Prof. Dr. Erich Eisenriegler

My main scientific interest is in:

Boundaries in Critical Systems
Due to the macroscopic correlation length in a critical system, the effects of boundaries penetrate deeply into the bulk. As a consequence the interaction induced between two boundaries such as those of two immersed particles has a mesoscopic range. Because of the “universality” of boundary critical phenomena, quantitative results for these systems can be obtained by studying simple field theoretic models. Examples of critical systems that I consider are long flexible polymer chains and binary fluid mixtures at their critical consolute point.

Polymers
I studied the interaction of polymer chains or dilute solutions thereof with boundaries by field-theoretic methods. This encompasses the adsorption of a single chain at a planar wall which can be viewed as a surface multicritical point phenomenon [1, 2] and the depletion interaction between two or more particles with spherical or ellipsoidal shape immersed in a solution of nonadsorbing polymers [3]. The shape of polymers [4] and the profile of the center of mass of a polymer near a hard wall [5] were other topics of interest.

Critical Fluid Mixtures
The interaction between mesoscopic particles induced by critical fluctuations such as those in a critical fluid mixture often go under the name “Critical Casimir Effect” in analogy to the corresponding effect induced by zero point fluctuations of the electromagnetic field. In a two-dimensional Ising system right at the critical point conformal invariance allows us to deduce exact results for the interaction of two particles with uniform boundary conditions for arbitrary distance to size ratio from the partition function of a corresponding annulus obtained by J. L. Cardy. Explicit results have been given for two circular disks [6] and two needles [7]. For spheres with a uniform boundary in arbitrary dimensions, for dumbbells and lenses, and for Janus-type particles in two dimensions, results have been obtained when they are close or distant, see [6] and [8].

Density Profiles and Response Function for Mixed Boundaries

Based on a conformal-invariance approach developed by T. W. Burkhardt, I. Guim, and T. Xue these authors obtained exact density profiles of the order parameter, the energy, and the stress tensor for the two-dimensional Ising universality class in the presence of mixed boundaries, see Ref. [9]. A simple application is for the Janus particle mentioned above. Recently we investigated an Ising model confined to a rectangle of arbitrary aspect ratio with mixed boundary conditions [10]. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions . or with free-spin boundary conditions f, exact results are obtained for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The Monte Carlo simulation results obtained by O. A. Vasilyev are in excellent agreement with these analytic predictions.
In systems with a mixed boundary of fixed up and down spins one finds situations in which local disordering enhances rather than weakens the local order [11]. This surprising behavior is related to the “zero lines” in these systems along which the order parameter profile vanishes.

Operator Expansions in Field Theory
Operator expansions are a useful tool to describe how a “small” object is affected by a “distant” perturbation. A well-known situation is how the average of two operators, i.e. the two-point function, is modified by the presence of distant boundaries. Here the small object formed by the two operators can be replaced by a series of operators Oj all centered at the midpoint between the two operator-positions (in the simplest case where the two operators are equal) and with prefactors that depend on their distance vector. On averaging the series, the operator O1 of lowest scaling dimension and its amplitude determine the leading modification of the two-point function due to the distant boundaries. While the average of O1 depends on the location and nature of the boundaries, the amplitude is independent of it, and can be determined from the most convenient configuration of the distant boundaries. Besides boundaries, distant operators qualify as perturbations as well. Inspired by this “operator-product expansion” other operator expansions have been found such as the “small particle expansion” [6]. Here the Boltzmann weight of a particle, for example a sphere, a dumbbell, a needle, or the Janus particle, can be expanded in an operator series in order to determine its interaction with distant boundaries [6–8]. In the same spirit one can expand an operator “close” to a boundary in a series of “boundary-operators” which are located at a point right in the boundary, which allows to determine the modification of density profiles near this boundary by distant perturbations. For a mixed boundary in two dimensions where the boundary condition switches between consecutive segments, different boundary operators arise depending on whether one expands about a boundary point in the interior of a segment or about a switching point, see Ref. [9]. For a boundary with a corner, both with equal or different boundary conditions on the two sides, one finds “corner operator expansions” in terms of operators located right at the apex of the corner, allowing to determine the modification of density profiles near the corner that arise from distant perturbations, see Ref. [10]. The enhanced order generated by disordering mentioned above follows in a straightforward way from an appropriate operator product expansion [11].