arXiv subjects

A. Pelissetto

Publications and source records attributed to A. Pelissetto.

34 records · Page 2Linked to original sources

arXiv2000

The effective potential of N-vector models: a field-theoretic study to O(ε^3)

We study the effective potential of three-dimensional O($N$) models. In statistical physics the effective potential represents the free-energy density as a function of the order parameter (Helmholtz free energy), and, therefore, it is related to the equation of state. In particular, we consider its small-field expansion in the symmetric (high-temperature) phase, whose coefficients are related to the zero-momentum $2j$-point renormalized coupling constants $g_{2j}$. For generic values of $N$, we calculate $g_{2j}$ to three loops in the field-theoretic approach based on the $ε$-expansion. The estimates of $g_{2j}$, or equivalently of $r_{2j}\equiv g_{2j}/g_4^{j-1}$, are obtained by a constrained analysis of the series that takes into account the exact results in one and zero dimensions.

cond-mat.stat-mech↗

arXiv2000

The N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: a six-loop study

We consider the Ginzburg-Landau Hamiltonian with a cubic-symmetric quartic interaction and compute the renormalization-group functions to six-loop order in d=3. We analyze the stability of the fixed points using a Borel transformation and a conformal mapping that takes into account the singularities of the Borel transform. We find that the cubic fixed point is stable for N>N_c, N_c = 2.89(4). Therefore, the critical properties of cubic ferromagnets are not described by the Heisenberg isotropic Hamiltonian, but instead by the cubic model at the cubic fixed point. For N=3, the critical exponents at the cubic and symmetric fixed points differ very little (less than the precision of our results, which is $\lesssim 1%$ in the case of $γ$ and $ν$). Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly-decaying scaling corrections with exponent $ω_2=0.010(4)$. For N=2, the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent $ω_2 = 0.103(8)$. These conclusions are confirmed by a similar analysis of the five-loop $ε$-expansion. A constrained analysis which takes into account that $N_c = 2$ in two dimensions gives $N_c = 2.87(5)$.

cond-mat.stat-mech↗

arXiv1999

Four-loop contributions to long-distance quantities in the two-dimensional nonlinear sigma-model on a square lattice: revised numerical estimates

We give the correct analytic expression of a finite integral appearing in the four-loop computation of the renormalization-group functions for the two-dimensional nonlinear sigma-model on the square lattice with standard action, explaining the origin of a numerical discrepancy. We revise the numerical expressions of Caracciolo and Pelissetto for the perturbative corrections of the susceptibility and of the correlation length. For the values used in Monte Carlo simulations, N=3, 4, 8, the second perturbative correction coefficient of the correlation length varies by 3%, 4%, 3% respectively. Other quantities vary similarly.

hep-lat↗

arXiv1999

Mean-field expansion for spin models with medium-range interactions

We study the critical crossover between the Gaussian and the Wilson-Fisher fixed point for general O(N)-invariant spin models with medium-range interactions. We perform a systematic expansion around the mean-field solution, obtaining the universal crossover curves and their leading corrections. In particular we show that, in three dimensions, the leading correction scales as $R^{-3}, R$ being the range of the interactions. We compare our results with the existing numerical ones obtained by Monte Carlo simulations and present a critical discussion of other approaches.

cond-mat.stat-mech↗

arXiv1998

Crossover scaling from classical to nonclassical critical behavior

We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-N limit of the O(N) phi^4 model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.

cond-mat.stat-mech↗

arXiv1998

Non-analyticity of the Callan-Symanzik beta-function of O(N) models

In the framework of the 1/N-expansion we show that the Callan-Symanzik beta- function associated with the four-point coupling g is non-analytic at its zero, i.e. at the fixed-point value g^* of g. This behavior can be interpreted by renormalization group arguments, and written in terms of scaling correction exponents. We obtain accurate determinations of g^* in 3-d and 2-d by exploiting two alternative approaches: the (4-d)-expansion and the high-temperature expansion. These results are compared with the available estimates by other approaches, such as the fixed-dimension perturbative expansion, Monte Carlo simulations, etc... We also present results for the n-point renormalized coupling constants that parameterize the behavior of the effective potential in the high- and low- temperature phases.

hep-lat↗

arXiv1998

Low-temperature effective potential of the Ising model

We study the low-temperature effective potential of the Ising model. We evaluate the three-point and four-point zero-momentum renormalized coupling constants that parametrize the expansion of the effective potential near the coexistence curve. These results are obtained by a constrained analysis of the $ε$-expansion that uses accurate estimates for the two-dimensional Ising model.

cond-mat.stat-mech↗

arXiv1998

Four-point renormalized coupling constant and Callan-Symanzik beta-function in O(N) models

We investigate some issues concerning the zero-momentum four-point renormalized coupling constant g in the symmetric phase of O(N) models, and the corresponding Callan-Symanzik beta-function. In the framework of the 1/N expansion we show that the Callan- Symanzik beta-function is non-analytic at its zero, i.e. at the fixed-point value g^* of g. This fact calls for a check of the actual accuracy of the determination of g^* from the resummation of the d=3 perturbative g-expansion, which is usually performed assuming analyticity of the beta-function. Two alternative approaches are exploited. We extend the ε-expansion of g^* to O(ε^4). Quite accurate estimates of g^* are then obtained by an analysis exploiting the analytic behavior of g^* as function of d and the known values of g^* for lower-dimensional O(N) models, i.e. for d=2,1,0. Accurate estimates of g^* are also obtained by a reanalysis of the strong-coupling expansion of lattice N-vector models allowing for the leading confluent singularity. The agreement among the g-, ε-, and strong-coupling expansion results is good for all N. However, at N=0,1, ε- and strong-coupling expansion favor values of g^* which are sligthly lower than those obtained by the resummation of the g-expansion assuming analyticity in the Callan-Symanzik beta-function.

cond-mat↗

arXiv1998

Effective potential in three-dimensional O(N) models

We consider the effective potential in three-dimensional models with O(N) symmetry. For generic values of N, and in particular for the physically interesting cases N=0,1,2,3, we determine the six-point and eight-point renormalized coupling constants which parametrize its small-field expansion. These estimates are obtained from the analysis of their $ε$-expansion, taking into account the exact results in one and zero dimensions, and, for the Ising model (i.e. N=1), the accurate high-temperature estimates in two dimensions. They are compared with the available results from other approaches. We also obtain corresponding estimates for the two-dimensional O($N$) models.

cond-mat.stat-mech↗

arXiv1997

The two-point correlation function of three-dimensional O(N) models: critical limit and anisotropy

In three-dimensional O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice non-linear O(N) sigma models, 1/N-expansion, field-theoretical methods within the phi^4 continuum formulation. In non-rotational invariant physical systems with O(N)-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent rho, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N=\infty one finds rho=2. We show that, for all values of $N\geq 0$, $ρ\simeq 2$. Non-Gaussian corrections to the universal low-momentum behavior of G(x) are evaluated, and found to be very small.

cond-mat↗

arXiv1997

On the evaluation of universal non-perturbative constants in O(N) sigma models

We investigate the relation between on-shell and zero-momentum non-perturbative quantities entering the parametrization of the two-point Green's function of two-dimensional non-linear O(N) sigma models. We present accurate estimates of ratios of mass-scales and renormalization constants, obtained by an analysis of the strong-coupling expansion of the two-point Green's function. These ratios allow to connect exact known on-shell results with typical zero-momentum lattice evaluations. Our results are supported by the 1/N-expansion.

hep-lat↗

arXiv1995

Four-point renormalized coupling constant in O(N) models

The renormalized zero-momentum four-point coupling $g_r$ of $O(N)$-invariant scalar field theories in $d$ dimensions is studied by applying the $1/N$ expansion and strong coupling analysis. The $O(1/N)$ correction to the $β$-function and to the fixed point value $g_r^*$ are explictly computed. Strong coupling series for lattice non-linear $σ$ models are analyzed near criticality in $d=2$ and $d=3$ for several values of $N$ and the corresponding values of $g_r^*$ are extracted. Large-$N$ and strong coupling results are compared with each other, finding a good general agreement. For small $N$ the strong coupling analysis in 2-d gives the best determination of $g^*_r$ to date (or comparable for $N=2,3$ with the available Monte Carlo estimates), and in 3-d it is consistent with available $ϕ^4$ field theory results.

hep-lat↗

arXiv1995

Four-point renormalized coupling constant in O(N) models

The renormalized zero-momentum four-point coupling $g_r$ of O(N)-invariant scalar field theories in $d$ dimensions is studied by applying the 1/N expansion and strong coupling analysis. The O(1/N) correction to the $β$-function and to the fixed point value $g_r^*$ are explictly computed. Strong coupling series for lattice non-linear sigma models are analyzed near criticality in d=2 and d=3 for several values of $N$ and the corresponding values of $g_r^*$ are extracted. Large-N and strong coupling results are compared with each other, finding a good general agreement. For small N the strong coupling analysis in 2-d gives the best determination of $g^*_r$ to date (or comparable for N=2,3 with the available Monte Carlo estimates), and in 3-d it is consistent with available $ϕ^4$ field theory results.

hep-lat↗

arXiv1992

Possible Failure of Asymptotic Freedom in Two-Dimensional $RP^2$ and $RP^3$ $σ$-Models

We have simulated the two-dimensional $RP^2$ and $RP^3$ $σ$-models, at correlation lengths up to about 220 (resp.\ 30), using a Wolff-type embedding algorithm. We see no evidence of asymptotic scaling. Indeed, the data rule out the conventional asymptotic scaling scenario at all correlation lengths less than about $10^{9}$ (resp.\ $10^{5}$). Moreover, they are consistent with a critical point at $β\approx 5.70$ (resp.\ 6.96), only 2\% (resp.\ 5\%) beyond the largest $β$ at which we ran. Preliminary studies of a mixed $S^{N-1}/RP^{N-1}$ model (i.e. isovector + isotensor action) show a similar behavior when $β_T \to \infty$ with $β_V$ fixed $\ltapprox 0.6$, while they are consistent with conventional asymptotic freedom along the lines $β_T/β_V$ fixed $\ltapprox 2$. Taken as a whole, the data cast doubt on (though they do not completely exclude) the idea that $RP^{N-1}$ and $S^{N-1}$ $σ$-models lie in the same universality class.

hep-lat↗

arXiv1992

Random Walks with Long-Range Self-Repulsion on Proper Time

We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. Analytic results on the exponent $ν$ are obtained. They are in good agreement with Monte Carlo simulations in two dimensions. A numerical study of the scaling functions and of the efficiency of the algorithm is also presented.

hep-lat↗

arXiv1992

Dynamic Critical Behaviour of Wolff's Algorithm for $RP^N$ $σ$-Models

We study the performance of a Wolff-type embedding algorithm for $RP^N$ $σ$-models. We find that the algorithm in which we update the embedded Ising model à la Swendsen-Wang has critical slowing-down as $z_χ\approx 1$. If instead we update the Ising spins with a perfect algorithm which at every iteration produces a new independent configuration, we obtain $z_χ\approx 0$. This shows that the Ising embedding encodes well the collective modes of the system, and that the behaviour of the first algorithm is connected to the poor performance of the Swendsen-Wang algorithm in dealing with a frustrated Ising model.

hep-lat↗