Field theory of critical behavior in a driven diffusive system

We use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field. A fixed point stable belowd _c=5 is found to govern the critical behavior. Scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in =5–d. Spatial correlations are found to be very anisotropic with elongated correlations along the external field. Long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents.     

The broken supersymmetry phase of a self-avoiding random walk

We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea _cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, (a), to be a supersymmetric order parameter for this transition. We prove that (a)(a _c–a) (–log(a _c–a))^1/2 asaa _c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.Research partially supported by NSF Grants DMS 91-2096 and DMS 91-96161.     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

Renormalization group approach to three-state spin models in two dimensions

The real space renormalization group approach is used to study spin one anisotropic models with dipolar and quadrupolar interactions on the triangular lattice. The method is tested for the three state Potts model and the Blume-Emery-Griffiths model leading to results which are basically in agreement with other treatments of these models. The phase diagram for a quadrupolar model (anisotropic Potts model) is obtained and the influence of an external magnetic field on the transition temperature to the quadrupolar phase is determined. The fixed point which controls a transition to the phase with Ising dipolar order andXY-type quadrupolar order is found.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich     

Gaussian,non-Gaussian critical fluctuations in the Curie-Weiss model

It is known that at the critical temperature the Curie-Weiss mean-field model has non-Gaussian fluctuations and that internal fluctuations can be Gaussian. Here we compute the distribution of theq-mode magnetization fluctuations as a function of the temperature, the wave vectorq, and a fading out external field. We obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out. We discuss also the susceptibility as the limitq tending to zero.     

Dynamic transition and hysteresis scaling in Heisenberg ferromagnet

Based on the mean-field treatment and Monte Carlo simulation, we studied the nature of the dynamic phase transition of two and three-dimensional magnetic films in Heisenberg model. The time averaged magnetization components ( mx , my, mz),the average hysteresis-loop area components A for magnetic films with different thickness have been calculated. The dynamic transition phase diagrams from to Q=0 for the 2D and 3D cases have been obtained. The relaxation times for different values of magnetic field, temperature, thickness of the films and the orientation number of spin have been simulated. It is found that the loop area follows the scaling relation, A-A0H0 T-, and the exponents and increase with increasing thickness, while the exponent decreases with increasing thickness. It was observed that the phase boundary line shrinks inward in the H0-T plane with decreasing value of the frequency of the magnetic field and thickness of multilayer film. The phase diagrams were explained by the competition between the relaxation time and the period of the external magnetic field. Moreover, it has been indicated that the dynamical behaviors for 2D and 3D cases derived by both mean-field method and Monte Carlo method in this work are consistent.     

Construction of a non-trivial planar field theory with ultraviolet stable fixed point

We study a ₄ ⁴ planar euclidean quantum field theory with propagator 1/p ^2–/2,>0. With the help of the tree expansion of Gallavotti and Nicolò [1], this non-renormalizable theory is shown to have a non-trivial ultraviolet-stable fixed point at negative coupling constant. The vicinity of the fixed point is discussed.     

Correlations in the Kosterlitz-Thouless phase of the two-dimensional Coulomb gas

The particle and charge correlations of the two-dimensional Coulomb gas are studied in the dielectric phase. A term-by-term analysis of the low-fugacity expansions suggests that the large-distance behaviors of the particle correlations are governed by multipolar interactions, similar to what happens in a system of permanent dipoles. These behaviors are compatible with the asymptotic structure of the BGY hierarchy equations; on the other hand, a new identity for the dielectric constant is used to show that the four-particle correlations decay as the dipole-dipole potential 1/r² when two neutral pairs are separated by a large distancer. Near the zero-density critical point of the Kosterlitz-Thouless transition, we resum the low-fugacity expansions of both 1/ and the charge correlation C(r). We thus retrieve the coupling constant flow equations of the renormalization group as well as the effective interaction energy of the iterated mean-field theory by Kosterlitz and Thouless. The coupling constant at the RG fixed point is then identified with 1/. The nonanalyticity of 1/ at the transition turns out to coincide with the divergence of the low-fugacity series for this quantity. The leading term in the large-distance behavior of C(r) is found to be the same as for external charges. Moreover, we exhibit the subleading terms which also contribute to 1/.     

Static and dynamic critical phenomena of the two-dimensionalq-state Potts model

Theq-state Potts model on the square lattice is studied by Monte Carlo simulation forq=3, 4, 5, 6. Very good agreement is obtained with exact results of Kiharaet al. and Baxter for energy and free energy at the critical point. Critical exponent estimates forq=3 are0.4,0.1,1.45, in rough agreement with high-temperature series extrapolation and real space renormalization-group methods. The transition forq=5, 6 is found to be a very weakly first-order transition, i.e., pronounced pseudocritical phenomena occur, specific heat, susceptibility, etc. (nearly) diverge at the first-order transition temperature. Dynamics is associated to the model in the same way as for the kinetic Ising model, and the nonlinear slowing down of the order parameter and of the energy is studied. The dynamic exponent is estimated to be (=zv)1.9. Within our accuracy noq dependence is detected. The relaxation is found to be consistent with dynamic scaling predictions, and dynamic scaling functions associated with the nonlinear relaxation are estimated.     

The exact phase diagram of the quantumXY spin glass model in a transverse field

The infinite rangeXY spin glass model in a transverse field is investigated by means of the Trotter-Suzuki approach. The exact phase diagram is obtained showing that a spin glass transition takes place for non-zero values of the transverse field up to a critical value _c =1.440.01. The present numerically exact calculations are in good agreement with our previous approximate results and they clear remaining discrepancies from previous work.     

Analytic relativistic model for a superdense star

A new analytic relativistic model has been obtained for superdense stars by solving the Einstein field equations for the spherically symmetric and static case. The model stands all the tests of physical reality. The density,, remains positive under all conditions and decreases smoothly from the center to the surface of the structure. The pressure,P, the ratioP/ anddP/d decrease with decreasing density. For all the finite values of pressure, the configurations are stable under radial perturbation. FordP/d 1, the maximum mass of neutron star model is 4.17, and the surface and central red shifts are 0.63 and 1.60, respectively. For an infinite central pressure, the surface red shift is 1.61. The structures are bound and the binding coefficients increase with the increasing mass.     

A Mathematical Representation of Quantizing the Motion of Relativistic Electrons in a Magnetic Field

It is shown that the quadratic component of the kinetic energy of continuous longitudinal motion of relativistic electrons in the external magnetic field is varied continuously between 0 and 2(2m _e c ²_B H) within each Landau energy level, undergoing an abrupt change at the boundaries of the levels. This results in the fact that in the quantum limit of a superstrong magnetic field where all electrons are at the zero Landau level, the maximum quadratic component of the kinetic energy of free longitudinal electron motion along the direction of the magnetic field is twice as high as the maximum quadratic component of the kinetic energy of its bound transverse motion.     

The functional renormalization group and O(4) scaling

The critical behavior of the chiral quark-meson model is studied within the Functional Renormalization Group (FRG). We derive the flow equation for the scale-dependent thermodynamic potential at finite temperature and density in the presence of a symmetry-breaking external field. We perform a set of approximations to formulate and solve the FRG flow equation in the presence of fermionic degrees of freedom and test their influence on the O(4) critical properties expected in the quark-meson model. Within this scheme, the critical scaling behavior of the order parameter, its transverse and longitudinal susceptibilities as well as the correlation lengths near the chiral phase transition are computed for vanishing baryon density. We focus on the scaling properties of these observables at non-vanishing external field when approaching the critical point from the symmetric as well as from the broken phase. We confront our numerical results with the Widom–Griffiths form of the magnetic equation of state, obtained by a systematic ε expansion of the scaling function.     

Trimodal random-field spin- Ising systems in a transverse field

The trimodal random-field spin- Ising system in a transverse field is investigated by combining the pair approximation with the discretized path-integral representation by introducing a parameter p to simulate the fractions of the spins not exposed to the external longitudinal magnetic field. The variation of the critical reduced transverse field and longitudinal magnetic field with the parameter p is studied for different coordination numbers and it is found that the system does not exhibit any tricritical points for p>0.22. The phase diagrams with respect to the external longitudinal random-field and the second-order phase transition temperature are obtained for given values of the transverse field, coordination numbers and the parameter p. It is found that for appropriate values of the system parameters the system does present tricritical points and reentrant phase transitions, which may be caused by the competition between the quantum effects and randomness.     

Perturbed sine-Gordon model: S=1/2 anisotropicxy chain with a longitudinal magnetic field

The thermodynamic properties and the longitudinal dynamical spin-correlation functionS ^zz (q, ) of theS=1/2 anisotropicxy chain with a longitudinal magnetic field are calculated in the continuum limit. For low fields we have an out-of-plane distortion of the sine-Gordon solitons present in the zero field case, while for high fields a completely different regime, characterized by a central peak forS ^zz (q, ), is found.Also Unità di Firenze GNSM-CISM     

Nonuniform van der Waals theory

The liquid-vapor interface of a confined fluid at the condensation phase transition is studied in a combined hydrostatic/mean-field limit of classical statistical mechanics. Rigorous and numerical results are presented. The limit accounts for strongly repulsive short-range forces in terms of local thermodynamics. Weak attractive longer-range ones, like gravitational or van der Waals forces, contribute a self-consistent mean potential. Although the limit is fluctuationfree, the interface is not a sharp Gibbs interface, but its structure is resolved over the range of the attractive potential. For a fluid of hard balls with –r ^–6 interactions the traditional condensation phase transition with critical point is exhibited in the grand ensemble: A vapor state coexists with a liquid state. Both states are quasiuniform well inside the container, but wall-induced inhomogeneities show up close to the boundary of the container. The condensation phase transition of the grand ensemble bridges a region of negative total compressibility in the canonical ensemble which contains canonically stable proper liquid-vapor interface solutions. Embedded in this region is a new, strictly canonical phase transition between a quasiuniform vapor state and a small droplet with extended vapor atmosphere. This canonical transition, in turn, bridges a region of negative total specific heat in the microanonical ensemble. That region contains subcooled vapor states as well as superheated very small droplets which are microcanonically stable.     

Slow flow of a conducting fluid past a non-conducting porous sphere with variable permeability

Slow flow of a conducting fluid past a non-conducting porous sphere of variable permeability in presence of a uniform radial magnetic field is studied. The drag experienced by the sphere is shown graphically and compared to that for a non-conducting fluid.Notation velocity vectors of the porous matrix and the conducting fluid - P, p pressures in the porous material and the free fluid - K permeability at a point of the porous medium - viscosity - v (=) kinematic viscosity - magnetic induction - current density - (r, , ) spherical coordinates - dimensionless constant - conductivity of the liquid The authors remain thankful to the referee for his valuable comments and helpful suggestions for improvement of the quality of the paper.     

Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

Screened potential and quarkonia properties at high temperatures

We perform a quark model calculation of the quarkonia b and c spectra using smooth and sudden string breaking potentials. The screening parameter is scale dependent and can be related to an effective running gluon mass that has a finite infrared fixed point. A temperature dependence for the screening mass is motivated by lattice QCD simulations at finite temperature. Qualitatively different results are obtained for quarkonia properties close to a critical value of the deconfining temperature when a smooth or a sudden string breaking potential is used. In particular, with a sudden string breaking potential quarkonia radii remain almost independent of the temperature up to the critical point, only well above the critical point the radii increase significantly. Such a behavior will impact the phenomenology of quarkonia interactions in medium, in particular for scattering dissociation processes.     

Self-energy-functional approach: Analytical results and the Mott-Hubbard transition

The self-energy-functional approach proposed recently is applied to the single-band Hubbard model at half-filling to study the Mott-Hubbard metal-insulator transition within the most simple but non-trivial approximation. This leads to a mean-field approach which is interesting conceptually: Trial self-energies from a two-site single-impurity Anderson model are used to evaluate an exact and general variational principle. While this restriction of the domain of the functional represents a strong approximation, the approach is still thermodynamically consistent by construction and represents a conceptual improvement of the linearized DMFT which has been suggested previously as a handy approach to study the critical regime close to the transition. It turns out that the two-site approximation is able to reproduce the complete (zero and finite-temperature) phase diagram for the Mott transition. For the critical point at T = 0, the entire calculation can be done analytically. This calculation elucidates different general aspects of the self-energy-functional theory. Furthermore, it is shown how to deal with a number of technical difficulties which appear when the self-energy functional is evaluated in practice.Received: 3 November 2003, Published online: 23 December 2003PACS: 71.10.-w Theories and models of many-electron systems - 71.15.-m Methods of electronic structure calculations - 71.30. + h Metal-insulator transitions and other electronic transitions