1-1 Calculational Techniques in Condensed Matter Theory

The toolbox of an experienced condensed matter theorist is divided into two major compartments, each one crowded with calculational techniques. One of the two compartments bears the label universal tools, the other one precision instruments.

The universal-tools compartment contains an assortment of general methods for the calculation of observable quantities of interest in condensed matter physics. Among them are general methods for a particular purpose. General methods for the calculation of dynamic correlation functions which are applicable to arbitrarily selected model systems, for example, belong in that compartment. Also found there are multi-purpose methods with a wide range of applicability. Universal tools must have a certain robustness against conditions that may invalidate their applicability. However, they are not meant to yield exact results on such a wide territory of applications.

The general calculational or computational techniques that are commonly used in condensed matter theory may be categorized as follows:

Methods with extrinsic limitations, such as computer simulations, Green's function methods, the recursion method, or finite-size studies. In these methods, the limitations are set by the amount of calculational effort or computational power invested in them.

Methods with intrinsic limitations, such as mean field theory, linear spin-wave theory (harmonic approximation), the random-phase approximation (within the framework of Green's function methods), the n-pole approximation (within the framework of the recursion method), have built-in limitations that cannot be overcome within their own respective scope.

The precision instruments, stored in the other compartment of the theorist's toolbox, are a collection of special methods. They have been designed for the exact solution of particular problems. The small but precious collection of exactly solved models in statistical mechanics and solid state physics is their main source of origin.

Once a challenging problem has been solved by a special method, it is by no means guaranteed that the solution can be reproduced by a general method. Nevertheless, it is usually instructive and illuminating to test the performance of universal tools on problems that have previously been solved by specially designed precision instruments. Many a general method has its root in a special method designed for the exact solution of a specific problem. What makes it a general method are applications to problems of a similar nature, where it is subject to intrinsic or extrinsic limitations. For example, mean field theory may be regarded as a special method for the solution of certain model systems with long-range interaction, and spin-wave theory derives its legitimacy from those special situations in which anharmonicities can be ignored in magnetic excitations.

For a thorough study of a broadly defined topic we need universal tools and precision instruments, i.e. general methods with their wide range of applicability and special methods that provide deeper insight for particular circumstances. We must combine systematics in breadth with systematics in depth in order to gain the best possible understanding of the topic under scrutiny. The use in isolation of (i) general methods with severe intrinsic limitations or (ii) special methods applicable only under highly non-generic circumstances is likely to invite misleading conclusions. Systematics in both directions is key to an understanding of the least accessible territory.

Exact solutions are usually out of reach except for particular circumstances and by special methods. The particular circumstances are always describable in terms of a simplification of the problem. There are basically two types of simplifications that may bring an exact solution to within reach:

Simplifications due to a special type of interaction between the degrees of freedom. The free-particle limit or special types of infinite-range interaction are obvious examples.

Simplifications due to a special state of a model system which otherwise exhibits generic behavior. A typical example is the ordered ground state of the Heisenberg ferromagnet.

Do these particular, simplifying circumstances translate into an improved performance of applicable general methods as well? The answer depends on the specifics of the general method under consideration. (i) Green's function methods pose the problem of approximating the infinite hierarchy of equations of motion in a controlled and systematic way. That is notoriously difficult even for weakly coupled degrees of freedom. However, in the noninteracting limit, that hierarchy reduces to a closed set of equations, from which the exact solution can readily be extracted. Simplifications due to a special state of the system do not, in general, result in a more tractable hierarchy of equations of motion. The reason is that the state of the system remains unspecified in the hierarchy of equations of motion. (ii) In the recursion method, the properties of generic systems manifest themselves in highly complex patterns exhibited by the sequences of continued-fraction coefficients, as we shall see. For noninteracting degrees of freedom, the amount of simplification in those sequences is comparable to that in the Green's function approach. However, the recursion method is decidedly better equipped to handle situations in which the simplification is due to a special state of the system. The reason is that the specification of the state has its impact on every continued-fraction coefficient as it is evaluated in the recursive calculational procedure.