Anomalous Chiral Surface Topological Order

• Anomalous chiral surface topological order is a phenomenon where robust, symmetry-protected surface states emerge from bulk Berry curvature and WZW action, defining its topological nature.
• It reveals that gauged WZW terms and the ensuing chiral (triangle) anomaly under external fields drive observable effects such as quantized Hall responses and chiral currents.
• The interplay between paired Berry monopoles, effective field theories, and thermal/rotational perturbations links theoretical constructs to measurable phenomena in quantum materials.

Anomalous chiral surface topological order refers to the emergence of robust, symmetry-protected surface states with nontrivial topological and anomalous transport properties that cannot be realized in strictly two-dimensional systems, manifesting at the boundary of higher-dimensional topological phases such as Weyl semimetals, topological insulators, or symmetry-protected topological (SPT) phases. These surface states arise when the bulk topology, often characterized by nontrivial Berry curvature or SPT invariants, induces protected boundary anomalies—most notably, the chiral (triangle) anomaly and related anomalous transport phenomena. The interplay between Berry curvature, topology, external fields, and thermal or rotational perturbations yields a rich landscape of surface responses, rendering the anomalous chiral surface topological order a central concept bridging condensed matter and high-energy physics.

1. Geometric Origin: Berry Curvature and Emergence of the Wess–Zumino–Witten (WZW) Term

The starting point for anomalous chiral surface topological order is the presence of Berry curvature on the Fermi surface, as found in systems with degenerate band touchings or Weyl points. Locally, near a linearized level crossing at the Fermi surface edge, the system maps to an effective relativistic dispersion with a monopole configuration of Berry curvature in momentum space—quantized, for instance, at $g = 1/2$.

The Berry phase contribution to the single-particle action is, in the adiabatic limit,

$$S_B = \int dt\, \vec{\mathcal{A}}(\vec{p}) \cdot \dot{\vec{p}}$$

where $\vec{\mathcal{A}}(\vec{p})$ is the Berry connection. By extending the time-dependent path $\vec{p}(t)$ to a two-dimensional homotopy with an auxiliary parameter $s$, this dynamical phase can be rendered as a local term: the Wess–Zumino–Witten (WZW) term,

$$S_B = \frac{1}{2k_F^2} \int_D d^2x\, \epsilon^{ijk} c_{ij} p_k$$

where $D$ is a suitable two-dimensional manifold embedded in momentum space. This term captures the nontrivial winding (topological charge) of the Berry phase over the Fermi surface and is fundamentally analogous to the WZW terms that describe chiral anomalies in relativistic field theory.

2. Gauged WZW Term and the Chiral (Triangle) Anomaly

When external electromagnetic gauge fields are coupled via minimal substitution, $p \to p + A$ ($A$ the external vector potential), the Berry-curved Fermi surface's WZW term becomes "gauged." The action picks up dependence on the field-strength tensor $F = dA$, leading to

$$S_B = \frac{g\, n_1}{2k_F^2} \int_{D\times \mathbb{R}^3} (p + A) \wedge (dp + 2F) \wedge (dp + 2F)$$

where $n_1$ is the density of Fermi-surface quasiparticles and integration includes a homotopy coordinate. Varying this action with respect to $A$ yields an anomalous nonconservation of the chiral current,

$$\partial_\mu J^\mu_R \sim \frac{g}{2k_F^2} F \wedge F \sim \vec{E}\cdot \vec{B}$$

which is the signature of the chiral (triangle) anomaly: a topological obstruction to maintaining both chiral and gauge symmetry quantum mechanically in the presence of electromagnetic fields. This is linked to $E \cdot B$ and is the condensed matter realization of the Adler–Bell–Jackiw anomaly.

3. Conservation Laws and Topological Order: Paired Berry Monopoles

A key feature is that physical fermion number is strictly conserved, even though the separate chiral (left/right) currents exhibit anomalous divergences. This is because, on the Fermi surface, Berry curvature (monopole) always appears in pairs of opposite charge ($g$ and $-g$). For each chiral anomaly at one Fermi patch, there is a compensating anti-anomaly at another, ensuring that anomaly cancellation conditions are met globally. This paired structure underpins a nontrivial topological order in which the global winding of the Berry phase (encoded in the WZW action) guarantees robust, anomaly-induced chiral transport at the boundary.

4. Thermal and Rotational Effects: Chiral Vortical Effect and Gravitational Anomaly

Beyond zero-temperature and nonrotating limits, temperature and rotation fundamentally modify the anomalous surface response. When the Fermi fluid is placed in a slowly rotating frame (angular velocity $\omega$), the chiral vortical effect (CVE) leads to temperature- and rotation-driven anomalous currents. Including fluid velocity $v$ and chemical potential $\mu$ via $A \to A + \mu v$, the WZW-influenced current reads

$$\delta J_R \sim g \left[ F^2 + 4\mu_R F\wedge dv + 4\mu_R^2 (dv)^2 \right]$$

and the axial vortical current at low temperature is corrected as

$\delta J^A_\text{vortical} \sim \frac{T^2}{24}$

which is tied to the gravitational anomaly (the nonconservation of the axial current in a gravitational field due to quantum effects). Tadpole-like diagrams account for these $T^2$ corrections, analogous to phenomena in superfluid $^3$He-A, and connect the low-temperature chiral surface anomalous response to underlying spacetime topology.

5. Physical Consequences and Manifestations in Quantum Materials

The theory predicts several robust, anomaly-induced phenomena observable in materials whose Fermi surfaces host nontrivial Berry curvature:

• Quantum Anomalous Hall Effect: In magnetic topological insulators, domain walls where the Dirac mass changes sign localize chiral edge states, leading to quantized Hall conductance that can be directly linked to the surface anomaly (Wang et al., 2014). The phase space for thin films is governed by parameters that control the Dirac mass and exchange field, with chiral edge states at the boundary or domain wall.
• Chiral Surface States in Weyl Semimetals: Fermi arcs, or chiral Dirac/Weyl surface modes, emerge as protected, boundary-localized consequences of net Berry monopole charge in the bulk (Hattori et al., 2016). Their number and chirality are set by the monopole content.
• Topological Crystalline Insulator Surfaces: Interacting surface states can form symmetry-preserving gapped phases with anomalous symmetry fractionalization not possible in 2D. These are evidenced by anyons carrying fractional quantum numbers under mirror symmetry (e.g., $M^2 = -1$ for all anyons in a certain $\mathbb{Z}_2$ topological order) (Qi et al., 2015), highlighting the importance of the bulk-boundary correspondence and bulk topological invariants.

6. Theoretical Framework Linking Topology, Anomaly, and Surface Order

The occurrence of anomalous chiral surface topological order is theoretically underpinned by:

• Bulk–Boundary Correspondence: The existence of a Berry curvature-induced WZW term in the bulk mandates the presence of anomalous currents or symmetry fractionalization at the boundary. The anomaly inflow mechanism embodies this, ensuring that the "would-be" nonconserved current at the surface is globally compensated.
• Effective Field Theory: The field theory structure encodes both electromagnetic and gravitational anomaly responses, with the WZW term and its coupling to external fields fundamentally controlling the surface transport.
• Robustness and Universality: The presence of anomalous chiral surface states is not incidental—rather, it is an inevitable property whenever the bulk hosts topological structures as described. The anomaly is unaffected by weak disorder, small gap variations, or other perturbations that preserve the topological class.

7. Bridge to Experiment and Broader Context

Experiments on quantum materials—such as conductance plateaus in quantum anomalous Hall systems, nonlocal transport in topological insulators, and temperature- or field-driven crossovers—are direct manifestations of anomalous chiral surface topological order. Temperature corrections to anomalous currents, the interplay of electromagnetic and gravitational responses, and the quantization of surface transport reflect the theoretical constructs arising from Berry curvature-induced WZW terms and their anomaly-induced consequences.

This synthesis demonstrates that the emergence of anomalous chiral surface topological order is a fundamental response intrinsically tied to the topological nature of the Fermi surface, the global structure of Berry curvature, and the unavoidable quantum anomalies in systems spanning both condensed matter and high-energy physics contexts (Zahed, 2012).