Saddle-point Equations Research Articles

Soliton quantization and internal symmetry.

We apply the method of collective coordinate quantization to a model of solitons in two spacetime dimensions with a global $U(1)$ symmetry. In particular we consider the dynamics of the charged states associated with rotational excitations of the soliton in the internal space and their interactions with the quanta of the background field (mesons). By solving a system of coupled saddle-point equations we effectively sum all tree-graphs contributing to the one-point Green's function of the meson field in the background of a rotating soliton. We find that the resulting one-point function evaluated between soliton states of definite $U(1)$ charge exhibits a pole on the meson mass shell and we extract the corresponding S-matrix element for the decay of an excited state via the emission of a single meson using the standard LSZ reduction formula. This S-matrix element has a natural interpretation in terms of an effective Lagrangian for the charged soliton states with an explicit Yukawa coupling to the meson field. We calculate the leading-order semi-classical decay width of the excited soliton states discuss the consequences of these results for the hadronic decay of the $\Delta$ resonance in the Skyrme model.

Finite N analysis of matrix models for an n-Ising spin on a random surface

The saddle point equation described by the eigenvalues of N × N Hermitian matrices is analyzed for a finite N case and the scaling relation for the large N is considered. The critical point and the critical exponents of matrix model are obtained by the finite N scaling. The one-matrix model and the two-matrix model are studied in detail. Small N behavior for n-Ising model on a random surface is investigated.

Metastable states of an asymmetric spin-glass with one stored pattern

The number of metastable states in a spin-glass with asymmetric Gaussian couplings (J ij ≠J ji ) is studied. In order to store one pattern a generalized stability condition on the local energies is considered for a part of the system (α=n/N) only. The average number of metastable states is calculated analytically as a function of the overlapm in the thermodynamic limit. For the contribution of the storage restriction an approximation is introduced to simplify the calculations. The corresponding saddlepoint equations are solved numerically for two distribution functions of the local energies. For several values of the parameters we find two bands with an exponential number of metastable states. At a critical value αcrit the second band disappears. The critical value for the storage parameter is calculated numerically. As expected we find αcrit→1 for δ→0.

The replica-symmetric solution without replica trick for the Hopfield model

We derive the saddle-point equations for the order parameters of the Hopfield model in the case of replica symmetry without using the replica trick, but assuming that the Edwards-Anderson parameter is a self-averaging quantity.

Order-parameter flow in the fully connected Hopfield model near saturation.

We present an exact dynamical theory, valid on finite time scales, to describe the fully connected Hopfield model near saturation in terms of deterministic flow equations for order parameters. Two transparent assumptions allow us to perform a replica calculation of the distribution of intrinsic-noise components of the alignment fields. Numerical simulations support our assumptions and indicate that our equations describe the shape of the intrinsic-noise distribution and the macroscopic dynamics correctly in the region where replica symmetry is stable. In equilibrium our theory reproduces the saddle-point equations obtained in the thermodynamic analysis by Amit, Gutfreund, and Sompolinsky [Phys. Rev. A 32, 1007 (1985); Phys. Rev. Lett. 55, 1530 (1985)], the only difference being the absence in the present formalism of negative entropies at low temperatures.

Schwinger-boson theory of the quantum XXZ model.

The quantum XXZ model is studied with the help of the Schwinger-boson formalism. The technique is particularly suited to describe the disordered state of low-dimensional quantum magnets. The Hubbard-Stratonovich transformation is employed to construct the saddle-point equations of the theory. The solution of these equations in one dimension yields the dependence of the Haldane gap as a function of the XY anisotropy.

Dynamics of fully connected attractor neural networks near saturation.

We present an exact dynamical theory, valid on finite time scales, to describe the fully connected Hopfield model near saturation in terms of deterministic flow equations for order parameters. Two transparent assumptions allow us to perform a replica calculation of the distribution of intrinsic noise components of the alignment fields. Numerical simulations indicate that our equations describe the dynamics correctly in the region where replica symmetry is stable. In equilibrium our theory reproduces the saddle-point equations obtained in the thermodynamic analysis by Amit et al.

Asymptotically exact mean-field theory for the Anderson model including double occupancy.

The Anderson impurity model for finite values of the Coulomb repulsion $U$ is studied using a slave boson representation for the empty and doubly occupied $f$-level. In order to avoid well known problems with a naive mean field theory for the boson fields, we use the coherent state path integral representation to first integrate out the double occupancy slave bosons. The resulting effective action is linearized using {\bf two-time} auxiliary fields. After integration over the fermionic degrees of freedom one obtains an effective action suitable for a $1/N_f$-expansion. Concerning the constraint the same problem remains as in the infinite $U$ case. For $T \rightarrow 0$ and $N_f \rightarrow \infty$ exact results for the ground state properties are recovered in the saddle point approximation. Numerical solutions of the saddle point equations show that even in the spindegenerate case $N_f = 2$ the results are quite good.

Large-N quantum gauge theories in two dimensions

The partition function of a two-dimensional quantum gauge theory in the large-N limit is expressed as the functional integral over some scalar field. The large-N saddle point equation is presented and solved. The free energy is calculated as the function of the area and of the Euler characteristic. There is no non-trivial saddle point at genus g>0. The existence of a non-trivial saddle point is closely related to the weak coupling behavior of the theory. Possible applications of the method to higher dimensions are briefly discussed.

The 1/ d expansion for the quantum mechanical n-body problem application for directed polymers in a random medium

In this paper we give a closed form expression for the 1/ d corrections to the leading d → ∞ approximation for the ground-state energy of an n-body quantum mechanical problem in a general potential (where d is the number of spatial dimensions). We then extend the results to the problem of directed polymers in a random medium where the limit n → 0 has to be taken in the corresponding n-body problem. In this case the effective two-body potential measures the strength of correlations of the quenched disorder. In the n → 0 limit a replica-symmetry-breaking solution to the saddle-point equations turns out to be the correct solutionin the strong-coupling regime, and we obtain the 1/ d corrections to the corresponding energy which constitute the gaussian fluctuations about this solution as well as provide information about its basin of stability.

Specific details of the mean-field slave-boson solution of the Hubbard model.

The single-band Hubbard model has been studied on a simple square lattice using the slave-boson representation. The magnetic phase diagram was derived numerically from solutions of the saddle-point equations. With increase in the on-site Coulomb repulsion U we find a phase transition to a ferromagnetic state that is fully spin polarized even for finite U

Multi-instanton valleys in the O(3) sigma model.

An explicit construction is given for valleys in the configuration space of an arbitrary number of instanton-anti-instanton pairs for the O(3) nonlinear $\ensuremath{\sigma}$ model in two dimensions. The $\ensuremath{\sigma}$-model action is supplemented by an explicit conformal-symmetry-breaking term and the multi-instanton valleys are used to examine the relative importance of single- and multi-instanton contributions to anomalous scattering. Numerical solutions to the saddle-point equations for the cross section are presented which support the existence of a critical energy above which the dilute instanton gas approximation breaks down. The relevance of these results to the problem of multi-instanton corrections to processes which violate baryon number in the standard model is discussed.

Strings with discrete target space

We investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line Z which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. Introducing a quantum string field Ψ x( l) depending on the x and the length l of the closed string, we give a formal definition of the string field theory in terms of a functional integral. The classical string background is found as a solution of the saddle-point equation which is equivalent to the loop equation we have previously considered. The continuum limit exists in the vicinity of the singular points of this equation. We show that for given target space there are many ways to achieve the continuum limit; they are related to the multicritical points of the ensemble of surfaces without embedding. Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the non-interacting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential.

Crumpled glass phase of randomly polymerized membranes in the large d limit

Tethered phantom membranes with quenched disorder in the internal preferred metric are studied in the limit of large embedding space dimension d↦∞. We find that the instability of the flat phase previously demonstrated via ϵ-expansion is towards a spin-glass-like phase which we call the crumpled glass phase. We propose a spin-glass order parameter that characterizes this phase and derive the free energy which describes the crumpled, flat and crumpled glass phases is described by local tangents which vanish on average, but display a nonzero Edwards-Anderson spin-glass order parameter. From the saddle point equations at large d we obtain the equation of state, phase diagram and the exponents characterizing these phases. We estimate the effects of the higher order corrections in the 1/d expansion by utilizing previous results for pure membranes. We use Flory arguments to calculate the wandering exponents and discuss the relevance of self-avoidance in the crumpled glass phase.

Saddlepoint approximations for the Bartlett-Nanda-Pillai trace statistic in multivariate analysis

SUMMARY The null distribution of the Bartlett-Nanda-Pillai trace statistic is shown to be equivalent to the conditional distribution of canonical sufficient statistics in a multiparameter exponential family setting. As such, sequential and double saddlepoint approximations are used to approximate the distribution. The existence of tabulated percentiles for the distribution allows for comparison of the accuracy of the various saddlepoint approximations. The ability of the approximations to accommodate large numbers of nuisance parameters is also revealed with numerical studies. The Bartlett-Nanda-Pillai trace statistic is one of the four principal test statistics used in the multivariate analysis of variance; p-values of tests based on it are computed from its null distribution. We show in ? 2 that this null distribution is equivalent to a conditional distribution involving components of the canonical sufficient statistic in the context of a certain multiparameter regular exponential family. Such equivalence allows for the possibility of saddlepoint approximation for both the null density and distribution function so that highly accurate approximations for p-values can be easily obtained. Section 3 reviews the various saddlepoint methods that we shall base approximations upon. Sequential saddlepoint methods introduced by Fraser, Reid & Wong (1991) are used to compute a sequential density approximation as well as distributional approximations based on the Lugannani & Rice (1980) formula and the r* approximation of Bamdorff-Nielsen (1986, 1991). Double saddlepoint analogues to the above methods are also computed. A double saddlepoint density approximation (Bamdorff-Nielsen & Cox, 1979) takes the explicit form in (3-8) below as a modification to a rescaled beta density. This explicit form is due to the explicit solution (4.11) that arises from the double saddlepoint equation. Double saddlepoint distributional approximations based on Skovgaard (1987) and the r* approximation (Bamdorff-Nielsen, 1986, 1991) are also computed. The numerical comparisons of ? 6 reveal that the double saddlepoint distributional approximations of Skovgaard (1987) and Bamdorff-Nielsen (1986, 1991) are grossly

Fast and accurate approximate double bootstrap confidence intervals

We propose two new methods for the construction of approximate iterated bootstrap confidence intervals. Our methods follow a saddlepoint approach, but employ approximate rather than exact solutions to the nonlinear saddlepoint equations that arise in applying that technique. The main advantages of the new techniques are that they yield confidence intervals with high coverage accuracy in small- to moderately-sized samples and that they achieve this accuracy with very substantial reductions in computation time compared to other methods

Asymptotic evaluation of three-dimensional wave packets in parallel flows

This paper examines the three-dimensional wave packets which are generated by an initially localized pulse disturbance in an incompressible parallel flow and described by a double Fourier integral in the wavenumber space. It aims to clear up some confusion arising from the asymptotic evaluation of this integral by the method of steepest descent. In this asymptotic analysis, the calculation of the eigenvalues can be facilitated by making use of the Squire transformation. It is demonstrated that the use of the Squire transformation introduces branch points in the saddle-point equation that links the physical coordinates to the saddle-point value, regardless of whether the flow is viscous or inviscid. It is shown that the correct branch should be chosen according to the principle of analytic continuation. The saddle-point values for the three-dimensional problem should be considered to be the analytic continuation of those for the two-dimensional case where the saddle-point values can be uniquely determined. The three-dimensional wave packets in an inviscid wake flow are examined; their behaviour at large time is calculated asymptotically by the method of steepest descent in terms of the two-dimensional eigenvalue relation.

Matrix field theory for disordered local pairing superconductors with two glass order parameters and spin-flip pairbreaking processes

We derive the exact matrix field theory for a replicated grassmannian representation of a local pairing superconducting disorder ensemble including three superconducting order parameters and the spin-flip pairbreaking mechanism. Disorder is assumed to be gaussian distributed. We find by exactly solving the saddle-point equation the criterion for a vanishing gap 〈Δ〉≦τσ−1+τΔ−1, where 〈Δ〉 denotes the averaged superconducting order parameter, τσ−1 the spin-flip scattering rate, and τΔ−1 the scattering rate corresponding to correlations of Re(Δ−〈Δ〉). Taken at Δ=0, our field theory, which is exact in all orders of τσ−1, contains new terms in addition to those of theO(τσ−1) model derived by Efetov et al. Our formulation transfers correctly to all orders the invariances of the action into symmetries of the matrix field theory. The saddle point approximation is outlined and it is shown how singular corrections to the saddle point density of states arise atEF in a gapless superconductor. Finally singular corrections in the two particle propagator, the density correlation function and the conductivity are calculated for 〈Δ〉=0 in one loop order. It turns out that these corrections can be entirely expressed by those of the single particle density of states.

Instanton approach to the conductivity of a disordered solid

For the ac-conductivity of a one-dimensional gaussian-disordered electron system a supersymmetric field theory is constructed. The saddle-point equation for the limit of deeply localized states is solved exactly. Calculating the fluctuations around this saddle point in lowest order of the frequency, we obtain the Mott formula σ( ω)∼ ω 2In 2 ω also in the limit of weak disorder in contrast to the work of Houghton, Schäfer and Wegner.

Replica-symmetric theory of nonlinear analogue neural networks

On the basis of the theory of the naive mean field model of spin glasses, analogue neural networks of the Hopfield type are investigated in the saturation limit. The saddle-point equations for the order parameters describing retrieval and spin glass phases of the networks are obtained by means of a statistical mechanical analysis within the framework of the replica-symmetric theory and are shown to undergo a modification to that of AGS theory, due to the absence of the Onsager reaction term in the TAP equation. Based on the equations, the memory storage capacity of the networks is analysed as a function of the analogue gain beta . A small increase in the critical storage capacity is found for finite values of beta , compared with that of the Ising model networks with corresponding inverse temperature, although the qualitative nature of the phase diagram is unchanged.