More on complexity of operators in quantum field theory

Recently it has been shown that the complexity of SU($n$) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten $p$-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as $k$-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU($n$) groups.Comment: Modified the Sec. 4.1, where we offered a powerful proof: if (1) the ket vector and bra vector in quantum mechanics contain same physics, or (2) adding divergent terms to a Lagrangian will not change underlying physics, then complexity in quantum mechanics must be bi-invariant

Principles and symmetries of complexity in quantum field theory

Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it continuous} systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU($n$) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in $k$-local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schr\"{o}dinger's equation in isolated systems - the quantum state evolves by the process of minimizing "computational cost."Comment: Published version; added a short introduction on Finsler geometr

Electrical Control of Magnetization in Charge-ordered Multiferroic LuFe2O4

LuFe2O4 exhibits multiferroicity due to charge order on a frustrated triangular lattice. We find that the magnetization of LuFe2O4 in the multiferroic state can be electrically controlled by applying voltage pulses. Depending on with or without magnetic fields, the magnetization can be electrically switched up or down. We have excluded thermal heating effect and attributed this electrical control of magnetization to an intrinsic magnetoelectric coupling in response to the electrical breakdown of charge ordering. Our findings open up a new route toward electrical control of magnetization.Comment: 14 pages, 5 figure

Portfolio Optimization on Multivariate Regime Switching GARCH Model with Normal Tempered Stable Innovation

We propose a Markov regime switching GARCH model with multivariate normal tempered stable innovation to accommodate fat tails and other stylized facts in returns of financial assets. The model is used to simulate sample paths as input for portfolio optimization with risk measures, namely, conditional value at risk and conditional drawdown. The motivation is to have a portfolio that avoids left tail events by combining models that incorporates fat tail with optimization that focuses on tail risk. In-sample test is conducted to demonstrate goodness of fit. Out-of-sample test shows that our approach yields higher performance measured by Sharpe-like ratios than the market and equally weighted portfolio in recent years which includes some of the most volatile periods in history. We also find that suboptimal portfolios with higher return constraints tend to outperform optimal portfolios

Spatiomechanical Modulation of EphB4-Ephrin-B2 Signaling in Neural Stem Cell Differentiation.

Interactions between EphB4 receptor tyrosine kinases and their membrane-bound ephrin-B2 ligands on apposed cells play a regulatory role in neural stem cell differentiation. With both receptor and ligand constrained to move within the membranes of their respective cells, this signaling system inevitably experiences spatial confinement and mechanical forces in conjunction with receptor-ligand binding. In this study, we reconstitute the EphB4-ephrin-B2 juxtacrine signaling geometry using a supported-lipid-bilayer system presenting laterally mobile and monomeric ephrin-B2 ligands to live neural stem cells. This experimental platform successfully reconstitutes EphB4-ephrin-B2 binding, lateral clustering, downstream signaling activation, and neuronal differentiation, all in a configuration that preserves the spatiomechanical aspects of the natural juxtacrine signaling geometry. Additionally, the supported bilayer system allows control of lateral movement and clustering of the receptor-ligand complexes through patterns of physical barriers to lateral diffusion fabricated onto the underlying substrate. The results from this study reveal a distinct spatiomechanical effect on the ability of EphB4-ephrin-B2 signaling to induce neuronal differentiation. These observations parallel similar studies of the EphA2-ephrin-A1 system in a very different biological context, suggesting that such spatiomechanical regulation may be a common feature of Eph-ephrin signaling