One-dimensional Field Theory Research Articles

Smooth one-dimensional topological field theories are vector bundles with connection

Smooth one-dimensional topological field theories are vector bundles with connection

Entanglement entropy in conformal quantum mechanics

We consider sets of states in conformal quantum mechanics associated to generators of time evolution whose orbits cover different regions of the time domain. States labelled by a continuous global time variable define the two-point correlation functions of the theory seen as a one-dimensional conformal field theory. Such states exhibit the structure of a thermofield double built on bipartite eigenstates of generators of non-global time evolution. In terms of the correspondence between radial conformal symmetries in Minkowski space-time and time evolution in conformal quantum mechanics proposed in [1, 2] these generators coincide with conformal Killing vectors tangent to worldlines of Milne and diamond observers at constant radius. The temperature of the thermofield double states in conformal quantum mechanics reproduces the temperatures perceived by such diamond and Milne observers. We calculate the entanglement entropy associated to the thermofield double states and obtain a UV divergent logarithmic behaviour analogous to known results in two-dimensional conformal field theory in which the entangling boundary is point-like.

Interpolating Wilson loops and enriched RG flows

We study new 1/24 BPS circular Wilson loops in ABJ(M) theory, which are defined in terms of several parameters that continuously interpolate between previously known 1/6 BPS loops (both bosonic and fermionic) and 1/2 BPS fermionic loops. We compute the expectation value of these operators up to second order in perturbation theory using a one-dimensional effective field theory approach. Within dimensional regularization, we find non-trivial β-functions for the parameters, which are marginally relevant deformations triggering RG flows from a UV fixed point represented by the 1/6 BPS bosonic loop to an IR fixed point represented by a 1/2 BPS fermionic loop. Generically, along all flows at least one supercharge of the theory is preserved, so that we refer to them as enriched RG flows. In particular, fixed points are connected through 1/6 BPS fermionic operators. This holds at framing zero, which is a consequence of the regularization scheme employed. We also establish the validity of the g-theorem, relating the expectation values of the Wilson loops corresponding to the UV and IR fixed points of the flow, and discuss the one-dimensional defect SCFT living on the Wilson loop contour.

Topological quantum field theories from Hecke algebras

We construct one-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional typesH3H_3,E6E_6andE7E_7.

Verification of the area law of mutual information in a quantum field simulator

Theoretical understanding of the scaling of entropies and the mutual information has led to significant advances in the research of correlated states of matter, quantum field theory, and gravity. Measuring von Neumann entropy in quantum many-body systems is challenging as it requires complete knowledge of the density matrix. In this work, we measure the von Neumann entropy of spatially extended subsystems in an ultra-cold atom simulator of one-dimensional quantum field theories. We experimentally verify one of the fundamental properties of equilibrium states of gapped quantum many-body systems, the area law of quantum mutual information. We also study the dependence of mutual information on temperature and the separation between the subsystems. Our work is a crucial step toward employing ultra-cold atom simulators to probe entanglement in quantum field theories.

Two-point functions of composite twist fields in the Ising field theory

All standard measures of bipartite entanglement in one-dimensional quantum field theories can be expressed in terms of correlators of branch point twist fields, here denoted by and . These are symmetry fields associated to cyclic permutation symmetry in a replica theory and having the smallest conformal dimension at the critical point. Recently, other twist fields (composite twist fields), typically of higher dimension, have been shown to play a role in the study of a new measure of entanglement known as the symmetry resolved entanglement entropy. In this paper we give an exact expression for the two-point function of a composite twist field that arises in the Ising field theory. In doing so we extend the techniques originally developed for the standard branch point twist field in free theories as well as an existing computation due to Horváth and Calabrese of the same two-point function which focused on the leading large-distance contribution. We study the ground state two-point function of the composite twist field and its conjugate . At criticality, this field can be defined as the leading field in the operator product expansion of and the disorder field µ. We find a general formula for and for (the derivative of) its analytic continuation to positive real replica numbers greater than 1. We check our formula for consistency by showing that at short distances it exactly reproduces the expected conformal dimension.

Thermodynamics and criticality of su(m) spin chains of Haldane-Shastry type.

We study the thermodynamics and critical behavior of su(m) spin chains of Haldane-Shastry type at zero chemical potential, both in the A_{N-1} and BC_{N} cases. We evaluate in closed form the free energy per spin for arbitrary values of m, from which we derive explicit formulas for the energy, entropy, and specific heat per spin. In particular, we find that the specific heat features a single Schottky peak, whose temperature is well approximated for m≲10 by the corresponding temperature for an m-level system with uniformly spaced levels. We show that at low temperatures the free energy per spin of the models under study behaves as that of a one-dimensional conformal field theory with central charge c=m-1 (with the only exception of the Frahm-Inozemtsev chain at zero value of its parameter). However, from a detailed study of the ground-state degeneracy and the low-energy excitations, we conclude that these models are only critical in the antiferromagnetic case, with a few exceptions that we fully specify.

Constrained systems, generalized Hamilton-Jacobi actions, and quantization

<p style='text-indent:20px;'>Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton–Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern–Simons theory, where the HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern–Simons theory and the toy model for 7D Chern–Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [<xref ref-type="bibr" rid="b21">21</xref>]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.</p>

Parafermionization, bosonization, and critical parafermionic theories

We formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality at k = 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory when k > 2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.

On the S-matrix of Liouville theory

The S-matrix for each chiral sector of Liouville theory on a cylinder is computed from the loop expansion of correlation functions of a one-dimensional field theory on a circle with a non-local kinetic energy and an exponential potential. This action is the Legendre transform of the generating function of semiclassical scattering amplitudes. It is derived from the relation between asymptotic in- and out-fields. Its relevance for the quantum scattering process is demonstrated by comparing explicit loop diagrams computed from this action with other methods of computing the S-matrix, which are also developed.

Chaotic solitons in driven sine-Gordon model

Profiles of static solitons in one-dimensional scalar field theory satisfy the same equations as trajectories of a fictitious particle in multidimensional mechanics. We argue that the structure and properties of the solitons are essentially different if the respective mechanical motions are chaotic. This happens in multifield models and models with spatially dependent potential. We illustrate our findings using one-field sine-Gordon model in external Dirac comb potential. First, we show that the number of different "chaotic" solitons grows exponentially with their length, and the growth rate is related to the topological entropy of the mechanical system. Second, the field values of stable solitons form a fractal; we compute its box-counting dimension. Third, we demonstrate that the distribution of field values in the fractal is related to the metric entropy of the analogous mechanical system.

Two-loop corrections to the large-order behavior of correlation functions in the one-dimensional N -vector model

For a long time, the predictive limits of perturbative quantum field theory have been limited by our inability to carry out loop calculations to arbitrarily high order, which become increasingly complex as the order of perturbation theory is increased. This problem is exacerbated by the fact that perturbation series derived from loop diagram (Feynman diagram) calculations represent asymptotic (divergent) series which limits the predictive power of perturbative quantum field theory. Here, we discuss an ansatz which could overcome these limits, based on the observations that (i) for many phenomenologically relevant field theories, one can derive dispersion relations which relate the large-order growth (the asymptotic limit of "infinite loop order") with the imaginary part of arbitrary correlation functions, for negative coupling ("unstable vacuum"), and (ii) one can analyze the imaginary part for negative coupling in terms of classical field configurations (instantons). Unfortunately, the perturbation theory around instantons, which could lead to much more accurate predictions for the large-order behavior of Feynman diagrams, poses a number of technical as well as computational difficulties. Here, we study, to further the above mentioned ansatz, correlation functions in a one-dimensional (1D) field theory with a quartic self-interaction and an O(N) internal symmetry group, otherwise known as the 1D N-vector model. Our focus is on corrections to the large-order growth of perturbative coefficients, i.e., the limit of a large number of loops in the Feynman diagram expansion. We evaluate, in momentum space, the two-loop corrections for the two-point correlation function, and its derivative with respect to the momentum, as well as the two-point correlation function with a wigglet insertion. Also, we study the four-point function.

Forces between kinks in ϕ8 theory

We investigate the dynamics of the kinks that emerge in a one-dimensional scalar field theory with an octic potential containing a quartic minimum and two quadratic minima. We show analytically that kink-antikink and kink-kink pairs interact with a force that scales with the fourth power of the inter-kink distance, and calculate its strength. This is done using two different techniques. The first employs a collective coordinate method to approximately solve the equation of motion for the profile of an accelerating kink. The second is based on modifying the potential to one that is able to support static solutions containing multiple kinks. We show that the two methods give consistent results. All calculations are supported by numerical work that confirms the validity of our results.

Optimal selection of the linear induction motor spacing for the medium-low speed maglev vehicle

ABSTRACT Based on the one-dimensional field theory for linear induction motor (LIM), the analytical model of two adjacent series motors was built, then the relationships between spacing and traction, current, voltage, etc., were derived. The theoretical calculation results were directly verified by finite element simulation and indirectly verified by experiments. Eventually, the selection method for LIM spacing in medium-low speed maglev was proposed. The results show that the spacing affects the backward travelling wave of LIM1 (the first LIM) and the forward travelling wave of LIM2 (the second LIM), thus affects the traction of the motors; the variation range of LIM2 traction influenced by spacing can be as great as 1kN; a reasonable spacing can effectively improve the traction. The selection method of the spacing can provide technical guidance for the design of the medium-low speed maglev, and effectively improve the traction performance of the vehicles.

Family of potentials with power law kink tails

We provide examples of a large class of one-dimensional higher-order field theories with kink solutions which asymptotically have a power law tail either at one end or at both ends. In particular, we provide examples of a family of potentials which admit a kink as well as a mirror kink solution where all four ends of the two kinks, or only two extreme ends of the two kinks have a power law tail (while the two ends facing each other have an exponential tail). The remaining case includes the situation when the ends facing each other have a power law tail while the two extreme ends of the two kinks have an exponential tail. Further, we show that for a kink with a power law tail at either one end or at both ends, there is no gap between the zero mode and the continuum of the corresponding stability equation. This is in contrast with the kinks with an exponential tail at both ends, in which case there is always a gap between the zero mode and the continuum.

One-dimensional Toy Cosmological Model

In this work, we explore a one-dimensional universe, a toy model based on the relativistic particle theory, seen as a one-dimensional field theory coupled to gravity. The gravitational field is represented by an einbein. Working in a cosmological context, we suppose as gravitational sources a set of scalar fields, and invoking ideas from finite temperature field theory, we find that under certain assumptions this toy model leads to regimes similar to inflation.

Random Field Parameter Estimation of Service Bridge Component and Comparative Analysis of Estimation Methods

Three 36-year-old beams are taken as research objects to estimate the fluctuation scale θ of structural dimensions and material property parameters of reinforced concrete structure in service. First, the protective layer thickness Cd and the concrete compressive strength fc of service bridge component are measured, and the ordinary kriging method is used to interpolate the measured data and gather effective data. Second, the θ values of Cd and fc are estimated by two methods, namely autocorrelation function (ACF) and semivariogram function (SVF). Thereafter, the spatial distribution of R of each corroded steel bar is studied on the basis of measured values of the pitting factor R at different positions of 246 corroded steel bars. On the basis of one-dimensional random field theory, each beam is discretized into several small elements. According to the fluctuation scale obtained by different estimation methods, the random variables are transformed into spatial correlation variables, and the parameters of each discrete element are obtained. Then the bending capacity of each discrete beam element and the corresponding whole beam are obtained on the basis of the value of parameters of each discrete element. Finally, the two estimation methods are compared and analyzed by comparing the theoretical moment value and the bearing capacity test results of the corroded beams. Results show that the mean values of θ of fc and Cd obtained by the SVF method are 2.242 and 2.467, respectively, which are larger than that obtained by the ACF method. Moreover, the fluctuation intensity of concrete compressive strength is smaller with an increase of θ value, and no firm correlation is identified between the spatial distributions of each pit on the surface of a corroded steel reinforcement. The bending moments of three beams from the SVF method are in better agreement with the experimental values than those predicted from the ACF method. In addition, the average relative error of the theoretical bending moment value of all beams based on the SVF estimation is 6.52%, which is 33.19% lower than that obtained by ACF method. Therefore, the SVF method is more effective in describing the spatial properties of material and geometrical parameters compared with the ACF method.

Tangent-space methods for uniform matrix product states

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

Entanglement Content of Quasiparticle Excitations.

We investigate the quantum entanglement content of quasiparticle excitations in extended many-body systems. We show that such excitations give an additive contribution to the bipartite von Neumann and Rényi entanglement entropies that takes a simple, universal form. It is largely independent of the momenta and masses of the excitations and of the geometry, dimension, and connectedness of the entanglement region. The result has a natural quantum information theoretic interpretation as the entanglement of a state where each quasiparticle is associated with two qubits representing their presence within and without the entanglement region, taking into account quantum (in)distinguishability. This applies to any excited state composed of finite numbers of quasiparticles with finite de Broglie wavelengths or finite intrinsic correlation length. This includes particle excitations in massive quantum field theory and gapped lattice systems, and certain highly excited states in conformal field theory and gapless models. We derive this result analytically in one-dimensional massive bosonic and fermionic free field theories and for simple setups in higher dimensions. We provide numerical evidence for the harmonic chain and the two-dimensional harmonic lattice in all regimes where the conditions above apply. Finally, we provide supporting calculations for integrable spin chain models and other interacting cases without particle production. Our results point to new possibilities for creating entangled states using many-body quantum systems.

Universal entanglement spectra of gapped one-dimensional field theories

We discuss the entanglement spectrum of the ground state of a gapped (1+1)-dimensional system in a phase near a quantum phase transition. In particular, in proximity to a quantum phase transition described by a conformal field theory (CFT), the system is represented by a gapped Lorentz invariant field theory in the "scaling limit" (correlation length $\xi$ much larger than microscopic 'lattice' scale $a$), and can be thought of as a CFT perturbed by a relevant perturbation. We show that for such (1+1) gapped Lorentz invariant field theories in infinite space, the low-lying entanglement spectrum obtained by tracing out, say, left half-infinite space, is precisely equal to the physical spectrum of the unperturbed gapless, i.e. conformal field theory defined on a finite interval of length $L_\xi=$ $\log(\xi/a)$ with certain boundary conditions. In particular, the low-lying entanglement spectrum of the gapped theory is the finite-size spectrum of a boundary conformal field theory, and is always discrete and universal. Each relevant perturbation, and thus each gapped phase in proximity to the quantum phase transition, maps into a particular boundary condition. A similar property has been known to hold for Baxter's Corner Transfer Matrices in a very special class of fine-tuned, namely integrable off-critical lattice models, for the entire entanglement spectrum and independent of the scaling limit. In contrast, our result applies to completely general gapped Lorentz invariant theories in the scaling limit, without the requirement of integrability, for the low-lying entanglement spectrum. - The finite-size spectra of the entanglement Hamiltonian and of the physical Hamiltonian on an interval of length R exhibit entirely different behaviors upon crossover from the critical regime $R \ll \xi$ to the gapped regime $R \gg \xi$.