Letters in Mathematical Physics最新文献

In this article, we establish the partial semi-classical Weyl’s laws for a bounded domain model and a harmonic trap model, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive the lowest free energy states for the subband models describing non-interacting 2DEGs.

It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the -diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend -diagrams and arise in the Deodhar decomposition of the Grassmannian.

We study the mixed Hessian of the dispersionless Toda (tau )-function for the one-harmonic (s)-fold symmetric conformal map (f(w)=rbigl (w+zeta w^{1-s}bigr )), where (r>0) is the exterior conformal radius and (zeta >0) is the dimensionless shape parameter. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral statements are formulated for its weighted symmetry-block realizations on a fixed Hilbert space. In that realization, the first spectral transition occurs at the analytic threshold (zeta _c=(s-1)^{s-1}/s^s), where the dominant square-root singularity of the inverse map reaches the circle (|z|=r), rather than at the geometric threshold (zeta _{textrm{univ}}=1/(s-1)), where univalence fails. After symmetry decomposition and weighted realization, each block develops exactly one logarithmically diverging eigenvalue as (zeta uparrow zeta _c), while the remaining spectrum stays bounded and converges to a compact limit. The instability is therefore rank one in every symmetry sector of the weighted block theory. We then continue the scalar Gram generating functions beyond (zeta _c). They are generalized hypergeometric functions on the slit plane (mathbb {C}{setminus }[zeta _c^2,infty )), their branch-point expansion contains the logarithmic term responsible for the divergence, and in the range (1le ple s) they admit Cauchy–Stieltjes and Jacobi-matrix realizations. In particular, the continued scalar quantities remain regular at (zeta _{textrm{univ}}), so the analytic spectral transition strictly precedes the geometric breakdown of univalence.

We review the construction of the spectral localiser (due to Loring and Schulz-Baldes) from a KK-theoretic perspective. We first give a KK-theoretic argument providing a spectral flow expression for the even or odd index pairing in terms of the “infinite volume” spectral localiser. Our approach towards this first step is more direct, treats the even and odd cases on an equal footing, and has the advantage that the construction of the spectral localiser becomes immediately apparent from the computation of the index pairing via a Kasparov product. In a second step of “spectral truncation”, we then describe how this spectral flow expression can be computed in terms of the signature of the “finite volume” spectral localiser. Throughout, we do not require invertibility of the operator representing the K-homology class, and the even index pairing then obtains an additional contribution coming from the Fredholm index.

We introduce a classical analog of the algebraic Heun operator associated with a classical Leonard pair. Given two observables X and Y satisfying the classical counterpart of the Askey–Wilson relations, we define a classical Heun observable W as the most general bilinear combination of X, Y, and their Poisson bracket. We prove that, when W is taken as Hamiltonian, the observables X and Y satisfy quartic differential equations whose generic solutions are elliptic functions of second order. This result provides a universal algebraic mechanism transforming the elementary dynamics associated with classical Leonard pairs into elliptic dynamics and yields an algebraic explanation of a classical observation of Manning on the connection between the Heun equation and elliptic solvability. The construction is illustrated on three examples: an extension of the Pöschl–Teller system, the Zhukovsky–Volterra gyrostat, and a relativistic (A_1) model related to the classical Askey–Wilson algebra.

We consider the transition between short-range entangled (SRE) and long-range entangled states of infinite quantum spin chains which is induced by local measurements. Specifically, we assume that the initial state is in a non-trivial symmetry-protected topological phase of a compact local symmetry group (mathcal {G}). We show that the on-site measurements of an appropriate local charge on intervals of increasing lengths transform the initial SRE state into a family of states with increasingly long-range correlations. In particular, the post-measurement states cannot be uniformly short-range entangled. In the case where the initial state is obtained from a product state using a quantum cellular automaton, we construct the infinite volume post-measurement state and exhibit almost-local observables that are maximally correlated.

Weingarten functions provide a tool for computing Haar measure matrix integrals of polynomials in the matrix entries. An important property of Weingarten functions is their particularly simple large N limits. In 2017, Benoît Collins and Sho Matsumoto studied when this limit holds for Weingarten functions associated with integrals of products of 2n matrix entries, as (n rightarrow infty ), together with the matrix size N [11]. They showed that the large N limit is uniformly achieved as long as (n=o(N^{4/7})), a result which already has applications to strong asymptotic freeness [2, 3]. However, their result is not optimal. They conjectured that their result should actually hold up to (n=o(N^{2/3})) which is optimal. We prove this conjecture for the matrix groups (G in {textrm{U}(N)), (textrm{O}(N)), (textrm{Sp}(N)}). The proof proceeds by introducing a Markov process on permutations (pairings) which we call the unitary (orthogonal) Weingarten process. We believe this process may have further applications to the theory of Weingarten functions. We also prove two new bounds regarding the large N limit of the Weingarten function in the regimes when (n=o(N^{4/5})), and (n=o(N)).

The entanglement bootstrap program has generated new quantum numbers associated with degrees of freedom living on gapped domain walls between topological phases in two dimensions. Most fundamental among these are the so-called parton quantum numbers, which give rise to a zoo of composite sectors. In this note, we propose a categorical description of partons. Along the way, we make contact with ideas from generalized symmetries and SymTFT.

In [4], two families of classical and quantum integrable hierarchies associated to arbitrary Cohomological Field Theories (CohFTs) were introduced: the meromorphic differential and twisted double ramification hierarchies. For trivial CohFT, the authors established a connection with the untwisted Double Ramification (DR) hierarchy. In this paper, we extend this study to the Hodge CohFT and prove an analogous correspondence with the untwisted DR hierarchy. This yields nontrivial identities between Hodge integrals over the DR cycle, the twisted DR cycle and the cycle of meromorphic differentials.

In this paper, we investigate the existence of internal modes of vortices in the gauged (mathbb {C}P^1) sigma model. We develop a clean geometric formalism that highlights the symmetries of the Jacobi operator, obtained from the second variation of the energy functional. The formalism and subsequent results fundamentally rely on the Bogomol’nyi decomposition of the energy functional and can therefore be extended to other models with such a decomposition. We prove the existence of at least one shape mode for a general (mathbb {C}P^1) vortex solution on (mathbb {R}^2) and find numerically the shape modes and corresponding frequencies of a radially symmetric vortex. A surprising result is that the shape mode eigenvalues are very close to the scattering threshold, suggesting weakly bound shape modes could be characteristic of the (mathbb {C}P^1) model.