nLab edge mode

Idea

In solid state physics, edge modes are excitations (of electrons or quasi-particles) seen at the boundary or at a corner of a crystalline materials, where the dynamics may be different from that in the crystal’s bulk.

A notable example are edge modes of topological insulators: As their name suggests, these topological phases of matter are in fact insulating in their bulk, but on their boundary they behave like conductors/metals, due to “edge modes” which may carry electric current.

In correspondence of how the effective field theory for fractional quantum Hall systems in the bulk is (Maxwell-)Chern-Simons theory (cf. at abelian Chern-Simons theory), so the corresponding effective field theory for the FQH edge modes is Floreanini-Jackiw theory (Wen 1992 §2.5, 1995 §3.3).

Related concepts

• bulk-boundary correspondence

• Floreanini-Jackiw theory

References

For the time being see any of the main references at topological insulator and at fractional quantum Hall systems.

See also:

• Wikipedia: Edge states

In fractional quantum Hall systems

On edge modes in fractional quantum Hall systems:

Original articles:

• Bertrand I. Halperin: Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982) 2185 [doi:10.1103/PhysRevB.25.2185]

• Xiao-Gang Wen: Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B 41 (1990) 12838 [doi:10.1103/PhysRevB.41.12838]

• Michael Stone: Edge waves in the quantum Hall effect, Annals of Physics 207 1 (1991) 38-52 [doi:10.1016/0003-4916(91)90177-A]

• Xiao-Gang Wen: Theory of Edge States in Fractional Quantum Hall Effects, International Journal of Modern Physics B 06 10 (1992) 1711-1762 [doi:10.1142/S0217979292000840]

• Xiao-Gang Wen: Topological order and edge excitations in fractional quantum Hall states, Advances in Physics 44 5 (1995) 405-437 [arXiv:cond-mat/9506066, doi;10.1080/00018739500101566]

Review:

• A. M. Chang: Chiral Luttinger liquids at the fractional quantum Hall edge, Rev. Mod. Phys. 75 (2003) 1449 [doi:10.1103/RevModPhys.75.1449]

• Eduardo Fradkin: Physics at the Edge, Chapter 15 in: Field Theories of Condensed Matter Physics, Cambridge University Press (2013) [ISBN: 9781139015509, doi:10.1017/CBO9781139015509, pdf]

• David Tong: Edge Modes, chapter 6 of: The Quantum Hall Effect, lecture notes (2016) [arXiv:1606.06687, course webpage, pdf, pdf]

In the generality of multi-component (K-matrix) FQH systems:

• Ana Lopez, Eduardo Fradkin: Effective field theory for the bulk and edge states of quantum Hall states in unpolarized single layer and bilayer systems, Phys. Rev. B 63 (2001) 085306 [doi:10.1103/PhysRevB.63.085306, arXiv:cond-mat/0008219]

• Zeno Bacciconi: Fractional Quantum Hall edge dynamics from a Quantum Optics perspective [arXiv:2111.05858]

On edge mode spectral flow:

• Wen 1990 §III

• Jürg Fröhlich, T. Kerler; pp 395–396, 400–402 in: Universality in quantum Hall systems, Nuclear Physics B 354 2–3 (1991) 369-417 [doi:10.1016/0550-3213(91)90360-A]

• Ivan P. Levkivskyi, Jürg Fröhlich, Eugene V. Sukhorukov; pages 9–10, 15–18: Theory of fractional quantum Hall interferometers, Phys. Rev. B 86 245105 (2012) [doi:10.1103/PhysRevB.86.245105, arXiv:1005.5703]

• Charles-Edouard Bardyn, Michele Filippone, Thierry Giamarchi; pages 3–4: Bulk Pumping in 2D Topological Phases, Phys. Rev. B 99 035150 (2019) [doi:10.1103/PhysRevB.99.035150, arXiv:1807.01710]

• Bacciconi 2021 §1.4.4 \& §1.2.4

On edge mode tunneling:

• Claudio Chamon, Xiao-Gang Wen: Resonant tunneling in the fractional quantum Hall regime, Phys. Rev. Lett. 70 (1993) 2605 [doi:10.1103/PhysRevLett.70.2605]

• Claudio Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, Xiao-Gang Wen: Two point-contact interferometer for quantum Hall systems, Phys. Rev. B 55 (1997) 2331 [doi:10.1103/PhysRevB.55.2331]

• D. Christian Glattli: Tunneling Experiments in the Fractional Quantum Hall Effect Regime, in: The Quantum Hall Effect, Progress in Mathematical Physics 45, Birkhäuser (2005) [doi:10.1007/3-7643-7393-8_5, pdf]

• G. Dolcetto, S. Barbarino, D. Ferraro, N. Magnoli, M. Sassetti: Tunneling between helical edge states through extended contacts, Phys. Rev. B 85 (2012) 195138 [arXiv:1203.4486, doi:10.1103/PhysRevB.85.195138]

• M. Ruelle et al. Time-domain braiding of anyons, Science 389 (2025) 6755 [doi:10.1126/science.adm7695]

In view of the bulk-edge correspondence:

• Johannes Kellendonk, Thomas Richter, Hermann Schulz-Baldes: Edge current channels and Chern numbers in the integer quantum Hall effect, Rev. Math. Phys. 14 1 (2002) 87–119 [doi:10.1142/S0129055X02001107]

• Johannes Kellendonk, Hermann Schulz-Baldes: Quantization of edge currents for continuous magnetic operators, J. Funct. Anal. 209 (2004) 388-413 [doi:10.1016/S0022-1236(03)00174-5, arXiv:math-ph/0405021]

• Johannes Kellendonk, Hermann Schulz-Baldes: Boundary maps for $C^\ast$-crossed products with $\mathbb{R}$ with an application to the quantum Hall effect, Commun. Math. Phys. 249 (2004) 611-637 [doi:10.1007/s00220-004-1122-7, arXiv:math-ph/0405022]

Monograph:

• Emil Prodan, Hermann Schulz-Baldes: Bulk and Boundary Invariants for Complex Topological Insulators – From K-Theory to Physics, Springer (2016) [doi:10.1007/978-3-319-29351-6]

Experimental detection:

• Ramon Guerrero-Suarez et al.: Universal Anyon Tunneling in a Chiral Luttinger Liquid, Nature Physics 21 (2025) 1787–1793 [doi:10.1038/s41567-025-03039-9, arXiv:2502.20551 cond-mat.mes-hall]

  “Despite many years of experimental investigation fractional quantum Hall edge modes remain enigmatic with significant discrepancies between experimental observations and detailed predictions of chiral Luttinger liquid theory.”

In the generality of higher dimensional integer quantum Hall systems and relating to area-preserving diffeomorphisms regularized by $\mathfrak{u}(\infty)$:

• Dimitra Karabali, V. Parameswaran Nair: The effective action for edge states in higher dimensional quantum Hall systems, Nucl. Phys. B 679 (2004) 427-446 [doi:10.1016/j.nuclphysb.2003.11.020, arXiv:hep-th/0307281]

Including non-relativistic corrections (Newton-Cartan geometry):

• William J. Wolf, James Read, Nicholas Teh: Edge modes and dressing fields for the Newton-Cartan quantum Hall effect, Found Phys 53, 3 (2023) [arXiv:2111.08052, doi:10.1007/s10701-022-00615-4]

See also:

• Samuel Bieri, Jürg Fröhlich: Effective field theory and tunneling currents in the fractional quantum Hall effect, Annals of Physics 327 4 (2012) 959-993 [doi:10.1016/j.aop.2011.10.012, arXiv:1107.5012]

• J. Fernando Barbero G., Bogar Díaz, Juan Margalef-Bentabol, Eduardo J.S. Villaseñor: Edge observables of the Maxwell-Chern-Simons theory, Physical Review D, 106 (2022) 025011 [doi:10.1103/PhysRevD.106.025011, arXiv:2204.06073]

  (via Maxwell-Chern-Simons theory)

In topological insulators

On edges modes in topological insulators:

• Lukasz Fidkowski, T. S. Jackson, Israel Klich, Edge modes in band topological insulators [arXiv:1101.0320]

See also:

• Gil Young Cho, Yuan-Ming Lu, Joel E. Moore: Gapless edge states of background field theory and translation-symmetric $\mathbb{Z}_2$ spin liquids, Phys. Rev. B 86 125101 (2012) [arXiv:10.1103/PhysRevB.86.125101]

Via Chern-Simons theory & other TQFT

On edge modes via the (abelian) WZW model on the boundart of (abelian) Chern-Simons theory (cf. also AdS3-CFT2 and CS-WZW correspondence):

• Nicola Maggiore: From Chern-Simons to Tomonaga-Luttinger, Int. J. Mod. Phys. A 33 (2018) 1850013 [doi:10.1142/S0217751X18500136, arXiv:1712.08744]

• Michael Levin: Protected edge modes without symmetry, Phys. Rev. X 3 021009 (2013) [doi:10.1103/PhysRevX.3.021009, arXiv:1301.7355]

• Irais Rubalcava-Garcia, §3.6 in: Constructing the theory at the boundary, its dynamics and degrees of freedom [arXiv:2003.06241]

• Thomas G. Mertens, Qi-Feng Wu: Minimal Factorization of Chern-Simons Theory – Gravitational Anyonic Edge Modes [arXiv:2505.00501]

See also:

• Atsushi Ueda, Kansei Inamura, Kantaro Ohmori: Chiral Edge States Emerging on Anyon-Net [arXiv:2408.02724]