Nontrivial Minimum Research Articles

Gluon condensates and effective gluon mass

Lattice simulations along with studies in continuum QCD indicate that non-perturbative quantum fluctuations lead to an infrared regularisation of the gluon propagator in covariant gauges in the form of an effective mass-like behaviour. In the present work we propose an analytic understanding of this phenomenon in terms of gluon condensation through a dynamical version of the Higgs mechanism, leading to the emergence of color condensates. Within the functional renormalisation group approach we compute the effective potential of covariantly constant field strengths, whose non-trivial minimum is related to the color condensates. In the physical case of an $SU(3)$ gauge group this is an octet condensate. The value of the gluon mass obtained through this procedure compares very well to lattice results and the mass gap arising from alternative dynamical scenarios.

Finding Nontrivial Minimum Fixed Points in Discrete Dynamical Systems: Complexity, Special Case Algorithms and Heuristics

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial-time algorithm for approximating a solution to this problem to within the factor n^(1 - epsilon) for any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics. A full version of the manuscript, source code and data are available at: https://github.com/bridgelessqiu/NMIN-FPE

A0 Condensation, Nielsen’s Identity and Effective Potential of Order Parameter

In high temperature SU(2) gluodynamics, the condensation of the zero component gauge field potential A_0 =const and its gauge-fixing dependence are investigated. A_0 is mutually related with Polyakov's loop <L>. The two-loop effective potential W(A_0,xi) is recalculated in the background relativistic R_xi gauge. It depends on the parameter xi, has a nontrivial minimum and satisfies Nielsen's identity. These signs mean gauge invariance of the condensation phenomenon. Following the idea of Belyaev, we express W(A_0,xi) in terms of <L>. The obtained effective potential of order parameter differs from that derived by this author. It is independent of xi and has a nontrivial minimum position. Hence the A_0 condensation follows. We show that the equation relating A_0 and (A_0)|_(classical) coincides with the special characteristic orbit in the (A)$-plain along which the W(A_0,xi) is xi-independent. In this way the link between these two gauge invariant descriptions is established. The minimum value of the Polyakov loop is calculated. Comparison with results of other authors is given.

Phase structure of the Euclidean three-dimensional O(1) ghost model

We have treated the Euclidean three-dimensional O(1) ghost model with a modified version of the effective average action (EAA) renormalization group (RG) method, developed by us. We call it Fourier–Wetterich RG approach and it is used to investigate the occurrence of a periodic condensate in terms of the functional RG. The modification involves additional terms in the ansatz of the EAA, corresponding to the Fourier-modes of the periodic condensate. The RG flow equations are derived keeping the terms up to the fourth order of the gradient expansion (GE), however the numerical calculations are conducted in the second order (or next-to-leading order, NLO) of the GE. The expansion of the flow equations around the nontrivial minimum of the local potential takes into account properly the vertices induced by the periodic condensate even if the wave function renormalization is set to be field-independent. The numerical analysis reveals several different phases with three multicritical points.

Miscibility in coupled dipolar and non-dipolar Bose–Einstein condensates

We perform a full three-dimensional study on miscible-immiscible conditions for coupled dipolar and non-dipolar Bose–Einstein condensates (BEC), confined in anisotropic traps. In view of recent experimental studies, our focus was the atomic erbium–dysprosium (168Er–164Dy) and dysprosium–dysprosium (164Dy–162Dy) mixtures. The miscibility is quantified by the overlap of the two-component densities, using an appropriate defined parameter. By verifying that stable regimes for pure-dipolar coupled BECs are only possible in pancake-type traps, we obtain some non-trivial local minimum biconcave-shaped states with density oscillations in both components. For non-dipolar systems with repulsive interactions, we show that immiscible stable configurations are also possible in cigar-type geometries. The main role of the trap aspect ratio and inter-species contact interaction for the miscibility is verified for different configurations, from non-dipolar to pure dipolar systems.

Cardy formula for 4d SUSY theories and localization

We study 4d mathcal{N}=1 supersymmetric theories on a compact Euclidean manifold of the form S1 × ℳ3. Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold ℳ3. Taking the limit of shrinking S1, we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of [1]. The limit is given in terms of an effective potential for the holonomies around the S1, whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by Tr(R), while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if Tr(R) ≤ 0.

Minimum light transmission in graphene in the presence of a magnetic field

We show that, on general theoretical grounds, transmission of light in graphene always presents a non-vanishing minimum value independently of any material and physical condition, the transmission coefficient being higher in the presence of a substrate, and getting increasing when QED corrections higher than α come into play. Explicit numerical calculations for typical cases are carried out when an external magnetic field is applied to the sample, showing that, in epitaxial graphene, a threshold effect exists leading to a non-trivial minimum transmission, for a non-vanishing light frequency, only for field values larger than a critical one, both in the large and in the intermediate chemical potential regime. Such a threshold effect manifests even in the maximum Faraday rotation polarization of light, which is substantially controlled by the applied magnetic field. Instead, more transmission minima in suspended graphene enters in the considered light frequency region for increasing magnetic field, displaying an effective shift of frequency bands where the sample gets more or less absorptive with a suitable tuning of the external field. Two transition regions in different magnetic field ranges are found, where the shift effect towards higher frequency values occurs both in the transmission coefficient and in the Faraday rotation angle. Potential technological application of the results presented are envisaged.

Inflation from broken scale invariance.

We construct a model of inflation based on a low-energy effective theory of spontaneously broken global scale invariance. This provides a shift symmetry that protects the inflaton potential from quantum corrections. Since the underlying scale invariance is noncompact, arbitrarily large inflaton field displacements are readily allowed in the low-energy effective theory. A weak breaking of scale invariance by almost marginal operators provides a nontrivial inflaton minimum, which sets and stabilizes the final low-energy value of the Planck scale. The underlying scale invariance ensures that the slow-roll approximation remains valid over large inflaton displacements, and yields a scale invariant spectrum of perturbations, as required by the cosmic microwave background observations.

Band structure effects on the nonlinear optical response of bilayer graphene

Recently, Rabi-like oscillations that occur far from resonance were predicted in monolayer graphene. In bilayer graphene, when the trigonal warping effect is taken into account, this new Rabi frequency shows a zero non-trivial minimum as a function of the strength of the applied electric field in addition to the trivial minimum at zero field. The zero non-trivial minimum occurs where the ‘leg pocket’ of the Fermi surface develops, described in the pioneering work of McCann et al. [Eur. Phys. J. Special Topics 148, 91 (2007)]. Thereafter, the anomalous Rabi frequency varies linearly with the square of the intensity of the applied field consistent with a bilayer system without trigonal warping. It is seen that this anomalous Rabi frequency is affected much more by trigonal warping than the conventional Rabi frequency. The induced current is also significantly affected by the trigonal warping. A fully numerical solution of the optical Bloch equations completely corroborates the analytical findings and provides a basis for the approximation schemes employed.

Thermal Yang–Mills theory in the Einstein universe

We study the stability of a non-Abelian chromomagnetic vacuum in Yang–Mills theory in Euclidean Einstein universe S1 × S3. We assume that the gauge group is a simple compact group G containing the group SU(2) as a subgroup and consider static covariantly constant gauge fields on S3 taking values in the adjoint representation of the group G and forming a representation of the group SU(2). We compute the heat kernel for the Laplacian acting on fields on S3 in an arbitrary representation of SU(2) and use this result to compute the heat kernels for the gluon and the ghost operators and the one-loop effective action. We show that the only configuration of the covariantly constant Yang–Mills background that is stable is the one that contains only spinor (fundamental) representations of the group SU(2); all other configurations contain negative modes and are unstable. For the stable configuration we compute the asymptotics of the effective action, the energy density, the entropy and the heat capacity in the limits of low/high temperature and small/large volume and show that the energy density has a non-trivial minimum at a finite value of the radius of the sphere S3.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Erratum: Gluon condensation and scaling exponents for the propagators in Yang-Mills theory [Phys. Rev. D83, 045014 (2011)

We investigate the infrared (strong-coupling) regime of SU(N)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field strength invariant F_{\mu {\nu}}F^{\mu {\nu}} from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the beta function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared imposes a new upper bound on the infrared exponents of the propagators. For the scaling solution, this bound reads kappa_c < 23/38 which, together with Zwanziger's horizon condition kappa_c> 1/2, defines a rather narrow window for this critical exponent. Current estimates from functional methods indeed satisfy these bounds.

Gluon condensation and scaling exponents for the propagators in Yang-Mills theory

We investigate the infrared (strong-coupling) regime of SU($N$)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field-strength invariant ${F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}$ from nonperturbative gauge correlation functions and find a nontrivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the $\ensuremath{\beta}$ function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared imposes a new upper bound on the infrared exponents of the propagators. For the scaling solution, this bound reads ${\ensuremath{\kappa}}_{c}l23/38$ which, together with Zwanziger's horizon condition ${\ensuremath{\kappa}}_{c}g1/2$, defines a rather narrow window for this critical exponent. Current estimates from functional methods indeed satisfy these bounds.

Anisotropic finite-size scaling of an elastic string at the depinning threshold in a random-periodic medium

We numerically study the geometry of a driven elastic string at its sample-dependent depinning threshold in random-periodic media. We find that the anisotropic finite-size scaling of the average square width $\overline{w^2}$ and of its associated probability distribution are both controlled by the ratio $k=M/L^{\zeta_{dep}}$, where $\zeta_{dep}$ is the random-manifold depinning roughness exponent, $L$ is the longitudinal size of the string and $M$ the transverse periodicity of the random medium. The rescaled average square width $\overline{w^2}/L^{2\zeta_{dep}}$ displays a non-trivial single minimum for a finite value of $k$. We show that the initial decrease for small $k$ reflects the crossover at $k \sim 1$ from the random-periodic to the random-manifold roughness. The increase for very large $k$ implies that the increasingly rare critical configurations, accompanying the crossover to Gumbel critical-force statistics, display anomalous roughness properties: a transverse-periodicity scaling in spite that $\overline{w^2} \ll M$, and subleading corrections to the standard random-manifold longitudinal-size scaling. Our results are relevant to understanding the dimensional crossover from interface to particle depinning. Received: 20 October 2010, Accepted: 1 December 2010; Edited by: A. Vindigni; Reviewed by: A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon, France; DOI: 10.4279/PIP.020008

Casimir energy and brane stability

We investigate the role of Casimir energy as a mechanism for brane stability in five-dimensional models with the fifth dimension compactified on an S 1 / Z 2 orbifold, which includes the Randall–Sundrum two brane model (RS1). We employ a ζ -function regularization technique utilizing the Schwinger proper time method and the Jacobi theta function identity to calculate the one-loop effective potential. We show that the combination of the Casimir energies of a scalar Higgs field, the three generations of Standard Model fermions and one additional massive non-SM scalar in the bulk produces a non-trivial minimum of the potential. In particular, we consider a scalar field with a coupling in the bulk to a Lorentz violating vector particle localized to the compactified dimension. Such a scalar may provide a natural means of fine tuning needed for stabilization of the brane separation. Finally, we briefly review the possibility that Casimir energy plays a role in generating the currently observed epoch of cosmological inflation by examining a simple five-dimensional anisotropic metric.

Can heterogeneity of the near-wellbore rock cause extrema of the Darcian fluid inflow rate from the formation (the Polubarinova-Kochina problem revisited)?

Darcian steady 2-D flow to a point sink (vertical well) placed eccentrically with respect to two circles demarcating zones of contrasting permeability is studied by the methods of complex analysis and numerically by MODFLOW package. In the analytical approach, two conjugated Laplace equations for a characteristic flow function are solved by the method of images, i.e. the original sink is mirrored about two circles that generates an infinite system of fictitious sinks and source. The internal circle of the annulus models formation damage (gravel pack) near the well and the ring-shaped zone represents a pristine porous medium. On the external circle the head (pressure) is fixed and on the internal circle streamlines are refracted. The latter is equivalent to continuity of pressure and normal component of specific discharge that is satisfied by the choice of the intensity and loci of fictitious sinks. Flow net and dependence of the well discharge on eccentricity are obtained for different annulus radii and permeability ratios. A non-trivial minimum of the discharge is discovered for the case of the ring domain permeability higher than that of the internal circle. In the numerical solution, a finite difference code is implemented and compared with the analytical results for the two-conductivity zone. Numerical solution is also obtained for an aquifer with a three-conductivity zonation. The case of permeability exponentially varying with one Cartesian coordinate within a circular feeding contour is studied analytically by series expansions of a characteristic function obeying a modified Helmholtz equation with a point singularity located eccentrically inside the feeding contour. The coefficients of the modified Bessel function series are obtained by the Sommerfeld addition theorem. A trivial minimum of the flow rate into a small-radius well signifies the trade-off between permeability variation and short-cutting between the well and feeding contour.

Structure of a simple fluid in the vicinity of the critical point: Approximate integral equation theory of liquids

An approximate Ornstein–Zernike equation [J. Chem. Phys. 114, 9496 (2001)] is used in order to analyze properties of a simple fluid in the vicinity of the critical point. The heat capacity and compressibility of the fluid are considered involving their relations with the density and temperature derivatives of the thermal potential (or the correlation function). A new structural critical exponent approaching zero value of the first nontrivial minimum of the total correlation function is defined. Using our approximate equation and that derived by Duh, Henderson, and Haymet [J. Chem. Phys. 103, 2625 (1995); 104, 6742 (1996)] a set of the thermodynamic parameters along the isotherm T*=1.310, which is very close to the critical, is calculated. Computer simulation data are also given for comparison.

STABILIZATION AND RADION IN de SITTER BRANE-WORLD

We consider the stabilization of de Sitter brane-world. The scalar field bulk-brane theory produces the nontrivial minimum of modulus potential where temporal radion is realized. The hierarchy problem (between Planck and electroweak scales) may be solved. However, the interpretation of radion is not so clear as in AdS brane-world. In particular, the introduction of two times physics or pair-creation of bulk spaces or identification of one of spatial coordinates with imaginary time (nonzero temperature) may be required.

Minimization of the Scalar Potential in Softly Broken SU(2)×U(1)N=1 SYM with Higgs Matter

We consider a particular extension of the scalar two-Higgs potential of the Minimal Supersymmetric Standard Model to an arbitrary number of Higgs doublets. The extended potential has an absolute and only nontrivial minimum that is found exactly. It appears that this minimum is a simple analog to the well-known minimum of the MSSM. This result is applied to a low-energy Higgs potential of the finite Grand Unification Theory where six Higgs doublets are present.

On the use of classical configurations as subcritical bubbles

The role of classical configuration as subcritical bubbles in the physics of the early universe is analyzed. It is shown that the subcritical bubbles in the scalar background can only be viewed classical objects, i.e. they obey the classical equations of motion, if their amplitude is small, so that the change of the number of quanta is not too rapid. Also it is required that the number of quanta is large enough. In particular analyses with Gaussian bubbles of the size of the size of the correlation length and amplitude of the non-trivial minimum of the scalar potential seem not to be generally valid but should be done with more care.

TWO-FORM GRAVITY AND THE GENERATION OF SPACE-TIME

In the framework of the two-form gravity, which is classically equivalent to the Einstein gravity, the one-loop effective potential for the conformal factor of metric is calculated in the finite volume and in the finite temperature by choosing a temporal gauge condition. There appears a quartically divergent term which cannot be removed by the renormalization of the cosmological term and we find there is only one nontrivial minimum in the effective potential. If the cutoff scale has a physical meaning, e.g. the Planck scale coming from string theory, this minimum might explain why the space-time is generated, i.e. why the classical metric has a nontrivial value.