[Paper Review] Geometrical effects in orbital magnetic susceptibility

Yang Gao, Shengyuan A. Yang+1 more|arXiv (Cornell University)|Nov 2, 2014

Topological Materials and Phenomena3 citations

TL;DR

This paper presents a gauge-invariant classification of orbital magnetic susceptibility in solids using the wave-packet semiclassical approach, identifying two new geometrical contributions: a dominant geometrical susceptibility from Berry curvature and quantum metric, and a Fermi surface energy polarization term. These arise from single-band physics and compete with Peierls-Landau and Pauli paramagnetism, with the only true interband contribution being Van Vleck susceptibility.

ABSTRACT

Within the wave-packet semiclassical approach, the Bloch electron energy is derived to second order in the magnetic field and classified into gauge-invariant terms with clear physical meaning, yielding a fresh understanding of the complex behavior of orbital magnetism. The Berry curvature and quantum metric of the Bloch states give rise to a geometrical magnetic susceptibility, which can be dominant when bands are filled up to a small energy gap. There is also an energy polarization term, which can compete with the Peierls-Landau and Pauli magnetism on a Fermi surface. All these, and an additional Langevin susceptibility, can be calculated from each single band, leaving the Van Vleck susceptibility as the only term truly from interband coupling.

Motivation & Objective

• To provide a gauge-invariant classification of orbital magnetic susceptibility in crystalline solids using the wave-packet semiclassical formalism.
• To identify and isolate geometrical contributions to magnetic susceptibility originating from intrinsic band geometry, such as Berry curvature and quantum metric.
• To clarify the physical origin and dominance conditions of the geometrical susceptibility, especially near band gaps.
• To demonstrate that the Fermi surface energy polarization term arises from single-band physics and competes with conventional paramagnetic responses.
• To show that only the Van Vleck susceptibility requires genuine interband coupling, while other terms can be computed from individual bands.

Proposed method

• Derives the wave-packet energy up to second order in the magnetic field using a gauge-invariant wave-packet formalism.
• Introduces vertical and horizontal mixing concepts to describe first-order corrections to Bloch states under magnetic perturbation.
• Classifies the second-order energy correction into gauge-invariant terms: geometrical susceptibility (involving Berry curvature and quantum metric), energy polarization, Langevin, and Van Vleck contributions.
• Expresses all terms in momentum space using the Fermi-Dirac distribution and band structure, enabling direct computation from single Bloch bands.
• Uses the minimal coupling Hamiltonian and extends the formalism to include non-minimal couplings such as Zeeman terms.
• Applies the formalism to atomic systems and free particles to identify the Langevin limit and validate the new terms.

Experimental results

Research questions

• RQ1 What are the gauge-invariant contributions to orbital magnetic susceptibility beyond the standard Peierls-Landau and Pauli mechanisms?
• RQ2 How do the Berry curvature and quantum metric contribute to the magnetic response in gapped systems?
• RQ3 Can a Fermi surface contribution to magnetic susceptibility arise from energy polarization, and how does it compete with other terms?
• RQ4 Under what conditions does the geometrical susceptibility dominate over conventional diamagnetic or paramagnetic responses?
• RQ5 Which terms in the susceptibility require interband coupling, and which can be computed from single-band properties?

Key findings

• The geometrical susceptibility, arising from the Berry curvature and quantum metric, becomes dominant in systems with small band gaps and is especially significant in topological insulators and 2D semimetals.
• The energy polarization term, originating from the Fermi surface, contributes to the magnetic susceptibility and competes with Peierls-Landau and Pauli paramagnetism.
• The Langevin susceptibility emerges naturally in the atomic limit and is recovered as a special case of the general formalism.
• All terms except the Van Vleck susceptibility can be computed from a single Bloch band, highlighting the role of single-band geometry in orbital magnetism.
• The geometrical susceptibility term vanishes in the atomic insulator limit due to its dependence on inter-lattice hopping, confirming its non-local, band-geometric origin.
• The formalism is extendable beyond minimal coupling, allowing inclusion of spin-Zeeman terms and other non-minimal couplings.