The analysis of complex physical systems hinges on the ability to extract the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture- and training-dependent learned "relevant" features bear to standard objects of physical theory. Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the information-theoretic notion of relevance defined in the Information Bottleneck (IB) formalism of compression theory, and the field-theoretic relevance of the Renormalization Group. We show analytically that for statistical physical systems described by a field theory the "relevant" degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an example of constructively incorporating physical interpretability in applications of deep learning in physics.
I present an example of how to analytically optimize a multiscale entanglement renormalization ansatz for a finite antiferromagnetic Heisenberg chain. For this purpose, a quantum-circuit representation is taken into account, and we construct the exactly entangled ground state so that a trivial IR state is modified sequentially by operating separated entangler layers (monodromy operators) at each scale. The circuit representation allows us to make a simple understanding of close relationship between the entanglement renormalization and quantum integrability. We find that the entangler should ma…
Using a real-space renormalization-group approximation, we study the anisotropic quantum Heisenberg model on hierarchical lattices, with interactions following aperiodic sequences. Three different sequences are considered, with relevant and irrelevant fluctuations, according to the Luck-Harris criterion. The phase diagram is discussed as a function of the anisotropy parameter $Δ$ (such that $Δ=0$ and $Δ=1$ correspond to the isotropic Heisenberg and Ising models, respectively). We find three different types of phase diagrams, with general characteristics: the isotropic Heisenberg plane is alway…
We use a renormalization group method, similar to that developed for random spin chains, to infer information about the layouts of cellular wireless networks.
We study the critical behavior of frustrated systems by means of Pade-Borel resummed three-loop renormalization-group expansions and numerical Monte Carlo simulations. Amazingly, for six-component spins where the transition is second order, both approaches disagree. This unusual situation is analyzed both from the point of view of the convergence of the resummed series and from the possible relevance of non perturbative effects.
We compare the roles of fidelity and entanglement in characterizing renormalization group flows and quantum phase transitions. It turns out that the scaling parameter extracted from fidelity for different ground states succeeds to capture nontrivial information including stable and unstable fixed points, whereas the von Neumann entropy as a bipartite entanglement measure (or equivalently, majorization relations satisfied by the spectra of the reduced density matrix along renormalization group flows) often fails, as far as the intrinsic irreversibility-information loss along renormalization gro…
We derive a new renormalization group to calculate a non-trivial critical exponent of the divergent correlation length which gives a universality classification of essential singularities in infinite-order phase transitions. This method resolves the problem of a vanishing scaling matrix. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well-known Berezinski\uı-Kosterlitz-Thouless tra…
In this article we apply a recently invented analytical real-space renormalization group formulation which is based on numerical concepts of the density matrix renormalization group. Within a rigorous mathematical framework we construct non-perturbative renormalization group transformations for the spin-1/2 XXX Heisenberg model in the finite temperature regime. The developed renormalization group scheme allows for calculating the renormalization group flow behaviour in the temperature dependent coupling constant. The constructed renormalization group transformations are applied within the ferr…
We propose a renormalization group treatment of stochastically growing networks. As an example, we study percolation on growing scale-free networks in the framework of a real-space renormalization group approach. As a result, we find that the critical behavior of percolation on the growing networks differs from that in uncorrelated nets.
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of the model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although specia…
The product-wavefunction renormalization group method, which is a novel numerical renormalization group scheme proposed recently,is applied to one-dimensional quantum spin chains in a magnetic field. We draw the zero-temperature magnetization curve of the spin chains, which excellently agrees with the exact solution in the whole range of the field.
Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents $η$ and $δ$, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant …
We study the anomalous scaling of the structure functions of a scalar field advected by a random Gaussian velocity field, the Kraichnan model, by means of Functional Renormalization Group techniques. We analyze the symmetries of the model and derive the leading correction to the structure functions considering the renormalization of composite operators and applying the operator product expansion.
We reexamine the functional renormalization-group theory of wetting transitions. As a starting point of the analysis we apply an exact equation describing renormalization group flow of the generating functional for irreducible vertex functions. We show how the standard nonlinear renormalization group theory of wetting transitions can be recovered by a very simple truncation of the exact flow equation. The derivation makes all the involved approximations transparent and demonstrates the applicability of the approach in any spatial dimension d\geq 2. Exploiting the non-uniqueness of the renormal…
Application of asymptotic freedom to the ultraviolet stability in Euclidean quantum field theories is revisited and illustrated through the hierarchical model making also use of a few technical developments that followed the original works of Wilson on the renormalization group.
We report a real-space renormalization group (RSRG) algorithm, which is formulated through Baxter's corner transfer matrix (CTM), for two-dimensional (d = 2) classical lattice models. The new method performs the renormalization group transformation according to White's density matrix algorithm, so that variational free energies are minimized within a restricted degree of freedom m. As a consequence of the renormalization, spin variables on each corner of CTM are replaced by a m-state block spin variable. It is shown that the thermodynamic functions and critical exponents of the q = 2, 3 Potts …
A class of exact infinitesimal renormalization group transformations is proposed and studied. These transformations are pure changes of variables (i.e., no integration or elimination of some degrees of freedom is required) such that a saddle point approximation is more accurate, becoming, in some cases asymptotically exact as the transformations are iterated. The formalism provides a simplified and unified approach to several known renormalization groups. It also suggests some new ways in which renormalization group methods might successfully be applied. In particular, an exact gauge covariant…
Using the renormalization group approach, the Coulomb gas and the coset techniques, the effect of slightly relevant perturbations is studied for the second parafermionic field theory with the symmetry $Z\_{5}$. New fixed points are found and classified.
We introduce two versions of a renormalization group scheme for the equal load sharing fiber bundle model. The renormalization group is based on formulating the fiber bundle model in the language of damage mechanics. A central concept is the work performed on the fiber bundle to produce a given damage. The renormalization group conserves this work. In the first version of the renormalization group, we take advantage of ordering the strength of the individual fibers. This procedure, which is the simpler one, gives EXACT results -but cannot be generalized to other fiber bundle models such as the…
The flow equations of the Functional Renormalization Group are applied to the O(N)-symmetric scalar theory, for N=1 and N=4, in four Euclidean dimensions, d=4, to determine the effective potential and the renormalization function of the field in the broken phase. In our numerical analysis, the infrared limit, corresponding to the vanishing of the running momentum scale in the equations, is approached to obtain the physical values of the parameters by extrapolation. In the N=4 theory a non-perturbatively large value of the physical renormalization of the longitudinal component of the field is o…
We give a holographic explanation how the renormalization group approach to singular perturbations in non-linear differential equations proposed by Chen, Goldenfeld and Oono is indeed equivalent to a renormalization group method in quantum field theories proposed by Gell-Mann and Low via AdS/CFT correspondence.
By adding a linear term to a renormalization-group equation in a system exhibiting infinite-order phase transitions, asymptotic behavior of running coupling constants is derived in an algebraic manner. A benefit of this method is presented explicitly using several examples.
In this thesis we use renormalization group methods to study the critical behaviour of the staggered F-model. This model is a six vertex model that can be used to calculate the thermodynamic properties of a facet of a BCC crystal in the (100) direction.
We study the renormalization group flow of the O(N) non-linear sigma model in arbitrary dimensions. The effective action of the model is truncated to fourth order in the derivative expansion and the flow is obtained by combining the non-perturbative renormalization group and the background field method. We investigate the flow in three dimensions and analyze the phase structure for arbitrary N. The corresponding results about the critical properties of the models will serve as a reference for upcoming simulations with the Monte-Carlo renormalization group.
Using the renormalization group approach, the Coulomb gas and the coset techniques, the effect of slightly relevant perturbations is studied for the second parafermionic field theory with the symmetry $Z_{N}$, for N odd. New fixed points are found and classified.
We investigate the one-dimensional polaron problem and calculate the ground state energy and the effective mass. We use a real-time renormalization group method and compare our results with first and second order perturbation theory, with Feynman's variational principle and with the method of Lee, Low and Pines.
We use the renormalization group theory to study the directed bond percolation (Gribov process) near its second-order phase transition between absorbing and active state. We present a numerical calculation of the renormalization group functions in the $ε$-expansion where $ε$ is a deviation from the upper critical dimension $d_c = 4$. Within this procedure anomalous dimensions $γ$ are expressed in terms of irreducible renormalized Feynman diagrams and thus the calculation of renormalization constants could be entirely skipped. The renormalization group is included by means of the $R$ operation,…
We review what we have learned about the "Renormalization-Group peculiarities" which were discovered about twenty years ago by Griffiths and Pearce, and which questions they asked are still widely open. We also mention some related developments.
We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin c…
We propose a simple derivation of renormalization group equations and Callan-Symanzik equations as decoupling theorems of the structures underlying effective field theories.
