Statistical Field Theory Research Articles

Marginal speed confinement resolves the conflict between correlation and control in collective behaviour

Speed fluctuations of individual birds in natural flocks are moderate, due to the aerodynamic and biomechanical constraints of flight. Yet the spatial correlations of such fluctuations are scale-free, namely they have a range as wide as the entire group, a property linked to the capacity of the system to collectively respond to external perturbations. Scale-free correlations and moderate fluctuations set conflicting constraints on the mechanism controlling the speed of each agent, as the factors boosting correlation amplify fluctuations, and vice versa. Here, using a statistical field theory approach, we suggest that a marginal speed confinement that ignores small deviations from the natural reference value while ferociously suppressing larger speed fluctuations, is able to reconcile scale-free correlations with biologically acceptable group’s speed. We validate our theoretical predictions by comparing them with field experimental data on starling flocks with group sizes spanning an unprecedented interval of over two orders of magnitude.

Nonextensive statistical field theory

We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter q. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic q-dependent nonextensive critical exponents for O(N) vector models, valid for all loop orders and |q−1|<1. Then emerges the new nonextensive O(N)q universality class. Both exponents ηq and zq reduce to those when q=1 for N→∞. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as q is closer to 1. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations.

Quantum field theories, Markov random fields and machine learning

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning within quantum field theory. Here, we will discuss how discretized Euclidean field theories, such as the ϕ 4 lattice field theory on a square lattice, are mathematically equivalent to Markov fields, a notable class of probabilistic graphical models with applications in a variety of research areas, including machine learning. The results are established based on the Hammersley-Clifford theorem. We will then derive neural networks from quantum field theories and discuss applications pertinent to the minimization of the Kullback-Leibler divergence for the probability distribution of the ϕ 4 machine learning algorithms and other probability distributions.

Resistance distance distribution in large sparse random graphs

We consider an Erdős–Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

Eliminating nonrenormalizability helps prove scaled affine quantization of φ44 is nontrivial

Following a modest comparison between canonical and affine quantization, which points to positive features in the affine procedures. We prove through Monte Carlo analysis that the covariant Euclidean scalar field theory, [Formula: see text], where [Formula: see text] denotes the power of the interaction term and [Formula: see text] where [Formula: see text] is the spatial dimension and 1 adds imaginary time, such that [Formula: see text] can be acceptably quantized using scaled affine quantization and the resulting theory is nontrivial, unlike what happens using canonical quantization.

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan–von Neumann–Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan–von Neumann–Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $\Phi$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(\Phi)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

Remarks on SUq (2) fermions

In this paper, we show that several applications of [Formula: see text] fermions to statistical mechanics and quantum field theory, previously discussed in literature, are based on a wrong statement about the connection between deformed and undeformed fermion operators. Then we exclude various classes of ansatz and we put some constraints about the form of such relation.

Bootstrapping Bloch bands

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

Lattice renormalization of quantum simulations

With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with Qiskit noiseless simulators for a $2+1$D discrete non-Abelian $D_4$ gauge theory with two spatial plaquettes.

Casimir-Like Effect from Thermal Field Fluctuations

Landau-Ginzburg ϕ4 field theory is usually applied to systems for understanding continuous phase transitions at critical points. Here, we analyse thermal field using similar free energy description in statistical field theory perspective, and study fluctuations in such a field with particular focus on realizing thermal Casimir effect. Initially, we qualitatively describe emergence of Casimir-like effect using mean-field approximation and further derive it using coarse-graining of perturbative renormalization procedure in the vicinity of Gaussian-fixed point. These results may lead to further the understanding of Casimir effect from scalar fields without employing the concept of zero-point energy in a cosmological sense.

Strong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus

We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.

Exact equilibrium distributions in statistical quantum field theory with rotation and acceleration: Dirac field

We derive the general exact forms of the Wigner function, of mean values of conserved currents, of the spin density matrix, of the spin polarization vector and of the distribution function of massless particles for the free Dirac field at global thermodynamic equilibrium with rotation and acceleration, extending our previous results obtained for the scalar field. The solutions are obtained by means of an iterative method and analytic continuation, which lead to formal series in thermal vorticity. In order to obtain finite values, we extend to the fermionic case the method of analytic distillation introduced for bosonic series. The obtained mean values of the stress-energy tensor, vector and axial currents for the massless Dirac field are in agreement with known analytic results in the special cases of pure acceleration and pure rotation. By using this approach, we obtain new expressions of the currents for the more general case of combined rotation and acceleration and, in the pure acceleration case, we demonstrate that they must vanish at the Unruh temperature.

Simple analytical theory for micelles with widespread radial distribution of charged heads

Modeling aggregation of ionic micelles in solution is an important task that has been attracting attention for years. The molecular-thermodynamic micellization models describe the electrostatic interactions within the Poisson-Boltzmann theory that does not take into account fluctuations of the surfactant heads in the micelle. In this work we apply the statistical field theory to spherical micelles where the charged heads of surfactant molecules undergo fluctuations in the diffuse coronae. From this theory, a simple equation is derived for the electrostatic contribution to the standard free energy of aggregation. The results from the new theory are compared with the numerical results and analytical approximations of the Poisson-Boltzmann theory for the spheres of varying charge and curvature in the ranges relevant to micelles and over a broad interval of solution salinity. The new electrostatic free energy term is employed in the classical molecular-thermodynamic model to describe spherical micelles in the aqueous solution of ionic surfactant with added salt. We introduce an empirical correction that takes into account the effect of fluctuations on the hydrophobic part of the aggregation free energy. Modeling aggregation equilibrium for the spherical SDS micelles in the aqueous solution of salt, we illustrate the performance of the new theory. We compare our results with the structural MD-data and experimental hydrodynamic radii to show that our model with the fluctuating surfactant heads can reflect a realistic picture of the micellar structure and gives reasonable size-estimates for the spherical ionic micelles if proper charge distributions are considered.

A statistical field approach to capital accumulation

This paper presents a model of capital accumulation for a large number of heterogeneous producer–consumer agents in an exchange space in which interactions depend on agents’ positions. Agents in the exchange space are subject to both attractive and repulsive forces: exchanges drive agents closer, but crowd out more distant agents. The formalism used in this paper was developed earlier by the authors and is based on statistical field theory. It allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and microeconomic features of the individual level. Our results show that the dynamics of capital accumulation and the agents’ positions in the exchange space are correlated. Interactions in the exchange space induce phases within the system that depend on the relative strength of the repulsive force. When the repulsive force is strong, the system is in a phase of regulated exchanges. An initial central position both favours and fastens capital accumulation in average, and high levels of initial capital drive agents towards the centre. Yet, this phase displays mild competition and a broad-based although slow improvement in exchange terms. In this phase, random shocks can redistribute capital and initiate a virtuous circle of capital accumulation. When the repulsive force is low, a phase of deregulated exchanges emerges, in which capital distribution is less homogeneous and competition among agents harshens. Increased mobility accelerates capital accumulation for high initial capital producers, whereas low initial capital producers are now evicted from the exchange space as their prices and revenues deteriorate. Thus, a threshold effect appears. Above a certain level of initial capital, agents benefit from and remain in a central position. Below this level, they remain at the periphery of the exchange space.

Unraveling the effects of multiscale network entanglement on empirical systems

Complex systems are large collections of entities that organize themselves into non-trivial structures, represented as networks. One of their key emergent properties is robustness against random failures or targeted attacks —i.e., the networks maintain their integrity under removal of nodes or links. Here, we introduce network entanglement to study network robustness through a multiscale lens, encoded by the time required for information diffusion through the system. Our measure’s foundation lies upon a recently developed statistical field theory for information dynamics within interconnected systems. We show that at the smallest temporal scales, the node-network entanglement reduces to degree, whereas at extremely large scales, it measures the direct role played by each node in keeping the network connected. At the meso-scale, entanglement plays a more important role, measuring the importance of nodes for the transport properties of the system. We use entanglement as a centrality measure capturing the role played by nodes in keeping the overall diversity of the information flow. As an application, we study the disintegration of empirical social, biological and transportation systems, showing that the nodes central for information dynamics are also responsible for keeping the network integrated.

Generative neural samplers for the quantum Heisenberg chain.

Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This paper tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat, and susceptibility for the isotropic XXX and the anisotropic XY chain are in good agreement with Monte Carlo results within the same approximation scheme.

The Search for the Universality Class of Metric Quantum Gravity

On the basis of a limited number of reasonable axioms, we discuss the classification of all the possible universality classes of diffeomorphisms invariant metric theories of quantum gravity. We use the language of the renormalization group and adopt several ideas which originate in the context of statistical mechanics and quantum field theory. Our discussion leads to several ideas that could affect the status of the asymptotic safety conjecture of quantum gravity and give universal arguments towards its proof.

Conformal correlation functions in four dimensions from quaternionic Lauricella system

Correlation functions in four-dimensional Euclidean conformal field theories are expressed in terms of representations of the conformal group SL(2,H), H being the field of quaternions, on the configuration space of points. The representations are obtained in terms of a Lauricella system derived using quaternions. It generalizes the two-dimensional case, wherein the N-point correlation function is expressed in terms of solutions of Lauricella system on the configuration space of N points on the complex plane, furnishing representation of the conformal group SL(2,C).

Quantum field-theoretic machine learning

We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the $\phi^{4}$ scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the $\phi^{4}$ theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the $\phi^{4}$ theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.