Infinite State Research Articles

Superfluidity in the one-dimensional Bose-Hubbard model

We study superfluidity in the one-dimensional Bose-Hubbard model using a variational matrix product state technique. We determine the superfluid density as a function of the Hubbard parameters by calculating the energy cost of phase twists in the thermodynamic limit. As the system is critical, correlation functions decay as power laws and the entanglement entropy grows with the bond dimension of our variational state. We relate the resulting scaling laws to the superfluid density. We compare two different algorithms for optimizing the infinite matrix product state and develop a physical explanation why one of them (VUMPS) is more efficient than the other (iDMRG). Finally, we comment on finite-temperature superfluidity in one dimension and how our results can be realized in cold-atom experiments.

A new approach to modeling and simulation of the nonlinear, fractional behavior of Li-ion battery cells

Two nonlinear infinite dimensional models of a Lithium-ion battery are introduced. The proposed models interpolate linear fractional models which are obtained from a widely spread equivalent circuit model. The linear models are parameterized by means of so-called dynamic impedance measurements, i.e. impedance measurements in several operating points. In each operating point, the fractional input output behavior is recovered by an infinite dimensional state space description. For numerical implementation the introduced models require for an appropriate approximation. Therefore, a Krylov subspace method and a finite element approach are proposed. The nonlinear lumped parameter models are validated on the basis of experimental data.

Cooling classical many-spin systems using feedback control

We propose a technique for polarizing and cooling finite many-body classical systems using feedback control. The technique requires the system to have one collective degree of freedom conserved by the internal dynamics. The fluctuations of other degrees of freedom are then converted into the growth of the conserved one. The proposal is validated using numerical simulations of classical spin systems in a setting representative of nuclear magnetic resonance experiments. In particular, we were able to achieve 90% polarization for a lattice of 1000 classical spins starting from an unpolarized infinite temperature state.

Dynamical signatures of the one-dimensional deconfined quantum critical point

We study the critical scaling and dynamical signatures of fractionalized excitations at two different deconfined quantum critical points (DQCPs) in an S = 1/2 spin chain using the time evolution of infinite matrix product states. The scaling of the correlation functions and the dispersion of the conserved current correlations explicitly show the emergence of enhanced continuous symmetries at these DQCPs. The dynamical structure factors in several different channels reveal the development of deconfined fractionalized excitations at the DQCPs. Furthermore, we find an effective spin–charge separation at the DQCP between the ferromagnetic (FM) and valence bond solid (VBS) phases, and identify two continua associated with different types of fractionalized excitations at the DQCP between the X-direction and Z-direction FM phases. Our findings not only provide direct evidence for the DQCP in one dimension but also shed light on exploring the DQCP in higher dimensions.

Queueing Network Controls via Deep Reinforcement Learning

Novel advanced policy gradient (APG) methods, such as trust region policy optimization and proximal policy optimization (PPO), have become the dominant reinforcement learning algorithms because of their ease of implementation and good practical performance. A conventional setup for notoriously difficult queueing network control problems is a Markov decision problem (MDP) that has three features: infinite state space, unbounded costs, and long-run average cost objective. We extend the theoretical framework of these APG methods for such MDP problems. The resulting PPO algorithm is tested on a parallel-server system and large-size multiclass queueing networks. The algorithm consistently generates control policies that outperform state-of-art heuristics in literature in a variety of load conditions from light to heavy traffic. These policies are demonstrated to be near optimal when the optimal policy can be computed. A key to the successes of our PPO algorithm is the use of three variance reduction techniques in estimating the relative value function via sampling. First, we use a discounted relative value function as an approximation of the relative value function. Second, we propose regenerative simulation to estimate the discounted relative value function. Finally, we incorporate the approximating martingale-process method into the regenerative estimator.

Learning-based airborne sensor task assignment in unknown dynamic environments

In sensor management, the existing researches rely on traditional system modeling and strive to maximize the information superiority. In fact, on the one hand, complex environmental disturbance, incomplete information or uncooperative behavior in air combat missions often bring out unknown system evolution; on the other hand, to take full advantage of sensor effectiveness is of course essential, but more importantly, the detection security is the primary guarantee. This paper proposes the airborne sensor task assignment problem in unknown dynamic environments. Different from traditional methods that minimize the estimation error covariance or information entropy based on system dynamic model, our scheme needs to maximize agent survival while maintaining the necessary sensor detection without such model support. In assignment implementation, it is not straightforward to apply existing reinforcement learning methods, but design the state space and rewards ingeniously to meet the actual combat requirements. First, instead of selecting the locations of agents and targets as fundamental and infinite state variables, we consider the situation variables, such as target threat ranking together with cumulative radiation and information acquisition indication of sensors, which are all discrete state variables to reduce computational burden. Second, the reward structure is also designed based on the complex constraints of the mission, which is to encourage lower assignment risk and relatively full utilization of sensing, while penalizing too dangerous continuance assignment and inadequate assignment revenue. Simulations show that our proposed scheme achieves the desirable mission completion rate and the acceptable target tracking accuracy.

Quantum criticality in spin-1/2 anisotropic XY model with staggered Dzyaloshinskii–Moriya interaction

By utilizing the infinite time evolving block decimation method in infinite matrix product state representation, the quantum criticality and critical exponents varying are investigated in the spin-1/2 anisotropic XY chain with staggered Dzyaloshinskii–Moriya interaction. The phase diagram is obtained from the entanglement measurement, where a XY phase line δ=0 separates the Néel phase. Along this critical line, the central charge c=1 is extracted from the finite entanglement and the finite correlation length. In addition, the characteristic critical exponents are obtained from the local transverse magnetization, nonlocal transverse Néel order, and the correlation length, respectively. It is found that all the critical exponents are varying continuously along the phase transition line δ=0, and the ratios of critical exponents imply that the phase transition is in conformity with the weak universality. The linear relations of the critical exponents are able to illustrate the dependence between the critical exponents and the Dzyaloshinskii–Moriya interaction.

Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes

We consider a Markovian multiserver queueing system with two customer classes, preemptive priorities, and reneging. We formulate this system as an infinite level-dependent quasi-birth-death process (LDQBD). We introduce an algorithm that endogenously truncates the level and calculates lower and upper bounds on stationary probabilities of this LDQBD such that the gap between the bounds can be any desired amount. Our algorithm can be applied to any LDQBD for which the rate matrices become elementwise nonincreasing above some level. This appears to be the first algorithm that provides bounds on stationary probabilities for an infinite-level LDQBD. To obtain these bounds, the algorithm first obtains lower and upper bounds on the rate matrices of the LDQBD using a novel method, which can be applied to any LDQBD. We then extend this algorithm to approximate performance measures of the system of interest and calculate exact lower and upper bounds for those that can be expressed as probabilities, such as the probability that an incoming low-priority customer will wait. We generate a wide range of instances with up to 100 servers and compare the solution times and accuracy of our algorithm with two existing algorithms. These numerical experiments indicate that our algorithm is faster than the other two algorithms for a given accuracy requirement. We investigate the impact of changing service rates on the proportion of low-priority customers served and their wait time, and we demonstrate how ignoring one of these measures can possibly mislead decision makers. Summary of Contribution: We contribute to operations research by modeling a practically important queueing system and developing an algorithm to accurately compute performance measures for that system. We also contribute to computer science by providing error and complexity analysis for the algorithm to solve a broad class of two-dimensional Markov chains with infinite state space.

Discovering causal structure with reproducing-kernel Hilbert space ε-machines.

We merge computational mechanics' definition of causal states (predictively equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation-a finite- or infinite-state kernel ϵ-machine-is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Planck equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high-dimensional data.

Intrinsic topological magnons in arrays of magnetic dipoles

We study a simple magnetic system composed of periodically modulated magnetic dipoles with an easy axis. Upon adjusting the geometric modulation amplitude alone, chains and two-dimensional stacked chains exhibit a rich magnon spectrum where frequency gaps and magnon speeds are easily manipulable. The blend of anisotropy due to dipolar interactions between magnets and geometrical modulation induces a magnetic phase with fractional Zak number in infinite chains and end states in open one-dimensional systems. In two dimensions it gives rise to topological modes at the edges of stripes. Tuning the amplitude in two-dimensional lattices causes a band touching, which triggers the exchange of the Chern numbers of the volume bands and switches the sign of the thermal conductivity.

Distributed control of linear partial integro-differential equations based on the input-output linearisation approach

In this paper, the input-output linearisation control approach is extended to distributed parameter systems whose dynamical behaviour is described by a partial integro-differential equation. The design of the infinite dimensional state feedback controller is achieved using the late lumping approach, i.e., using the partial integro-differential equation model without any prior reduction or approximation. Thus, based on the notion of the characteristic index as a generalisation of the relative degree, a distributed state feedback controller is designed by evaluating the successive time derivatives of the controlled output. The designed controller yields in closed loop a first order lumped parameter system where the time constant is a design parameter that fixes the desired dynamical behaviour. The stability of the closed loop system is investigated, based on semi-group theory, by employing the perturbation theorem of the bounded linear operators, and the sufficient condition for exponential stability in L2-norm is derived. This condition yields the upper bound for the design parameter, i.e., the time constant. Both output tracking and stabilisation capabilities of the developed state feedback are demonstrated through numerical simulation by considering three application examples: Volterra, Fredholm and Fredholm-Volterra PIDEs. The effectiveness of the developed controller is shown by simulation.

$ L^p $-exact controllability of partial differential equations with nonlocal terms

<p style='text-indent:20px;'>The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces, <inline-formula><tex-math id="M3">\begin{document}$ 1<p<\infty $\end{document}</tex-math></inline-formula>. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.</p>

An Improved Anti-Jamming Method Based on Deep Reinforcement Learning and Feature Engineering

To improve the performance of anti-jamming communication in dynamic and adversarial jamming environment, an improved anti-jamming method is proposed based on deep reinforcement learning and feature engineering. Different from the existing studies that use computer vision of deep learning based on the infinite state of spectrum waterfall, the proposed method relays on analyzing spectrum differences between adjacent time slots which contains information and features of jamming patterns. First, anti-jamming strategy is trained by countering the jammer which carries out a random jamming patterns switching strategy. Second, an improved state space is introduced by containing historical spectrum of communication and jamming signal between adjacent time slots, which can help an anti-jamming agent effectively extract the features of jamming patterns to reduce computational complexity. In addition, an improved reward function based on channel switch cost is improved for considering propagation characteristics which may cause communication performance lost. Taking advantage of both feature engineering and deep reinforcement learning, an improved anti-jamming method is proposed to improve reliable anti-jamming performance. Compared with the traditional CNN-based deep reinforcement learning anti-jamming method, simulation results show that the improved method can obtain better performance and lower computational complexity.

How treewidth helps in verification

Treewidth, or similar measures, have been a quite useful technique in the verification of infinite state systems. We illustrate, through examples, how such measures help in analyzing a system.

Lieb's Theorem and Maximum Entropy Condensates

Coherent driving has established itself as a powerful tool for guiding a many-body quantum system into a desirable, coherent non-equilibrium state. A thermodynamically large system will, however, almost always saturate to a featureless infinite temperature state under continuous driving and so the optical manipulation of many-body systems is considered feasible only if a transient, prethermal regime exists, where heating is suppressed. Here we show that, counterintuitively, in a broad class of lattices Floquet heating can actually be an advantageous effect. Specifically, we prove that the maximum entropy steady states which form upon driving the ground state of the Hubbard model on unbalanced bi-partite lattices possess uniform off-diagonal long-range order which remains finite even in the thermodynamic limit. This creation of a `hot' condensate can occur on any driven unbalanced lattice and provides an understanding of how heating can, at the macroscopic level, expose and alter the order in a quantum system. We discuss implications for recent experiments observing emergent superconductivity in photoexcited materials.

Two-wire junction of inequivalent Tomonaga-Luttinger liquids

We develop two novel numerical schemes to study the conductance of the two-wire junction of inequivalent Tomonaga-Luttinger Liquids. In the first scheme we use the static current-current correlation function across the junction to extract the linear conductance through a relation that is derived via the bosonization method. In the second scheme we apply a bias and evaluate the time-dependent current across the junction to obtain the current-voltage characteristic. The conductance is then extracted from the small bias result within the linear response regime. Both schemes are based on the infinite size matrix product state to minimize the finite-size effects. Due to the lack of the translational invariance, we focus on a finite-size window containing the junction. For time-independent calculations, we use infinite boundary conditions to evaluate the correlations within the window. For time-dependent calculations, we use the window technique to evaluate the local currents within the window. The numerical results obtained by both schemes show excellent agreement with the analytical predictions.

Analog Neural Computing With Super-Resolution Memristor Crossbars

Memristor crossbar arrays are used in a wide range of in-memory and neuromorphic computing applications. However, memristor devices suffer from non-idealities that result in the variability of conductive states, making programming them to a desired analog conductance value extremely difficult as the device ages. In theory, memristors can be a nonlinear programmable analog resistor with memory properties that can take infinite resistive states. In practice, such memristors are hard to make, and in a crossbar, it is confined to a limited set of stable conductance values. The number of conductance levels available for a node in the crossbar is defined as the crossbar's resolution. This paper presents a technique to improve the resolution by building a super-resolution memristor crossbar with nodes having multiple memristors to generate r-simplicial sequence of unique conductance values. The wider the range and number of conductance values, the higher the crossbar's resolution. This is particularly useful in building analog neural network (ANN) layers, which are proven to be one of the go-to approaches for forming a neural network layer in implementing neuromorphic computations.

Critical exponents of block-block mutual information in one-dimensional infinite lattice systems.

We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. Asthelattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.

Entropy changes in crystalline material under phase transition and symmetry breaking

Heat removal from a crystalline material at its critical temperature results in phase transitions which are associated with spontaneous symmetry breaking whereby the final state exhibits infinite degenerate states. Calculations of entropy changes in such systems are not addressed in classical thermodynamics as the system is driven away from equilibrium due to the asymmetric energy landscape of the system. Here, we present a novel mathematical formulation that allows us to calculate entropy changes in such systems while arguing that heat applied to such a system results in an increase in entropy along with the excitation of Goldstone modes. These ideas offer a novel theoretical framework towards understanding the phenomenon of entropy changes in systems driven away from equilibrium.

On Quantum States with a Finite-Dimensional Approximation Property

We consider a class (convex set) of quantum states containing all finite rank states and infinite rank states with the sufficient rate of decreasing of eigenvalues (in particular, all Gaussian states). Quantum states from this class are characterized by the property (called the FA-property) that allows to obtain several results concerning finite-dimensional approximation of basic entropic and information characteristics of quantum systems and channels. We obtain a simple sufficient condition of the FA-property. We show that this property implies finiteness of the von Neumann entropy, but leave unsolved the question concerning the converse implication. We obtain uniform approximation results for characteristics depending on a pair (channel, input state) and for characteristics depending on a pair (channel, input ensemble). We establish the uniform continuity of the above characteristics as functions of a channel w.r.t. the strong convergence provided that the FA-property holds either for the input state or for the average state of input ensemble.