Non-local reparametrization action in coupled Sachdev-Ye-Kitaev models

We continue the investigation of coupled Sachdev-Ye-Kitaev (SYK) models without Schwarzian action dominance. Like the original SYK, at large N and low energies these models have an approximate reparametrization symmetry. However, the dominant action for reparametrizations is non-local due to the presence of irrelevant local operator with small conformal dimension. We semi-analytically study different thermodynamic properties and the 4-point function and demonstrate that they significantly differ from the Schwarzian prediction. However, the residual entropy and maximal chaos exponent are the same as in Majorana SYK. We also discuss chain models and finite N corrections.

Light-ray moments as endpoint contributions to modular Hamiltonians

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J(n) of modular weight n over a spacelike surface passing through x = 0. For |n| ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator J(n) and its descendants. For perturbations on null planes only moments of J(n) itself contribute.

Optimal Emission Reduction and Pricing in the Tourism Supply Chain Considering Different Market Structures and Word-of-Mouth Effect

In the context of carbon tax policy and word-of-mouth, local operators and tour operators in the tourism supply chain need to determine optimal wholesale price, carbon reduction level, and retail price of tour packages strategies. To address these decision-making issues, while considering the word-of-mouth effect, our paper considers a local operator determining wholesale price and carbon reduction level of the tour package and a tour operator determining retail price of the tour package. According to different bargaining powers, we study three scenarios: the local operator leading Stackelberg (LL), the tour operator leading Stackelberg (TL), and the static Nash game (NG). We develop three theoretical models and present some insights. We find that tourist’s sensitivity to word-of-mouth has positive (negative) impacts on optimal wholesale price, carbon reduction level, retail price, demand, and profits if the impact of word-of-mouth is positive (negative), while the impact of word-of-mouth is always having positive impacts on optimal decisions, demand, and profits. Interestingly, the NG market structure contributes the most environmentally-friendly products but mostly hurts the environment. The local operator under LL can obtain the largest profit, which is even larger than the profit of the tour operator, while the tour operator under NG and TL can obtain more profit than the local operator.

A survey on numerical methods for spectral Space-Fractional diffusion problems

The survey is devoted to numerical solution of the equation $ {\mathcal A}^\alpha u=f $, 0 < α<1, where $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of $ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator $ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh.The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40].Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits

We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously prove that if a circuit has an ultralocal soliton moving to the left (right), the entanglement of local operators initially supported on even (odd) sites saturates to a constant value and its dynamics can be computed exactly. Importantly, this does not bound the growth of complexity in chiral circuits, where solitons move only in one direction, say to the left. Indeed, in this case we observe numerically that operators on the odd sublattice have unbounded entanglement. Finally, we present a closed-form expression for the local-operator entanglement entropies in circuits with ultralocal solitons moving in both directions. Our results hold irrespectively of integrability.