Jacob Shapiro

Office: 603 Fine Hall
Email: shapiro@math.princeton.edu

Mailing address:
Fine Hall
Department of Mathematics
Princeton NJ 08540

I am an assistant professor in the Department of Mathematics at Princeton University. I completed my PhD in Theoretical Physics at ETH Zürich in 2018 under the supervision of Gian Michele Graf and was then a postdoc in the math department of Columbia University and physics department of Princeton University. More detailed info available in my CV (Oct 2023)pdf.

In Spring 2024 I am teaching MAT 595 / PHY 508: Topics in Mathematical Physics: Mathematical Aspects of Condensed Matter Physics and MAT330: Complex Analysis and co-organizing the mathematical physics and probability seminars.

Research

My research is in mathematical physics. I study problems in analysis and probability that originate in condensed matter physics and statistical mechanics. I am particularly interested in disordered topological insulators, Anderson localization, random band matrices, the $O(N)$ model in classical statistical mechanics and random height functions.


Selected works

A complete list of publications and preprints is available on Google Scholar; see also ORCID iD iconorcid.

Strongly disordered topological insulators

Topological insulators are novel materials which insulate in the bulk of the sample, however, are very good conductors along their boundaries. One may associate with such materials an index, a discrete-valued number computed from their quantum mechanical Hamiltonian which is experimentally stable (macroscopic quantization). This stability is mathematically explained by identifying a topological space of quantum mechanical Hamiltonians obeying certain constraints (e.g. insulators) and showing that the set of path-connected components of this space of Hamiltonians is not a singleton. In interesting cases it is $\mathbb{Z}$ or $\mathbb{Z}_2$. In physics these materials are usually studied while assuming perfect translation invariance, whence the application of classical tools from algebraic topology (such as classification of vector bundles) is made possible. However, the assumption of translation invariance is not realistic, and furthermore, certain aspects of these phenomena need strong disorder (in the persence of Anderson localization). I have worked on the classification problem in this regime, as well as the bulk-edge correspondence, specializing on understanding these questions using Fredholm index theory:

The tight-binding reduction and topological equivalence

A classical theme in condensed matter physics is the tight-binding reducation, namely, that elliptic operators on $L^2(\mathbb{R}^d)$ are well-approximated by discrete operators on $\ell^2(\mathbb{Z}^d)$, in the sense that the long-time dynamics of low-energy states agree. This should happen, e.g., when the continuum Hamiltonian is made of a series of wells centered on $\mathbb{Z}^d$, which become arbitrarily deep. It is a challenge to make this statement mathematically precise in the presence of disorder and magnetic fields. This latter feature is crucial in order to understand whether the tight-binding limit commutes with calculating topological indices of topological insulators (i.e., whether the continuum and discrete descriptions are topologically equivalent). In order to understand the magnetic tight-binding limit, microscopic questions of magnetic double-well eigenvalue splitting must be understsood as well.

The BKT transition in 2D $O(2)$ models and height-function delocalization

In classical statistical mechanics, the $O(N)$ model describes a ferromagnet where each lattice site takes on a spin value in the $(N-1)$-sphere. When the temperature is sufficiently high, correlations of the spin between distant lattice sites decay exponentially in the distance. In one and two space dimensions, when $N>1$, the Mermin-Wagner theorem says that at arbitrarily low temperatures correlations must still decay. However, the question is the rate. In two space dimensions, for $N=2$, it has been discovered by Berezinskii–Kosterlitz–Thouless that the spin-spin correlation function behaves like the characteristic function of the Gaussian free field (i.e. the spin-wave approximation), and so the decay is merely polynomial. The proof of this fact dates back to the celebrated work of Fröhlich and Spencer from the early 80s. It is however still interesting to find new perspectives on this phenomenon, and to connect it to the theory of height-function delocalization. Coming from the perspective of the latter, the delocalization phase is again the harder one to establish.
  • Depinning in the integer-valued Gaussian field and the BKT phase of the 2D Villain model
    Michael Aizenman, Matan Harel, Ron Peled, Jacob Shapiro
    arXiv:2110.09498v2.

Localization for random band matrices

Random band matrices are $N \times N$ matrices whose entries are random around a diagonal band of width $W$ and zero outside of this band. A main conjecture on such models is whether a localization-delocalization (in the sense of Anderson localization) transition occurs as both $N,W\to\infty$, but where the limit is taken so that, $W \sim \sqrt{N}$.
  • Dynamical Localization for Random Band Matrices up to $W \ll N^{1/4}$
    Giorgio Cipolloni, Ron Peled, Jeffrey Schenker, Jacob Shapiro
    arXiv:2206.05545.



Manuscripts

Pedagogic manuscripts



Presentations

Other "Jacob Shapiro"'s

  • Jacob N. Shapiro: Professor of Politics and International Affairs at Princeton.
  • Jacob Z. Shapiro: Professor of Mathematics at University of Dayton (actually also doing mathematical physics, but more into microlocal analysis).