbitrary $N$.
• Apply the formalism to model Hamiltonians with multifold linear band crossings to compare geometric properties.
• Demonstrate the method’s broader applicability to compute physical observables and study quantum dynamics without eigenstate construction.

Experimental results

Research questions

• RQ1 How can quantum geometric tensors be computed in $N$-band systems without relying on parameter-dependent eigenstates?
• RQ2 What is the generalization of the two-band Berry curvature formula to arbitrary $N$-band systems?
• RQ3 How do geometric and topological properties differ in systems with identical energy spectra but distinct symmetries?
• RQ4 What role does particle-hole symmetry play in shaping the quantum geometry of $N$-band systems?
• RQ5 Can the formalism be used to compute observables or study dynamics without explicit eigenstate computation?

Key findings

• The paper derives a gauge-invariant expression for the $N$-band Berry curvature in terms of the Hamiltonian vector, generalizing the two-band formula to arbitrary $N$.
• The formalism enables the computation of the quantum metric in $N$-band systems through Bloch vector expansions, avoiding singularities in eigenstates.
• For particle-hole symmetric systems, the formalism reveals that the Berry curvature and quantum metric are constrained by the symmetry structure.
• Model Hamiltonians with identical linear band crossings but different symmetries exhibit distinct geometric and topological properties, as revealed by the formalism.
• The method allows for the direct computation of physical observables and quantum dynamics without constructing eigenstates, enhancing numerical and analytical feasibility.
• The use of eigenprojectors and Bloch vectors provides a robust, singularity-free alternative to standard eigenstate-based approaches in quantum geometry.