Symplectic symmetry of quadratic-band-touching Hamiltonians in two dimensions

This paper identifies the internal low-energy symmetry of two-dimensional quadratic-band-touching Hamiltonians as the unitary symplectic group $USp(2N)$, constructs the corresponding rotationally invariant interacting theory, and demonstrates that for lattice systems like honeycomb, this symmetry reduces to the unitary group $U(N)$ through the intersection of symplectic and orthogonal symmetries.

The Gist

Imagine you are a detective trying to understand the hidden rules that govern a bustling city of tiny particles called electrons. In most cities (materials), these electrons move in a very specific, predictable way, like cars on a highway. But in some special materials, like a specific type of graphene, the electrons meet at a "traffic jam" point where they can move in strange, quadratic ways (their speed depends on the square of their momentum, not just linearly). This is called a Quadratic Band Touching (QBT) point.

The authors of this paper, Igor Herbut and Samson Ling, discovered a hidden "secret language" or symmetry that these electrons speak when they are at this special traffic jam.

Here is the story of their discovery, broken down into simple concepts:

1. The Old Rulebook: The "Orthogonal" Group

For a long time, physicists knew that when electrons move in a straight line (like in standard graphene), they follow a set of rules called O(2N) symmetry.

• The Analogy: Imagine a group of dancers (the electrons). In the "Old Rulebook," the dancers can swap places with anyone else, spin around, and change partners in a way that looks like a perfect, rigid geometric shape (an orthogonal group). It's like a square dance where everyone has to stay in a specific grid.

2. The New Discovery: The "Symplectic" Group

The authors realized that when the electrons are at the Quadratic Band Touching point (where their movement is curved, not straight), the rules change completely. The symmetry isn't the square dance anymore; it's a Unitary Symplectic Group (USp).

• The Analogy: Think of the dancers now as a pair of synchronized swimmers. They still move together, but they have a special "handshake" rule. If one swimmer moves left, the other must move right in a very specific, locked-in way to keep the water (the system) balanced. This "locked-in" relationship is the Symplectic symmetry. It's a more complex, tighter bond than the old square dance.

3. The "Two-Handshake" Interaction

In the old world (linear movement), if the electrons wanted to interact (like bumping into each other), there was essentially only one way for them to do it while respecting the rules.

• The New Twist: In this new Symplectic world, the authors found there are two distinct ways the electrons can interact and still respect the rules.
• The Metaphor: Imagine two people trying to shake hands. In the old world, there was only one way to shake hands (right hand to right hand). In this new world, they can shake hands in two different ways (maybe right-to-right or left-to-left) and still be considered "polite" (symmetric). This gives the material more flexibility to change its state.

4. Breaking the Rules: The "Spontaneous Break"

Sometimes, these electrons decide to break the symmetry. They might all suddenly decide to line up in a specific direction, creating a new state of matter (like a magnet or a superconductor).

• The Outcome: The authors showed that when this happens, the big group of dancers (USp) splits into two smaller, independent groups (USp × USp).
• The Metaphor: Imagine a large choir singing in perfect harmony. Suddenly, they split into two separate choirs. The left side sings one song, and the right side sings a different song. They are no longer one big group; they have broken into two smaller, self-contained groups. This is what happens when the material becomes a superconductor or an insulator.

5. The "Honeycomb" Compromise

The paper also looks at real-world materials like honeycomb lattices (the structure of graphene). In these materials, the electrons experience both the old straight-line rules and the new curved rules at the same time.

• The Overlap: When you mix the "Old Rulebook" (Orthogonal) and the "New Rulebook" (Symplectic), they don't cancel each other out. Instead, they overlap to create a third set of rules: the U(N) symmetry.
• The Metaphor: Imagine you have two different languages. If you try to speak both at once, you don't get gibberish; you accidentally invent a new, simpler language that combines the best parts of both. This is the "overlap" the authors found. It turns out that for real materials like graphene, the ultimate symmetry is this new, simpler language.

Why Does This Matter?

This isn't just abstract math. Understanding these hidden symmetries helps scientists predict how these materials will behave.

• Predicting New States: It tells us exactly what kinds of "ordered states" (like superconductors or magnets) can exist in these materials.
• Designing Materials: If we want to build a new type of computer chip or a super-efficient battery, knowing these "dance rules" helps us engineer materials that can switch between states easily.

In Summary:
The paper is like finding a new set of traffic laws for a specific type of city. The authors discovered that when electrons move in a "curved" way, they follow a complex, synchronized dance (Symplectic symmetry) that allows for two types of interactions. When they break this dance, they split into two groups. And when you mix this with the old "straight-line" rules found in real materials, you get a unique, simplified set of rules that governs the material's behavior. It's a fundamental discovery about how the universe organizes itself at the smallest scales.

Technical Summary

1. Problem Statement

The paper addresses the internal low-energy symmetries of massless fermionic systems in two spatial dimensions ($2D$).

• Context: In standard $2+1$ dimensional Dirac materials (like graphene), the low-energy excitations are linear in momentum. It is a known mathematical fact that the massless Dirac Hamiltonian possesses an enlarged internal symmetry group, $O(2N)$, where $N$ is the number of fermion flavors (valleys/spins), provided the dispersion is linear (odd under momentum reversal).
• Gap: Many $2D$ systems, such as Bernal-stacked bilayer graphene, checkerboard lattices, and Kagome lattices, exhibit Quadratic Band Touching (QBT) points. In these systems, the low-energy dispersion is quadratic (even under momentum reversal). The authors investigate whether an analogous enlarged symmetry exists for these even-parity Hamiltonians, distinct from the $O(2N)$ symmetry of Dirac systems.

2. Methodology

The authors employ a field-theoretic approach using Majorana fermion representations to analyze the symmetry structure of the single-particle Hamiltonian.

• Formalism: They define a general low-energy action for $N$ two-component complex fermions in $2D$. The non-interacting Lagrangian is expressed in terms of Pauli matrices acting on the spinor space and operators $F_i(\hat{p})$ representing the momentum dependence.
• Parity Analysis: The core of the method involves classifying the operators $F_i(\hat{p})$ based on their behavior under momentum reversal ($\hat{p} \to -\hat{p}$):
  • Odd functions: Correspond to Dirac-like (linear) dispersion.
  • Even functions: Correspond to QBT (quadratic) dispersion.
• Majorana Transformation: The complex fermion field $\Psi$ is decomposed into real Majorana components ($\chi_1, \chi_2$). The authors analyze the transpose properties of the resulting matrices in the Lagrangian to identify the invariant transformation groups.
• Group Theory: They utilize Clifford algebra representations and the properties of symplectic and orthogonal groups to identify the generators of the symmetry. They specifically look for conditions where the generators satisfy the defining relations of the Unitary Symplectic group, $USp(2N)$.
• Interaction Construction: They construct the most general quartic interaction terms invariant under the identified symmetry and spatial rotations ($O(2)$).
• Symmetry Breaking Analysis: They analyze the effects of finite chemical potential and spontaneous symmetry breaking (mass generation) on the symmetry group.

3. Key Contributions and Results

A. Emergence of $USp(2N)$ Symmetry

The primary result is the identification of the Unitary Symplectic group, $USp(2N)$, as the internal symmetry of massless QBT Hamiltonians in $2D$.

• Mechanism: When the dispersion functions $F_i(\hat{p})$ are even under momentum reversal, the Majorana representation of the Lagrangian involves matrices that are antisymmetric and imaginary.
• Generator Count: The number of generators for this symmetry is $N(2N+1)$, which matches the dimension of the Lie algebra of $USp(2N)$. This contrasts with the $O(2N)$ symmetry of Dirac systems, which has $N(2N-1)$ generators.
• Irreducible Representations (Irreps): The authors classify all fermion bilinears ($\chi^T Y \chi$) into irreps of $USp(2N)$:
  1. Adjoint: Dimension $N(2N+1)$ (Generators of the symmetry).
  2. Mass Terms: Dimension $N(2N-1)-1$. These correspond to gapped states (insulators or superconductors).
  3. Nematic Terms: Two sets of dimension $N(2N-1)$. These break spatial rotational symmetry ($O(2)$) while preserving internal symmetry.

B. Interacting Theory

The authors construct the minimal interacting field theory respecting $USp(2N)$ and spatial $O(2)$ symmetry.

• Interaction Terms: Unlike the Dirac case (where a unique $O(2N)$-invariant interaction exists, equivalent to the Gross-Neveu model), the QBT case allows two independent interaction terms that respect the full symmetry.
  • One term involves the singlet mass channel.
  • The second term is a sum of two Fierz-independent quartic terms that combine to form a $USp(2N)$ invariant.
• Renormalization Group (RG) Implications: If these couplings are infrared-relevant, the system can flow to a ground state where the symmetry is either preserved or spontaneously broken to $USp(N) \times USp(N)$.

C. Symmetry Breaking Patterns

• Chemical Potential: A finite chemical potential breaks $USp(2N)$ down to $U(N)$, recovering the "original" flavor symmetry.
• Mass Generation:
  • If a mass term (other than the singlet) acquires a vacuum expectation value, the symmetry breaks as $USp(2N) \to USp(N) \times USp(N)$, resulting in $N^2$ Goldstone bosons.
  • If a nematic order parameter develops, both internal and spatial rotational symmetries are broken.

D. Overlap of Symmetries in Lattices (Honeycomb Example)

The paper addresses systems like graphene where the Hamiltonian contains both odd (linear) and even (quadratic) terms in the momentum expansion.

• Result: The symmetry of the full Hamiltonian is the intersection (overlap) of the $O(2N)$ symmetry (from odd terms) and the $USp(2N)$ symmetry (from even terms).
• Mathematical Identity: The authors prove that this overlap is exactly the unitary group:
  $$O(2N) \cap USp(2N) = U(N)$$
  This explains why the larger symplectic or orthogonal symmetries do not survive in realistic lattice models with mixed parity terms; the symmetry reduces to the standard $U(N)$.

4. Significance

• Theoretical Novelty: This is the first demonstration that a symplectic group ($USp$) emerges as the fundamental symmetry of a quantum Hamiltonian in condensed matter physics. It establishes a rigorous parallel between Dirac fermions ($O$ symmetry) and QBT fermions ($Sp$ symmetry).
• Classification of Phases: The work provides a complete classification of possible ordered phases (insulators, superconductors, nematics) for QBT systems based on group theory. For example, in bilayer graphene ($N=4$), it predicts 27 distinct symmetry-breaking mass terms.
• Interaction Physics: By identifying two independent interaction terms, the paper suggests a richer phase diagram for QBT systems compared to Dirac systems, potentially hosting novel critical points or competing orders.
• Resolution of Lattice Symmetry: The derivation of the $U(N)$ overlap clarifies why specific lattice symmetries (like in graphene) do not exhibit the full $O(2N)$ or $USp(2N)$ symmetries despite having Dirac or QBT points, resolving a potential confusion in the literature regarding symmetry protection in lattice models.

Conclusion

Herbut and Ling establish that the internal symmetry of $2D$ quadratic-band-touching Hamiltonians is the unitary symplectic group $USp(2N)$. This symmetry dictates the structure of fermion bilinears, allows for two independent interaction channels, and leads to specific symmetry-breaking patterns. Furthermore, they demonstrate that in realistic lattice models containing both linear and quadratic terms, the symmetry reduces to the intersection $U(N)$, unifying the understanding of symmetries across different classes of 2D semimetals.