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

Abstract: The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, $USp(2N)$. All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp(2N)$ symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to $USp(N) \times USp(N)$. The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is $O(2N) \bigcap USp(2N) = U(N)$.

• The paper demonstrates that 2D QBT Hamiltonians exhibit a USp(2N) symmetry distinctly different from the traditional O(2N) symmetry in Dirac systems.
• It classifies fermionic bilinears into precise irreducible representations, identifying novel mass and nematic order parameters.
• The study develops an interacting theory with two Fierz-inequivalent quartic terms, predicting new quantum phases and symmetry-breaking patterns.

Symplectic Symmetry in Two-Dimensional Quadratic-Band-Touching Hamiltonians

Overview

The paper "Symplectic symmetry of quadratic-band-touching Hamiltonians in two dimensions" (2604.21524) undertakes a comprehensive symmetry analysis of two-dimensional quadratic-band-touching (QBT) systems. It establishes that the internal low-energy symmetry of a class of QBT Hamiltonians is the unitary symplectic group $USp(2N)$, in marked contrast to the $O(2N)$ symmetry familiar from Dirac Hamiltonians with linear dispersion. The authors provide an explicit construction of the interacting theory respecting this symmetry and analyze the behavior of fermion bilinears and symmetry-breaking patterns. Notably, the study reveals that the interplay between orthogonal and symplectic symmetries on certain lattices leads to an emergent $U(N)$ symmetry, highlighting nontrivial consequences for both the classification of order parameters and the structure of interacting field theories in low-dimensional materials.

Symmetry Analysis: From Dirac to Quadratic-Band-Touching

For massless Dirac Hamiltonians in $(2+1)$d, the emergent symmetry at low energies is $O(2N)$, provided the kinetic terms are odd functions of momentum. This symmetry enhancement arises because the Dirac Hamiltonians can be rewritten in a real Majorana basis, revealing invariances under large orthogonal transformations. In QBT systems, where the single-particle terms are instead even functions of momentum, such as in Bernal-stacked bilayer graphene or on the checkerboard and Kagome lattices, the symmetry group fundamentally changes.

The paper identifies that when all terms in the single-particle Hamiltonian are even under momentum inversion, the symmetry enlarges not to an orthogonal group, but to the unitary symplectic group $USp(2N)$. This conclusion is derived by explicit construction of the fermion bilinear forms and examination of their commutation relations, confirming that the symmetry generators close under the $USp(2N)$ Lie algebra.

Classification of Fermionic Bilinears

All local fermion bilinears are classified as irreducible representations (irreps) of the $USp(2N)$ symmetry:

• The $N(2N+1)$-dimensional adjoint representation comprises the symmetry generators themselves.
• Mass terms, corresponding to Dirac mass-like bilinears, fall into two irreps: a $USp(2N)$ singlet and an irrep of dimension $O(2N)$0.
• There are additional $O(2N)$1-dimensional representations corresponding to "nematic" order parameters, which, when condensed, break both the internal $O(2N)$2 symmetry and spatial rotational symmetry.

This classification is in contrast to the Dirac case, where the mass terms and their structure under $O(2N)$3 are more constrained.

Interacting Theory at Quadratic Band Touching

Constructing the minimal interacting theory invariant under $O(2N)$4 and spatial rotations, the authors show that, unlike in the Dirac case, two independent Fierz-inequivalent four-fermion interaction terms are allowed. The general interaction Lagrangian thus consists of two marginally relevant or irrelevant couplings, which can be formally written as:

$O(2N)$5

where $O(2N)$6 describes the QBT kinetic term, and $O(2N)$7, $O(2N)$8, $O(2N)$9 are bilinear composites discussed in the text, with $U(N)$0 jointly forming the second $U(N)$1-invariant quartic term. This richer interaction structure implies new infrared fixed points and possible quantum phases beyond those expected in $U(N)$2-invariant models.

Symmetry Breaking Patterns and Physical Implications

Introducing a finite chemical potential reduces the $U(N)$3 symmetry to the subgroup $U(N)$4. The symmetry breaking pattern with a nonzero expectation value of a general mass bilinear is

$U(N)$5

for $U(N)$6 even and $U(N)$7. This yields $U(N)$8 Goldstone modes, consistent with the dimension of the coset. The identification of all mass and nematic bilinears fixes the enumeration of possible gapped insulating and superconducting order parameters, as well as those associated with rotational symmetry breaking (nematics). For instance, in the $U(N)$9 case (spin and valley-degenerate QBT as in physical Bernal-stacked bilayer graphene), the construction identifies 15 insulating and 12 superconducting mass order parameters—figures matching previously established results in the literature, but here derived by group-theoretic means.

Lattice Models and Emergent Symmetry

Expanding the effective Hamiltonian about QBT points on honeycomb or similar lattices inherently produces both even and odd terms under momentum inversion. The symmetries of these two sectors are $(2+1)$0 and $(2+1)$1, respectively. The intersection of these groups on the full lattice Hamiltonian, computed explicitly, leads to a residual symmetry group $(2+1)$2. This result has direct implications for understanding symmetry-protected degeneracy and stability in multi-valley or multi-component materials, as well as for the universality classes of possible interaction-driven quantum critical points.

Conclusion

This work establishes that the natural symmetry of two-dimensional QBT Hamiltonians is the unitary symplectic group $(2+1)$3. The explicit classification of fermionic bilinears, enumeration of possible interaction terms, and analysis of spontaneous symmetry breaking patterns provide a robust theoretical framework for exploring collective phenomena in bilayer graphene and related systems. On general lattices with both even and odd momentum terms, the composite symmetry reduces further to $(2+1)$4. The identification of $(2+1)$5 as a fundamental symmetry group in QBT physics represents a significant advance in the field-theoretic description of low-energy electron systems in two dimensions. These results inform future studies of quantum criticality, topological order, and symmetry breaking in strongly correlated or topological condensed matter systems, and suggest directions for the construction of novel effective field theories capturing non-relativistic band structures.