Then-vector Model Research Articles

Wilson–Polchinski exact renormalization group equation forO (N) systems: leading and next-to-leading orders in the derivative expansion

With a view to study the convergence properties of the derivative expansion of the exactrenormalization group (RG) equation, I explicitly study the leading and next-to-leadingorders of this expansion applied to the Wilson–Polchinski equation in the case of theN-vector model with thesymmetry O (N). As a test,the critical exponents η and ν as well as thesubcritical exponent ω (and higher ones) are estimated in three dimensions for values ofN rangingfrom 1 to 20. Icompare the results with the corresponding estimates obtained in preceding studies or treatments ofother O (N) exact RG equations at second order. The possibility of varyingN allows the derivative expansion method to be better valued. The values obtained from theresummation of high orders of perturbative field theory are used as standards to illustratethe eventual convergence in each case. Particular attention is drawn to the preservation (ornot) of the reparameterization invariance.

Random walk representations and the mayer expansion in theN-vector model

We consider a random walk representation of non-abelian statistical models, and apply it to represent the free energy in terms of the correlation functions of random walks. This enables us to find an analytic region of the free energy with respect to the inverse temperature. This method can be applied to the block spin transformations.

Large-n limit of the Heisenberg model: Random external field and random uniaxial anisotropy

The thermodynamic equivalence of the large-n limit of then-vector model in a random external field and the corresponding disordered spherical model is proved. An analytic expression for the free energy and a phase diagram of the large-n limit of then-vector model with random uniaxial anisotropy are obtained by rigorous argument. The ferromagnetic order in the large-n limit is proved to be stable against the switching on of an arbitrarily small random anisotropy.

Minimal renormalization without ?-expansion: Amplitude functions in three dimensions

The minimal subtraction scheme and the Borel resummation method are used to calculate the amplitudes of renormalized correlation functions aboveTc for then-vector model in three dimensions. Accurate representations are given for the effective amplitudes of the renormalized expressions of the correlation length, of the susceptibility and of the specific heat forn=1, 2, 3. The resummed higher-order contributions turn out to yield only small corrections to the low-order approximations. These results complement a recent calculation of renormalization-group functions and provide the basis for accurate analyses of the critical behavior in three dimensions including the amplitude functions.

Anomalous dimensions of high-gradient operators in then-vector model in 2+? dimensions

The anomalous dimensions of operators with an arbitrary number of gradients are determined for then-vector model ind=2+e dimensions in one-loop order. For those operators which do not vanish ind=2 dimensions all anomalous dimensions can be given explicitly. Among the scalar operators (underO(n) andO(d)) with 2s derivatives there is an operator with the full dimensiony=2(1−s)+ɛ(1+s(s−1)/(n−2))+O(ɛ2). Thus similarly as for theQ-matrix model investigated by Kravtsov, Lerner, and Yudson, large positive corrections in one-loop order are obtained for then-vector model. Possible consequences of the corrections are discussed.

High-gradient operators of the unitary matrix-model

A complete classification of all rotationally invariant operators of the two-dimensional unitary matrix model composed of gradients of the fieldQ and their anomalous dimensions are given in one-loop order. Similarly as in the orthogonal case and for then-vector model the leading correction of operators with2n factors δQ grows withn(n−1).

Series expansion solution of the Wegner-Houghton renormalisation group equation

The momentum independent projection of the Wegner-Houghton renormalisation group equation is solved with power series expansion. Convergence rate is analyzed for then-vector model. Further evidence is presented for the first order nature of the chiral symmetry restoration at finite temperature in QCD with 3 light flavors.

The spherical-model limit in a random field

The spherical-model limitn → ∞ of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then → ∞ and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.

Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.

Validity of the long-range expansion in then-vector model

Validity of the long-range expansion in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-vector model

Convexity violations for noninteger parameters in certain lattice models

The convexity of the free energy is studied for several lattice models in situations in which a parameter which is normally a positive integer takes on noninteger real values. Examples include the numbern of components in then-vector model, the number of states in the Potts model, and the dimensionality of the lattice. In a typical case there is a critical value of the parameter such that convexity is preserved when the parameter exceeds the critical value, but can be violated for appropriate Hamiltonians whenever the parameter is less than the critical value, but not a positive integer. In several cases the critical value of the parameter increases with the size of the system, thus raising questions about the significance of a continuous variation of the parameter in the thermodynamic limit.

Erratum: Polymer statistics, then-vector model, and thermodynamic stability

Erratum: Polymer statistics, the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-vector model, and thermodynamic stability

The spherical limit forn-vector correlations

We investigate then→ ∞ limit of then-vector model single-spin and pairspin correlation functions. In this limit we show that the correlation functions become those of the corresponding spherical model.

Approach to two dimensions in then-vector model

The specific heat of then-vector model to lowest order in 1/n is analyzed when the dimensionality varies continuously betweend=3 andd=2. The changeover from three-dimensional to two-dimensional behavior is related to the location of the bare transition temperature with respect to the critical region.