arXiv subjects

Yangang Chen

Publications and source records attributed to Yangang Chen.

4 recordsLinked to original sources

arXiv2021

Towards Spacetime Entanglement Entropy for Interacting Theories

Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin's entropy to non-Gaussian theories where Wick's theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.

hep-th↗

arXiv2019

Monotone Mixed Finite Difference Scheme for Monge-Ampère Equation

In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge-Ampère equation. In order to accomplish this, we convert the Monge-Ampère equation to an equivalent Hamilton-Jacobi-Bellman (HJB) equation. Based on the HJB formulation, we apply the standard 7-point stencil discretization, which is second order accurate, to the grid points wherever monotonicity holds, and apply semi-Lagrangian wide stencil discretization elsewhere to ensure monotonicity on the entire computational domain. By dividing the admissible control set into six regions and optimizing the sub-problem in each region, the computational cost of the optimization problem at each grid point is reduced from $O(M^2)$ to $O(1)$ when the standard 7-point stencil discretization is applied and to $O(M)$ otherwise, where the discretized control set is $M \times M$. We prove that our numerical scheme satisfies consistency, stability, monotonicity and strong comparison principle, and hence is convergent to the viscosity solution of the Monge-Ampère equation. In the numerical results, second order convergence rate is achieved when the standard 7-point stencil discretization is applied monotonically on the entire computation domain, and up to order one convergence is achieved otherwise. The proposed mixed scheme yields a smaller discretization error and a faster convergence rate compared to the pure semi-Lagrangian wide stencil scheme.

math.NA↗

arXiv2019

Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.

q-fin.CP↗

arXiv2014

Entanglement contour

In the context of characterizing the structure of quantum entanglement in many-body systems, we introduce the entanglement contour, a tool to identify which real-space degrees of freedom contribute, and how much, to the entanglement of a region A with the rest of the system B. The entanglement contour provides a complementary, more refined approach to characterizing entanglement than just considering the entanglement entropy between A and B, with several concrete advantages. We illustrate this in the context of ground states and quantum quenches in fermionic quadratic systems. For instance, in a quantum critical system in $D = 1$ spatial dimensions, the entanglement contour allows us to determine the central charge of the underlying conformal field theory from just a single partition of the system into regions A and B, (using the entanglement entropy for the same task requires considering several partitions). In $D \geq 2$ dimensions, the entanglement contour can distinguish between gapped and gapless phases that obey a same boundary law for entanglement entropy. During a local or global quantum quench, the time-dependent contour provides a detailed account of the dynamics of entanglement, including propagating entanglement waves, which offers a microscopic explanation of the behavior of the entanglement entropy as a function of time.

cond-mat.str-el↗