Harmonically trapped ideal quantum gases

We solve the problem of a Bose or Fermi gas in d-dimensions trapped by δ ⩽ d mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature T _c if and only if d + δ > 2, along with a jump in the specific heat at T _c if and only if d + δ > 4. Specific heats for both gas types precisely coincide as functions of temperature when d + δ = 2. The trapped system behaves like an ideal free quantum gas in d + δ dimensions. For δ = 0 we recover all known thermodynamic properties of ideal quantum gases in d dimensions, while in 3D for δ = 1, 2 and 3 one simulates behavior reminiscent of quantum wells, wires anddots, respectively. Good agreement is found between experimental critical temperatures for the trapped boson gases ₃₇ ⁸⁷Rb, ₃ ⁷Li, ₃₇ ⁸⁵Rb, ₂ ⁴He, ₁₉ ⁴¹K and the known theoretical expression which is a special case for d = δ = 3, but only moderate agreement for ₁₁ ²⁷Na and ₁ ¹H. Received 17 July 2002 / Received in final form 14 October 2002 Published online 21 January 2003 RID="a" ID="a"e-mail: mdgg@hp.fciencias.unam.mx     

Relaxation of quasi-stationary states in long range interacting systems and a classification of the range of pair interactions

Systems of particles interacting with long range interactions present generically ”quasi-stationary states” (QSS), which are approximately time-independent out of equilibrium states. In this proceedings, we explore the generalization of the formation of such QSS and their relaxation from the much studied case of gravity to a generic pair interaction with the asymptotic form of the potential v(r) ∼ 1/r ^γ with γ > 0 in d dimensions. We compute analytic estimations of the relaxation time calculating the rate of two body collisionality in a virialized system approximated as homogeneous. We show that for γ < (d − 1/2), the collision integral is dominated by the size of the system, while for γ > (d − 1/2), it is dominated by small impact parameters. In addition, the lifetime of QSS increases with the number of particles if γ < d − 1 (i.e. the force is not integrable) and decreases if γ > d − 1. Using numerical simulations we confirm our analytic results. A corollary of our work gives a ”dynamical” classification of interactions: the dynamical properties of the system depend on whether the pair force is integrable or not.     

1/d corrections for interacting spinless fermions: One-particle properties

One-particle properties of the spinless fermion model with repulsion at half filling are calculated within an approach correct to first order in the inverse of the lattice dimensiond. Continuity of the limitd requires a scaling of the nearest-neighbour hopping proportional to and of the nearest-neighbour interaction proportional to 1/d. Due to this scaling the Hartree approximation becomes exact in infinite dimensions. We show that 1/d corrections comprise the Fock diagram and the local correlation diagram in the self-consistent Dyson equation. This approach is applied to simple-cubic systems in dimensiond=1, 2 and 3. Ground state properties and the charge-density wave phase diagram are calculated. AtT=0 the inclusion of 1/d terms gives only small corrections to the leading Hartree contribution ind=2, 3. ForT>0, however, the 1/d corrections are important. They lead to a non-negligible reduction of the critical temperature. Ind=1 the 1/d corrections are very large, but they do not succeed in removing the spurious phase transition atT>0. The 1/d approach provides a good and tractable approximation ind=3 and probably ind=2, which allows also further systematic improvement.     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

Global strings in extra dimensions: The full map of solutions,matter trapping,and the hierarchy problem

We consider (d ₀ + 2)-dimensional configurations with global strings in two extra dimensions and a flat metric in d ₀ dimensions, endowed with a warp factor e ^2γ depending on the distance l from the string center. All possible regular solutions of the field equations are classified by the behavior of the warp factor and the extradimensional circular radius r(l). Solutions with r → ∞ and r → const > 0 as l → ∞ are interpreted in terms of thick brane-world models. Solutions with r → 0 as l → l _c > 0, i.e., those with a second center, are interpreted as either multibrane systems (which are appropriate for large enough distances l _c between the centers) or as Kaluza-Klein-type configurations with extra dimensions invisible due to their smallness. In the case of the Mexican-hat symmetry-breaking potential, we build the full map of regular solutions on the (ɛ, Γ) parameter plane, where ɛ acts as an effective cosmological constant and Γ characterizes the gravitational field strength. The trapping properties of candidate brane worlds for test scalar fields are discussed. Good trapping properties for massive fields are found for models with increasing warp factors. Kaluza-Klein-type models are shown to have nontrivial warp factor behaviors, leading to matter particle mass spectra that seem promising from the standpoint of hierarchy problems. The text was submitted by the authors in English.     

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability . We show that if d<s<2d, then the walk is transient, and if s≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d≥ 1, if d>s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network. Received: 27 October 2000 / Accepted: 29 November 2001     

Painlevé analysis and integrability of the damped anharmonic oscillator equation

The Painlevé analysis is applied to the anharmonic oscillator equation . The following three integrable cases are identified: (i)C=0,d ²=25A/6,A>0,B arbitrary, (ii)d ²=9A/2,B=0,A>0,C arbitrary and (iii)d ²=−9A/4,C=2B ²/(9A),A<0,C<0,B arbitrary. The first two integrable choices are already reported in the literature. For the third integrable case the general solution is found involving elliptic function with exponential amplitude and argument.     

Single-Site Approximation for Reaction-Diffusion Processes

We consider the branching and annihilating random walk and with reaction rates σ and λ, respectively, and hopping rate D, and study the phase diagram in the λ/D,σ/D) plane. According to standard mean-field theory, this system is in an active state for all σ/D≥0, and perturbative renormalization suggests that this mean-field result is valid for d>2; however, nonperturbative renormalization predicts that for all d there is a phase transition line to an absorbing state in the λ/D,σ/D) plane. We show here that a simple single-site approximationreproduces with minimal effort the nonperturbative phase diagram both qualitatively and quantitatively for all dimensions d>2. We expect the approach to be useful for other reaction-diffusion processes involving absorbing state transitions.     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Effect of thickness on the structure, morphology and optical properties of sputter deposited Nb2O5 films

Nb₂O₅ films with the thickness (d) ranging from 55 to 2900 nm were deposited on BK-7 substrates at room temperature by a low frequency reactive magnetron sputtering system. The structure, morphology and optical properties of the films were investigated by X-ray diffraction, atomic force microscopy and spectrophotometer, respectively. The experimental results indicated that the thickness affects drastically the structure, morphology and optical properties of the film. There exists a critical thickness of the film, d_cri =2010 nm. The structure of the film remains amorphous as d < d_cri. However, it becomes crystallized as d > d_cri. The root mean square of surface roughness increases with increasing thickness as d > 1080 nm. Widths and depths of the holes on film surface increase monotonously with increasing thickness, and widths of the holes are larger than 1000 nm for the crystalline films. Refractive index increases with increasing thickness as d < d_cri, while it decreases with increasing thickness as d > d_cri. In addition, the extinction coefficient increases with increasing thickness as d > d_cri.     

A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and <₀, where ₀ is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.     

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

The anomalous dimensions of operators with an arbitrary number of gradients are determined for then-vector model ind=2+ 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)+(1+s(s–1)/(n–2))+O( ²). 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.Dedicated to Professor W. Brenig on the occasion of his 60th birthdayWork supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft     

Interface formation and a structural phase transition for the spherical model of ferromagnetism

A detailed analysis is reported examining the local magnetic susceptibility (r), in relation to the correlation functionG(R) and correlation length , of a spherical model ferromagnet confined to geometry =L ^{d –d} × ^d ( ^d 2,d>2) under a continuous set oftwisted boundary conditions. The twist parameter in this problem may be interpreted as a measure of the geometry-dependent doping level of interfacial impurities (or antiferromagnetic seams) in theextended system at various temperatures. For _j 0, jd-d, no seams are present except at infinity, whereas if _j = 1/2, impurity saturation occurs. For 0 < _j < 1/2 the physical domain _phys =D ^{d –d} × ^d (D>L), defining the region between seams containing the origin, depends on temperature above a certain threshold (T>T ₀). Below that temperature (T>T ₀), seams are frozen at the same position (DL/2,d-d'=1), revealing a smoothly varying largescale structural phase transition.     

Tails in harnesses

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa _s ^t , wheres ∈ Z ^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z _d ⁺ define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a _s ^t =a _s ^t . Then by induction, wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a _s ^t diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a _s ^t diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a _s ^t , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a _s ^t converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima _s ^t )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a _s ^t )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a _s ^t )=E-decay(ν).     

A note on γ traces for the Wilson action in the continuum limit

In d=4 and d=2 dimensions we calculate averages of certain products of matrices with respect to closed lattice paths of length L. The approach to the asymptotic behaviour for large L is considered and found to be quite different in d=4 and d=2 dimensions.Institute für Theoretische Physik der Universität Hamburg, F.R.G.     

Geometric analysis of φ4 fields and Ising models. Parts I and II

We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) ⁴ Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the ⁴ field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the universality of hyperscaling.A. P. Sloan Foundation Research Fellow. Supported in part by the National Science Foundation Grant PHY-78-25390-A02     

Renormalization group study of theA+B→⊘ diffusion-limited reaction

TheA+B diffusion-limited reaction, with equal initial densitiesa(0)=b(0)=n ₀, is studied by means of a field-theoretic renormalization group formulation of the problem. For dimensiond>2 an effective theory is derived, from which the density and correlation functions call be calculated. We find the density decays in time as ford<4, with =n ₀ –Cn ₀ ^d/2 +..., whereC is a universal constant andC is nonuniversal. The calculation is extended to the case of unequal diffusion constantsD _A D _B , resulting in a new amplitude but the same exponent. Ford2 a controlled calculation is not possible, but a heuristic argument is presented that the results above give at least the leading term in an =2–d expansion. Finally, we address reaction zones formed in the steady state by opposing currents ofA andB particles, and derive scaling properties.     

Direction-direction correlations of oriented polymers

We calculate the direction-direction correlations between the tangent vectors of an oriented self-avoiding walk (SAW). LetJ (x) andJ ^v (0) be components of unit-length tangent vectors of an oriented SAW, at the spatial pointsx and 0, respectively. Then for distances |x| much less than the average distance between the endpoints of the walk, the correlation function ofJ (x) withJ ^v (0) has, ind dimensions, the form . The dimensionless amplitudek(d) is universal, and can be calculated exactly in two dimensions by using Coulomb gas techniques, where it is found to bek(2)=12/25 ². In three dimensions, the -expansion to second order in together with the exact value ofk(2)in two dimensions allows the estimatek(3)=0.0178±0.0005. In dimensionsd4, the universal amplitudek(d) of the direction-direction correlation functions of an oriented SAW is the same as the universal amplitude of the direction-direction correlation functions of an oriented random walk, and is given byk(d)= ²(d/2)/(d–2) ^d .     

Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on . In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is , and thereby prove a version of the Alexander–Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.     

From dilute to dense self-avoiding walks on hypercubic lattices

A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd ^–1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf1–(2d)^–1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).