Mathematical Physics, Analysis and Geometry最新文献

We study a class of quantum Hamiltonian models describing a family of N two-level systems (spins) coupled with a structured boson field of positive mass, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences. Spin–spin interactions which do not modify the total number of excitations are also included. Generalizing previous results in the single-spin case, we provide explicit expressions for the self-adjointness domain and the resolvent of these models, both of them carrying an intricate dependence on the spin–field and spin–spin coupling via a family of concatenated propagators. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors via normalizable ones, for example an ultraviolet cutoff.

We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix C is minimized over the set of all bipartite coupling states (rho ^{AB}) with fixed reduced density matrices (rho ^A) and (rho ^B) of size m and n. The minimum quantum optimal transport cost (textrm{T}^Q_{C}(rho ^A,rho ^B)) can be efficiently computed using semidefinite programming. In the case (m=n) the cost (textrm{T}^Q_{C}) gives a semidistance if and only if C is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if C satisfies the above conditions, then (sqrt{textrm{T}^Q_{C}}) induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix (C^Q) to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for (textrm{T}^Q_{C^Q}) for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Monge–Kantorovich distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity—the SWAP-fidelity—and compare its properties with the standard Uhlmann–Jozsa fidelity. We also discuss the quantum optimal transport for general d-partite systems.

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible (*)-structures, encompassing the quasitriangular case.

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

We analyze the symmetric simple partial exclusion process, which allows at most (alpha ) particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter (theta in {mathbb {R}}). We prove that the hydrodynamic behavior is given by the heat equation and depending on the value of (theta ), the equation is supplemented with different boundary conditions. Setting (alpha = 1) we find the results known in Baldasso et al. (J Stat Phys 167(5):1112–1142, 2017) and Bernardin et al. (Markov Processes Relat. Fields 25:217–274, 2017) for the symmetric simple exclusion process.

Using Riemann–Hilbert methods, we establish a Tracy–Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.

In this paper, we study the HC-model with a countable set (mathbb Z) of spin values on a Cayley tree of order (kge 2). This model is defined by a countable set of parameters (that is, the activity function (lambda _i>0), (iin mathbb Z)). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:

• Let (Lambda =sum _ilambda _i). For (Lambda =+infty ) there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);

• For (Lambda <+infty ), the uniqueness of TIGM is proved;

• Let (Lambda _textrm{cr}(k)=frac{k^k}{(k-1)^{k+1}}). If (0<Lambda le Lambda _textrm{cr}), then there is exactly one TPGM that is TIGM;

• For (Lambda >Lambda _textrm{cr}), there are exactly three TPGMs, one of which is TIGM.

We show that Rieffel’s quantum Gromov–Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov–Hausdorff distance applied to the associated state spaces.

In this paper we consider the Blume–Emery–Griffiths model on Cayley trees. We reduce the problem of describing the splitting Gibbs measures of the Blume–Emery–Griffiths model to the description of the solutions of some algebraic equation. Also, we analyse the set of translation-invariant splitting Gibbs measures for a two parametric BEG model on Cayley trees.

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice (mathbb {Z}^d) in dimensions (d>4), in the vicinity of the critical point, and prove an upper bound (|x|^{-(d-2)}exp [-c|x|/xi ]), where the correlation length (xi ) has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions (d{>}4) has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.