10.5 Edge states

Concept of edge states

Edge states are electronic states that live at the boundaries of topological materials rather than in the bulk. They're important because they carry current without scattering off impurities, which makes them central to phenomena like the quantum Hall effect and to potential technologies like topological quantum computing.

These states aren't accidental. They're forced to exist by the topology of the bulk band structure, and they survive as long as certain symmetries remain intact.

Definition and characteristics

An edge state is a quantum state confined to the edge or surface of a topological material. Several properties set these states apart from ordinary surface states:

• Unidirectional propagation: Edge states travel in only one direction along a boundary, making them immune to backscattering from non-magnetic impurities.
• Spin-momentum locking: The electron's spin orientation is tied to its direction of motion. If it moves to the right, its spin points up; reverse the direction, and the spin flips.
• Linear (Dirac-like) dispersion: Their energy-momentum relation is linear, resembling massless Dirac fermions rather than the parabolic dispersion of ordinary electrons.
• High conductivity: Because topological protection suppresses scattering, edge states carry current with very low dissipation.

Topological origin

Edge states don't appear by accident. They arise because the bulk band structure has a non-trivial topology, characterized by invariants like the Chern number or the $\mathbb{Z}_2$ index.

When two regions with different topological invariants share a boundary, the bulk energy gap must close at that interface. That gap closing produces the edge states. As long as the bulk topology doesn't change (for instance, the gap doesn't close and reopen throughout the bulk), the edge states persist. This is what gives them their famous robustness against perturbations.

Bulk-boundary correspondence

This is the principle that connects what happens in the bulk to what you see at the edge. Specifically, the number of edge modes at a boundary equals the difference in topological invariants between the two sides.

For example, if a Chern insulator with Chern number $C = 1$ borders a trivial insulator with $C = 0$, there will be exactly one chiral edge mode at the interface. This principle applies broadly across Chern insulators, $\mathbb{Z}_2$ topological insulators, and higher-order topological insulators. It's the reason you can predict edge state behavior from bulk properties alone, without solving the full boundary problem.

Types of edge states

Different topological phases host different kinds of edge states, depending on which symmetries are present or broken.

Chiral edge states

Chiral edge states propagate in only one direction along the boundary. They appear in systems where time-reversal symmetry is broken, such as:

• Quantum Hall systems: A 2D electron gas in a strong perpendicular magnetic field.
• Chern insulators: Materials that exhibit a quantum Hall effect without an external magnetic field (the anomalous quantum Hall effect).

Because all the edge modes move the same way, there's no partner mode to scatter into. This makes backscattering essentially impossible and produces the precisely quantized Hall conductance seen in the integer quantum Hall effect.

Helical edge states

Helical edge states come in counter-propagating pairs with opposite spin. They appear in 2D topological insulators (also called quantum spin Hall insulators).

• One spin species moves clockwise around the edge; the other moves counterclockwise.
• These pairs form Kramers pairs, guaranteed by time-reversal symmetry.
• Spin-momentum locking means that flipping the propagation direction also flips the spin.
• Non-magnetic impurities can't backscatter an electron without flipping its spin, so transport is dissipationless as long as time-reversal symmetry holds.

Majorana edge states

Majorana modes are quasiparticles that are their own antiparticles. They appear as zero-energy states localized at the ends of 1D topological superconductors.

• They obey non-Abelian statistics: exchanging two Majorana modes changes the quantum state of the system in a way that depends on the order of exchanges, not just whether an exchange happened.
• This property makes them candidates for topological qubits, which would be inherently resistant to local sources of decoherence.
• Experimentally, they've been pursued in hybrid semiconductor-superconductor nanowire systems (e.g., InSb or InAs nanowires proximitized by an s-wave superconductor), though unambiguous confirmation remains an active area of research.

Quantum Hall effect

The quantum Hall effect was the first experimental realization of a topological phase, and it remains the clearest demonstration of topological edge states.

Integer quantum Hall effect

When a 2D electron gas is placed in a strong perpendicular magnetic field at low temperature, the electrons organize into discrete Landau levels. The Hall resistance becomes quantized at plateaus:

$R_{xy} = \frac{h}{ne^2}$

where $n$ is an integer and $h/e^2 \approx 25.8 \, \text{k}\Omega$. Between Landau levels, bulk states are localized by disorder, and current flows only through chiral edge channels. Each filled Landau level contributes one edge channel, and the topological invariant (Chern number) equals $n$.

Fractional quantum Hall effect

At even stronger magnetic fields and lower temperatures in very clean samples, plateaus appear at fractional filling factors:

$\nu = \frac{p}{q}$

where $p$ and $q$ are integers (e.g., $\nu = 1/3, \, 2/5$). This effect arises from strong electron-electron interactions and is described by the formation of composite fermions. The quasiparticles carry fractional charge (e.g., $e/3$) and obey anyonic statistics, which are neither bosonic nor fermionic. Edge states in the fractional regime are more complex, often involving multiple edge channels with fractional charges.

Edge states in QHE

In both integer and fractional quantum Hall systems, edge states form chiral 1D channels running along the sample boundary. Key features:

• Counterpropagating modes are spatially separated on opposite edges, so backscattering requires tunneling across the entire sample.
• Each channel carries a quantized current, producing the precise Hall conductance.
• These channels can be probed through multi-terminal transport measurements and used for electron interferometry (e.g., Fabry-Pérot and Mach-Zehnder interferometers built from quantum Hall edge states).

Topological insulators

Topological insulators are materials that are insulating in the bulk but host conducting states at their boundaries, protected by time-reversal symmetry.

2D topological insulators

Also called quantum spin Hall insulators, these host helical edge states with spin-momentum locking. The key experimental realizations:

• HgTe/CdTe quantum wells: The first confirmed 2D topological insulator (König et al., 2007). When the HgTe layer exceeds a critical thickness (~6.3 nm), the bands invert and helical edge states appear.
• InAs/GaSb heterostructures: An alternative platform where band inversion occurs between the electron-like band in InAs and the hole-like band in GaSb.

Each edge contributes a quantized conductance of $2e^2/h$ (factor of 2 from the two spin channels in a Kramers pair).

3D topological insulators

In three dimensions, the boundary states form a 2D conducting surface with a single Dirac cone dispersion. Well-studied examples include:

• $\text{Bi}_2\text{Se}_3$: Has a relatively large bulk gap (~0.3 eV) and a single Dirac cone at the $\Gamma$ point.
• $\text{Bi}_2\text{Te}_3$ and $\text{Sb}_2\text{Te}_3$: Similar properties, widely studied for thermoelectric applications as well.

Surface states exhibit spin-momentum locking and suppressed backscattering, confirmed by spin-resolved ARPES experiments.

Edge states vs. surface states

Both are topologically protected, but their different dimensionalities mean different experimental approaches and different potential applications.

Experimental observations

Transport measurements

Transport experiments are the most direct way to detect edge states. The typical approach:

1. Fabricate a multi-terminal Hall bar device from the topological material.
2. Measure longitudinal and Hall resistance as a function of magnetic field (for QHE) or gate voltage (for TIs).
3. Look for quantized conductance plateaus, which signal edge state transport.

Non-local measurements can distinguish edge transport from bulk transport: if current injected at one pair of contacts produces a voltage at distant contacts in a pattern consistent with edge channels, that's strong evidence. Aharonov-Bohm interferometry can further probe the coherence and phase of edge states.

Spectroscopic techniques

• Scanning tunneling spectroscopy (STS): Measures the local density of states with atomic resolution. You can directly see how the density of states changes as you move from the bulk to the edge.
• Angle-resolved photoemission spectroscopy (ARPES): Maps the band structure in momentum space. For 3D TIs, ARPES directly reveals the Dirac cone of the surface states.
• Optical and Raman spectroscopy: Probe transitions involving edge states and their coupling to phonons.

STM and ARPES studies

STM provides real-space images of edge states, showing where the electronic density is concentrated. ARPES provides the complementary momentum-space picture, mapping out the dispersion relation. Together, they give a complete view:

• STM can image standing wave patterns at edges and step edges, revealing the wavelength and decay length of edge states.
• Spin-resolved ARPES directly confirms spin-momentum locking by measuring the spin polarization of photoelectrons as a function of momentum.

Applications of edge states

Quantum computing

Majorana edge states are the leading candidates for topological qubits. The idea is that quantum information stored in pairs of spatially separated Majorana modes is inherently protected from local perturbations, because no local measurement can determine the state of a single Majorana. This could dramatically reduce error rates compared to conventional qubits.

Quantum Hall edge states have also been explored as platforms for quantum gates, where braiding of anyonic quasiparticles (in the fractional regime) could implement fault-tolerant quantum logic.

Spintronics

The spin-momentum locking of topological insulator edge and surface states makes them efficient at converting between spin currents and charge currents. Specific applications include:

• Spin-orbit torque devices: A charge current through a TI surface generates a spin accumulation that can switch the magnetization of an adjacent ferromagnet.
• Spin field-effect transistors: Helical edge states could serve as spin-polarized channels controlled by gate voltages.
• Dissipationless spin transport: In quantum spin Hall insulators, spin currents flow along edges without energy loss (at low temperatures).

Low-power electronics

Because topological edge states resist scattering, devices based on them could operate with significantly less power dissipation. Research directions include:

• Topological insulator-based transistors with improved on/off ratios
• Quantum anomalous Hall effect edge channels as low-dissipation interconnects
• Hybrid architectures combining topological materials with conventional CMOS technology

The main barrier is that most of these effects currently require cryogenic temperatures. Room-temperature operation remains a major goal.

Theoretical models

Haldane model

The Haldane model (1988) showed that a quantum Hall state can exist without an external magnetic field. It uses a honeycomb lattice with:

1. Ordinary nearest-neighbor hopping (real-valued).
2. Complex next-nearest-neighbor hopping that breaks time-reversal symmetry.
3. A staggered sublattice potential that breaks inversion symmetry.

The complex hopping generates a non-zero Chern number, producing chiral edge states and quantized Hall conductance. This model is the prototype for Chern insulators and was experimentally realized in ultracold atomic gases in optical lattices (Jotzu et al., 2014).

Kane-Mele model

Kane and Mele (2005) extended the Haldane model by including spin-orbit coupling and restoring time-reversal symmetry. The result is two copies of the Haldane model (one for each spin), with opposite Chern numbers. The total Chern number is zero, but the system is characterized by a $\mathbb{Z}_2$ topological invariant.

This model predicts helical edge states and the quantum spin Hall effect. While the spin-orbit coupling in real graphene turned out to too weak to observe this phase, the model provided the theoretical foundation for the entire field of $\mathbb{Z}_2$ topological insulators.

Bernevig-Hughes-Zhang (BHZ) model

The BHZ model (2006) predicted the quantum spin Hall effect in HgTe/CdTe quantum wells, which was confirmed experimentally the following year. The key physics:

1. In thin quantum wells, the band ordering is normal (s-type band above p-type band).
2. Beyond a critical thickness (~6.3 nm), the bands invert due to strong spin-orbit coupling in HgTe.
3. This band inversion changes the $\mathbb{Z}_2$ invariant from trivial to non-trivial, producing helical edge states.

The model is written as an effective 4-band Hamiltonian and has been generalized to describe band inversion in other material systems.

Edge state protection

Time-reversal symmetry

Time-reversal symmetry (TRS) is the primary protector of helical edge states in quantum spin Hall insulators. Here's how it works:

• TRS guarantees Kramers degeneracy at time-reversal invariant momenta: every state has a partner with opposite spin and momentum.
• For a single pair of helical edge states, backscattering would require flipping the spin. But TRS forbids elastic scattering between Kramers partners.
• Magnetic impurities or applied magnetic fields break TRS and can open a gap in the edge states, destroying their protection.

This is why experiments on 2D TIs must be performed at low temperatures with non-magnetic samples.

Particle-hole symmetry

Particle-hole symmetry (PHS) is relevant primarily in superconducting topological systems. It ensures that for every state at energy $E$, there's a corresponding state at $-E$. This symmetry pins Majorana modes to exactly zero energy, since a Majorana mode is its own particle-hole conjugate.

PHS, combined with TRS, defines the chiral symmetry classes in the periodic table of topological insulators and superconductors (the "tenfold way" classification).

Topological protection mechanisms

Several mechanisms work together to protect edge states:

• Bulk-boundary correspondence guarantees that edge states exist whenever the bulk has non-trivial topology.
• A large bulk gap prevents thermal excitation of bulk carriers that could hybridize with edge states. For $\text{Bi}_2\text{Se}_3$, this gap is ~0.3 eV.
• Spatial separation of counterpropagating modes (on opposite edges in QHE, or with opposite spins in QSH) suppresses scattering.
• Topological invariants (Chern number, $\mathbb{Z}_2$ index) are integers that can't change continuously. A perturbation must be strong enough to close the bulk gap to destroy the edge states.

Challenges and future directions

Material design

Most known topological insulators have small bulk gaps, limiting applications to cryogenic temperatures. Current research focuses on:

• Finding or engineering materials with larger bulk gaps for room-temperature operation
• Exploring higher-order topological insulators, which host states on hinges or corners rather than entire edges or surfaces
• Designing heterostructures that combine topological materials with superconductors or ferromagnets to create new phases
• Developing materials with tunable topological properties (e.g., via strain, electric fields, or chemical doping)

Room temperature applications

The central challenge is that thermal energy at room temperature ($k_BT \approx 25 \, \text{meV}$) can exceed the bulk gap of many topological materials, activating bulk carriers that short-circuit the edge transport. Strategies to address this include increasing spin-orbit coupling strength, finding materials with intrinsically larger gaps, and exploring strongly correlated topological phases where interactions enhance the gap.

Integration with existing technologies

For topological materials to move beyond the lab, they need to be compatible with existing semiconductor fabrication processes. Open problems include:

• Developing reliable methods for growing high-quality thin films of topological materials on standard substrates
• Making low-resistance electrical contacts to edge states without disrupting them
• Scaling topological qubit architectures and interfacing them with classical control electronics
• Characterizing and controlling edge states in nanoscale devices where finite-size effects become important