A Personal History of the Hastings-Michalakis Proof of Hall Conductance Quantization
AA Personal History of the Hastings-Michalakis Proof of Hall ConductanceQuantization
Matthew B. Hastings
This is a personal history of the Hastings-Michalakis proof of quantum Hall conductance quanti-zation.
The Hall conductance quantization was an open problem in mathematical physics for a long time. A fundamentalphysical argument for the quantization was given by Laughlin[1], but finding a mathematical proof for an interactingsystem of electrons remained open until [2]. In this note, I give the history of this proof in brief. Recently, a pairof opinion pieces appeared in Nature and Nature Reviews Physics discussing this specific paper. Since those piecesdo not accurately convey the history, it may be of some interest to give the broader history, emphasizing what newmathematical tools and ideas were needed and why those tools and ideas were developed. The focus will be on theproof of [2] itself, so references will unfortunately be incomplete.Of course, Hall conductance quantization had been proven in various forms before. The result was proven for freefermions[3] using noncommutative geometry techniques and had been proven for interacting systems by Avron andSeiler under an additional averaging assumption[4], but there was no proof for an interacting system without thisaveraging assumption. This elegant averaging proof was extremely important for the Hall effect proof of [2]; roughly,the Hall effect proof of [2] involved replacing the average curvature of a connection due to adiabatic evolution asconsidered in [4] with the curvature of a quasi-adiabatic evolution operator, as I’ll explain more below.The story of the proof of Hall conductance quantization proof of [2] starts, for me, years earlier around 2002-2003.It starts with the proof of the higher dimensional Lieb-Schultz-Mattis theorem (LSM)[5], which is where the toolsneeded to prove Hall conductance quantization were developed. I had read a very insightful article by Misguich andLhuiller[6] on why they believed such an LSM theorem should exist; up to that point it had only been proven inone-dimension. This article was an inspiration to find a proof of such an LSM theorem, and I developed the tool ofquasi-adiabatic continuation to do this.In brief, the Lieb-Schultz-Mattis theorem in one dimension proves that a system of spin-1 / SU (2)symmetric Hamiltonian and translation invariance cannot have a unique ground state with a spectral gap. Furtherextensions with Affleck[7] generalized the theorem and clarified the implications of the theorem. Roughly, these one-dimensional spin systems must either break some discrete symmetry (in this case, translation symmetry), in whichcase the ground state becomes degenerate or close to degenerate; or the system may break a continuous symmetry(such as ferromagnetically ordering), in which case there is a continuous excitation spectrum; or the system may forma one-dimensional spin liquid with spinon excitations, again having a continuous excitation spectrum. Attempts toextend the theorem beyond one dimension[8] ran into trouble. In a sense, the variational argument of Lieb, Schultz,and Mattis was too “violent” in its effects on the ground state.Misguisch and Lhuillier however argued that such a theorem should hold in higher dimensions, noting that in addi-tion to the above possibilities, the system could have some kind of topological order, bringing a beautiful connectionto topology. Indeed, in that case the system would still be unable to have a unique gapped ground state but for adifferent reason: there would be a topological degeneracy.The tool of quasi-adiabatic continuation introduced in [5] to prove the LSM theorem was more “gentle” thanprevious variational attempts, and was able to diagnose the presence of topological order. This tool served as ageneral way to get precise theorems out of physical arguments involving slowly inserting gauge flux into some physicalsystem; it is worth mentioning the very important work of Oshikawa here too[9] who had the idea that the Lieb-Schultz-Mattis theorem might be related to flux insertion. Misguich and Lhuillier realized that on physical groundsone might expect the gap to not depend much on flux for a liquid-like state (though counter-examples involving firstorder phase transitions at particular values of the flux can be constructed) making flux insertion seem indeed like agood way to generate variational states; quasi-adiabatic continuation allowed one to make this idea precise so thateven if the gap did close one could “pretend” it did not for some purposes.Quasi-adiabatic continuation allowed one to show that, given a gapped local Hamiltonian, the change in the groundstate under a small change in the Hamiltonian could be given by a local and Hermitian operator acting on the groundstate but much of the use of this technique is to apply this operator even in a case where the gap might close. Thisoperator played a key role in the Hall proof.The proof of the LSM theorem in higher dimensions[5] introduced other tools in addition to quasi-adiabatic con-tinuation, tools that would be needed for Hall conductance quantization. This paper re-discovered Lieb-Robinsonbounds, and gave the first proof of these bounds in a way that was independent of local Hilbert space dimension.This alternative proof, which at the time might have seemed merely to improve some constants since the local Hilbertspace dimension in those applications typically was small (2 or 4), in fact turned out to be an essential ingredient a r X i v : . [ phy s i c s . h i s t - ph ] S e p in the Hall conductance proof. This alternative proof method has also become a general template for proving manyvariations of Lieb-Robinson bounds such as in systems with both short and long range interactions or systems withapproximately commuting terms in the Hamiltonian.The Lieb-Robinson bounds generally show how excitations can spread in a lattice system, showing that there isan approximate light-cone in these systems limiting how fast the system can respond to any perturbation. Also, thispaper [5] proved the exponential decay of correlations in gapped lattice systems with a local Hamiltonian; this proofis an example of how many of the proofs involving Lieb-Robinson bounds in gapped systems go. First, one usessome analytic technique relying on the spectral gap to relate some quantity (in this case, a correlation function) toa commutator (in this case, a Green’s function given by an expectation value of a commutator) and then one usesLieb-Robinson bounds to control that commutator.It soon became apparent to me that this general tool of quasi-adiabatic continuation could be used to make Hallconductance quantization arguments precise and to remove the averaging assumption. Rather than needing a singleflux insertion as in the LSM theorem which serves to detect a topologically ordered ground state, one would needtwo flux insertions to detect a curvature in the response of the ground state. It was clear to me that the topologicalarguments of Thouless and collaborators[10] could be made general using this tool. These arguments of Thouless et.al. served as a precursor to the averaging argument of Avron and Seiler[4]. To handle a many-body system withoutaveraging assumptions, one would need to compute the curvature of evolution under this quasi-adiabatic continuationoperator.At a technical level, the LSM theorem relied on a “virtual flux”, and it was also clear that the Hall conductanceproof would then need two virtual fluxes. The virtual flux plays the following role. One considers a system on atorus, and one wishes to understand the response of the system to some flux. Here we imagine Aharonov-Bohm fluxesthreading one or two directions of the torus. To give a picture, let us regard the torus as a square with opposite sidesidentified; introduce x, y coordinates, periodic with periods L x , L y . By making a gauge choice, one can consider thisflux as being implemented by a gauge field which is nonvanishing on some vertical line such as x = 0 (or similarly,a flux may be inserted on some horizontal line). In general, the ground state Ψ could have some very complicatedresponse to the flux, but one wishes to show that there is some state Ψ θ which is an approximate eigenstate of theHamiltonian with a flux θ inserted that has the following two properties: far from the line x = 0, the reduced densitymatrix of Ψ θ is (almost) the same as that of Ψ , while near the line x = 0, the reduced density matrix of Ψ θ is(almost) related to that of Ψ by some gauge transformation by angle θ .To construct Ψ θ , we use quasi-adiabatic continuation; while quasi-adiabatic continuation is defined to match adia-batic evolution in the case that the ground state has a spectral gap, we apply quasi-adiabatic continuation even if thegap closes! (If the gap closes, the state produced by quasi-adiabatic continuation might not be the ground state but itstill may be useful as a variational state or for other purposes.) Quasi-adiabatic continuation changes the state onlynear the line, giving the first property immediately. To get the second property, imagine also inserting an oppositeflux along the line x = L x /
2; then, by locality of quasi-adiabatic continuation, this second insertion has no effect near x = 0 (so, even if the insertion near x = L x / pretend it is done for the purposes of computingproperties near x = 0), but the combined effect of the two insertions is a gauge transformation since the net flux iszero.This property of the state produced by quasi-adiabatic continuation, that locally it is related to the ground state bya gauge transformation, is related to the “gentleness” of this method and it is why the curvature of the quasi-adiabaticcontinuation operator could be used to prove the Hall conductance quantization. The averaging method of Avron andSeiler for Hall conductance considered the curvature of adiabatic evolution as a function of two fluxes, θ, φ , defininga so-called “flux torus”. The curvature at θ = φ = 0 could be related to the Hall conductance and the average of thecurvature could be proven to be quantized by topological methods. Using the quasi-adiabatic continuation operatorinstead, this “gentleness” meant that the average of the curvature of the quasi-adiabatic evolution operator indeedwas (almost) equal to the curvature at θ = φ = 0; indeed, the curvature depended only weakly on flux angle in thatcase.The conceptual ingredients were then all in place to prove the Hall conductance quantization. However, these kindsof proofs tended to be rather lengthy, involving an enormous number of triangle inequalities and optimization overmany parameters. The Hall conductance proof promised to be even more lengthy. One problem is that one needed toshow that evolution under this quasi-adiabatic evolution operator also had locality properties, i.e., that this operatoralso had a Lieb-Robinson bound (here is where the dimension independent Lieb-Robinson bounds were necessary).I was at Los Alamos National Laboratory at the time and I chanced after that to receive a grant to hire a postdocand I had an applicant, Spiros Michalakis, coming from a mathematical physics group. So, it seemed like he wouldbe good to work with on finishing the detailed estimates. We succeeded, and we found ultimately a rather clean andsimple proof. The key ingredient in the proof indeed was quasi-adiabatic continuation.One key to making a clean proof was to use a modified “exact” form of quasi-adiabatic continuation, originallyintroduced by Tobias Osborne[11] in 2007, a few years after [5]; Osborne realized that some of the approximationsmade in the quasi-adiabatic continuation operator in the LSM theorem could be made exact at the cost of turningsome other exponentially small errors into merely super-polynomially small errors, and that the needed Lieb-Robinsonbound for this operator held. I realized that an old result in analysis[12] showed that these errors could be made“almost exponentially small” so that one could still find fairly tight bounds in this way. [1] Robert B Laughlin. Quantized hall conductivity in two dimensions. Physical Review B , 23(10):5632, 1981.[2] Matthew B Hastings and Spyridon Michalakis. Quantization of hall conductance for interacting electrons on a torus.
Communications in Mathematical Physics , 334(1):433–471, 2015.[3] Jean Bellissard, Andreas van Elst, and Hermann Schulz-Baldes. The noncommutative geometry of the quantum hall effect.
Journal of Mathematical Physics , 35(10):5373–5451, 1994.[4] Joseph E Avron and Ruedi Seiler. Quantization of the hall conductance for general, multiparticle schr¨odinger hamiltonians.
Physical review letters , 54(4):259, 1985.[5] Matthew B Hastings. Lieb-schultz-mattis in higher dimensions.
Physical review b , 69(10):104431, 2004.[6] Gregoire Misguich and Claire Lhuillier. Some remarks on the lieb-schultz-mattis theorem and its extension to higherdimensions. arXiv preprint cond-mat/0002170 , 2000.[7] Ian Affleck and Elliott H Lieb. A proof of part of haldane’s conjecture on spin chains.
Letters in Mathematical Physics ,12(1):57–69, 1986.[8] Ian Affleck. Spin gap and symmetry breaking in cuo 2 layers and other antiferromagnets.
Physical Review B , 37(10):5186,1988.[9] Masaki Oshikawa. Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice.
Physical review letters , 84(7):1535, 2000.[10] David J Thouless, Mahito Kohmoto, M Peter Nightingale, and Marcel den Nijs. Quantized hall conductance in a two-dimensional periodic potential.
Physical review letters , 49(6):405, 1982.[11] Tobias J Osborne. Simulating adiabatic evolution of gapped spin systems.
Physical review a , 75(3):032321, 2007.[12] AE Ingham. A note on fourier transforms.