Topological charge of fermions and Landau theory of Fermi liquid

This paper proposes that the electric charge of a fermion is equivalent to a topological charge, a correspondence that underpins the topological stability of the Fermi surface and extends the applicability of Landau's Fermi liquid theory to non-Fermi liquids and crystalline insulators via the Luttinger theorem.

The Gist

Imagine you are trying to understand a crowded dance floor. In the world of physics, this dance floor is made of electrons (tiny charged particles) moving around inside a metal. For decades, scientists have used a famous set of rules called Landau's Fermi Liquid Theory to describe how these electrons behave.

This paper by G.E. Volovik argues that these rules work not just because of math, but because of topology—a branch of math that studies shapes and how they hold together even when you stretch or twist them. Think of topology like a coffee mug and a donut: to a topologist, they are the same because they both have exactly one hole. You can't turn a mug into a sphere without tearing it; similarly, certain properties of electrons are "stitched" into their nature and cannot be easily broken.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The "Topological ID Card" of Electrons

In a normal metal, electrons fill up energy levels like water filling a bucket. There is a specific line called the Fermi Surface that separates the "full" bucket (occupied states) from the "empty" space above it.

Volovik suggests that every electron state has a hidden Topological ID Card.

• The Analogy: Imagine every seat in a stadium has a ticket. Some seats are occupied (Ticket = 1), and some are empty (Ticket = 0).
• The Magic: In this theory, these tickets aren't just random numbers; they are topological charges. Just like a knot in a string stays a knot until you cut the string, these "1s" and "0s" are stable. You can't smoothly change an occupied seat to an empty one without a dramatic event (a "phase transition").
• Why it matters: This proves that the number of "quasiparticles" (the dancing electrons we see) is exactly the same as the number of actual electrons. The topological ID card guarantees the count never lies, even if the electrons start interacting and fighting with each other.

2. The Unbreakable Fence (The Fermi Surface)

The boundary between the occupied seats and the empty ones is the Fermi Surface.

• The Analogy: Think of this surface as a fence made of steel. Even if you shake the ground (add heat) or push the fence (add interactions), the fence doesn't disappear. It might wiggle, but it stays there because it is held up by the topological ID cards on either side.
• The Result: This explains why the Luttinger Theorem (a rule stating the size of the Fermi surface tells you how many electrons are in the system) works so well. Even if the electrons get messy and the "pole" in the math turns into a "zero," the topological fence remains intact. The rule holds true because the shape of the fence is protected by topology.

3. The "Flat Band" and the Super-Condensed Crowd

The paper discusses a special scenario where the electrons get so crowded that they form a Flat Band.

• The Analogy: Imagine a hill where people usually slide down. In a normal metal, the hill is steep. But in a "Flat Band," the ground becomes perfectly flat. Everyone is standing at the exact same energy level, like a massive crowd standing shoulder-to-shoulder on a flat plain.
• The Consequence: When everyone is on this flat plain, the "density of states" (how many people can fit in a small space) becomes huge.
• The Superconductivity Connection: Usually, to get electrons to dance together in a superconductor (flow without resistance), you need very cold temperatures. But on this flat plain, because there are so many electrons packed together, they can start dancing together (superconducting) even at room temperature.
• Real-world hints: The author mentions experiments with graphite (pencil lead) where hints of room-temperature superconductivity have been seen. He suggests these might be tiny "islands" of this flat-band magic hiding on the surface of the material.

4. Crystals as Elastic Rubber Sheets

The paper also looks at Topological Insulators (materials that conduct electricity on the surface but act like insulators inside).

• The Analogy: Think of a crystal lattice (the grid of atoms) not as a rigid skeleton, but as a stretchy rubber sheet. The paper uses "elasticity tetrads" to describe how this sheet stretches and twists.
• The Gauge Field: It turns out that the way this rubber sheet deforms acts like a "gauge field" (a force field). This connects the physical shape of the crystal to the flow of electricity.
• The Strong CP Problem: This is a deep mystery in particle physics (why the universe doesn't behave differently if you swap matter for antimatter and flip left/right). Volovik suggests that the topological rules governing these crystals might offer a clue to solving this cosmic puzzle, much like how the shape of a donut dictates its properties.

Summary: The Big Picture

Volovik is telling us that the behavior of electrons isn't just about forces and collisions; it's about shape and stability.

1. Stability: The rules of Fermi liquids work because electrons have topological "ID cards" that prevent them from changing their count.
2. Resilience: The Fermi surface is a topological fence that survives even strong interactions.
3. Opportunity: If we can engineer materials where electrons flatten out into a "flat band," we might unlock the holy grail of physics: Superconductivity at room temperature, which would revolutionize energy transmission, electronics, and transportation.

In short, the paper argues that the universe's "glue" holding these electron systems together is a topological knot that is incredibly hard to untie, and understanding this knot is the key to unlocking the next generation of technology.

Technical Summary

1. Problem Statement

The paper addresses the fundamental theoretical justification for the Landau theory of Fermi liquids (LFL) and the validity of the Luttinger theorem in interacting many-body systems.

• The Core Issue: In standard Landau theory, the number of quasiparticles is assumed to coincide with the number of physical particles. However, in strongly interacting systems (non-Landau Fermi liquids or Mott insulators), the Green's function may lose its pole (quasiparticle residue $Z \to 0$), leading to questions about the stability of the Fermi surface and the conservation of particle number.
• The Gap: There is a need to demonstrate that the stability of the Fermi surface and the validity of the Luttinger theorem (which relates the Fermi surface volume to particle density) are not merely phenomenological but are rooted in topological invariance within momentum-frequency space.
• Extension: The paper also seeks to extend these topological concepts to crystalline insulators and explore the mechanism of flat band formation as a pathway to high-temperature superconductivity.

2. Methodology

The author employs a topological field theory approach in momentum-frequency space $(p, \omega)$, utilizing Green's functions and topological invariants.

• Spectral Asymmetry Index: The analysis begins by defining the spectral asymmetry index $\nu$ as the difference between the number of negative and positive energy states. This is reformulated using the Green's function $G(\omega, p)$:
  $$\nu = \text{Tr} \int \frac{d\omega}{2\pi i} G \partial_\omega G^{-1}$$
• Topological Invariants:
  • $N_\omega(p)$: A local topological invariant defined for each momentum state $p$. It acts as a generalized occupation number, taking integer values (0 or 1) for fermions.
  • $N_1$: A topological invariant defined on a contour enclosing the Fermi surface in momentum space, ensuring the topological stability of the surface itself.
• Adiabatic Continuity: The paper argues that interacting systems can be reached from non-interacting ones via adiabatic transformations. Since topological invariants can only change via discrete jumps (topological quantum phase transitions), the integer nature of the particle number is preserved even if the spectral properties (poles vs. zeros) change.
• Elasticity Tetrads: For crystalline insulators, the author introduces elasticity tetrads ($E^a_\mu = \partial_\mu X^a$), treating crystal lattice deformations as a "translational gauge field." This allows the construction of topological actions analogous to gauge theories.

3. Key Contributions

A. Topological Justification of Landau Theory

• Quasiparticle Identity: The paper establishes that the topological invariant $N_\omega(p)$ plays the exact role of the quasiparticle occupation number $n(p)$ in Landau theory.
• Conservation Law: The total topological charge (sum of $N_\omega(p)$ over all $p$) is equivalent to the total number of fermions in the system. This proves that the conservation of fermion number is equivalent to the conservation of topological charge.
• Robustness: The Luttinger theorem remains valid even in "non-Landau" Fermi liquids where the Green's function pole transforms into a zero (as in Mott insulators), because the topological invariant $N_1$ remains unchanged during this transformation.

B. Flat Band Formation and Superconductivity

• Khodel-Shaginyan Mechanism: The paper analyzes the transition from a standard Fermi liquid to a flat band state under strong electron-electron interactions. In this state, the energy dispersion $\epsilon(p) = 0$ over a finite region of momentum space.
• Topological Interpretation: The formation of a flat band is described as a topological quantum phase transition where the $2\pi$ winding number of the original Fermi surface splits into $\pi$ windings around the boundaries of the flat band. This is analogous to cosmic strings bounding domain walls (Kibble-Lazarides-Shafi mechanism).
• Room-Temperature Superconductivity: The author argues that flat bands possess an extremely high density of states. Unlike BCS theory where the transition temperature $T_c$ is exponentially suppressed, in flat bands, $T_c$ scales linearly with the coupling constant. This suggests a mechanism for room-temperature superconductivity, potentially explaining experimental hints in graphite systems and hydrides.

C. Topology of Crystalline Insulators

• Generalized Luttinger Theorem: For insulators, the topological invariant $N_\omega$ corresponds to the number of fully occupied Brillouin zones. The paper derives a topological action involving elasticity tetrads and gauge fields, showing that the electron density is topologically tied to the volume of occupied bands.
• New Topological Actions: The paper introduces a hierarchy of topological actions for insulators involving Chern-Simons, Wess-Zumino, and $\Theta$-terms. These actions couple the electromagnetic gauge field $A_\mu$ with translational gauge fields (elasticity tetrads) and torsion fields.
• Strong CP Problem: The $\Theta$-term in the topological action (Eq. 21/28) is linked to the strong CP problem in Quantum Chromodynamics (QCD). The author suggests that topological quantization in momentum space could provide a solution where the CP-violating parameter remains zero due to topological stability.

4. Results

• Mathematical Proof: The derivation shows that the number of particles $N$ is strictly equal to the sum of topological invariants $\sum_p N_\omega(p)$, validating the Landau hypothesis for interacting systems.
• Stability: The Fermi surface is proven to be topologically stable as the boundary between regions where $N_\omega(p)=1$ and $N_\omega(p)=0$.
• Flat Band Topology: The flat band is identified as a composite topological object in momentum space, distinct from standard Fermi surfaces but governed by the same topological principles.
• Experimental Correlation: The theoretical framework supports experimental observations of room-temperature superconductivity in graphite interfaces and hydrides, attributing them to flat band mechanisms driven by van Hove singularities and strong correlations.

5. Significance

• Unification: The paper unifies the Landau theory of Fermi liquids, the Luttinger theorem, and the physics of topological insulators under a single framework of momentum-space topology.
• Beyond Perturbation Theory: It provides a non-perturbative explanation for the stability of Fermi liquids and the validity of sum rules in strongly correlated systems where standard perturbation theory fails.
• New Pathways for Superconductivity: By linking flat band formation to topological phase transitions, the paper offers a theoretical basis for the search for room-temperature superconductors, moving beyond the limitations of conventional BCS theory.
• Fundamental Physics: The connection between the topological invariants of condensed matter systems and the strong CP problem in high-energy physics (QCD) suggests deep, previously unexplored links between condensed matter and particle physics.

In conclusion, Volovik demonstrates that the "particle number" in quantum many-body systems is fundamentally a topological charge. This insight not only solidifies the theoretical foundation of Fermi liquid theory but also opens new avenues for understanding exotic phases of matter like flat bands and topological insulators.