534 research outputs found
Summability of the perturbative expansion for a zero-dimensional disordered spin model
We show analytically that the perturbative expansion for the free energy of
the zero dimensional (quenched) disordered Ising model is Borel-summable in a
certain range of parameters, provided that the summation is carried out in two
steps: first, in the strength of the original coupling of the Ising model and
subsequently in the variance of the quenched disorder. This result is
illustrated by some high-precision calculations of the free energy obtained by
a straightforward numerical implementation of our sequential summation method.Comment: LaTeX, 12 pages and 4 figure
The stability of a cubic fixed point in three dimensions from the renormalization group
The global structure of the renormalization-group flows of a model with
isotropic and cubic interactions is studied using the massive field theory
directly in three dimensions. The four-loop expansions of the \bt-functions
are calculated for arbitrary . The critical dimensionality and the stability matrix eigenvalues estimates obtained on the basis of
the generalized Pad-Borel-Leroy resummation technique are shown
to be in a good agreement with those found recently by exploiting the five-loop
\ve-expansions.Comment: 18 pages, LaTeX, 5 PostScript figure
On the definition of a unique effective temperature for non-equilibrium critical systems
We consider the problem of the definition of an effective temperature via the
long-time limit of the fluctuation-dissipation ratio (FDR) after a quench from
the disordered state to the critical point of an O(N) model with dissipative
dynamics. The scaling forms of the response and correlation functions of a
generic observable are derived from the solutions of the corresponding
Renormalization Group equations. We show that within the Gaussian approximation
all the local observables have the same FDR, allowing for a definition of a
unique effective temperature. This is no longer the case when fluctuations are
taken into account beyond that approximation, as shown by a computation up to
the first order in the epsilon-expansion for two quadratic observables. This
implies that, contrarily to what often conjectured, a unique effective
temperature can not be defined for this class of models.Comment: 32 pages, 5 figures. Minor changes, published versio
Dynamical field theory for glass-forming liquids, self-consistent resummations and time-reversal symmetry
We analyse the symmetries and the self-consistent perturbative approaches of
dynamical field theories for glassforming liquids. In particular, we focus on
the time-reversal symmetry (TRS), which is crucial to obtain
fluctuation-dissipation relations (FDRs). Previous field theoretical treatment
violated this symmetry, whereas others pointed out that constructing symmetry
preserving perturbation theories is a crucial and open issue. In this work we
solve this problem and then apply our results to the mode-coupling theory of
the glass transition (MCT). We show that in the context of dynamical field
theories for glass-forming liquids TRS is expressed as a nonlinear field
transformation that leaves the action invariant. Because of this nonlinearity,
standard perturbation theories generically do not preserve TRS and in
particular FDRs. We show how one can cure this problem and set up
symmetry-preserving perturbation theories by introducing some auxiliary fields.
As an outcome we obtain Schwinger-Dyson dynamical equations that automatically
preserve FDRs and that serve as a basis for carrying out symmetry-preserving
approximations. We apply our results to MCT, revisiting previous field theory
derivations of MCT equations and showing that they generically violate FDR. We
obtain symmetry-preserving mode-coupling equations and discuss their advantages
and drawbacks. Furthermore, we show, contrary to previous works, that the
structure of the dynamic equations is such that the ideal glass transition is
not cut off at any finite order of perturbation theory, even in the presence of
coupling between current and density. The opposite results found in previous
field theoretical works, such as the ones based on nonlinear fluctuating
hydrodynamics, were only due to an incorrect treatment of TRS.Comment: 54 pages, 21 figure
Stability of a cubic fixed point in three dimensions. Critical exponents for generic N
The detailed analysis of the global structure of the renormalization-group
(RG) flow diagram for a model with isotropic and cubic interactions is carried
out in the framework of the massive field theory directly in three dimensions
(3D) within an assumption of isotropic exchange. Perturbative expansions for RG
functions are calculated for arbitrary up to the four-loop order and
resummed by means of the generalized Pad-Borel-Leroy technique.
Coordinates and stability matrix eigenvalues for the cubic fixed point are
found under the optimal value of the transformation parameter. Critical
dimensionality of the model is proved to be equal to that
agrees well with the estimate obtained on the basis of the five-loop
\ve-expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B342, 284
(1995)] resummed by the above method. As a consequence, the cubic fixed point
should be stable in 3D for , and the critical exponents controlling
phase transitions in three-dimensional magnets should belong to the cubic
universality class. The critical behavior of the random Ising model being the
nontrivial particular case of the cubic model when N=0 is also investigated.
For all physical quantities of interest the most accurate numerical estimates
with their error bounds are obtained. The results achieved in the work are
discussed along with the predictions given by other theoretical approaches and
experimental data.Comment: 33 pages, LaTeX, 7 PostScript figures. Final version corrected and
added with an Appendix on the six-loop stud
Slow dynamics in critical ferromagnetic vector models relaxing from a magnetized initial state
Within the universality class of ferromagnetic vector models with O(n)
symmetry and purely dissipative dynamics, we study the non-equilibrium critical
relaxation from a magnetized initial state. Transverse correlation and response
functions are exactly computed for Gaussian fluctuations and in the limit of
infinite number n of components of the order parameter. We find that the
fluctuation-dissipation ratios (FDRs) for longitudinal and transverse modes
differ already at the Gaussian level. In these two exactly solvable cases we
completely describe the crossover from the short-time to the long-time
behavior, corresponding to a disordered and a magnetized initial condition,
respectively. The effects of non-Gaussian fluctuations on longitudinal and
transverse quantities are calculated in the first order in the
epsilon-expansion and reliable three-dimensional estimates of the two FDRs are
obtained.Comment: 41 pages, 9 figure
Divergent Perturbation Series
Various perturbation series are factorially divergent. The behavior of their
high-order terms can be found by Lipatov's method, according to which they are
determined by the saddle-point configurations (instantons) of appropriate
functional integrals. When the Lipatov asymptotics is known and several lowest
order terms of the perturbation series are found by direct calculation of
diagrams, one can gain insight into the behavior of the remaining terms of the
series. Summing it, one can solve (in a certain approximation) various
strong-coupling problems. This approach is demonstrated by determining the
Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling
constants. An overview of the mathematical theory of divergent series is
presented, and interpretation of perturbation series is discussed. Explicit
derivations of the Lipatov asymptotic forms are presented for some basic
problems in theoretical physics. A solution is proposed to the problem of
renormalon contributions, which hampered progress in this field in the late
1970s. Practical schemes for summation of perturbation series are described for
a coupling constant of order unity and in the strong-coupling limit. An
interpretation of the Borel integral is given for 'non-Borel-summable' series.
High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD
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 in the case of
and ). Moreover, the irrelevant interaction bringing from the symmetric to
the cubic fixed point gives rise to slowly-decaying scaling corrections with
exponent . For N=2, the isotropic fixed point is stable and
the cubic interaction induces scaling corrections with exponent . These conclusions are confirmed by a similar analysis of the
five-loop -expansion. A constrained analysis which takes into account
that in two dimensions gives .Comment: 29 pages, RevTex, new refs added, Phys. Rev. B in pres
Relaxation phenomena at criticality
The collective behaviour of statistical systems close to critical points is
characterized by an extremely slow dynamics which, in the thermodynamic limit,
eventually prevents them from relaxing to an equilibrium state after a change
in the thermodynamic control parameters. The non-equilibrium evolution
following this change displays some of the features typically observed in
glassy materials, such as ageing, and it can be monitored via dynamic
susceptibilities and correlation functions of the order parameter, the scaling
behaviour of which is characterized by universal exponents, scaling functions,
and amplitude ratios. This universality allows one to calculate these
quantities in suitable simplified models and field-theoretical methods are a
natural and viable approach for this analysis. In addition, if a statistical
system is spatially confined, universal Casimir-like forces acting on the
confining surfaces emerge and they build up in time when the temperature of the
system is tuned to its critical value. We review here some of the theoretical
results that have been obtained in recent years for universal quantities, such
as the fluctuation-dissipation ratio, associated with the non-equilibrium
critical dynamics, with particular focus on the Ising model with Glauber
dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting
in a film is discussed within the Gaussian model.Comment: Talk delivered at Statphys23, Genova, Italy, July 9-13, 2007. 8
pages, 7 figure
Coordinated optimization of visual cortical maps (I) Symmetry-based analysis
In the primary visual cortex of primates and carnivores, functional
architecture can be characterized by maps of various stimulus features such as
orientation preference (OP), ocular dominance (OD), and spatial frequency. It
is a long-standing question in theoretical neuroscience whether the observed
maps should be interpreted as optima of a specific energy functional that
summarizes the design principles of cortical functional architecture. A
rigorous evaluation of this optimization hypothesis is particularly demanded by
recent evidence that the functional architecture of OP columns precisely
follows species invariant quantitative laws. Because it would be desirable to
infer the form of such an optimization principle from the biological data, the
optimization approach to explain cortical functional architecture raises the
following questions: i) What are the genuine ground states of candidate energy
functionals and how can they be calculated with precision and rigor? ii) How do
differences in candidate optimization principles impact on the predicted map
structure and conversely what can be learned about an hypothetical underlying
optimization principle from observations on map structure? iii) Is there a way
to analyze the coordinated organization of cortical maps predicted by
optimization principles in general? To answer these questions we developed a
general dynamical systems approach to the combined optimization of visual
cortical maps of OP and another scalar feature such as OD or spatial frequency
preference.Comment: 90 pages, 16 figure
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