Finite-size scaling and surface tension from effective one dimensional systems

We develop a method for precise asymptotic analysis of partition functions near first-order phase transitions. Working in a (+1)-dimensional cylinder of volumeL×...×L×t, we show that leading exponentials int can be determined from a simple matrix calculation providedtv logL. Through a careful surface analysis we relate the off-diagonal matrix elements of this matrix to the surface tension andL, while the diagonal matrix elements of this matrix are related to the metastable free energies of the model. For the off-diagonal matrix elements, which are related to the crossover length from hypercubic (L=t) to cylindrical (t=) scaling, this includes a determination of the pre-exponential power ofL as a function of dimension. The results are applied to supersymmetric field theory and, in a forthcoming paper, to the finite-size scaling of the magnetization and inner energy at field and temperature driven first-order transitions in the crossover region from hypercubic to cylindrical scaling.Research partially supported by the A. P. Sloan Foundation and by the NSF under DMS-8858073Research partially supported by the NSF under DMS-8858073 and DMS-9008827     

Ward identities for interacting electronic systems

A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and Wölfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical wave transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-Wölfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel.     

An exact result for bimetallic interface energies

An exact result is derived for the interface energy of a jellium interface. An approximate model of the bimetallic interface is then used to find approximate values of the adhesive energies of jellium interfaces and specific metal pairs.     

Universality of entropy scaling in one dimensional gapless models

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.     

Log-Sobolev inequalities for infinite one dimensional lattice systems

It is shown that a unique Gibbs measure of infinite spin system with short range interaction on one dimensional lattice satisfies log-Sobolev inequality.Supported by SFB 237     

Transmission coefficient of interacting few-body system in one dimensional space

We present a novel way of defining transmission coefficient of one spatial dimensional few interacting electrons system. The formalism is based on the probability interpretation of unitarity of physical scattering S-matrix. The relation of our formalism to the well-established method for describing the conducting properties of non-interacting systems, Landauer-Büttiker formula, is discussed. The transport properties of interacting two-electron system is also discussed as a specific example of our formalism.     

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈n₀〉/N and specific heat C_N around the transition temperature T_c (i.e., t = T/T_c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ? 0.068.     

A model for cluster confinement in one dimensional many-body systems

A variant of the one-dimensional “toy model” for QCD of Horowitz et al. is presented, which uses a density dependent rather than the many-body force originally proposed by Lenz et al. Our approach suggests, already within the RPA framework, the occurrence of a phase transition simulating the one from quark to hadronic matter, similar to the transition foreseen by Horowitz et al. However our force is easy to handle in three dimensions as well, but in such a case, of course, the correspondence between our and the “toy model” remains to be verified.     

On surface states of one dimensional semiinfinite systems

Surface state energies are computed for a one dimensional semi-infinite system, which accommodates simultaneously (1) surface distortion (2) adsorbed impurity (3) finite distance of separation between vacuum potential discontinuity and surface atom and (4) negative (attractive) bulk potential. The locations of surface state energies in the band gaps are indicated; further, nature of variations of surface state energies with relevant parameters, covering a wide range, is studied. The results are discussed critically; besides, they are compared with similar ones obtained on the basis of two other models, one due to Aerts and the other due to Neuberger and Rutherford Fischer.The authors are thankful to Prof. G. B. Mitra, Head of the Department of Physics, for his endless interest and encouragement. They are further grateful to the esteemed referee who reviewed the paper and suggested so kindly several modifications.     

Friedel theorem for one dimensional relativistic spin-1/2 systems

The Friedel sum rule is generalized to relativistic systems of spin-1/2 particles in one dimension. The change of the total energy due to the presence of an impurity is studied. The relation of the sum rule with the relativistic Levinson theorem is presented. Density oscillations in such systems are discussed. Since the Friedel theorem has been of major importance in understanding the impurity scattering in materials, the present results may be useful to explain some phenomena in one dimensional atomic chain, quantum wire, and fermionic many body systems.     

Coupled charge density waves in nearly one dimensional systems

A mean field theory of coupling between charge density waves CDW in linear chain systems is described. It is shown that interchain coupling can stabilize CDW's and lead to a semiconducting behavior. The model is applied to semiconducting TCNQ salts.     

Structure of gibbs states of one dimensional Coulomb systems

We present a method of computing the Coulomb forces on particles in an infinite configuration of charges in one dimension. The resolution of the apparent nonuniqueness in this problem leads to a structural proof of the translation symmetry breaking in jellium, at all temperatures, and to a related phenomenon of phase nonuniqueness in the two component system. The appropriate generalizations of the DLR and KMS conditions for these states are discussed.Supported in part by U.S. National Science Foundation, grant No. PHY-78 25390 A01     

An exact result for the 1D random Ising model in a transverse field

It is shown exactly that for an N-site cyclic chain with hamiltonian H = ?Σ^N_i=1(γ_iS^x_i + J_iS^z_iS^z_iS^z_i+1), the gap in the excitation spectrum goes to zero when N → ∞ at the “critical point” given by the relation Π^N_i=1Γ_i = Π^N_i=1J_i.     

An exactly conservative particle method for one dimensional scalar conservation laws

A particle scheme for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities. Particle interaction is resolved locally, satisfying exact conservation of area. Shocks stay sharp and propagate at correct speeds, while rarefaction waves are created where appropriate. The method is variation diminishing, entropy decreasing, exactly conservative, and has no numerical dissipation away from shocks. Solutions, including the location of shocks, are approximated with second order accuracy. Source terms can be included. The method is compared to CLAWPACK in various examples, and found to yield a comparable or better accuracy for similar resolutions.     

Microcanonical density functionals for critical systems: An exact one-dimensional example

Free-energy functionals suitable for describing realistic, nonuniform systems near criticality are discussed with emphasis on the advantages of a local formalism. It is proposed to investigatemicro canonical functionals in which both the usual order-parameter (or magnetization) density m(r)and the local energy density (r), which has independent critical fluctuations, are employed. This approach is tested by an exact calculation of the microcanonical functional [{m}, {}] in the continuum limit for a one-dimensional Ising model. Remarkably, the microcanonical functional is found to be local irrespective of the proximity to the critical point (located at zero temperature and zero field). Furthermore, its form relates closely to the scaling postulate advanced earlier by de Gennes and Fisher and displays features of conformal covariance.