Auxiliary Field Approach Research Articles

Damping in a row of locally-resonant inclusions: Dynamic homogenization and scattering of transient shear waves

The scattering of scalar waves by a periodic row of inclusions is theoretically and numerically investigated. The wavelength in the background medium is assumed to be much larger than the typical sizes of the inclusions. The latter are also much softer than the matrix, yielding localized resonances within the microstructure. Previous works in the inviscid case have concerned: (i) the derivation of effective resonant jump conditions, that are non local in time (Touboul et al. (2020) [41]); (ii) the introduction of auxiliary fields along the interface, providing a time-domain formulation of the scattering problem (Touboul et al. (2020) [40]). The present contribution extends the analysis to dissipative cases, which allows to be closer from real devices. The effective jump conditions with damping are obtained, both in the frequency domain and in the time domain. An exact plane-wave solution is proposed. A balance of energy is written, and new auxiliary fields are introduced. Practical implementation of the simulation methods is discussed. Then, numerical experiments are proposed to validate the auxiliary-field approach. The effect of dissipation is examined, and the relevance of the homogenized simulations in comparison with full-field simulations of transient waves is assessed. As an application, a numerical experiment of Coherent Perfect Absorption is finally proposed: at critical values of the attenuation parameter and close to the resonant frequencies, the waves impacting the dissipative resonant interface are fully absorbed.

B -meson light-cone distribution amplitude from Euclidean quantities

A new method for the model-independent determination of the light-cone distribution amplitude (LCDA) of the $B$-meson in heavy quark effective theory (HQET) is proposed by combining the large momentum effective theory (LaMET) and the numerical simulation technique on the Euclidean lattice. We demonstrate the autonomous scale dependence of the non-local quasi-HQET operator with the aid of the auxiliary field approach, and further determine the perturbative matching coefficient entering the hard-collinear factorization formula for the $B$-meson quasi-distribution amplitude at the one-loop accuracy. These results will be crucial to explore the partonic structure of heavy-quark hadrons in the static limit and to improve the theory description of exclusive $B$-meson decay amplitudes based upon perturbative QCD factorization theorems.

Improvement, generalization, and scheme conversion of Wilson-line operators on the lattice in the auxiliary field approach

Nonlocal quark bilinear operators connected by link paths are used for studying parton distribution functions (PDFs) and transverse momentum-dependent PDFs of hadrons using lattice QCD. The nonlocality makes it difficult to understand the renormalization and improvement of these operators using standard methods. In previous work, we showed that by introducing an auxiliary field on the lattice, one can understand an on-axis Wilson-line operator as the product of two local operators in an extended theory. In this paper, we provide details about the calculation in perturbation theory of the factor for conversion from our lattice-suitable renormalization scheme to the MS-bar scheme. Extending our work, we study Symanzik improvement of the extended theory to understand the pattern of discretization effects linear in the lattice spacing, $a$, which are present even if the lattice fermion action exactly preserves chiral symmetry. This provides a prospect for an eventual $O(a)$ improvement of lattice calculations of PDFs. We also generalize our approach to apply to Wilson lines along lattice diagonals and to piecewise-straight link paths.

An auxiliary field approach for computing optical resonances in dispersive media

We report on an auxiliary field approach for solving nonlinear eigenvalue problems occurring in nano-optical systems with material dispersion. The material dispersion can be described by a rational function for the frequency-dependent permittivity, e.g., by a Drude-Lorentz model or a rational function fit to measured material data. The approach is applied to compute plasmonic resonances of a metallic grating.

On the power divergence in quasi gluon distribution function

Recent perturbative calculation of quasi gluon distribution function at one-loop level shows the existence of extra linear ultraviolet divergences in the cut-off scheme. We employ the auxiliary field approach, and study the renormalization of gluon operators. The non-local gluon operator can mix with new operators under renormalization, and the linear divergences in quasi distribution function can be into the newly introduced operators. After including the mixing, we find the improved quasi gluon distribution functions contain only logarithmic divergences, and thus can be used to extract the gluon distribution in large momentum effective theory.

Accurate simulation of the finite density lattice Thirring model

We present a study of the finite density lattice Thirring model in 1+1 dimensions using the world-line/fermion-bag algorithm. The model has features similar to QCD and provides a test case for exploring the accuracy of various methods of solving sign problems. In the massless limit and with open boundary conditions we show that the sign problem is an artifact of the auxiliary field approach and is completely eliminated in the fermion bag approach. With periodic boundary conditions the sign problem is mild in the fermion bag method. We present accurate results for various quantities in the model that can be used as a benchmark for comparison with other methods of solving sign problems.

Solution to sign problems in models of interacting fermions and quantum spins.

We show that solutions to fermion sign problems that are found in the formulation where the path integral is expanded in powers of the interaction in continuous time can be extended to systems involving fermions interacting with dynamical quantum spins. While these sign problems seem unsolvable in the auxiliary field approach, solutions emerge in the world-line representation of quantum spins. Combining this idea with meron-cluster methods, we are able to further extend the class of models that are solvable. We demonstrate these solutions to sign problems by considering several examples of strongly correlated systems that contain the physics of semimetals, insulators, superfluidity, and antiferromagnetism.

A real space auxiliary field approach to the BCS-BEC crossover

The BCS to BEC crossover in attractive Fermi systems is a prototype of weak to strong coupling evolution in many body physics. While extensive numerical results are available, and several approximate methods have been developed, most of these schemes are unsuccessful in the presence of spatial inhomogeneity. Such situations call for a real space approach that can handle large spatial scales and retain the crucial thermal fluctuations. With this in mind, we present comprehensive results of a real space auxiliary field approach to the BCS to BEC crossover in the attractive Hubbard model in two dimensions. The scheme reproduces the Hartree-Fock-Bogoliubov ground state, and leads to a $T_c$ scale that agrees with quantum Monte Carlo estimates to within a few percent. We provide results on the $T_c$, amplitude and phase fluctuations, density of states, and the momentum resolved spectral function over the entire interaction and temperature window. We suggest how the method generalises successfully to the presence of disorder, trapping, and population imbalance.

Higher-order nonlinearity of refractive index: the case of argon.

The nonlinear coefficients, n4, of the time-dependent refractive index for argon are calculated in the non-resonant optical regime. Second-order polynomial fitting of DC-Kerr, γ((2))(-ω; ω, 0, 0), electric field induced second harmonic generation (ESHG), γ((2))(-2ω; ω, ω, 0), and static second-order hyperpolarizability, γ((2))(0; 0, 0, 0), is performed using an auxiliary electric field approach to obtain the corresponding fourth-order optical properties. A number of basis sets are investigated for the fourth-order hyperpolarizability processes at 800 nm at coupled cluster singles and doubles level of theory, starting with the t-aug-cc-pV5Z basis set and expanding that basis set by adding diffuse functions and polarization functions. Comparison shows that the results obtained with the t-aug-cc-pV5Z basis are in very good agreement with the results obtained using the q-aug-cc-pV5Z, t-aug-cc-pV6Z, and q-aug-cc-pV6Z basis sets. To calculate the nonlinear refractive index n4, an approximate formula is suggested which expresses the related degenerate six-wave mixing coefficient, γ((4))(-ω; ω, -ω, ω, -ω, ω), in terms of the DC-Kerr, γ((4))(-ω; ω, 0, 0, 0, 0), ESHG, γ((4))(-2ω; ω, ω, 0, 0, 0), and the static fourth-order hyperpolarizability coefficients. The higher-order nonlinear refractive index n4 is found to be positive over the wavelengths 300 nm-2000 nm. In the infrared spectral range, the obtained values of n4 are in qualitative agreement with the results of Kramers-Kronig-based calculations.

Fermion bag solutions to some sign problems in four-fermion field theories

Lattice four-fermion models containing N flavors of staggered fermions, that are invariant under Z2 and U(1) chiral symmetries, are known to suffer from sign problems when formulated using the auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. In this talk, we show that the sign problems disappear when the models are formulated in the fermion bag approach, allowing us to solve them rigorously for the first time.

Fermion bag solutions to some sign problems in four-fermion field theories

Lattice four-fermion models containing $N$ flavors of staggered fermions, which are invariant under ${Z}_{2}$ and $U(1)$ chiral symmetries, are known to suffer from sign problems when formulated using the auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. Here we show that the sign problems disappear when the models are formulated in the fermion bag approach, allowing us to solve them rigorously for the first time.

Auxiliary-field approach to dilute Bose gases with tunable interactions

We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields related to the normal and anomalous condensate densities. We derive the loop expansion of the effective action in the composite-field propagators. The lowest-order auxiliary field (LOAF) theory is a conserving mean-field approximation consistent with the Goldstone theorem without some of the difficulties plaguing approximations such as the Hartree and Popov approximations. LOAF predicts a second-order phase transition. We give a set of Feynman rules for improving results to any order in the loop expansion in terms of composite-field propagators. We compare results of the LOAF approximation with those derived using the Popov approximation. LOAF allows us to explore the critical regime for all values of the coupling constant, and we determine various parameters in the unitarity limit.

A higher-derivative Lee-Wick standard model

The Lee-Wick Standard Model assumes a minimal set of higher-derivative quadratic terms that produce a negative-norm partner for each Standard Model particle. Here we introduce additional terms of one higher order in the derivative expansion that give each Standard Model particle two Lee-Wick partners: one with negative and one with positive norm. These states collectively cancel unwanted quadratic divergences and resolve the hierarchy problem as in the minimal theory. We show how this next-to-minimal higher-derivative theory may be reformulated via an auxiliary field approach and written as a Lagrangian with interactions of dimension four or less. This mapping provides a convenient framework for studies of the formal and phenomenological properties of the theory.

Accuracy of auxiliary field approach for baryons

We provide a check of the accuracy of the auxiliary field formalism used to derive the Effective Hamiltonian for baryons in the Field Correlator Method. To this end we compare the solutions for the Effective Hamiltonian with those obtained from the solution of the Salpeter equation. Comparing these results gives a first estimate of the systematic uncertainty due to the use of the auxiliary field formalism for baryons.

Energy and momentum of light in dielectric media

The conservation of energy, linear momentum, and angular momentum of the electromagnetic field in linear dielectric media with arbitrary dispersion and absorption is studied in the framework of an auxiliary field approach in which the electric and magnetic fields are complemented by a material field. This material field depends on a continuous variable omega, and describes harmonic motions of the charges with eigen frequency omega. It carries an electric dipole moment and couples as such to the electric field. The equations of motion of the model are equivalent to Maxwell's equations in an arbitrary dispersive and absorbing dielectric and imply that several quantities are conserved. These quantities may be interpreted as the energy, momentum, and angular momentum of the total system, and can be viewed as the sum of the corresponding quantities of the field and matter subsystems. The total momentum turns out to be equal to the Minkowski momentum plus a dispersive contribution. The total energy and total momentum of a wave packet both travel with the group velocity, while the ratio of total momentum and total energy is given by the phase velocity.

Path integrals, diffusion onSU(2) and the fully frustrated antiferromagnetic spin cluster

We study the path-integral treatment of the quantum mechanics of a fully frustrated cluster of spins: a cluster in which every pair of spins is coupled equally by antiferromagnetic Heisenberg interactions. Such clusters are interesting partly because they are the building blocks of geometrically frustrated spin systems. Using a Hubbard–Stratonovich transformation to decouple the interactions, the Boltzmann factor for the spin cluster is written in terms of the time-evolution operator for a single spin in a stochastically varying magnetic field. The time-evolution operator follows a random walk in SU(2): by switching from a Langevin to a Fokker–Planck description of this walk and computing the probability distribution of its end-point, we arrive at an expression for the partition function as a finite sum of single integrals. The calculation provides an analytically tractable illustration of the auxiliary field approach, as used in quantum Monte Carlo calculations, and may potentially be extended to treat more complex frustrated spin systems.

On a New Formulation of the Real-Time Propagator

The development of theoretical tools for the study of dynamical phenomena of many-particle systems on the quantum level is a fundamental challenge since many decades. A lot of efforts have been invested on Feynman's path integral approach, however, no computationally tractable method for investigating realistic systems could be developed up to now. In this paper we propose an alternative representation of the real-time many-body evolution operator formulated within the framework of the auxiliary field formalism. Our goal is to derive a new auxiliary field functional integral representation, in which the large oscillations of the functional integrand are reduced, in order to render the auxiliary field approach more attractive for real-time computation. This objective is attained using a generalized version of the method of Gaussian equivalent representation of Efimov and Ganbold [Phys. Stat. Sol. 168 (1991) 165], which eliminates the low-order fluctuations of the auxiliary field from the interaction functional.

Computation within the auxiliary field approach

Recently, the classical auxiliary field methodology has been developed as a new simulation technique for performing calculations within the framework of classical statistical mechanics. Since the approach suffers from a sign problem, a judicious choice of the sampling algorithm, allowing a fast statistical convergence and an efficient generation of field configurations, is of fundamental importance for a successful simulation. In this paper we focus on the computational aspects of this simulation methodology. We introduce two different types of algorithms, the single-move auxiliary field Metropolis Monte Carlo algorithm and two new classes of force-based algorithms, which enable multiple-move propagation. In addition, to further optimize the sampling, we describe a preconditioning scheme, which permits to treat each field degree of freedom individually with regard to the evolution through the auxiliary field configuration space. Finally, we demonstrate the validity and assess the competitiveness of these algorithms on a representative practical example. We believe that they may also provide an interesting possibility for enhancing the computational efficiency of other auxiliary field methodologies.

Auxiliary Field Functional Integral Representation of the Many-Body Evolution Operator

Finding an appropriate functional integral representation of the many-body evolution operator is a crucial task for performing efficient calculations of fermionic systems within the auxiliary field approach. In this paper we derive a new field representation of the imaginary-time evolution operator using the method of Gaussian equivalent representation of Efimov and Ganbold (1991, Physica Status Solidi168, 165). The goal is to obtain a functional integral representation, in which the main divergences caused by the tadpole Feynman diagrams are efficiently eliminated. These diagrams provide the main contributions to the ground state of the system under consideration, and therefore it is important to take them into account adequately, especially at lower temperatures. In addition, we show that the well-known mean field representation of the imaginary-time evolution operator is only the limiting case of the Gaussian equivalent representation in the small time-step regime.

QCD string in light-light and heavy-light mesons

The spectra of light-light and heavy-light mesons are calculated within the framework of the QCD string model, which is derived from QCD in the Wilson loop approach. Special attention is paid to the proper string dynamics that allows us to reproduce the straight-line Regge trajectories with the inverse slope being $2\ensuremath{\pi}\ensuremath{\sigma}$ for light-light and twice as small for heavy-light mesons. We use the model of the rotating QCD string with quarks at the ends to calculate the masses of several light-light mesons lying on the lowest Regge trajectories and compare them with the experimental data as well as with the predictions of other models. The masses of several low-lying orbitally and radially excited heavy-light states in the D, ${D}_{s},$ B, and ${B}_{s}$ mesons spectra are calculated in the einbein (auxiliary) field approach, which has proven to be rather accurate in various calculations for relativistic systems. The results for the spectra are compared with the experimental and recent lattice data. It is demonstrated that an account of the proper string dynamics encoded in the so-called string correction to the interquark interaction leads to an extra negative contribution to the masses of orbitally excited states that resolves the problem of the identification of the $D(2637)$ state recently claimed by the DELPHI Collaboration. For heavy-light system we extract the constants $\overline{\ensuremath{\Lambda}},$ ${\ensuremath{\lambda}}_{1},$ and ${\ensuremath{\lambda}}_{2}$ used in heavy quark effective theory and find good agreement with the results of other approaches.