Universal non-Hermitian skin effect in two and higher dimensions

Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a non-Hermitian hamiltonian are localized at the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodic-boundary spectrum of the hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic non-Hermitian hamiltonian, and, unlike in one dimension, is compatible with all spatial symmetries. We propose two new types of skin effect in two and higher dimensions: the corner-skin effect where all eigenstates are localized at one corner of the system, and the geometry-dependent-skin effect where skin modes disappear for systems of a particular shape, but appear on generic polygons. An immediate corollary of our theorem is that any non-Hermitian system having exceptional points (lines) in two (three) dimensions exhibits skin effect, making this phenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.

Operator expansions, layer susceptibility and two-point functions in BCFT

We show that in boundary CFTs, there exists a one-to-one correspondence between the boundary operator expansion of the two-point correlation function and a power series expansion of the layer susceptibility. This general property allows the direct identification of the boundary spectrum and expansion coefficients from the layer susceptibility and opens a new way for efficient calculations of two-point correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function 〈ϕiϕi〉 of the O(N) model at the extraordinary transition in 4 − ϵ dimensional semi-infinite space to order O(ϵ). The bulk operator product expansion of the two-point function gives access to the spectrum of the bulk CFT. In our example, we obtain the averaged anomalous dimensions of scalar composite operators of the O(N) model to order O(ϵ2). These agree with the known results both in ϵ and large-N expansions.

A multifractal boundary spectrum for $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$

We study $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$ SLE κ ( ρ ) curves, with $$\kappa $$ κ and $$\rho $$ ρ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed “angle” and determine the almost sure Hausdorff dimensions of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $$g_t$$ g t , by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

Compactness and hypercyclicity of co-analytic Toeplitz operators on de Branges-Rovnyak spaces

We study the compactness and the hypercyclicity of Toeplitz operators {T_{\bar \varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ(b). For the compactness of {T_{\bar \varphi ,b}}, we will see that the result depends on the boundary spectrum of b. We will prove that there are non trivial compact operators of the form {T_{\bar \varphi ,b}}, with ϕ ∈ H∞ ∩ C(𝕋), if and only if m(σ(b) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then {T_{\bar \varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ(𝔻) ∩ 𝕋 ≠ ∅.

On the existence of high-frequency boundary resonances in layered elastic media

We analyse the asymptotic behaviour of high-frequency vibrations of a three-dimensional layered elastic medium occupying the domain Ω =(− a , a ) 3 , a >0. We show that in both cases of stress-free and zero-displacement boundary conditions on the boundary of Ω a version of the boundary spectrum, introduced in Allaire and Conca (1998 J. Math. Pures. Appl. 77, 153–208. ( doi:10.1016/S0021-7824(98)80068-8 )), is non-empty and part of it is located below the Bloch spectrum. For zero-displacement boundary conditions, this yields a new type of surface wave, which is absent in the case of a homogeneous medium.