This chapter introduces a finite-temperature algorithm for the simulation of interacting electrons on a lattice. Because this algorithm was developed by Blankenbecler, Scalapino, and Sugar (1981; Scalapino and Sugar, 1981), it is sometimes called the BSS algorithm. The method uses a Hubbard-Stratonovich transformation to convert the interacting electron problem into a noninteracting one coupled to an imaginarytime- dependent auxiliary field. For this reason, it is also called the auxiliary-field method. We use here yet another name, the determinant method, which is fitting because the transformation to a problem of noninteracting electrons generates determinants as the statistical weights. The finite-temperature determinant algorithm is a general-purpose electron algorithm that enables computations of a wide variety of local observables and correlation functions. For a discussion of a zero-temperature determinant method, refer to Appendix I.

Theoretical framework

Feynman and Hibbs (1965) formulated quantum mechanics in terms of integrals over all paths in configuration space. In real time, each path contributes a phase to the integral that is determined by the classical action along the path. Two paths can interfere constructively or destructively. In the classical limit, only the stationary-phase path is important. Being characterized by many interfering paths, real-time quantum dynamics more than challenges importance sampling. Statistical mechanics, on the other hand, involves path integrals in imaginary time. Contributions to the integrals vary exponentially in magnitude but not in phase. Thus, the path integral is dominated by paths of large magnitude. The tasks of a quantum Monte Carlo method are identifying these important paths and sampling them efficiently.

In this chapter, we address the classicization of many-electron problems at finite temperatures via a Feynman path integral. The result is a method often called the determinant method as the weights of the paths can be expressed as determinants, hardly classical-looking weights, but ones quite suggestive of the antisymmetry of Fermion states. Sampling these weights efficiently and in a stable manner requires special techniques. We begin with a brief overview to motivate the general form of the classical representation and the weights we need to sample.

This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems.

Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers.

Quantum impurity models

A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.

Quantum Monte Carlo versions of the power method for finding ground states come with different names, including the projector method, diffusion Monte Carlo, and Green's function Monte Carlo. These quantum Monte Carlo methods are primarily adoptions of methods developed for classical problems. We summarize the basics of deterministic power methods, detail key features of Monte Carlo power methods, and put these concepts into the context of quantum ground state calculations. We conclude the chapter by outlining power methods that allow the computation of a few excited states. In the computation of excited states, we encounter sign problems, which are discussed inmore detail in Chapter 11. Themethods and techniques of the present chapter are very general and most usefully applied to systems of Bosons and to systems of Fermions and quantum spins that do not suffer from a sign problem.

Deterministic direct and inverse power methods

The power method is over a century old. As a deterministic method, it originally was combined with the deflation technique (Meyer, 2000; Stewart, 2001b) as a method to compute all the eigenvalues and eigenvectors of small matrices. Today, its deterministic use is limited primarily to special applications involving very large, sparse matrices. Many of these applications are in the study of quantum lattice models as the corresponding Hamiltonian matrices are typically very sparse. In these applications, we can perhaps initially store in computer memory the relatively small number of nonzero matrix elements and all the vectors. Soon, because of what is undoubtedly the now familiar exponentially increasing number of basis states that determines the order of the Hamiltonian matrix and hence the size of our vectors, we reach the point where computer memory restrictions allow us to store only a few vectors and we need to compute the matrix elements on the fly. When we have problems for which we cannot even store all the components of one vector, the power method's expression as a Monte Carlo procedure is our only option. With it, we can compute a few of the extremal eigenvalues, that is, a few of the largest or smallest eigenvalues.

The mathematical basis for the method is simple.

Classification of solids

Condensed matter physics and solid state physics usually refer to the same area of physics, but in principle the former title is broader. Condensed matter is meant to include solids, liquids, liquid crystals, and some plasmas in or near solids. This is the largest branch of physics at this time, and it covers a broad scope of physical phenomena. Topics range from studies of the most fundamental aspects of physics to applied problems related to technology

The focus of this book will be primarily on the quantum theory of solids. To begin, it is useful to start with the concept of a solid and then describe the two commonly used models that form the basis for modern research in this area. The word “solid” evokes a familiar visual picture well described by the definition in the Oxford Dictionary: “Of stable shape, not liquid or fluid, having some rigidity.” It is the property of rigidity that is basic to the early studies of solids. These studies focused on the mechanical properties of solids. As a result, until the nineteenth century the most common classification of solids involved their rigidity or mechanical properties. The Mohs hardness scale (talc – 1; calcite – 3; quartz – 7; diamond – 10) is a typical example. This is a useful but limited approach for classifying solids.

The advent of atomic theory brought more microscopic concepts about solids. Solids were viewed as collections of more or less strongly interacting atoms. From the point of view of atomic theory, a gas is described in terms of a collection of almost independent atoms, while a liquid is formed by atoms that are weakly interacting. This picture leads to a description of the formation of solids, under pressure or by freezing, in which the distances between atoms are reduced and, in turn, this causes them to interact more strongly. Molecular solids are formed by condensing molecular gases.

Hence, the development of atomic physics and chemical analysis led to a more detailed classification of solids according to chemical composition. Although for most studies of solids it is necessary to establish the identity of the constituent atoms, such a scheme provides limited insight into many of the basic concepts of condensed matter physics.