Saddle-point Solution Research Articles

Dynamics of order parameters of nonstoquastic Hamiltonians in the adaptive quantum Monte Carlo method.

We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic p-spin model with infinite-range interactions. The p-spin model has a first-order phase transition for p>2. In the case of p≥5, the p-spin model with antiferromagnetic XX interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a nonstoquastic Hamiltonian, resulting in a negative sign problem. To simulate the p-spin model with antiferromagnetic XX interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the antiferromagnetic XX interaction as fluctuations of the transverse magnetic field. A previous study [J. Inoue, J. Phys. Conf. Ser. 233, 012010 (2010)1742-659610.1088/1742-6596/233/1/012010] derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method, and we verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.

Non-perturbative probability distribution function for cosmological counts in cells

We present a non-perturbative calculation of the 1-point probability distribution function (PDF) for the spherically-averaged matter density field. The PDF is represented as a path integral and is evaluated using the saddle-point method. It factorizes into an exponent given by a spherically symmetric saddle-point solution and a prefactor produced by fluctuations. The exponent encodes the leading sensitivity of the PDF to the dynamics of gravitational clustering and statistics of the initial conditions. In contrast, the prefactor has only a weak dependence on cosmology. It splits into a monopole contribution which is evaluated exactly, and a factor corresponding to aspherical fluctuations. The latter is crucial for the consistency of the calculation: neglecting it would make the PDF incompatible with translational invariance. We compute the aspherical prefactor using a combination of analytic and numerical techniques. We demonstrate the factorization of spurious enhanced contributions of large bulk flows and their cancellation due to the equivalence principle. We also identify the sensitivity to the short-scale physics and argue that it must be properly renormalized. The uncertainty associated with the renormalization procedure gives an estimate of the theoretical error. For zero redshift, the precision varies from sub percent for moderate density contrasts to tens of percent at the tails of the distribution. It improves at higher redshifts. We compare our results with N-body simulation data and find an excellent agreement.

ε-Saddle point solution for a nonlinear robot coordination game

This paper presents an approximated or -saddle point solution for a nonlinear game proposed to solve the coordination of a set of planar robots. The dynamic equation of each individual robot is given by the polynomial approximation of the nonlinear model, this introduces interesting challenges in the coordination control problem. Due to the difficulties in finding an exact solution, we explore the possibilities of the so-called state-dependent Riccati equations to find a near or -saddle point solution that effectively drives of the robots to a particular coordinated movement. A pre-designed trajectory is given to one of the robots in order to achieve the coordination by the group of robots. The proposed solution is given as a polynomial Riccati-like state-dependent differential equation which utilizes a -linear form tensor representation for its polynomial part. A numerical example is presented to illustrate the effectiveness of the approach.

A dimension-reduction solution of free-time differential games for spacecraft pursuit-evasion

This paper presents a dimension-reduction method for solving the free-time pursuit-evasion game between two spacecraft near circular orbits. Theoretically, finding the saddle-point solution of the game results in solving a high-dimensional (twenty-four in our case) two-point boundary value problem (TPBVP), which is quite difficult and computationally intensive. By using the circular-orbit variational equations to model the relative states of two spacecraft near circular orbits, the 24-dimensional TPBVP is firstly transformed into solving a set of four-dimensional nonlinear equations. Then, a hybrid numerical algorithm is proposed to solve the equations, where the differential evolution algorithm is used to obtain an initial guess and the Newton's iteration method is used to find the accurate solution. Numerical results show that the proposed method is more efficient than previous methods, and that the proposed method are suitable for large-phasing-distance orbital pursuit-evasion problems.

Accelerated Distributed Primal-Dual Dynamics Using Adaptive Synchronization

This paper proposes an adaptive primal-dual dynamics for distributed optimization in multi-agent systems. The proposed dynamics incorporates an adaptive synchronization law that reinforces the interconnection strength between the coupled agents. By strengthening the synchronization between the primal variables of the coupled agents, the given law accelerates the convergence of the proposed dynamics to the saddle-point solution. The resulting dynamics is represented as a feedback-interconnected networked system that proves to be passive. The passivity properties of the proposed dynamics are exploited along with the LaSalle's invariance principle for hybrid systems, to establish asymptotic convergence and stability of the saddle-point solution. Further, the primal dynamics is analyzed for the rate of convergence and stronger convergence bounds are established, it is proved that the primal dynamics achieve accelerated convergence under the adaptive synchronization. The robustness of the proposed dynamics is quantified using $L_{2}$ -gain analysis and the correlation between the rate of convergence and robustness of the proposed dynamics are presented. The effectiveness of the proposed dynamics is demonstrated by applying it to solve distributed least squares and distributed support vector machines problems.

Dynamical structure factor of the triangular antiferromagnet: Schwinger boson theory beyond mean field

We compute the zero temperature dynamical structure factor $S({\bf q},\omega)$ of the triangular lattice Heisenberg model (TLHM) using a Schwinger boson approach that includes the Gaussian fluctuations ($1/N$ corrections) of the saddle point solution. While the ground state of this model exhibits a well-known 120$^{\circ}$ magnetic ordering, experimental observations have revealed a strong quantum character of the excitation spectrum. We conjecture that this phenomenon arises from the proximity of the ground state of the TLHM to the quantum melting point separating the magnetically ordered and spin liquid states. Within this scenario, magnons are described as collective modes (two spinon-bound states) of a spinon condensate (Higgs phase) that spontaneously breaks the SU(2) symmetry of the TLHM. Crucial to our results is the proper account of this spontaneous symmetry breaking. The main qualitative difference relative to semi-classical treatments ($1/S$ expansion) is the presence of a high-energy spinon continuum extending up to about three times the single-magnon bandwidth. In addition, the magnitude of the ordered moment ($m=0.224$) agrees very well with numerical results and the low energy part of the single-magnon dispersion is in very good agreement with series expansions. Our results indicate that the Schwinger boson approach is an adequate starting point for describing the excitation spectrum of some magnetically ordered compounds that are near the quantum melting point separating this Higgs phase from the {\it deconfined} spin liquid state.

Supersymmetry breaking phase in three dimensional large N gauge theories

Three dimensional supersymmetric gauge theories are often in a gapped phase, in which SUSY is spontaneously broken, if all the matter fields are massive and decoupled in the low energy. We study this phase in the large N limit using the localization technique for the theory on the ellipsoid, which interpolates the round three sphere and the flat space compactified on S1. We find a large N saddle point solution for the gauge theory with some massive matter fields. This solution gives a vanishing (generalized) Polyakov loop in the flat space limit, thus, it corresponds to the confining phase at the leading order in the 1/N expansion.

A Min–Max Approach to Event- and Self-Triggered Sampling and Regulation of Linear Systems

This paper presents both an event- and a self-triggered sampling and regulation scheme for continuous time linear dynamic systems by using zero-sum game formulation. A novel performance index is defined wherein the control policy is treated as the first player and the threshold for control input error due to aperiodic dynamic feedback is treated as the second player. The optimal control policy and sampling intervals are generated using the saddle point or Nash equilibrium solution, which is obtained from the corresponding game algebraic Riccati equation. To determine the optimal event-based sampling scheme, an event-triggering condition is derived by utilizing the worst case control input error as the threshold. To avoid the additional hardware for the event-triggering mechanism, a near optimal self-triggering condition is derived to determine the future sampling instants given the current state vector. To guarantee Zeno-free behavior in both the event- and self-triggered closed-loop systems, the minimum intersample times are shown to be lower bounded by a nonzero positive number. Asymptotic stability of the closed-loop system is ensured using Lyapunov stability analysis. Finally, simulation examples are provided to substantiate the analytical claims.

Saddle point solution of the Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu Ising model

Saddle point solution of the Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu Ising model

Saddle-point solutions to Dirichlet problems on the Sierpiński gasket

We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters.

Full-state autopilot-guidance design under a linear quadratic differential game formulation

Full-state single-loop and full-state two-loop autopilot-guidance architectures are derived under a linear quadratic differential game formulation. In the full-state single-loop case, the guidance command is injected directly to the actuator, whereas in the full-state two-loop case, it is the input to the autopilot loop. To prevent impractical end-game scenarios, where the states diverge to unacceptable values, a cost function that includes appropriate running cost terms on some of the states is proposed. The conditions for obtaining an equivalence relation between the full-state single-loop and full-state two-loop architectures are derived under a linear quadratic differential game formulation and the proposed cost function. Under such a formulation, the two full-state architectures are identical if and only if the number of guidance commands matches the number of available controllers. The guidance laws performance is illustrated using an interceptor missile having forward and aft controls in linear and nonlinear settings, while considering two types of evasion strategies. The first strategy is a linear controller based on the linear quadratic differential game solution. The second strategy is a “bang–bang” controller based on the optimal evasion solution. It is shown that the linear evasion strategy may not be suitable to represent a realistic evading strategy. In addition, the conditions for the existence of a saddle point solution are analyzed for the two full-state guidance laws.

Open-loop solution of a defender–attacker–target game: penalty function approach

A defender–attacker–target problem with non-moving target is considered. This problem is modelled by a pursuit-evasion zero-sum differential game with linear dynamics and quadratic cost functional. In this game, the pursuer is the defender, while the evader is the attacker. The objective of the pursuer is to minimise the cost functional, while the evader has two objectives: to maximise the cost functional and to keep a given terminal state inequality constraint. The open-loop saddle point solution of this game is obtained in the case where the transfer functions of the controllers for the defender and the attacker are of arbitrary orders.

BPS sphalerons in the F2 nonlinear sigma model

We construct static and also time-dependent solutions in a non-linear sigma model with target space being the flag manifold $F_2=SU(3)/U(1)^2$ on the four dimensional Minkowski space-time by analytically solving the second order Euler-Lagrange equation. We show the static solutions saturate an energy lower bound and can be derived from coupled first order equations though they are saddle point solutions. We also discuss basic properties of the time-dependent solutions.

'Quantum Equilibrium-Disequilibrium': Asset Price Dynamics, Symmetry Breaking, and Defaults as Dissipative Instantons

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” (QED) model. The QED model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from the QED dynamics. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.

Communication Complexity of Dual Decomposition Methods for Distributed Resource Allocation Optimization

Dual decomposition methods are among the most prominent approaches for finding primal/dual saddle point solutions of resource allocation optimization problems. To deploy these methods in the emerging Internet of things networks, which will often have limited data rates, it is important to understand the communication overhead they require. Motivated by this, we introduce and explore two measures of communication complexity of dual decomposition methods to identify the most efficient communication among these algorithms. The first measure is $\epsilon$ -complexity, which quantifies the minimal number of bits needed to find an $\epsilon$ -accurate solution. The second measure is $b$ -complexity, which quantifies the best possible solution accuracy that can be achieved from communicating $b$ bits. We find the exact $\epsilon$ - and $b$ -complexity of a class of resource allocation problems where a single supplier allocates resources to multiple users. For both the primal and dual problems, the $\epsilon$ -complexity grows proportionally to $\log _2(1/\epsilon)$ and the $b$ -complexity proportionally to $1/2^b$ . We also introduce a variant of the $\epsilon$ - and $b$ -complexity measures where only algorithms that ensure primal feasibility of the iterates are allowed. Such algorithms are often desirable because overuse of the resources can overload the respective systems, e.g., by causing blackouts in power systems. We provide upper and lower bounds on the convergence rate of these primal feasible complexity measures. In particular, we show that the $b$ -complexity cannot converge at a faster rate than $\mathcal {O}(1/b)$ . Therefore, the results demonstrate a tradeoff between fast convergence and primal feasibility. We illustrate the result by numerical studies.

Multi-scaling in the critical phenomena in the quenched disordered systems

The Landau–Ginzburg–Wilson Hamiltonian with random temperature for the phase transition in disordered systems from the Griffiths phase to ordered phase is reexamined. From the saddle point solutions, especially the excited state solutions, it is shown that the system self-organizes into blocks coupled with their neighbors like superspins, which are emergent variables. Taking the fluctuation around these saddle point solutions into account, we get an effective Hamiltonian, including the emergent superspins of the blocks, the fluctuation around the saddle point solutions, and their couplings. Applying Stratonovich–Hubbard transformation to the part of superspins, we get a Landau–Ginzburg–Wilson Hamiltonian for the blocks. From the saddle point equations for the blocks, we can get the second generation blocks, of which sizes are much larger than the first generation blocks. Repeating this procedure again and again, we get many generations of blocks to describe the asymptotic behavior. If a field is applied, the effective field on the superspins is multiplied greatly and proportional to the block size. For a very small field, the effective field on the higher generation superspins can be so strong to cause the superspins polarized radically. This can explain the extra large critical isotherm exponent discovered in the experiments. The phase space of reduced temperature vs. field is divided into many layers , in which different generation blocks dominate the critical behavior. The sizes of the different generation emergent blocks are new relevant length scales. This can explain a lot of puzzles in the experiments and the Monte Carlo simulation.

Nonlinear Control for Spacecraft Pursuit-Evasion Game Using the State-Dependent Riccati Equation Method

Spacecraft pursuit-evasion game is formulated as a two-player zero-sum differential game. The Hill reference frame is used to describe the dynamics of the game. The goal is to derive feedback control law which forms a saddle point solution to the game. Both spacecraft use continuous-thrust engines. The linear quadratic differential game theory is applied to derive a linear control law. This theory is extended to derive a nonlinear control law using the state-dependent Riccati equation method. A state-dependent coefficient matrix is developed for this purpose. Various scenarios are considered for comparing the performance of the linear and nonlinear laws. In all the scenarios, the efficacy of the nonlinear control law is found to be superior to that of the linear control law with computation cost similar to the linear law.

One dimensionalization in the spin-1 Heisenberg model on the anisotropic triangular lattice

We investigate the effect of dimensional crossover in the ground state of the antiferromagnetic spin-$1$ Heisenberg model on the anisotropic triangular lattice that interpolates between the regime of weakly coupled Haldane chains ($J^{\prime}\! \!\ll\!\! J$) and the isotropic triangular lattice ($J^{\prime}\!\!=\!\!J$). We use the density-matrix renormalization group (DMRG) and Schwinger boson theory performed at the Gaussian correction level above the saddle-point solution. Our DMRG results show an abrupt transition between decoupled spin chains and the spirally ordered regime at $(J^{\prime}/J)_c\sim 0.42$, signaled by the sudden closing of the spin gap. Coming from the magnetically ordered side, the computation of the spin stiffness within Schwinger boson theory predicts the instability of the spiral magnetic order toward a magnetically disordered phase with one-dimensional features at $(J^{\prime}/J)_c \sim 0.43$. The agreement of these complementary methods, along with the strong difference found between the intra- and the interchain DMRG short spin-spin correlations; for sufficiently large values of the interchain coupling, suggests that the interplay between the quantum fluctuations and the dimensional crossover effects gives rise to the one-dimensionalization phenomenon in this frustrated spin-$1$ Hamiltonian.

Unbinding slave spins in the Anderson impurity model

We show that a generic single-orbital Anderson impurity model, lacking for instance any kind of particle-hole symmetry, can be exactly mapped without any constraint onto a resonant level model coupled to two Ising variables, which reduce to one if the hybridisation is particle-hole symmetric. The mean-field solution of this model is found to be stable to unphysical spontaneous magnetisation of the impurity, unlike the saddle-point solution in the standard slave-boson representation. Remarkably, the mean-field estimate of the Wilson ratio approaches the exact value $R_\text{W}=2$ in the Kondo regime.

Non-minimal derivative coupling in Palatini cosmology: acceleration in chaotic inflation potential

NMDC-Palatini cosmology in slow-roll regime is of our interests. We present flat FLRW cosmological NMDC-Palatini field equations and acceleration condition. Late time trajectory is approximated. Chaotic inflation potential is considered here as it is viable in a range of negative coupling constant as constrained by CMB results. Phase portraits show that the NMDC-Palatini gives new saddle-point solutions. The NMDC-Palatini effect decreases the acceleration-allowed region and forbid acceleration at small-field values.