Valley-Gauge Symmetry Breaking

• Valley-gauge-symmetry breaking is a phenomenon where the valley degree of freedom or gauge symmetry (or both) is disrupted, resulting in distinctive physical states in particle physics and 2D materials.
• It leverages mechanisms such as the Coleman–Weinberg potential in field theory and explicit symmetry perturbations in 2D systems to control electronic and gauge properties.
• Practical realizations include anomalous valley Hall effects in TMDC heterostructures and spin-selective valley symmetry breaking in twisted graphene, paving the way for advances in valleytronics and superconductivity.

Valley-gauge-symmetry breaking refers to a set of symmetry-lowering mechanisms whereby either the internal "valley" degree of freedom or a continuous gauge symmetry (or both) are broken—either spontaneously or via explicit perturbations—giving rise to distinctive physical phenomena. This concept pervades multiple contexts, including particle physics (where "valley" denotes flat directions in scalar potentials giving rise to gauge symmetry breaking) and condensed matter, especially 2D materials, where valley refers to distinct extrema in momentum space that can be manipulated by symmetry engineering, interfacial effects, or interaction-driven instabilities.

1. Valley, Gauge, and the Notion of Symmetry Breaking

The term "valley" has two prominent technical meanings: in gauge theory, it refers to flat directions in a scalar potential (flat directions are sometimes called "valleys"), and in solid-state/2D materials, it describes inequivalent momentum-space points (such as K and K′ in hexagonal lattices) with nearly degenerate energy. Gauge symmetry refers to the continuous group invariance of a quantum field theory, such as SU($N_c$). Valley-gauge-symmetry breaking thus encompasses phenomena where either the valley degree of freedom, the gauge invariance, or their interplay is disrupted.

The distinction is essential: in particle physics, spontaneous symmetry breaking via a valley can induce "gauge symmetry breaking," whereas in materials science, "valley symmetry breaking" often involves discrete point-group operations or the breaking of time-reversal and mirror symmetries that relate valleys in reciprocal space (Appelquist et al., 2021, Zhu et al., 2024, Wrzos et al., 27 Nov 2025, Gonzalez et al., 2021).

2. Spontaneous Valley-Induced Gauge Symmetry Breaking in Field Theory

The archetype in field theory is a gauge theory (e.g., SU($N_c$)) coupled to scalars and fermions in the large-$N_c$ limit. The scalar potential may possess flat directions—valleys—along which the potential is degenerate at tree level. Explicitly, for the Appelquist–Street–Wijewardhana (ASW) model, the tree-level scalar potential is

$$V_\text{tree}(\phi) = \hat{h}\,\text{Tr}(\phi^\dagger \phi \phi^\dagger \phi) + \tilde{f} [\text{Tr} (\phi^\dagger \phi)]^2,$$

which becomes flat in certain directions when $h+f=0$ after RG flow (Appelquist et al., 2021). The valley direction is parametrized by a real field $v$: $\phi_{i\alpha}=v\delta_{i\alpha}$. Radiative corrections (Coleman–Weinberg mechanism) generate a one-loop effective potential along this valley, of the form $$V_\text{eff}(v) = A v^4 \ln(v^2/\Lambda_\text{IR}^2)$$, leading to a spontaneous nonzero vacuum expectation value at $v^2 = e^{-1/2}\Lambda_\text{IR}^2$.

The resulting VEV

$$\langle \phi_{i\alpha} \rangle = v\,\delta_{i\alpha}$$

breaks SU($N_c$) down to SU($N_c-N_s$) × SU($N_s$)$_\text{diag}$ × U(1), giving mass to $2N_cN_s-N_s^2$ gauge bosons. Among scalars, most become massive, while one flat direction (the dilaton) is lifted by quantum effects and becomes parametrically light: $$m_\text{dil}^2 = 4(4\pi v)^2[8x_sh_*^2 + (3/4)(1+x_s)\lambda_*^2]/N_c,$$ where $h_*,\lambda_*$ are weak-coupling IR fixed-point values (Appelquist et al., 2021).

3. Valley Symmetry Breaking in 2D Materials: Role of Crystal Symmetries

In 2D hexagonal lattice materials (e.g., TMDCs, graphene, CrOBr, FeCl$_2$, AgCrP$_2$Se$_6$), valleys refer to inequivalent $K$ and $K'$ points in the Brillouin zone. Valley-gauge-symmetry breaking here involves breaking crystal symmetries that "enforce" valley degeneracy or make the net Berry curvature cancel. Any symmetry—such as inversion $P$, time-reversal $T$, twofold rotations $C_2$, or vertical mirrors—that relates $K\leftrightarrow K'$ or flips the sign of the Berry curvature $\Omega(\mathbf{k})$ precludes a finite valley-polarized response.

Removing these enforced symmetries (by magnetic order, interfacial stacking, or electric fields) yields: $$\Omega_n(K) = -\Omega_n(K'),\qquad\Omega_n(K)\neq 0,$$ and allows a nonzero valley Hall conductivity

$$\sigma_{xy}^v = \frac{e^2}{\hbar}\sum_n\int_\mathrm{BZ} f_n(\mathbf{k})\Omega_n(\mathbf{k})\tau_k d^2k,$$

where $\tau_k = \pm 1$ is the valley index. This behavior underpins the anomalous valley Hall effect (AVHE) (Zhu et al., 2024).

4. Valley-Gauge Potentials and Interfacial Engineering

In van der Waals heterostructures—such as MnPS$_3$/TMDC stacks—valley degeneracy is lifted via a combination of broken inversion and mirror symmetries, magnetic order (antiferromagnetic or ferromagnetic exchange), and layer registry. The low-energy model incorporates valley (gauge) potentials: $$H_\text{int}(k; \tau, s_z) = H_0(k) + \Delta_0\tau I_2 + \Delta_1\tau s_z \sigma_z + \Delta_2 s_z I_2,$$ with $\tau = \pm 1$ (valley), $s_z$ (spin), embedded in the TMDC Dirac-like basis. The parameters $\Delta_0$ and $\Delta_1$ derive from asymmetric interlayer hopping and SOC-assisted coupling; $\Delta_2$ is a proximity-induced spin splitting (Wrzos et al., 27 Nov 2025).

For different stacking registries (S1, S2) and magnetic orientations, the valley splitting can be tuned continuously. In S2 stacking with broken $C_3$ and $\sigma_h$, DFT calculations yield conduction-band splittings up to $9$–$15$ meV and valence-band splittings up to $22$–$25$ meV, with further tunability via spin orientation and interlayer twist (Wrzos et al., 27 Nov 2025).

System	$\Delta_v^c$ (S1/S2) [meV]	$\Delta_v^v$ (S1/S2) [meV]
MnPS$_3$/MoS$_2$	1 / 9	2 / 14
MnPS$_3$/WS$_2$	2 / 12	3 / 18
MnPS$_3$/MoSe$_2$	1 / 11	2 / 16
MnPS$_3$/WSe$_2$	2 / 15	4 / 22

5. Spin-Selective Valley Symmetry Breaking and Emergent Gauge Fields in Graphene Moiré Systems

In twisted trilayer graphene (TTG), many-body interactions and band structure combine to induce spin-selective valley symmetry breaking. The primary mechanism is a valley-polarization order parameter for each spin, $\Delta_v^{(\sigma)}=P_-^{(\sigma)}$, constructed from imaginary next-nearest-neighbor flux loops. The system minimizes its energy by locking $\Delta_v^{(\uparrow)}=+\lvert\Delta_v\rvert$, $\Delta_v^{(\downarrow)}=-\lvert\Delta_v\rvert$: valley symmetry breaking is opposite for the two spin channels (Gonzalez et al., 2021).

This "spin-valley locking" leads to breakdown of $C_6\to C_3$ band symmetry, highly anisotropic (triangular) Fermi surfaces, and strong Kohn–Luttinger pairing instabilities. Moreover, the VS-breaking term generates an emergent Kane–Mele spin-orbit coupling ($\lambda_I \sim 1$ meV), protecting the intervalley singlet superconducting state against in-plane magnetic fields, resulting in Ising superconductivity (Gonzalez et al., 2021).

6. Representative Realizations and Control Strategies

Explicit symmetry breaking for valley manipulation and valley-gauge-symmetry breaking has been demonstrated and theoretically validated in a variety of systems:

• 2D Hexagonal Materials for AVHE: Synergy between space-group symmetries and external fields or magnetism enables toggling between valley-degenerate and valley-polarized phases in AgCrP$_2$Se$_6$, CrOBr, FeCl$_2$, and bilayer TcGeSe$_3$. The symmetry-based approach generalizes to arbitrary Brillouin zone points and materials design (Zhu et al., 2024).
• Heterostructures (MnPS$_3$/TMDCs): Valley splitting is highly sensitive to stacking geometry, interlayer twist, strain, and MnPS$_3$ spin orientation. S2 registry and out-of-plane AFM vectors maximize the effect, enabling valleytronic functionalities without the need for net magnetization or strong SOC (Wrzos et al., 27 Nov 2025).
• Twisted Graphene Systems: TTG under appropriate doping and screening produces a spin-selective valley-symmetry-broken ground state, enabling a tunable interplay between valley, spin, and moiré potential (Gonzalez et al., 2021).

The control of valley-gauge-symmetry breaking is thus enabled by (1) identification and removal of point-group and antiunitary symmetries that relate valleys and (2) engineering perturbations—magnetic, electric, or interfacial—to lift degeneracies or produce valley-dependent gauge potentials.

7. Theoretical Significance and Broader Connections

Valley-gauge-symmetry breaking provides a unifying framework for understanding a variety of condensed-matter and high-energy phenomena:

• In quantum field theory, valleys in scalar potentials underpin spontaneous (Coleman–Weinberg) gauge symmetry breaking, light dilaton mechanisms, and calculable mass hierarchies (Appelquist et al., 2021).
• In 2D materials, valley symmetry breaking allows for novel electronic responses (AVHE, spin-valley Hall) and for the design of valleytronic devices with electrical and magnetic control (Zhu et al., 2024, Wrzos et al., 27 Nov 2025).
• In moiré superlattices, interaction-driven spin-valley symmetry breaking generates emergent gauge fields and new routes to superconductivity and correlated states (Gonzalez et al., 2021).

This interdisciplinary reach underscores the fundamental and practical significance of valley-gauge-symmetry breaking in contemporary research.