Infinite Limit Research Articles

Bosonization and correlators of the one-dimensional Hubbard model

Abstract We present a simple derivation of the Luttinger liquid relation for the one-dimensional Hubbard model, considering a finite U and in the U = ∞ limit. We describe a simple solution of the Hubbard model in the infinite repulsion limit and use it to calculate the correlators of the model in this limit in a simple and physical way using the bosonization technique. We then calculate the asymptotics of the correlators of the model at arbitrary U through a single parameter, which can be calculated from the Bethe Ansatz solution. Our derivation of the critical exponents is simple and allows one to express different physical operators of the Hubbard model through the charge and spin Bose fields in a direct and physically transparent way.

Determination of the particle size distribution of cube-shaped colloidal perovskite quantum dots from photoluminescence spectra: A combined theoretical-experimental approach.

A theoretical-experimental approach is proposed to convert the photoluminescence spectra of colloidal perovskite quantum dot ensembles into accurate estimates for their intrinsic particle size distribution functions. Two main problems were addressed and properly correlated: the size dependence of the first excitonic transition in a single cube-shaped quantum dot and the inhomogeneous broadening of the fluorescence line shape due to the size nonuniformity of the chemically prepared quantum dot suspension in addition to the single-dot homogeneous broadening. By applying the reported methodology to CsPbBr3 quantum dot samples belonging to the strong and intermediate confinement regimes, the calculated size distributions exhibited close agreement with those obtained from transmission electron microscopy, with precise estimates for the average particle size and standard deviation. Specifically for strongly confined ultrasmall CsPbBr3 quantum dots, the presented spectroscopic model for size distribution computation is based on a new analytical expression for the size-dependent bandgap, which was developed within the framework of the finite-depth square-well effective mass approximation accounting for band nonparabolicity effects. Such a quantum mechanical approach correctly predicts the expected transition to the intermediate confinement regime in sufficiently large quantum dots, which are traditionally described by the well-known bandgap equation in the infinite potential barrier limit with a spatially correlated electron-hole wavefunction and nonparabolic carrier effective masses. The proposed calculation scheme originates from general theoretical considerations so that it can be readily adapted to semiconductor quantum dots of many other systems, from all inorganic metal halides to hybrid perovskite materials, regardless of the adopted chemical synthesis route.

Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms

In quantum chemistry, the variational quantum eigensolver (VQE) is a promising algorithm for molecular simulations on near-term quantum computers. However, VQEs using hardware-efficient circuits face scaling challenges due to the barren plateau problem. This raises the question of whether chemically inspired circuits from unitary coupled cluster (UCC) methods can avoid this issue. Here we provide theoretical evidence indicating they may not. By examining alternated dUCC ansätzes and relaxed Trotterized UCC ansätzes, we find that in the infinite depth limit, a separation occurs between particle-hole one- and two-body unitary operators. While one-body terms yield a polynomially concentrated energy landscape, adding two-body terms leads to exponential concentration. Numerical simulations support these findings, suggesting that popular 1-step Trotterized unitary coupled-cluster with singles and doubles (UCCSD) ansätze may not scale. Our results emphasize the link between trainability and circuit expressiveness, raising doubts about VQEs’ ability to surpass classical methods.

A twist at infinite distance in the CHL string

We analyze a space-time algebra of BPS states that emerges in the infinite distance limit in the moduli space of the nine-dimensional CHL string as the theory decompactifies to the ten-dimensional E8 × E8 heterotic string. We find an affine algebra as expected from the heterotic case, but in a twisted version.

A numerical evaluation of the Finite Monkeys Theorem

The Infinite Monkeys Theorem has long-established the eventual certainty of the complete works of William Shakespeare being reproduced by a monkey randomly pressing keys on a typewriter. This only considers the infinite limit, with either an infinite number of monkeys and/or an infinite time period of monkey labour. Here, we consider the Finite Monkeys Theorem and look at the probability of a given string being typed by one of a finite number of monkeys within a finite time allocation consistent with estimates for the lifespan of our universe. We also calculate the expected number of keystrokes until a target string would first be produced. Given the expected time until the heat death of the universe, we demonstrate that the widely-accepted conclusion from the Infinite Monkeys Theorem is, in fact, misleading in our finite universe. As such, this places the theorem in a class of probabilistic problems or paradoxes, including the St. Petersburg paradox, Zeno's dichotomy paradox and the Ross–Littlewood paradox wherein the infinite-resource conclusions directly contradict those obtained when considering limited resources, however sizeable.

End of the world brane networks for infinite distance limits in CY moduli space

Dynamical Cobordism provides a powerful method to probe infinite distance limits in moduli/field spaces parameterized by scalars constrained by generic potentials, employing configurations of codimension-1 end of the world (ETW) branes. These branes, characterized in terms of critical exponents, mark codimension-1 boundaries in the spacetime in correspondence of finite spacetime distance singularities at which the scalars diverge. Using these tools, we explore the network of infinite distance singularities in the complex structure moduli space of Calabi-Yau fourfolds compactifications in M-theory with a four-form flux turned on, which is described in terms of normal intersecting divisors classified by asymptotic Hodge theory. We provide spacetime realizations for these loci in terms of networks of intersecting codimension-1 ETW branes classified by specific critical exponents which encapsulate the relevant information of the asymptotic Hodge structure characterizing the corresponding divisors.

A Bhattacharyya-type Conditional Error Bound for Quadratic Discriminant Analysis

We give an upper bound on the conditional error of Quadratic Discriminant Analysis (QDA), conditioned on parameter estimates. In the case of maximum likelihood estimation (MLE), our bound recovers the well-known Chernoff and Bhattacharyya bounds in the infinite sample limit. We perform an empirical assessment of the behaviour of our bound in a finite sample MLE setting, demonstrating good agreement with the out-of-sample error, in contrast with the simpler but uninformative estimated error, which exhibits unnatural behaviour with respect to the sample size. Furthermore, our conditional error bound is applicable whenever the QDA decision function employs parameter estimates that differ from the true parameters, including regularised QDA.

On a class of linear quadratic Gaussian quantilized mean field games

An energy provider faced with energy generation risks and a large homogeneous pool of customers designs its energy price as a time-varying function of a risk-related quantile of the total energy demand, which generalizes pricing through the mean of the total energy demand. In the infinite population limit, we model the pricing problem with a class of linear quadratic Gaussian quantilized mean field games. For these quantilized mean field games, we show existence and uniqueness of an equilibrium which reveals the price trajectory, as well as an approximate Nash property when the quantilized mean field game’s feedback control functions are applied to the large but finite game and the rate of convergence of the Nash deviation to zero as a function of the population size and the quantile is provided. Finally, the use of this class of quantilized mean field games is illustrated in the context of equivalent thermal parameter models for households heater and an energy provider using solar generation.

Evolutionary Game-Theoretic Approach to the Population Dynamics of Early Replicators.

The population dynamics of early replicators has revealed numerous puzzles, highlighting the difficulty of transitioning from simple template-directed replicating molecules to complex biological systems. The resolution of these puzzles has set the research agenda on prebiotic evolution since the seminal works of Manfred Eigen in the 1970s. Here, we study the effects of demographic noise on the population dynamics of template-directed (non-enzymatic) and protein-mediated (enzymatic) replicators. We borrow stochastic algorithms from evolutionary game theory to simulate finite populations of two types of replicators. These algorithms recover the replicator equation framework in the infinite population limit. For large but finite populations, we use finite-size scaling to determine the probability of fixation and the mean time to fixation near a threshold that delimits the regions of dominance of each replicator type. Since enzyme-producing replicators cannot evolve in a well-mixed population containing replicators that benefit from the enzyme but do not encode it, we study the evolution of enzyme-producing replicators in a finite population structured in temporarily formed random groups of fixed size n. We argue that this problem is identical to the weak-altruism version of the n-player prisoner's dilemma, and show that the threshold is given by the condition that the reward for altruistic behavior is equal to its cost.

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We consider 5d KK dualities, that is multiple 5d gauge theories with the same 6d infinite coupling limit. We provide a prescription to associate 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 quivers to the 5d dual quivers, such that the 4d quivers are also dual to each other. The 4d dualities are infrared dualities which can be checked matching global symmetry anomalies and in certain cases can be proven using basic Seiberg dualities sequentially. We also consider dualities obtained by Higgsing in two different ways the same 5d theory, in some simple examples.

Nonempirical Prediction of the Length-Dependent Ionization Potential in Molecular Chains.

The ionization potential of molecular chains is well-known to be a tunable nanoscale property that exhibits clear quantum confinement effects. State-of-the-art methods can accurately predict the ionization potential in the small molecule limit and in the solid-state limit, but for intermediate, nanosized systems prediction of the evolution of the electronic structure between the two limits is more difficult. Recently, optimal tuning of range-separated hybrid functionals has emerged as a highly accurate method for predicting ionization potentials. This was first achieved for molecules using the ionization potential theorem (IPT) and more recently extended to solid-state systems, based on an ansatz that generalizes the IPT to the removal of charge from a localized Wannier function. Here, we study one-dimensional molecular chains of increasing size, from the monomer limit to the infinite polymer limit using this approach. By comparing our results with other localization-based methods and where available with experiment, we demonstrate that Wannier-localization-based optimal tuning is highly accurate in predicting ionization potentials for any chain length, including the nanoscale regime.

3He adsorbed on molecular hydrogen surfaces.

Using a diffusion Monte Carlo technique, we calculated the phase diagram of 3He adsorbed on a first solid layer of a molecular hydrogen isotope (H2, HD, and D2) on top of graphite. The results are qualitatively similar in all cases: a two-dimensional gas spanning from the infinite dilution limit to a second-layer helium density of 0.048 ± 0.004 Å-2. That gas is in equilibrium with a 7/12 commensurate structure, more stable than any incommensurate triangular solid of similar density. These findings are in reasonably good agreement with available experimental data.

Approaching periodic systems in ensemble density functional theory via finite one-dimensional models

Abstract Ensemble Density Functional Theory (EDFT) is a generalization of ground-state Density Functional Theory (GS DFT), which is based on an exact formal theory of finite collections of a system's ground and excited states. EDFT in various forms has been shown to improve the accuracy of calculated energy level differences in isolated model systems, atoms, and molecules, but it is not yet clear how EDFT could be used to calculate band gaps for periodic systems. We extend the application of EDFT toward periodic systems by estimating the thermodynamic limit with increasingly large finite one-dimensional "particle in a box" systems, which approach the uniform electron gas (UEG). Using ensemble-generalized Hartree and Local Spin Density Approximation (LSDA) exchange-correlation functionals, we find that corrections go to zero in the infinite limit, as expected for a metallic system. However, there is a correction to the effective mass, with results comparable to other calculations on 1D, 2D, and 3D UEGs, which indicates promise for non-trivial results from EDFT on periodic systems.

Discontinuous transition to chaos in a canonical random neural network.

We study a paradigmatic random recurrent neural network introduced by Sompolinsky, Crisanti, and Sommers (SCS). In the infinite size limit, this system exhibits a direct transition from a homogeneous rest state to chaotic behavior, with the Lyapunov exponent gradually increasing from zero. We generalize the SCS model considering odd saturating nonlinear transfer functions, beyond the usual choice ϕ(x)=tanhx. A discontinuous transition to chaos occurs whenever the slope of ϕ at 0 is a local minimum [i.e., for ϕ^{'''}(0)>0]. Chaos appears out of the blue, by an attractor-repeller fold. Accordingly, the Lyapunov exponent stays away from zero at the birth of chaos.

Explaining neural scaling laws

The population loss of trained deep neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains the origins of and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both dataset and model size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pretrained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents under modifications of task and architecture aspect ratio. Our work provides a taxonomy for classifying different scaling regimes, underscores that there can be different mechanisms driving improvements in loss, and lends insight into the microscopic origin and relationships between scaling exponents.

KPZ fluctuations in finite volume

These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects describing the relaxation to its stationary state of a finite system starting from a given initial condition. The second part is more specifically devoted to an approach allowing to express in a simple way the statistics of the current in the totally asymmetric simple exclusion process in terms of a contour integral on a compact Riemann surface, whose infinite genus limit leads to KPZ fluctuations in finite volume.

Stringy evidence for a universal pattern at infinite distance

Infinite distance limits in the moduli space of a quantum gravity theory are characterized by having infinite towers of states becoming light, as dictated by the Distance Conjecture in the Swampland program. These towers imply a drastic breakdown in the perturbative regimes of the effective field theory at a quantum gravity cut-off scale known as the species scale. In this work, we find a universal pattern satisfied in all known infinite distance limits of string theory compactifications, which relates the variation in field space of the mass of the tower and the species scale: ∇→mm·∇→ΛspΛsp=1d−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\\overrightarrow{\ abla}m}{m}\\cdotp \\frac{\\overrightarrow{\ abla}{\\Lambda}_{\ extrm{sp}}}{\\Lambda_{\ extrm{sp}}}=\\frac{1}{d-2} $$\\end{document} in d spacetime dimensions. This implies a more precise definition of the Distance conjecture and sharp bounds for the exponential decay rates. We provide plethora of evidence in string theory in diverse dimensions and with different levels of supersymmetry, and identify some sufficient conditions that allow the pattern to hold from a bottom-up perspective.

Polarization conversion and lateral shifts in multilayered structure with finitely-gapped topological surface states

Abstract The polarizatison conversion and the Goos–Hänchen (GH) shifts of the reflected electromagnetic wave for the multilayered structure made of topological insulator (TI) layers with finite surface energy gap are investigated. The transfer matrix formalism is adopted to analyze the reflection of electromagnetic wave through the multilayered structure, and the influences of surface energy gap, thickness and number of the TI layers are discussed. We find that maximum polarization conversion rate can be obtained with appropriate surface energy gap of TI, and within a certain range of finite energy gap, the polarization conversion effect is stronger than that for the case under the infinite surface energy gap limit. Greater polarization conversion rate for TI with small surface energy gap can be found than that for TI with larger energy gap in some range of layer numbers. At large incident angles the GH shifts vary considerably with the layer number for TI with relatively larger energy gap. Result of the combined influence of surface energy gap and layer number shows that, there exists both the positive and negative enhancement peaks of the GH shifts, and for smaller energy gap, fewer TI layers are required to obtain the transition between positive and negative GH shifts.

Comparing the finite and infinite limits of sequences and functions: A mathematical and phenomenological analysis and its implications in Spanish textbooks

The purpose of this article is to conduct a mathematical and phenomenological comparison of three concepts: (1) the finite limit of a function at a point, (2) the finite limit of a sequence, and (3) the infinite limit of a sequence. Additionally, we aim to analyse the presence of these concepts in Spanish textbooks. The methodology employed is exploratory and descriptive. Our mathematical comparison revealed differences in several areas, including the dependence between variables, the involved infinite processes, the types of infinity, the dimensioning for each variable, and the intuition of continuity in the interval. Our phenomenological comparison found a correspondence between phenomena using a formal approach, but differences in phenomena when using an intuitive approach. Finally, our analysis of textbooks revealed that all three limits are most commonly presented in the verbal representation system and definition format.Contribution: This study has contributed to the teaching and learning of the notion of limit.

Bound states from the spectral Bethe-Salpeter equation

We compute the bound state properties of three-dimensional scalar ϕ4 theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations. Published by the American Physical Society 2024