Communications in Mathematical Physics最新文献

For the focusing, energy critical wave equation in dimension 5, we construct multi-solitons with any number of solitons, any choice of signs, speeds, scaling parameters and translation parameters. This requires to revisit in depth previous constructions of multi-solitons based on a unidirectional approach, to fully take into account the dimension of the space and the possibility for solitons to move in any direction. Then, as a consequence of this more general construction and of the arguments developed in [20], the inelastic nature of any collision of solitons is proved under a non-cancellation assumption on the parameters.

This work introduces a surface observable for nonabelian four-dimensional BF theory with a cosmological term. The surface observable yields new 2-knot invariants that may extend beyond known examples such as the Alexander invariant. By BV pushforward, the surface observable induces an electric observable in nonabelian Yang–Mills theory, offering a concrete realization of ’t Hooft operators. An application to self-dual Yang–Mills theory is also discussed.

This paper explains the methods and algorithms we used to obtain explicit F-symbols, R-symbols, and pivotal coefficients of what we believe are all multiplicity-free pivotal complex fusion categories up to rank 7. Particular attention is given to how the individual steps for solving the consistency equations are done with Anyonica, a software package we developed for working with fusion categories. Gauge and automorphism equivalence are reviewed and a method to remove equivalent categories via a gauge-split basis is given. A census of all multiplicity-free fusion categories up to rank 7, along with all compatible braided and pivotal structures, is presented, and comments on their general properties are given.

At a magic relative twist angle, magic angle twisted bilayer graphene (MATBG) has an octet of flat bands that can host strong correlation physics when partially filled. A key theoretical discovery in MATBG is the existence of ferromagnetic Slater determinants as exact ground states of the corresponding flat band interacting (FBI) Hamiltonian. The FBI Hamiltonian describes the behavior of electrons that interact with each other in a high-dimensional space, and is constructed from the band structure of the non-interacting Bistritzer–MacDonald model at the chiral limit. A key property of the FBI Hamiltonian for MATBG is that it is frustration-free and can be written as a sum of non-commuting terms. In this work, we provide a complete characterization of the ground state manifold of the FBI Hamiltonian, proving that it is precisely the linear span of such ferromagnetic Slater determinants.

We prove the existence of a phase transition in dimension (d>1) in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range (gamma ^{-1}), with (gamma >0). This transition is “close”, for small positive (gamma ), to the one proved previously by Lebowitz and Penrose in the van der Waals limit (gamma downarrow 0). It is of the type of the liquid–vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. Previous results on phase transitions in continuum systems with stable potentials required the use of unphysical four-body interactions or special symmetries between the liquid and vapor. The pair interaction we consider is obtained by partitioning space into cubes of volume (gamma ^{-d}), and letting the Kac part of the pair potential be uniform in each cube and act only between adjacent cubes. The “short-range” part of the pair potential is quite general (in particular, it may or may not include a hard core), but restricted to act only between particles in the same cube. Our setup, the “boxed particle model”, is a special case of a general “spin” system, for which we establish a first-order phase transition using reflection positivity and the Dobrushin–Shlosman criterion.

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by (K=K_N) (with (K_Nuparrow infty ) as (Nuparrow infty )), the Kawasaki rates by (N^2) and space by 1/N. We start the process so that the continuum interface (Gamma _t) formed, between ‘high’ (rho _+) and ‘low’ (rho _-) density regions, is stationary that is, (Gamma _t) is ‘flat’. When the Glauber rates are balanced on ({mathbb {T}}^d), the interface (Gamma _t=Gamma ={v: v_1=0}) is immobile and the hydrodynamic limit is given by (rho (t,v) = rho _+) for (v_1in (0,1/2)) and (rho (t,v)= rho _-) for (v_1in (-1/2,0)) for all (tge 0), where (v=(v_1,ldots ,v_d)in {mathbb {T}}^d) identified with ([-1/2,1/2)^d). In the pre-limit setting, a boundary region of width (O(1/sqrt{K_N})) about the interface arises, where the density is more indeterminate than outside the region. Accordingly, we will scale the (v_1) coordinate in the fluctuation field by (sqrt{K_N}) so that the fluctuation field scaling limit will capture information ‘near’ the interface. We identify the fluctuation limit as a Gaussian field when (K_N= O(sqrt{log (N)})) in (dle 2). In the one dimensional case, the field limit is given by (textbf{e}(v_1) B_t) where (B_t) is a Brownian motion and (textbf{e}) is the normalized derivative of a decreasing ‘standing wave’ solution (phi ) of (partial ^2_{v_1} phi - V'(phi )=0) on ({mathbb {R}}), where (V') is the homogenization of the Glauber rates. In two dimensions, the limit is (textbf{e}(v_1)Z_t(v_2)) where (Z_t) is the solution of a one dimensional stochastic heat equation. The appearance of the function (textbf{e}(cdot )) in the limit field indicates that the interface fluctuation retains the shape of the transition layer (phi ).

We study the evolution of the Gross–Pitaevskii equation linearized around the Ginzburg–Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable (L^2)-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

Information stored in quantum states naturally decays under noise. Many earlier works have quantified and demonstrated lower bounds on decay rates of quantum relative entropy, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information when noise is sufficiently weak? In several scenarios, including when a quantum channel induces decay of relative entropy to its own fixed point, we find such multiplicative converse bounds. However, there are other cases in which the ratio can fall by an arbitrary extent. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.

The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mariño and Vafa as a means to calculate open Gromov–Witten invariants of toric Calabi–Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov–Witten invariants of log Calabi–Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov–Witten invariants.