Divide and conquer method for proving gaps of frustration free Hamiltonians

Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter $n$ is at most $o\left(\frac{\log(n)^{2+\epsilon}}{n}\right)$ for any positive $\epsilon$.Comment: This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1742-5468/aaa793, Journal of Statistical Mechanics: Theory and Experiment, March 201

Superadditivity of quantum relative entropy for general states

The property of superadditivity of the quantum relative entropy states that, in a bipartite system $\mathcal{H}_{AB}=\mathcal{H}_A \otimes \mathcal{H}_B$, for every density operator $\rho_{AB}$ one has $D( \rho_{AB} || \sigma_A \otimes \sigma_B ) \ge D( \rho_A || \sigma_A ) +D( \rho_B || \sigma_B)$. In this work, we provide an extension of this inequality for arbitrary density operators $\sigma_{AB}$. More specifically, we prove that $\alpha (\sigma_{AB})\cdot D({\rho_{AB}}||{\sigma_{AB}}) \ge D({\rho_A}||{\sigma_A})+D({\rho_B}||{\sigma_B})$ holds for all bipartite states $\rho_{AB}$ and $\sigma_{AB}$, where $\alpha(\sigma_{AB})= 1+2 || \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - \mathbb{1}_{AB} ||_\infty$.Comment: 14 pages. v3: Final version. The main theorem has been improved, adding a fourth step to its proof and also some remarks. v2: There was a flaw in the proof of the previous version. This has been corrected in this version. The constant appearing in the main Theorem has changed accordingl

A Multiscale Gibbs-Helmholtz Constrained Cubic Equation of State

This paper presents a radically new approach to cubic equations of state (EOS) in which the Gibbs-Helmholtz equation is used to constrain the attraction or energy parameter, a. The resulting expressions for a(T, p) for pure components and a(T, p, x) for mixtures contain internal energy departure functions and completely avoid the need to use empirical expressions like the Soave alpha function. Our approach also provides a novel and thermodynamically rigorous mixing rule for a(T, p, x).When the internal energy departure function is computed using Monte Carlo or molecular dynamics simulations as a function of current bulk phase conditions, the resulting EOS is a multiscale equation of state. The proposed new Gibbs-Helmholtz constrained (GHC) cubic equation of state is used to predict liquid densities at high pressure and validated using experimental data from literature. Numerical results clearly show that the GHC EOS provides fast and accurate computation of liquid densities at high pressure, which are needed in the determination of gas hydrate equilibria

Are Metabolic Pathways Nash Equilibria?

Flux balance analysis (FBA) has been the mainstay for understanding metabolic networks for many years. However, recently Lucia and DiMaggio have shown that metabolic pathways are more correctly modeled using game theory, specifically Nash Equilibrium, because it captures the natural competition among enzymes. The key ideas behind the Nash equilibrium approach to metabolic pathway analysis are that 1. Enzymes are treated as players in a multi-player game. 2. The objective or payoff function for each player is a constrained nonlinear programming (NLP) problem where each player (enzyme) minimizes the Gibbs free energy of the reaction it catalyzes subject to element mass balances. 3. The goal of the metabolic network is to find the best overall solution given the natural competition for nutrients among enzymes. The Nash equilibrium approach has many advantages over FBA and its many variants, constraint-based modeling (CBM), and kinetic approaches to determining fluxes and other information associated with any metabolic network. One can 1. include co-factors in modeling sub-networks. 2. model electrolyte solution behavior and incorporate charge balancing. 3. include feedback, allosteric, and other forms of inhibition. 4. explicitly include enzyme-substrate reactions as part of the model. 5. up-scale genetic information and consider mutations and/or re-engineered enzymes. 6. model up/down regulation of enzymes. In this talk, I will present the fundamentals of Nash Equilibrium as it applies to metabolic pathways and present a survey of results for a number of common pathway including glycolysis, the Krebs cycle, Acetone-Butanol-Ethanol (ABE) production, the mevalonate and methionine salvage pathways, and the ornithine cycle. I will also show that, in cases where experimental data is available, numerical predictions using the Nash Equilibrium approach show remarkably good agreement with experimental metabolite concentrations and other biological metrics such as turnover ratio

Multi-Scale, Gibbs-Helmholtz Constrained Cubic Equations of State

Cubic equations of state (EOS) are often desirable when fast, reliable determination of physical properties is required. However, it is well known that they generally perform poorly at high pressure, especially in the compressed liquid regime. In this talk I will describe a radically new approach to the development of cubic equations of state. Specifically, I show how to develop a new thermodynamically rigorous framework for pure component parameters in cubic equations of state (EOS) based on using the Gibbs-Helmholtz equation as a constraint. I will present both closed-form and integral multi-scale expressions for pure component parameters that directly incorporate molecular level information obtained from Monte Carlo or molecular dynamics simulations. I will also present a new mixing rule based on the Gibbs-Helmholtz equation for mixtures. The resulting new family of multi-scale, thermodynamically constrained cubic EOS is truly predictive and has the capability of directly accounting for molecular interactions in non-electrolyte systems (e.g., van der Waals forces) as well as electrostatic effects in weak/strong electrolyte solutions (i.e., charge-charge, charge-dipole, quadrupole, etc.) through the use of an appropriate potential energy function. High pressure behavior of many pure components and mixtures has been determined. Numerical results are compared with experimental data available in the literature for carbon dioxide, water, and carbon dioxide/water mixtures at high pressure and show excellent agreement. Numerical results are also compared to numerical results for other existing cubic EOS at high pressure and clearly show that the proposed Gibbs-Helmholtz constrained (GHC) cubic EOS is superior to all existing cubic EOS. Many geometric illustrations are used to elucidate key points

Tensor network representations from the geometry of entangled states

Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.Comment: 35 pages, 9 figure

Undecidability of the Spectral Gap in One Dimension

The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.Comment: 7 figure

A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs

We consider the spectral gap question for AKLT models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of $G$ with a chain of $n$ sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States (TNS) approach from a work by Abdul-Rahman et. al. 2020, we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.Comment: Discussion of additional results, and minor correction