Christoph P. Hofmann

Collocation method for fractional quantum mechanics

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on little sinc functions, which discretizes the Schrodinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a Wentzel–Kramers–Brillouin analysis is performed.

Interpretation of topologically restricted measurements in lattice σ-models

We consider models with topological sectors, and difficulties with their Monte Carlo simulation. In particular we are concerned with the situation where a simulation has an extremely long auto-correlation time with respect to the topological charge. Then reliable numerical measurements are possible only within single topological sectors. The challenge is to assemble such restricted measurements to obtain an approximation for the full-fledged result, which corresponds to the correct sampling over the entire set of configurations. Under certain conditions this is possible, and it provides in addition an estimate for the topological susceptibility χt. Moreover, the evaluation of χt might be feasible even from data in just one topological sector, based on the correlation of the topological charge density. Here we present numerical test results for these techniques in the framework of non-linear σ-models.

Low-Temperature Properties of Two-Dimensional Ideal Ferromagnets

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has never been addressed in a systematic way so far. In the present paper the lowtemperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry O(3) → O(2) and are defined on a square, honeycomb, ~

Effective Lagrangians for quantum many-body systems

A bstractThe low-energy and low-momentum dynamics of systems with a spontaneously broken continuous symmetry is dominated by the ensuing Nambu-Goldstone bosons. It can be conveniently encoded in a model-independent effective field theory whose structure is fixed by symmetry up to a set of effective coupling constants. We construct the most general effective Lagrangian for the Nambu-Goldstone bosons of spontaneously broken global internal symmetry up to fourth order in derivatives. Rotational invariance and spatial dimensionality of one, two or three are assumed in order to obtain compact explicit expressions, but our method is completely general and can be applied without modifications to condensed matter systems with a discrete space group as well as to higher-dimensional theories. The general low-energy effective Lagrangian for relativistic systems follows as a special case. We also discuss the effects of explicit symmetry breaking and classify the corresponding terms in the Lagrangian. Diverse examples are worked out in order to make the results accessible to a wide theoretical physics community.

Symmetry analysis of holes localized on a skyrmion in a doped antiferromagnet

We use the low-energy effective field theory for holes coupled to the staggered magnetization in order to investigate the localization of holes on a Skyrmion in a square lattice antiferromagnet. When two holes get localized on the same Skyrmion, they form a bound state. The quantum numbers of the bound state are determined by the quantization of the collective modes of the Skyrmion. Remarkably, for p-wave states the quantum numbers are the same as those of a hole-pair bound by one-magnon exchange. Two holes localized on a Skyrmion with winding number

Homogeneous versus spiral phases of hole-doped antiferromagnets : A systematic effective field theory investigation

n = 1

Low-Temperature Properties of Ferromagnetic Spin Chains in a Magnetic Field

or 2 may have s- or d-wave symmetry as well. Possible relations with preformed Cooper pairs of high-temperature superconductors are discussed.

Thermodynamics of Two-Dimensional Ideal Ferromagnets - Three-Loop Analysis

Using the low-energy effective field theory for magnons and holes--the condensed matter analog of baryon chiral perturbation theory for pions and nucleons in QCD--we study different phases of doped antiferromagnets. We systematically investigate configurations of the staggered magnetization that provide a constant background field for doped holes. The most general configuration of this type is either constant itself or represents a spiral in the staggered magnetization. Depending on the values of the low-energy parameters, a homogeneous phase, a spiral phase, or an inhomogeneous phase is energetically favored. The reduction of the staggered magnetization upon doping is also investigated.

Thermodynamics of O(N) antiferromagnets in 2+1 dimensions

The thermodynamic properties of ferromagnetic spin chains have been analyzed with a variety of microscopic methods over the years: Bethe ansatz, spin-wave theory, Schwinger-boson mean-field theory, Green functions and renormalization group methods. Surprisingly, in all these different studies, to the best of our knowledge, the manifestation of the spin-wave interaction in the low-temperature series for the thermodynamic quantities has been ignored. In the present work, we address this problem by following a different path, based on the systematic effective Lagrangian method. We evaluate the partition function up to two-loop order and derive the lowtemperature expansion of the energy density, entropy density, heat capacity, magnetization and susceptibility in the presence of a weak external magnetic field. Remarkably, the spin-wave interaction only manifests itself beyond two-loop order. In particular, there is no term of order T 2 in the low-temperature series of the free energy density. This is the analog of Dyson’s statement that, in the case of three-dimensional ideal ferromagnets, there is no term of order T 4 in the low-temperature series of the free energy density. The range of validity of our series is critically examined in view of the Mermin-Wagner theorem. We also compare our results with the condensed matter literature and point out that there are some misleading statements.

The constraint effective potential of the staggered magnetization in an antiferromagnet

Within the effective Lagrangian framework, we explicitly evaluate the partition function of two-dimensional ideal ferromagnets up to three loops at low temperatures and in the presence of a weak external magnetic field. The low-temperature series for the free energy density, energy density, heat capacity, entropy density, as well as the magnetization are given and their range of validity is critically examined in view of the Mermin-Wagner theorem. The calculation involves the renormalization and numerical evaluation of a particular three-loop graph which is discussed in detail. Interestingly, in the low-temperature series for the two-dimensional ideal ferromagnet, the spin-wave interaction manifests itself in the form of logarithmic terms. In the free energy density the leading such term is of order