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Jacob Shapiro

Publications and source records attributed to Jacob Shapiro.

At least 37 records · Page 2Linked to original sources
arXiv2025

Edge spectrum for truncated $\mathbb{Z}_2$-insulators

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a $\mathbb{Z}_2$-index: the Fu-Kane-Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in an earlier work by two of the authors for quantum Hall systems. It boils down to showing that the $\mathbb{Z}_2$-index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the $\mathbb{Z}_2$-index.

math-ph↗
arXiv2025

Essentially Commuting with a Unitary

Let $R$ be a unitary operator whose spectrum is the circle. We show that the set of unitaries $U$ which essentially commute with $R$ (i.e., $[U,R]\equiv UR-RU$ is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with $R$ and obey a non-triviality condition, and prove it is bijective with $\mathbb{Z}$.

math.FA↗
arXiv2024

Quantum tunneling and its absence in deep wells and strong magnetic fields

We present new results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to anti-symmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.

math-ph↗
arXiv2024

Absolutely continuous spectrum for truncated topological insulators

We show that if a topological insulator is truncated along a curve that separates the plane in two sufficiently large regions, then the edge system admits absolutely continuous spectrum. Our approach combines a recent version of the bulk-edge correspondence along curves that separates geometry and intrinsic conductance [DZ24], with a result about absolutely continuous spectrum for straight edges [BW22].

math-ph↗
arXiv2024

Topological Classification of Insulators: II. Quasi-Two-Dimensional Locality

We provide an alternative characterization of two-dimensional locality (necessary e.g. to define the Hall conductivity of a Fermi projection) using the spectral projections of the Laughlin flux operator. Using this abstract characterization, we define generalizations of this locality, which we term quasi-2D. We go on to calculate the path-connected components of spaces of unitaries or orthogonal projections which are quasi-2D-local and find a starkly different behavior compared with the actual 2D column of the Kitaev table, exhibiting e.g., in the unitary chiral case, infinitely many $\mathbb{Z}$-valued indices.

math-ph↗
arXiv2024

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is a one dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for $P$, and a limiting absorption resolvent estimate with sharp $h$-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

math.AP↗
arXiv2023

Semiclassical resolvent bounds for short range $L^\infty$ potentials with singularities at the origin

We consider, for $h, E > 0$, resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V - E$. Near infinity, the potential takes the form $V = V_L+ V_S$, where $V_L$ is a long range potential which is Lipschitz with respect to the radial variable, while $V_S = O(|x|^{-1} (\log |x|)^{-ρ})$ for some $ρ> 1$. Near the origin, $|V|$ may behave like $|x|^{-β}$, provided $0 \le β< 2(\sqrt{3} -1)$. We find that, for any $\tildeρ > 1$, there are $C, \, h_0 >0$ such that we have a resolvent bound of the form $\exp(Ch^{-2} (\log(h^{-1}))^{1 + \tildeρ})$ for all $h \in (0, h_0]$. The $h$-dependence of the bound improves if $V_S$ decays at a faster rate toward infinity.

math.AP↗
arXiv2023

Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems

We study non-interacting electrons in disordered one-dimensional materials which exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry classes. We define an appropriate topology on the space of Hamiltonians so that the so-called strong topological invariants become complete invariants yielding the one-dimensional column of the Kitaev periodic table, but now derived without recourse to K-theory. We thus confirm the conjecture regarding a one-to-one correspondence between topological phases of gapped non-interacting 1D systems and the respective Abelian groups $\{0\},\mathbb{Z},2\mathbb{Z},\mathbb{Z}_2$ in the spectral gap regime. The main tool we develop is an equivariant theory of homotopies of local unitaries and orthogonal projections. Moreover, we discuss an extension of the unitary theory to partial isometries, to provide a perspective towards the understanding of strongly-disordered, mobility-gapped materials.

math-ph↗
arXiv2023

Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation

We consider the initial-value problem for a one-dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges. The key ingredient of the proof is a high frequency resolvent estimate for an associated Helmholtz operator with a BV potential.

math.AP↗
arXiv2022

Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

We study the nonlinear Helmholtz equation $(Δ- λ^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $λ> 0$, $p \in \mathbb{N}$ odd, and more generally $(Δ_g + V - λ^2)u = N[u]$, where $Δ_g$ is the (positive) Laplace-Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1) > 4$ and $k > (n-1)/2$, for every $f \in H^k(S^{n-1}_ω)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} b(ω) + O(r^{-ε}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b \in H^k(S_ω^{n-1})$ and $ε> 0$. That is, the scattering matrix $f \mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work with Zhang, that proved a similar result with a loss of four derivatives.

math.AP↗
arXiv2022

Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(|x|) - E$ in dimension $n \ge 2$, where $h, \, E > 0$, and $V: [0, \infty) \to \mathbb{R}$ is $L^\infty$ and compactly supported. The weighted resolvent norm grows no faster than $\exp(Ch^{-1})$, while an exterior weighted norm grows $\sim h^{-1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.

math.AP↗
arXiv2022

Is the continuum SSH model topological?

The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on the values $0$ and $1$ labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is therefore an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation to lattices which depend on the tight-binding asymptotic parameter.

math-ph↗
arXiv2022

Fredholm Homotopies for Strongly-Disordered 2D Insulators

We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We furthermore provide a proof of the stability of the $\mathbb{Z}_2$ index in the mobility gap regime. These two-dimensional results serve as a model for the study of higher dimensional $\mathbb{Z}_2$ indices.

math-ph↗
arXiv2022

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

For a family of integer-valued height functions defined over the faces of planar graphs, we establish a relation between the probability of connection by level sets and the spin-spin correlations of the dual $O(2)$ symmetric spin models formulated over the graphs' vertices. The relation is used to show that in two dimensions the Villain spin model exhibits non-summable decay of correlations at any temperature at which the dual integer-restricted Gaussian field exhibits depinning. For the latter, we devise a new monotonicity argument through which the recent alternative proof by Lammers of the existence of a depinning transition in two-dimensional graphs of degree three, is extended to all doubly-periodic graphs, in particular to $\mathbb{Z}^2$. Essential use is made of the inequality of Regev and Stephens-Davidowitz, which allows also an alternative (to absolute-value FKG) proof of convergence of the height-function's distribution in the infinite-volume limit. Similar results are established for the $XY$ spin model and its dual Bessel random height function. Taken together these statements yield a new perspective on the Berezinskii-Kosterlitz-Thouless phase transition in $O(2)$ spin models, and complete a new proof of depinning in two-dimensional integer-valued height functions.

math.PR↗
arXiv2022

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along their boundary. This property is quantified by topological indices associated with either the bulk or the edge system. In the field of condensed matter physics, tight binding (discrete) approximate models, parametrized by hopping coefficients, have been used successfully to capture the topological behavior of TIs in many settings. However, whether such tight binding models capture the same topological features as the underlying continuum models of quantum physics has been an open question. We resolve this question in the context of the archetypal example of topological behavior in materials, the integer quantum Hall effect. We study a class of continuum Hamiltonians, $H^λ$, which govern electron motion in a two-dimensional crystal under the influence of a perpendicular magnetic field. No assumption is made on translation invariance of the crystal. We prove, in the regime where both the magnetic field strength and depth of the crystal potential are sufficiently large, $λ\gg1$, that the low-lying energy spectrum and eigenstates (and corresponding large time dynamics) of $H^λ$ are well-described by a scale-free discrete Hamiltonian, $H^{\rm TB}$; we show norm resolvent convergence. The relevant topological index is the Hall conductivity, which is expressible as a Fredholm index. We prove that for large $λ$ the topological indices of $H^λ$ and $H^{\rm TB}$ agree. This is proved separately for bulk and edge geometries. Our results justify the principle of using discrete models in the study of topological matter.

math-ph↗
arXiv2022

Lower Bound on Quantum Tunneling for Strong Magnetic Fields

We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the eigenvalue splitting--is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping probability between the wells, a key parameter in tight binding models of solid state physics.

math-ph↗