Symmetric Non-linear Σ-model Research Articles

1-Loop improved lattice action for the nonlinear σ-model

In this paper we show the Wilson effective action for the 2-dimensional O( N + 1)-symmetric lattice nonlinear σ-model computed in the 1-loop approximation for the nonlinear choice of blockspin /gf( x ), /gf( x ) = C /gf( x )/| C /gf( x ) where C is averaging of the fundamental field /gf( z ) over a square x of side a . The result for S eff is composed of the classical perfect action with a renormalized coupling constant β eff , an augmented contribution from a Jacobian, and further genuine 1-loop correction terms. Our result extends Polyakov's calculation which had furnished those contributions to the effective action which are of order 1n a a , where a is the lattice spacing of the fundamental lattice. An analytic approximation for the background field which enters the classical perfect action will be presented elsewhere [1].

Variational resummation for ϵ-expansions of critical exponents of nonlinear O( n)-symmetric σ-model in 2+ ϵ dimensions

We develop a method for extracting accurate critical exponents from perturbation expansions of the O( n)-symmetric nonlinear σ-model in D=2+ ϵ dimensions. This is possible by considering the ϵ-expansions in this model as strong-coupling expansions of functions of the variable ε ̃ ≡2(4−D)/(D−2) , whose first five weak-coupling expansion coefficients of powers of ε ̃ are known from ε-expansions of critical exponents in O( n)-symmetric φ 4-theory in D=4− ε dimensions.

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Simulation of statistical systems with not necessarily real and positive probabilities

A new method to determine expectation values of observables in statistical systems with not necessarily real and positive probabilities is proposed. It is tested in a numerical study of the two-dimensional O(3)-symmetric nonlinear σ-model with Symanzik's one-loop improved lattice action. This model is simulated as a polymer system with field dependent activities which can be made positive definite or indefinite by adjusting additive constants of the action. For a system with indefinite activities the new proposal is found to work. It is also verified that local observables are not affected by far-away polymers with indefinite activities when the system has no long-range order.

Some exact results for the O( N)-symmetric non-linear σ-model to O(1/ N)

With the non-linear σ-model on the lattice in view, we study its continuum version with a finite, sharp momentum cut-off. We find exact analytical results for its mass gap and magnetic susceptibility to two leading orders in 1/ N. Asymptotically, at weak coupling, our results converge to those of renormalization group improved weak coupling perturbation theory to two-loop order, and give the ratio of the mass to the Λ-parameter analytically. Thus our results provide an opportunity to study the rate at which asymptotic scaling is approached. Since this approach is slow for low values of N, we find a region of intermediate coupling strenghts, where our results describe non-asymptotic scaling of a well-defined and universally valid kind.

Formula omitted]-expansion of the non-linear σ-model: The first three orders

The two-point function of the O( N)-symmetric non-linear σ-model is expanded in 1 N , keeping terms of three leading orders. The mass gap and the magnetic susceptibility are obtained from the two-point function. They are evaluated on square lattices for N = 3 and N = 4. The systematic errors of the 1 N series truncated after the first, second, or third term are found by using recent high-precision Monte Carlo results as bench marks. For all three truncations, we find systematic errors which are smaller than the expected magnitude of neglected terms, both for the mass gap and for the susceptibility. This result is uniform in the inverse coupling β, and valid for N as small as 3. We conclude that the 1 N series approach the exact results as rapidly as one could ever hope for.

1/ N-expansion of the non-linear σ-model: The first three orders

The two-point function of the O( N)-symmetric non-linear σ-model is expanded in 1/ N, keeping terms of three leading orders. The mass gap and the magnetic susceptibility obtained from the two-point function are evaluated on square lattices for N = 3 and N = 4. Using recent high precision Monte carlo data as bench marks, the systematic error on the truncated 1/ N-expansion is smaller than the terms neglected to any of the three orders at hand. Actually, the non-linear σ-model with N ≥ 3 is little more than a Gaussian model in a one-component auxiliary field, as far as the mass gap and magnetic susceptibility are concerned.

Scaling versus asymptotic scaling in the non-linear σ-model in 2D. continuum version

The two-point function of the O( N)-symmetric non-linear σ-model in two dimensions is large- N expanded and renormalized, neglecting terms of O( 1 N 2 ) . At finite cut-off, universal, analytical expressions relate the magnetic susceptibility and the dressed mass to the bare coupling. Removing the cut-off, a similar relation gives the renormalized coupling as a function of the mass gap. In the weak-coupling limit these relations reproduce the results of renormalization group improved weak-coupling perturbation theory to two-loop order. The constant left unknown, when the renormalization group is integrated, is determined here. The approach to asymptotic scaling is studied for various values of N.

Gauge invariance and systems with second-class constraints

We show that under a certain assumption systems with second-class constraints can be regarded as gauge-fixed systems with first-class constraints. The massive Yang-Mills theory, a point particle on a sphere and theO(N) symmetric non-linear σ-model are considered as concrete examples.

Dyson-Schwinger equations for the non-linear σ-model

Dyson-Schwinger equations for the O( N)-symmetric non-linear σ-model are derived. They are polynomials in N, hence 1/ N-expanded ab initio. A finite, closed set of equations is obtained by keeping only the leading term and the first correction term in this 1/ N-series. These equations are solved numerically in two dimensions on square lattices measuring 50 × 50, 100 × 100, 200 × 200, and 400 × 400. They are also solved analytically at strong coupling and at weak coupling in a finite volume. In these two limits the solution is asymptotically identical to the exact strong- and weak-coupling series through the first three terms. Between these two limits, results for the magnetic susceptibility and the mass gap are identical to the Monte Carlo results available for N = 3 and N = 4 within a uniform systematic error of O(1/N 3) , i.e. the results seem good to O(1/N 2) , though obtained from equations that are exact only to O(1/N) . This is understood by seeing the results as summed infinite subseries of the 1/ N-series for the exact susceptibility and mass gap. We conclude that the kind of 1/ N-expansion presented here converges as well as one might ever hope for, even for N as small as 3.

The non-linear σ-model to [formula omitted

The O( N)-symmetric non-linear σ-model is expanded in 1/ N, keeping terms of ▪(1) and ▪(1/ N). The Feynman diagrams giving the mass gap and the magnetic susceptibility are evaluated on two-dimensional, square lattices. Our results approximate Monte Carlo results very well for all values of the inverse coupling β.

Formula omitted] of the non-linear σ-model

The two-point function of the O(N)-symmetric non-linear σ-model is expanded in 1/N, keeping both the leading term and the first correction term. From this function the mass gap and the magnetic susceptibility are obtained. After mass renormalization, the Feynman diagrams giving these quantities are evaluated on two-dimensional, square lattices. The results compare very favorably with Monte Carlo results for all values of the inverse coupling β. The first non-vanishing term in 1N-expansion of the four-point function is also obtained.

Resummation of the [formula omitted] of the non-linear σ-model by Dyson-Schwinger equations

Dyson-Schwinger equations for the O( N)-symmetric non-linear σ-model are derived and expanded in 1 N . A closed set of equations is obtained by keeping only the leading term and the first correction term in this expansion. These equations are solved numerically in two dimensions on square lattices of sizes 50×50 and 100×100. Results for the magnetic susceptibility and the mass gap are compared with predictions of the ordinary 1 N - expansion and with Monte Carlo results. The results obtained with the Dyson-Schwinger equations show the same scaling behavior as found in the Monte Carlo results. This is not the behavior predicted by the perturbative renormalization group.

Recent results for the 2-dimensional σ-model using Dyson-Schwinger equations

The Dyson-Schwinger equations for the O( N)-symmetric non-linear σ-model are found and expanded in 1 N . A closed set of equations is obtained by keeping only the leading term and the first correction term in this expansion. These equations are solved numerically in 2 dimensions on square lattices of sizes 50 × 50 and 100 × 100. Results for the magnetic susceptibility and the mass gap are compared with predictions of the ordinary 1 N - expansion and with Monte Carlo results. The results obtained with the Dyson-Schwinger equations show the same scaling behavior as found in the Monte Carlo results. This is not the behavior predicted by the perturbative renormalization group.

Understanding of the symmetric space σ models through the soliton connection

It is pointed out that a special class of soliton equations is actually like the symmetric σ-model. Through this understanding we derive a new set of local and nonlocal conservation laws for the SU(m + n) S(U(m) × U(n)) symmetric space non-linear σ-model.

Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N ⩾ 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N ⩾ 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N ⩾ 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.