The recently published article (Michalakis, S. Why is the Hall conductance quantized? Nat. Rev. Phys. 2, 392–393 (2020)1) does not accurately convey the full history of the proof of the quantization theorem2. Here I will primarily present my personal history before the work on this paper started, and then briefly discuss my work with Spyridon Michalakis, before mentioning some interesting work that resulted.
The proof results from combining two ideas. First, the elegant proof of Joseph Avron and Ruedi Seiler that the Hall conductance averaged over a ‘flux torus’ of different Aharonov-Bohm fluxes is quantized3. The second idea is to replace the curvature of a connection due to adiabatic evolution considered in3,4 with the curvature of a quasi-adiabatic evolution operator.
The story of the proof of ref.2 starts for me around 2002–2003, with my proof of the higher dimensional Lieb-Schultz-Mattis theorem (LSM) ref.5, where the tool of quasi-adiabatic continuation was introduced. This tool served as a general way to get rigorous theorems out of physical arguments6,7 involving flux insertion. This paper5 also introduced other tools needed for Hall conductance quantization, including the first proof of Lieb-Robinson bounds that were independent of local Hilbert space dimension.
It soon became apparent to me that this tool of quasi-adiabatic continuation could be used to prove Hall conductance quantization without using an averaging assumption. Rather than needing one Aharonov-Bohm flux insertion as in the LSM theorem, one would need two flux insertions to detect a curvature in the response of the ground state. This curvature would compute the Hall conductance, and due to the ‘gentleness’ of this method, the curvature depends only weakly on flux angle allowing one to remove the averaging assumption. At a technical level, the LSM theorem relied on a ‘virtual flux’, and it was clear that the Hall conductance proof would need two virtual fluxes.
The conceptual ingredients were then all in place to prove Hall conductance quantization. However, these proofs are still technically involved. I was at Los Alamos National Laboratory then and I received a grant to hire a postdoc and I had an applicant, Spiros Michalakis, from a mathematical physics group. So, it seemed like he would be good to work with on finishing the detailed estimates. We succeeded, and we found ultimately a rather clean and simple proof. The key ingredient in the proof indeed was quasi-adiabatic continuation.
Both of us made considerable effort to finish the proof. I will just mention one point of possible scientific interest. With two fluxes intersecting, one needed to show that evolution under this quasi-adiabatic continuation operator also had locality properties (here the dimension independent Lieb-Robinson bounds are needed). The key to a clean proof was using a modified ‘exact’ form of quasi-adiabatic continuation, originally introduced by Tobias Osborne8 in 2007, a few years after ref.5; Osborne realized that some approximations in quasi-adiabatic continuation could become exact at the cost of turning other exponentially small errors into super-polynomially small errors, and that the Lieb-Robinson bound for this operator held. Spiros and I independently noticed that an old result in analysis9 showed that these errors could be made ‘almost exponentially small’ so that one could still find fairly tight bounds.
Quasi-adiabatic continuation has found many other applications in mathematics and physics. To mention just a few, there are new proofs and results on the stability of matter10,11, and applications to entanglement entropy12. Hopefully these interesting results in the intersection of math, physics, and quantum information will continue!
Michalakis, S. Why is the Hall conductance quantized? Nat. Rev. Phys. 2, 392–393 (2020).
Hastings, M. B. & Michalakis, S. Quantization of hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015).
Avron, J. E. & Seiler, R. Quantization of the Hall conductance for general, multiparticle Schrodinger hamiltonians. Phys. Rev. Lett. 54, 259 (1985).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).
Hastings, M. B. Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004).
Misguich, G. and Lhuillier, C. Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions. Preprint at: https://arxiv.org/abs/cond-mat/0002170 (2000).
Oshikawa, M. Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice. Phys. Rev. Lett. 84, 1535 (2000).
Osborne, T. J. Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007).
Ingham, A. E. A note on Fourier transforms. J. Lond. Math. Soc. 1, 29–32 (1934).
Bravyi, S. & Hastings, M. B. A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609 (2011).
Michalakis, S. & Zwolak, J. P. Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322, 277–302 (2013).
Michalakis, S. Stability of the area law for the entropy of entanglement. Preprint at: https://arxiv.org/abs/1206.6900 (2012).
The authors declare no competing interests.
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Hastings, M.B. A different perspective on the history of the proof of Hall conductance quantization. Nat Rev Phys 2, 723 (2020). https://doi.org/10.1038/s42254-020-00255-5