Symmetric Field Theory Research Articles

A new Lagrangian dynamic reduction in field theory

For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first jet bundle of a principal bundle. It is shown that covariant and dynamic reduction lead to equivalent equations of motion. This is achieved by constructing a new Lagrangian defined on an infinite dimensional space which turns out to be gauge group invariant.

Almeida-Thouless transition below six dimensions

The existence of an Almeida-Thouless (AT) instability surface below the upper critical six dimensions is demonstrated in the generic replica symmetric field theory. Renormalization flows from around the zero-field fixed point are investigated. By introducing the temperature and magnetic-field dependence of the bare parameters, the fate of the AT line can be followed from mean field $(d=\ensuremath{\infty})$ down to $d=6\ensuremath{-}ϵ$.

Stationary Symmetric α-Stable Discrete Parameter Random Fields

We establish a connection between the structure of a stationary symmetric α-stable random field (0<α<2) and ergodic theory of non-singular group actions, elaborating on a previous work by Rosinski (Ann. Probab. 28:1797–1813, 2000). With the help of this connection, we study the extreme values of the field over increasing boxes. Depending on the ergodic theoretical and group theoretical structures of the underlying action, we observe different kinds of asymptotic behavior of this sequence of extreme values.

Non-perturbative calculations for the effective potential of the PT symmetric and non-Hermitian (-gφ4) field theoretical model

We investigate the effective potential of the PT symmetric (-gφ4) field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential, from which the predicted vacuum condensate vanishes exponentially as G→0+, in agreement with previous calculations. For the higher orders, we employ the invariance of the bare parameters under the change of the mass scale t to fix the transformed form totally equivalent to the original theory. The form so obtained up to G3 is new and shows that all the 1PI amplitudes are perturbative for both the \(G\ll1\) and the \(G\gg1\) region. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all G values. This unique formula is necessary, because the effective potential is the generating functional for all the one-particle irreducible (1PI) amplitudes that can be obtained via ∂nE/∂bn, and thus we can obtain an analytic calculation of the 1PI amplitudes. Again, the resummed form of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit to a previous calculation concerning bound states.

Quantum Hamiltonian for gravitational collapse

Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted to flat spatial slicing, we give a construction of the reduced Hamiltonian operator. This Hamiltonian, together with the null expansion operators presented in an earlier work, form a framework for studying gravitational collapse in quantum gravity. We describe a setting for its numerical implementation, and discuss some conceptual issues associated with quantum dynamics in a partial gauge fixing.

Is the droplet theory for the Ising spin glass inconsistent with the replica field theory?

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.

COMMENTS ON THE DEPENDENCE BETWEEN ELECTRIC CHARGE AND MAGNETIC CHARGE

By using the two four-dimensional potential formulation of electromagnetic (EM) field theory introduced in Ref. 1, we found that the SO(2) duality symmetric EM field theory can be reduced to the magnetic source-free case by a special choice of SO(2) parameter, we called this special case nature picture of the EM field theory, the reduction condition led to a result, i.e. the electric charge and magnetic charge are no longer independent. Some comments to Ref. 10 are also mentioned.

Replica field theory and renormalization group for the Ising spin glass in an external magnetic field.

We use the generic replica symmetric cubic field theory to study the transition of short-range Ising spin glasses in a magnetic field around the upper critical dimension. A novel fixed point is found from the application of the renormalization group. In the spin-glass limit, this fixed point governs the critical behavior of a class of systems characterized by a single cubic parameter. For this universality class, the spin-glass susceptibility diverges at criticality, whereas the longitudinal mode remains massive. The third mode, however, behaves unusually. The physical consequences of this unusual behavior are discussed, and a comparison with the conventional de Almeida-Thouless scenario is presented.

Spin-glass transition in a magnetic field: A renormalization group study

We study the transition of short range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number n to 0, and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin glass transitions in zero and non-zero field are analyzed in a common framework. The mean field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in non-zero field, at the de Almeida-Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order epsilon = 6-d. In the general theory we find a stable fixed-point associated to the spin glass transition in zero field. This fixed-point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed-point to describe the AT transition, in agreement with previous results of other authors.

The \({\rbb Z}_{\mibs N}\) Symmetric Quantum Lattice Field Theory The Quantum Group Symmetry, the Yang–Baxter Equation, and the Integrals of Motion

We study the quantum group symmetry of the \({\bb Z}_{N}\) symmetric lattice field theory, which is viewed as a discretization of the \(\widehat{s\ell (N)}\) affine Toda field theory. We reveal the quantum group structure, and construct explicitly the quantum local integrals of motion by use of the quantum dilogarithm function.

Quantum group symmetry and integrals of motion in the [formula omitted] symmetric lattice field theory

We study the quantum group symmetry of the Z 3 symmetric lattice field theory. We shall explicitly construct a generating function of the local integrals of motion, and show that it belongs to the kernel of adjoint action of the screening charges.

Finite-size scaling in the theory above the upper critical dimension

We derive exact results for several thermodynamic quantities of the O ( n ) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O ( n ) symmetric model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near Tc of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the field theory turn out to be nonuniversal whereas those of the lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions.

Finite-Size Effects in the φ4 Field and Lattice Theory Above the Upper Critical Dimension

We demonstrate that the standard O(n) symmetric φ4 field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d&gt;4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n →∞ and n=1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-dimensional Ising model are not conclusive.

SCALING VIOLATION IN O(N) VECTOR MODELS

We investigate O(N)-symmetric vector field theories in the double scaling limit. Our model describes branched polymeric systems in D dimensions, whose multicritical series interpolates between the Cayley tree and the ordinary random walk. We give explicit forms of residual divergences in the free energy, analogous to those observed in the strings in one dimension.

Coaxing charge from the vacuum

We examine a U(1)-symmetric field theory for two scalar fields in which thee tree-level potential is ununded from below. We calculate the finite-temperature effective potential and demonstrate that (1) radiative corrections stabilize the theory; (2) Q-balls appear at low temperatures; and (3) the U(1) symmetry is spontaneously broken at high temperatures and restored at low temperatures. We investigate the theory's dynamics in a sector of fixed charge, focusing on the cosmological production of Q-balls as relics of the early era of broken symmetry.

Neural networks with low levels of activity: Ising vs. McCulloch-Pitts neurons

The performance of neural networks used as associative memory for uncorrelated patterns with prescribed mean activity is analyzed within the replica symmetric mean field theory. The optimal representation of the possible states of the neutrons, active or inactive, is found to depend on the mean activity. For activity equal one half Ising neurons and for low activities McCulloch-Pitts neurons are optimal. In this optimal representation the noise due to noncondensed patterns is reduced.

Partially connected models of neural networks

A partially connected Hopfield neural network model is studied under the restriction that w, the ratio of connections per site to the size of the system, remains finite as the size N to infinity with the connection structure at each site being the same. The replica symmetric mean field theory equations for the order parameters are derived. The zero-temperature forms of these equations are then solved numerically for a few different 'local' connectivity architectures showing phase transitions at different critical storage ratios, alpha c, where the states which the authors are trying to store in the network become discontinuously unstable. They show that the information capacity per connection improves for partially connected systems.

Magnetic behavior of the Co2−xZnxTiO4 and Co2SnO4 systems

Our ac susceptibility (χ) measurements, carried out between 1.7 and 100 K, show five χ peaks, signaling five transition temperatures, in the disordered frustrated spinel ferrite Co2TiO4. This result, in conjunction with the replica symmetric mean field theory of vector spin glasses and earlier magnetization and neutron diffraction studies, indicates a separate freezing of A- and B-site spins. To our knowledge, such a phenomenon has not been observed before. Co2SnO4 and Co1.2Zn0.8TiO4 χ measurements, where three and one χ peaks are respectively observed, support this picture.

The principle of equivalence and a theory of gravitation

We examine a well-known thought experiment often used to explain why we should expect a ray of light to be bent by gravity; according to this the light bends downward in the gravitational field because this is just what an observer would see if there were no field and he were accelerating upward instead. We show that this description of the action of Newtonian gravity in a flat space-time corresponds to an old two-index symmetric tensor field theory of gravitation.

Solitons in cyclic symmetric field theories

One-dimensional solitons are found in a model of Z(2) and Z(3) cyclic symmetric vector field theories, as functions of the scalar x 4-kink solution.