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1.
The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used to show that the point of zero-temperature replica
symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of
size L = R √N corresponds to the onset of exponential in N growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures, a simple variational
upper bound on the true free energy of the R = ∞ version of the problem has been constructed and it has been shown that this approximation can recover the position of
the whole de-Almeida-Thouless line.
The text was submitted by the authors in English. 相似文献
2.
Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N?1)-dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the “trust region subproblem” or “constraint least square problem”. When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number $\mathcal{N}_{tot}$ of critical (stationary) points in the cost function landscape. In the first regime $\mathcal{N}_{tot}$ remains of the order N and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor. 相似文献
3.
This paper contains two results. The first is that for a theory of N chiral scalar superfields, for which the supersymmetry is not spontaneously broken, the effective potential vanishes at the same points as the tree potential. This is also verified in the one- and two- loop approximation. It is concluded that the absolute minima of the tree potential are also absolute minima of the effective potential, and hence that any degeneracy in the tree potential will persist to all orders in the radiative corrections. The second result is that, in the case of one chiral superfield, the one-loop contribution to the effective potential not only vanishes at the zeros of the tree potential, but is non-negative at all other points. This results is taken as an indication that the supersymmetry itself is not spontaneously broken by the radiative corrections. 相似文献
4.
Gaku Arakawa 《Annals of Physics》2004,311(1):152-169
In the present paper we shall study (2+1)-dimensional ZN gauge theories on a lattice. It is shown that the gauge theories have two phases, one is a Higgs phase and the other is a confinement phase. We investigate low-energy excitation modes in the Higgs phase and clarify relationship between the ZN gauge theories and Kitaev’s model for quantum memory and quantum computations. Then we study effects of random gauge couplings (RGC) which are identified with noise and errors in quantum computations by Kitaev’s model. By using a duality transformation, it is shown that time-independent RGC give no significant effects on the phase structure and the stability of quantum memory and computations. Then by using the replica methods, we study ZN gauge theories with time-dependent RGC and show that nontrivial phase transitions occur by the RGC. 相似文献
5.
Tamiaki Yoneya 《Nuclear Physics B》1982,205(1):130-140
Rigorous inequalities among magnetic-flux free energies of tori with varying diameters are derived in lattice gauge theories. From the inequalities, it follows that if the magnetic-flux free energy vanishes in the limit of large uniform dilatation of a torus, the free energy must always decrease exponentially with the area of the cross section of the torus. The latter property is known to be sufficient for permanent confinement of static quarks. As a consequence of this property, a lower bound V(R) ? const · R for the static quark-antiquark potential is obtained in three-dimensional U(N) lattice gauge theory for sufficiently large R. 相似文献
6.
We numerically investigate the aging dynamics of a monatomic Lennard-Jones glass, focusing on the topology of the potential energy landscape which, to this aim, has been partitioned in basins of attraction of stationary points (saddles and minima). The analysis of the stationary points visited during the aging dynamics shows the existence of two distinct regimes: (i) at short times the system visits basins of saddles whose energies and orders decrease with t; (ii) at long times the system mainly lies in basins pertaining to minima of slowly decreasing energy. The long time dynamics can be represented by a simple random walk on a network of minima with a jump probability proportional to the inverse of the waiting time. 相似文献
7.
《Nuclear Physics B》1986,268(1):161-178
We study the large-N limit of various (SU(N) gauge theories with chiral fermion content. Assuming that the leading N → ∞ behavior is given by a sum of planar diagrams, we find that the gauge interactions must fail to confine color in some models. Other models, assuming both a planar diagram limit and confinement, must contain massless composite fermions. 相似文献
8.
《Nuclear Physics B》2001,618(3):465-499
We compare the critical multipoint correlation functions for two-dimensional (massless) Dirac fermions in the presence of a random su(N) (non-Abelian) gauge potential, obtained by three different methods. We critically reexamine previous results obtained using the replica approach and in the limit of infinite disorder strength and compare them to new results (presented here) obtained using the supersymmetric approach to the N=2 case. We demonstrate that this ménage à trois of different approaches leads to identical results. Remarkable relations between apparently different conformal field theories (CFTs) are thereby obtained. We further establish a connection between the random Dirac fermion problem and the c=−2 theory of dense polymers. The presence of the c=−2 theory may be seen in all three different treatments of the disorder. 相似文献
9.
Fyodorov YV 《Physical review letters》2004,92(24):240601
Finding the mean of the total number N(tot) of stationary points for N-dimensional random energy landscapes is reduced to averaging the absolute value of the characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in a random environment. For N>1 our asymptotic analysis reveals a phase transition at some critical value mu(c) of a control parameter mu from a phase with a finite landscape complexity: N(tot) approximately e(N Sigma), Sigma(mu0 to the phase with vanishing complexity: Sigma(mu>mu(c))=0. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems. 相似文献
10.
《Nuclear Physics B》1988,302(3):403-422
We discuss supersymmetry breaking in the effective supergravity theories of four-dimensional N = 1 superstrings. Improving a recent suggestion, we introduce a superpotential modification that describes spontaneous supersymmetry breaking with vanishing cosmological constant and Str M2 = 0 at any minimum of the tree level potential. We prove that in a class of models there are classical minima at which also Str f(M2 = 0 for any function f. In particular, the whole one-loop cosmological constant vanishes. We propose a new boson-fermion symmetry of the spectrum, relating the graviton and the “dilatino”, which explains these remarkable properties. 相似文献
11.
《Nuclear Physics B》1998,525(3):671-696
We reexamine in detail the problem of fermions interacting with a non-Abelian random vector potential. Without resorting to the replica or supersymmetry approaches, we show that in the limit of infinite disorder strength the theory possesses an exact solution which takes the form of a logarithmic conformal field theory. We show that the proper treatment of the locality conditions in the SU(2) theory leads to the termination of the multifractal spectrum, or in other words to the termination of the infinite hierarchies of negative-dimensional operators that were thought to occur. Based on arguments of logarithmic degeneracies, we conjecture that such a termination mechanism should be present for general SU(N). Moreover, our results lead to the conclusion that the previous replica solution of this problem yields incorrect results. 相似文献
12.
13.
Nobuo Yoshida 《Journal of statistical physics》2010,138(4-5):598-618
We consider a discrete-time stochastic growth model on the d-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of “replica overlap”. The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit. 相似文献
14.
We consider the adsorption of an isolated, Gaussian, random, and quenched copolymer chain at an interface. We first propose
a simple analytical method to obtain the adsorption/depletion transition, by averaging over the disorder the partition function
instead of the free energy. The adsorption thresholds obtained by previous authors at a solid/liquid and at a liquid/liquid
interface for multicopolymer chains can be rederived using this method. We also compare the adsorption thresholds obtained
for bimodal and for Gaussian disorder; they only agree for small disorder. We focus on the specific case of an ideally flat
asymmetric liquid/liquid interface, and consider the situation where the chain is composed of monomers of two different chemical
species A and B. The replica method is developed for this case. We show that the Hartree approximation, coupled to a replica symmetry assumption,
leads to the same adsorption thresholds as obtained from our general method. In order to describe the properties of the adsorbed
(or depleted) chain, we develop a new approximation for long chains, within the framework of the replica theory. In most cases,
the behavior of a random copolymer chain can be mapped onto that of a homopolymer chain at an asymmetric attractive interface.
The values of the effective adsorption energy are different for a random and a periodic copolymer chain. Finally, we consider
the case of uncorrelated annealed disorder. The behavior of an annealed chain can be mapped onto that of a homopolymer chain
at an asymmetric non attractive interface; hence, an annealed chain cannot adsorb at an asymmetric interface.
Received 21 January 1999 相似文献
15.
Tamiaki Yoneya 《Nuclear Physics B》1978,144(1):195-218
We investigate the properties of Z(N) topological excitations in Wilson's lattice formulation of SU(N) Yang-Mills theories. We exhibit the Z(N) topological excitations as exact classical solutions on the lattice. After giving detailed qualitative discussions about the Z(N) excitations and their relevance to confinement, we investigate the Z(N) lattice gauge theories with the Wilson action and show that Z(2), Z(3) and Z(4) models are self-dual systems. (The self-duality of the Z(2) case has been known previously.) This property enables us to locate the critical points exactly in those systems under the assumption that the phase transition occurs at only one point in the coupling constant space. We then derive the effective action for the Z(N) topological excitations in the lattice SU(N) Yang-Mills theories in the steepest descent approximation. The critical coupling constants in the SU(N) models corresponding to the phase transition caused by the Z(N) excitations are estimated by using the information on the Z(N) models with the Wilson action. It is quite probable that the estimated value (for SU(3)) is an upper bound. This indicates that the Wilson model of the SU(3) gauge field can be effective action of the QCD gluons which exhibit permanent quark confinement and, at the same time, freedom up to the distance characterized by the energy, at least, ~1 TeV. 相似文献
16.
We studied the quantum phase transition revealed to occur in the ferrimagnetic mixed-spin (S,s)=(1,1/2) chain with positive crystal-field anisotropy D under an external magnetic field by using concepts from quantum information theory such as pairwise entanglement and purity, incorporated into the density matrix renormalization group algorithm. In this system, a magnetization plateau appears at 1/3 of the saturation magnetization for all values of D except for a critical value Dc where it vanishes. We obtained this value Dc=1.11445±0.00065 within the thermodynamic limit as local minima of the pairwise entanglement and the purity. Moreover, using this procedure we were able to investigate the second-order (or continuous) character of the quantum phase transition. 相似文献
17.
The critical behavior of weakly disordered systems with a p-component order parameter is described using field-theoretic methods. A renormalization-group analysis of the effective model replica Hamiltonian with an interaction potential breaking replica symmetry is performed directly for three-dimensional systems in the two-loop approximation. For the case of a one-step replica symmetry breaking, fixed points of renormalization group equations are found using the Padé-Borel summation technique. It is established that the critical behavior of weakly disordered systems is stable with respect to replica symmetry breaking and that the former scenario of the influence of structural defects on the critical behavior of these systems is realized. 相似文献
18.
A. Barra L. DeSanctis 《The European Physical Journal B - Condensed Matter and Complex Systems》2008,64(1):119-124
In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and
the Viana-Bray (VB) models, to show that, in the framework of the cavity method, the transition from the annealed regime to
a broken replica symmetry phase can be thought of as the failure of the saturability property (detailed explained along the paper) of the overlap fluctuations which act as the order parameters of the theory.
We show furthermore how this coincides with the lacking of the commutativity of the infinite volume limit with respect to
a, suitably chosen, vanishing perturbing field inducing the transition as prescribed by standard statistical mechanics. This
is another step towards a complete theory of disordered systems. As a well known consequence it turns out that the annealed
and the replica symmetric regions must coincide, implying that the averaged overlap is zero in this phase. Within our framework
the finding of the values of the critical point for the SK and line for the VB becomes available straightforwardly and the
method is of a large generality and applicable to several other mean field models 相似文献
19.
Manu P. Singh O. P. S. Negi B. S. Rajput 《International Journal of Theoretical Physics》2000,39(8):2029-2042
Breaking N = 2 SU(3) and N =2 SO(6) supersymmetric Yang–Mills theoriesto corresponding N = 1 theories by suitable tree-levelsuperpotentials, thehyperelliptic curves describing the Coulomb phase of these theories have beenobtained and it has been shown that the mass gap in the N = 1confining phaseof these theories vanishes when N = 1 parameters are properlytuned to approachthe highest critical points. 相似文献
20.
《Nuclear Physics B》1999,558(3):573-588
Some N = 1 gauge theories, including SQED and NF = 1 SQCD, have the property that, for arbitrary superpotentials, all stationary points of the potential V = F + D are D-flat. For others, stationary points of V are complex gauge transformations of D-flat configurations. As an implication, the technique to parametrize the moduli space of supersymmetric vacua in terms of a set of basic holomorphic G invariants can be extended to non-supersymmetric vacua. A similar situation is found in non-gauge theories with a compact global symmetry group. 相似文献