A pedestrian's guide to the topological phases of free fermions

This pedagogical lecture note provides a detailed many-body perspective on classifying free fermionic topological phases, covering symmetry-protected topological insulators, symmetry-free topological superconductors, and the stability of 1D phases under interactions.

The Gist

Imagine you are walking through a vast, magical forest. In this forest, the trees are not made of wood, but of tiny, invisible particles called fermions (like electrons). Usually, these particles just sit there or bounce around randomly. But sometimes, under very specific rules, they arrange themselves into a special, hidden pattern that gives the whole forest a unique "personality."

This paper is a guidebook for understanding these special patterns, which physicists call Topological Phases. The author, Frank Schindler, wants to explain this complex math using simple logic, like a tour guide showing you the sights without getting bogged down in the technical blueprints.

Here is the story of the forest, broken down into simple concepts.

1. What is a "Topological Phase"? (The Knot vs. The String)

Imagine you have a piece of string.

• Trivial Phase: If the string is just lying flat on the ground, you can pick it up, shake it, and lay it down again. It's "trivial."
• Topological Phase: Now, imagine that string is tied into a complex knot. You can pull and stretch the string all you want, but you cannot untie the knot without cutting the string. The "knot" is a property of the whole shape, not just a single point.

In the quantum world, electrons can form these "knots" in their collective behavior.

• SPT (Symmetry Protected Topological) Phases: These are knots that only stay tied as long as you follow a specific rule (a "symmetry"). If you break the rule (like turning off a magnetic field), the knot unties, and the material becomes normal.
• Topological Superconductors: These are knots that stay tied even if you break the rules. They are incredibly robust.

2. The Rules of the Game: "Free Fermions"

The author starts by looking at a simplified version of the forest where the particles don't talk to each other (no "interactions"). They only follow the Pauli Exclusion Principle: "No two electrons can sit in the same seat." This rule alone is enough to create some interesting patterns, even without the particles chatting.

The guidebook explores these patterns in different "dimensions" (sizes of the forest):

• 0D (A Single Dot): Imagine a single parking spot. The only thing that matters is: Is the spot empty or full? This is a simple "Yes/No" or "1/0" state.
• 1D (A Long Hallway): Imagine a long line of parking spots.
  • With a Rule (Charge Conservation): If we promise to keep the total number of cars constant, we find that in a 1D hallway, there are no special knots. Everything can be smoothly untangled.
  • Without Rules (No Symmetry): If we drop the rule about keeping the car count constant, we find a special knot called the Kitaev Chain.
• 2D (A Flat Field): Here, the particles can flow in circles. If you have a "Chern Insulator," the electrons flow like a river that never stops, creating a special current on the edges. This is a very strong knot that cannot be untied, even if you break the rules.
• 3D (A Cube): Surprisingly, in 3D, the "free" particles can't make these special knots again. The pattern resets.

The Pattern: The guidebook reveals a repeating cycle (like a clock):

• 0D: Special (Z)
• 1D: Nothing special (Free particles)
• 2D: Special (Z)
• 3D: Nothing special
• It repeats every two dimensions.

3. The Magic Trick: Majorana Fermions

To understand the 1D "Kitaev Chain," the author introduces a magical character: the Majorana Fermion.

Think of a normal electron as a pair of socks (left and right). A Majorana is a sock that is its own pair. It's half a particle.

• In the Kitaev Chain, these "half-particles" hide at the very ends of the hallway (the edges).
• The Bulk-Boundary Correspondence: The "bulk" (the middle of the hallway) is boring and empty. But the "boundary" (the ends) is where the magic happens. The existence of these "half-particles" at the ends proves that the middle of the hallway is in a special topological state.
• The Result: If you have one Kitaev chain, you have two "half-particles" at the ends. They form a "ghost" particle that can be either there or not. This creates a 2-fold degeneracy (two possible ground states). It's like having a light switch that is stuck halfway between On and Off.

4. Adding a New Rule: Time-Reversal Symmetry

The author then asks: "What if we add a rule that time runs backwards?" (Spinless Time-Reversal).

• Without this rule: We only had 2 states (On/Off).
• With this rule: The "half-particles" at the ends become even more stubborn. You can't just pair them up and cancel them out easily.
• Now, instead of just 2 states, you can have any number of these chains stacked together. If you have 1 chain, it's special. If you have 2, they cancel out (become trivial). If you have 3, it's special again.
• This creates an Infinite (Z) classification. You can have 1, 2, 3, 4... infinite types of these topological forests.

5. The Plot Twist: What if the Particles Talk? (Interactions)

So far, we assumed the particles didn't talk to each other. But in real life, electrons do interact. The author asks: "Do these magical knots survive if the particles start chatting?"

He tests this by stacking more and more Kitaev chains together and seeing if the "chatting" (interactions) can untie the knots.

• 1 Chain: Still safe. The knot holds.
• 2 Chains: Still safe (Time-reversal protects them).
• 3, 4, 5, 6, 7 Chains: Still safe! The interactions aren't strong enough to untie the complex knots formed by these many chains.
• 8 Chains: BAM! Here is the surprise. When you stack 8 chains, the interactions can finally untie the knot. The particles can rearrange themselves in a way that cancels out the topological protection.

The Final Lesson:

• Without interactions, the classification is Infinite (Z).
• With interactions, the classification shrinks to 8 (Z8).
• It's like a clock with 8 hours. If you have 1 to 7 chains, you are in a special state. But if you have 8, you are back to zero (trivial).

Summary Analogy

Imagine you are trying to tie a knot in a rope.

1. Free Fermions: You can tie knots in 2D and 0D, but not 1D or 3D.
2. No Symmetry: In 1D, you can tie a knot that splits the rope into two "ghost" ends.
3. Time-Reversal Symmetry: This makes the knot harder to untie. You can stack 1, 2, 3... infinite ropes, and they stay knotted.
4. Interactions (The Real World): If the rope fibers start sticking to each other, the knot becomes easier to untie. It turns out that if you stack 8 ropes, the fibers stick together so well that the knot disappears completely.

The Takeaway:
This paper is a "pedestrian's guide" (a walking tour) showing us that the quantum world has a hidden rhythm. Even though the math is complex, the underlying logic is a beautiful, repeating cycle of stability and instability, governed by how many "layers" of particles you stack and how they interact with each other. The number 8 is the magic number where the topological protection finally breaks down.

Technical Summary

1. Problem Statement

The paper addresses the classification of Symmetry-Protected Topological (SPT) phases in free fermion systems across various spatial dimensions and symmetry classes. Specifically, it seeks to:

• Systematically derive the topological classification of non-interacting fermions protected by U(1) charge conservation (topological insulators).
• Determine the classification of fermionic phases without U(1) symmetry (topological superconductors), focusing on fermion parity and spinless time-reversal symmetry.
• Investigate the stability of these free-fermion classifications when electron-electron interactions are introduced, specifically in 1D systems.

The author aims to provide a "pedestrian" (highly pedagogical, step-by-step) derivation of these results, moving from simple 0D cases to higher dimensions and finally to interacting systems, avoiding heavy quantum field theory machinery in favor of explicit Hamiltonian constructions and perturbation theory.

2. Methodology

The paper employs a "bottom-up" approach, utilizing the following technical tools:

• Hamiltonian Diagonalization: Starting with general quadratic fermion Hamiltonians, the author diagonalizes them into quasiparticle or Majorana bases to identify ground states and energy gaps.
• Band Theory and Continuum Limits: For systems with translational symmetry, the author uses Bloch Hamiltonians $H(k)$. To classify phases, the method involves analyzing gap-closing points in the Brillouin zone. The low-energy physics near these points is described by Dirac equations ($h(q) = \gamma \cdot q + m$).
• Clifford Algebra and Homotopy: The classification is reduced to determining the space of possible "mass terms" (gap-opening matrices) that satisfy specific symmetry constraints (anti-commutation with kinetic terms). The distinct phases correspond to the connected components of the manifold of these mass matrices, often related to homotopy groups (e.g., $\pi_0$ of orthogonal or unitary groups).
• Bulk-Boundary Correspondence: The existence of topological phases is verified by constructing models with Open Boundary Conditions (OBC) and counting the degeneracy of the ground state or the presence of Majorana zero modes localized at the edges.
• Perturbative Interaction Analysis: To test stability, the author introduces local interaction terms to the edge modes of 1D chains. By checking if these interactions can lift the ground state degeneracy while preserving symmetries (fermion parity and time-reversal), the author determines if the free-fermion classification holds or reduces.

3. Key Contributions and Results

A. Free Fermions with U(1) Symmetry (Charge Conservation)

The paper derives the classification of topological insulators protected by U(1) symmetry:

• 0D: Classified by the integer number of particles ($\mathbb{Z}$).
• 1D: The classification is trivial. The author demonstrates that any 1D band structure with a gap can be smoothly deformed to a trivial atomic limit without breaking U(1) symmetry, provided locality is maintained.
• 2D: Classified by the Chern number ($\mathbb{Z}$). The author derives this via the 2D Dirac equation, showing that the mass term space allows for distinct topological sectors corresponding to the number of chiral edge modes.
• 3D and Higher: The classification returns to trivial for 3D.
• Pattern: The results follow a Bott periodicity pattern of period 2: $\mathbb{Z}$ in even dimensions, trivial in odd dimensions.

B. Free Fermions without Symmetry (Topological Superconductors)

By relaxing U(1) symmetry (allowing pairing terms $\Delta$), the system is described by Majorana fermions.

• Symmetry: The only remaining symmetry is Fermion Parity ($P = (-1)^N$), which is intrinsic to all local fermion systems.
• 0D: Classified by $\mathbb{Z}_2$ (even vs. odd fermion parity).
• 1D: The classification is $\mathbb{Z}_2$. The author explicitly constructs the Kitaev chain, showing that the non-trivial phase possesses two degenerate ground states in OBC due to unpaired Majorana zero modes at the ends. Stacking two chains ($1+1=0$) allows the edge modes to couple and gap out, confirming the $\mathbb{Z}_2$ nature.
• 2D/3D: Briefly noted as $\mathbb{Z}$ (2D) and trivial (3D), shifting the periodicity to mod 8 (Bott periodicity for real classes).

C. Free Fermions with Spinless Time-Reversal Symmetry (TRS)

The paper introduces a spinless time-reversal operator $T$ where $T^2 = +1$.

• Effect on 1D: The constraint $T \gamma_{odd} T^{-1} = \gamma_{odd}$ and $T \gamma_{even} T^{-1} = -\gamma_{even}$ forbids the coupling of Majorana zero modes at the same edge for a single chain.
• Result: The classification upgrades from $\mathbb{Z}_2$ to $\mathbb{Z}$. An integer number $n$ of Kitaev chains results in $n$ protected Majorana zero modes at each edge. The author provides a rigorous proof using the anti-commutation of the time-reversal operator with the mass matrix to show that an odd number of zero modes cannot be gapped out without closing the bulk gap.

D. Stability to Interactions in 1D

The final section investigates how interactions modify the free-fermion classifications.

• Case 1: No Symmetry (Fermion Parity only)
  • Result: The $\mathbb{Z}_2$ classification remains stable ($\mathbb{Z}_2 \to \mathbb{Z}_2$).
  • Reasoning: Local interactions cannot lift the degeneracy between the even and odd parity ground states of a single Kitaev chain because they must preserve fermion parity.
• Case 2: Spinless Time-Reversal Symmetry
  • Result: The $\mathbb{Z}$ classification reduces to $\mathbb{Z}_8$ ($\mathbb{Z} \to \mathbb{Z}_8$).
  • Mechanism:
    • For $n < 8$, interactions cannot fully gap the edge modes while respecting TRS and locality.
    • For $n=8$, the author constructs a specific local interaction term involving 4 Majorana operators per edge (e.g., $\zeta_1\zeta_2\zeta_3\zeta_4 + \dots$) that commutes with TRS and splits the 16-fold degeneracy into a unique gapped ground state.
    • This demonstrates that 8 copies of the topological phase are trivial in the presence of interactions.

4. Significance

• Pedagogical Clarity: The paper fills a gap in the literature by providing explicit, non-abstract derivations of topological classifications. It avoids high-level category theory or advanced field theory, instead using concrete matrix algebra and perturbation theory to explain why the classifications are what they are.
• Clarification of SPT vs. Topological Order: It clearly distinguishes between SPT phases (which require symmetry and are short-range entangled) and intrinsic topological order (long-range entangled).
• Interaction Effects: The derivation of the $\mathbb{Z}_8$ reduction for 1D topological superconductors with TRS is a crucial result. It highlights that free-fermion classifications can be fragile; interactions can "trivialize" higher-order topological phases that appear stable in the non-interacting limit.
• Foundation for Further Study: By establishing the "pedestrian" groundwork, the notes serve as an accessible entry point for understanding the periodic table of topological insulators and superconductors, the role of Bott periodicity, and the phenomenon of symmetry fractionalization at boundaries.

In summary, Schindler's notes provide a rigorous yet accessible roadmap through the landscape of free fermion topological phases, demonstrating how symmetry and dimensionality dictate classification, and how interactions fundamentally alter the stability of these phases in one dimension.