2D Dirac Semimetals: Symmetry-Protected States

• 2D Dirac semimetals are quantum materials where conduction and valence bands meet at symmetry-protected Dirac points, giving rise to gapless linear dispersion.
• Their tight-binding models incorporate strong spin–orbit coupling without gapping the spectrum, distinguishing them from graphene and enabling controlled phase transitions.
• Symmetry breaking via perturbations can trigger transitions to trivial, topological, or Weyl phases, offering a versatile platform for designing quantum devices.

A two-dimensional Dirac semimetal is a quantum electronic material in which the valence and conduction bands touch at discrete points (Dirac points) in the two-dimensional Brillouin zone, with a linear band dispersion in their vicinity. Unlike graphene, where even infinitesimal spin–orbit coupling (SOC) opens a gap and makes the system a quantum spin Hall insulator, symmetry-engineered 2D Dirac semimetals can exhibit symmetry-protected, gapless Dirac points that are robust even in the presence of strong SOC. Their stabilization and unique behavior are governed by crystalline symmetries—especially nonsymmorphic space group operations—as well as time-reversal and, in some cases, inversion symmetries. The interplay of these crystalline symmetries enforces band degeneracies at high-symmetry points in momentum space, realizes a variety of topological phase transitions, and connects 2D Dirac semimetals to the boundaries between trivial and topological insulating phases.

1. Symmetry Protection and Distinction from Graphene and 3D Semimetals

The essential characteristic of 2D Dirac semimetals discussed in this framework is the presence of symmetry-protected, gapless Dirac points located at high-symmetry points in the Brillouin zone. In standard graphene, the Dirac points at the K and K′ points are protected only in the absence of SOC; any SOC, however weak, opens a gap and the system becomes a quantum spin Hall insulator. In contrast, 2D Dirac semimetals engineered via crystal symmetry retain Dirac points that are stable against SOC due to an enforced fourfold degeneracy. This stability arises from the joint action of time-reversal symmetry, inversion symmetry, and crucially, nonsymmorphic crystal symmetries—such as glide mirrors and screw axes—which force the bands to touch at specific momenta (Young et al., 2015).

Unlike 3D Dirac semimetals—where Dirac points may originate from band inversion away from high-symmetry points and rely on different symmetry and topological structures—the 2D counterparts realize a single class dictated by space group and band filling. In 2D, it is impossible to realize an isolated, symmetry-protected single Dirac point; instead, Dirac points must occur in pairs or higher multiples. This constraint stems from the crystalline symmetry algebra and the structure of the Brillouin zone.

2. Role of Spin–Orbit Coupling and Symmetry Constraints

The presence of SOC typically acts as a band gap opening perturbation. However, in the symmetry-protected 2D Dirac semimetal, SOC is incorporated into the tight-binding model in such a way that it preserves the gapless Dirac points. This is achieved by combining time-reversal, inversion, and nonsymmorphic symmetries (fractional translations combined with rotation or mirror operations) to enforce a Kramers degeneracy and further enforce a fourfold degeneracy at special points (for instance, the M point and the X points on a square lattice).

The model Hamiltonian that realizes this features the following structure:

$$H = 2t\,\tau_x \cos(k_x/2) \cos(k_y/2) + t_2(\cos k_x + \cos k_y) + t^{SO} \tau_z \left[\sigma_y \sin k_x - \sigma_x \sin k_y \right]$$

Here $\tau$ and $\sigma$ are Pauli matrices acting on sublattice and spin, respectively; $t$ is the nearest-neighbor hopping, $t_2$ breaks particle-hole symmetry, and $t^{SO}$ is the spin–orbit coupling strength. The structure of these terms, and the symmetry of the Hamiltonian, guarantee that Dirac points survive at high-symmetry momenta even with strong SOC (Young et al., 2015).

3. Nonsymmorphic Symmetries and Proof of Dirac Point Protection

Nonsymmorphic space group operations combine point group elements (such as mirrors or twofold rotations) with fractional lattice translations. The presence of such symmetries strongly constrains Bloch eigenstates along high-symmetry lines or at time-reversal invariant momenta, resulting in “band sticking”—i.e., the enforcement of connectivity and degeneracy between bands. For example, an operator of the form $\{g | t\}$—with $g$ a mirror or rotation and $t$ a half-lattice translation—implies, for Bloch wavevectors on the invariant subspace, that the Bloch eigenvalues must exchange as $k$ shifts by a reciprocal lattice vector. This “switching” mandates that bands cannot be disentangled at these special $k$ points, producing Dirac points at filling $4n+2$.

As a direct consequence, at this critical filling, the system’s bands are forced into accidental degeneracy (Dirac points) at high-symmetry points, creating a semimetal at the direct boundary between trivial and topological insulator phases. Breaking these nonsymmorphic symmetries gaps out the Dirac points, positioning the system into either trivial or quantum spin Hall (topological) insulator regimes. The Dirac semimetal phase in this construction always sits at a symmetry-dictated phase boundary.

4. Detailed Tight-Binding Model and Phase Tuning

The minimal tight-binding model employs a square lattice doubled into a $\sqrt{2}\times\sqrt{2}$ unit cell, with two sublattices “crinkled” out of the plane to implement nonsymmorphic symmetries. The most general Hamiltonian with these degrees of freedom, and additional symmetry-allowed perturbations, can be written as: $$\begin{aligned} H & = 2t\,\tau_x\cos(k_x/2)\cos(k_y/2) + t_2(\cos k_x + \cos k_y) \ & \quad + t^{SO}\tau_z(\sigma_y \sin k_x - \sigma_x \sin k_y) \ V_1 & = \Delta_1 \sin(k_x/2)\sin(k_y/2)\tau_x \ V_2 & = \Delta_2 \cos(k_x/2)\sin(k_y/2)\tau_y \ V_4 & = [m_1\,\sin((k_x+k_y)/2) + m_2\,\sin((k_x-k_y)/2)]\,\tau_y \end{aligned}$$ Here, $V_1$ and $V_2$ selectively break certain crystalline symmetries and so tune the system among three distinct Dirac semimetal phases—one with two equivalent Dirac points, another with two inequivalent Dirac points, and a third with three Dirac points. Notably, symmetry constraints prohibit the presence of a single symmetry-protected Dirac point in two dimensions.

Phase transitions can be triggered by externally breaking these protecting symmetries. The $V_4$ perturbation, for instance, breaks inversion symmetry and “selects” the sign of the gap in different regions of the Brillouin zone. When $m_1 > m_2$, the gapped phase is topological (quantum spin Hall insulator); for $m_1 < m_2$, it is trivial, with the Dirac semimetal at the phase boundary.

5. Symmetry Breaking, Weyl Points, and Line Nodes

The response of the 2D Dirac semimetal to symmetry-breaking perturbations is rich. If inversion symmetry is broken but a nonsymmorphic glide mirror is retained, the Dirac points are replaced by either Weyl (twofold) points or entire nodal lines, depending on which residual symmetries are preserved. For example, in the presence of a glide symmetry, a circular nodal line can form, centered at the former Dirac point momentum.

Consequently, by tuning external controls (strain, electric field, structural distortions), the 2D Dirac semimetal can be driven into a variety of semimetallic (Weyl, nodal line) or insulating (trivial, topological) phases. This controllability makes such systems attractive for realizing symmetry-tuned quantum phase transitions and for designing quantum devices with switchable transport regimes.

6. Physical Consequences and Applications

Symmetry-protected 2D Dirac semimetals, as constructed via nonsymmorphic space group operations, provide a fermionic system at the critical boundary of topological and trivial phases. Their robustness to SOC distinguishes them sharply from graphene and points to new platforms for studying relativistic electron dynamics and topological quantum effects in low-dimensional systems.

Key applications and implications include:

• Symmetry-enforced band touching leads to unique magneto-transport and optical responses, such as high mobility and protected edge states when in proximity to gapped (topological) phases.
• Breaking certain symmetries yields tunable edge or Fermi arc states, with potential for device applications in topological electronics and valleytronics.
• Being at a phase boundary, these systems are sensitive to symmetry-lowering perturbations, enabling external electric field or strain control over the topological character of the ground state.
• The foundational connection between nonsymmorphic symmetry, filling, and topological phase boundaries provides a paradigm for engineering further classes of quantum materials with tunable band topology (Young et al., 2015).

Feature	2D Graphene	2D Dirac Semimetals (nonsymmorphic)	3D Dirac Semimetal
SOC Effect	Opens small gap	Dirac points remain gapless	May open gap or deform
Protection Mechanism	Point group + 𝒯, inversion	Nonsymmorphic space group + 𝒯, inversion	Band inversion/topology
Number of protected Dirac points	2 (K, K′)	≥2 (cannot have only one)	Various
Response to symmetry breaking	Gaps open	Gaps, nodal lines, Weyl points possible	Varied

This framework, and especially the explicit tight-binding models and symmetry-based arguments, establish the fundamental criteria for the realization and manipulation of robust two-dimensional Dirac semimetals and pave the way for exploring new quantum phase transitions and device functionalities anchored in symmetry-enforced band topology.