Take a large piece of material and measure some of its macroscopic properties, for example its density, compressibility or magnetisation. Now divide it into two roughly equal halves, keeping the external variables like pressure and temperature the same. The macroscopic properties of each piece will then be the same as those of the whole. The same holds true if the process is repeated. But eventually, after many iterations, something different must happen, because we know that matter is made up of atoms and molecules whose individual properties are quite different from those of the matter which they constitute. The length scale at which the overall properties of the pieces begin to differ markedly from those of the original gives a measure of what is termed the correlation length of the material. It is the distance over which the fluctuations of the microscopic degrees of freedom (the positions of the atoms and suchlike) are significantly correlated with each other. The fluctuations in two parts of the material much further apart than the correlation length are effectively disconnected from each other. Therefore it makes no appreciable difference to the macroscopic properties if the connection is completely severed. Usually the correlation length is of the order of a few inter-atomic spacings. This means that we may consider really quite small collections of atoms to get a very good idea of the macroscopic behaviour of the material.
Take a large piece of material and measure some of its macroscopic properties, for example its density, compressibility or magnetisation. Now divide it into two roughly equal halves, keeping the external variables like pressure and temperature the same. The macroscopic properties of each piece will then be the same as those of the whole. The same holds true if the process is repeated. But eventually, after many iterations, something different must happen, because we know that matter is made up of atoms and molecules whose individual properties are quite different from those of the matter which they constitute. The length scale at which the overall properties of the pieces begin to differ markedly from those of the original gives a measure of what is termed the correlation length of the material. It is the distance over which the fluctuations of the microscopic degrees of freedom (the positions of the atoms and suchlike) are significantly correlated with each other. The fluctuations in two parts of the material much further apart than the correlation length are effectively disconnected from each other. Therefore it makes no appreciable difference to the macroscopic properties if the connection is completely severed. Usually the correlation length is of the order of a few inter-atomic spacings. This means that we may consider really quite small collections of atoms to get a very good idea of the macroscopic behaviour of the material.
If we turn down our light source from 100 million whatever photons a second to, say, just ten a second, we see the problem, if not the screen, very clearly. These last few photons of light head off, one by one, for the screen, some of them hitting the piece of card but the odd one going through a slit. Now, according to common sense, each should be able to land on the screen unaffected by whether the other slit is there-or not. But none of them, no matter how long we wait, will try to land on the dark areas seen in the interference pattern with lots of light. After all, if an individual photon could behave in this way, it might take a long time but, eventually, the screen would record a picture more like the one-slit experiment with the light diffusing evenly, than the interference one. And the individual photons recognise the contradiction. They continue to behave individually as ‘waves’ all on their own. Points where the light particle could go when just one slit is open become unavailable when the other slit is open. (Incidentally, the same thing, scientists have observed, will happen with any small particle-electrons, whole atoms or even molecules will do just as well.) In experiments, the individual particles of light seem to know about the other slit on the piece of card, indeed what is going on elsewhere in the universe. For, in principle at least, the distance between the slits can be any size, and the same interference effect will apply. (Although if the slits were even a metre apart, the screen might need to be on the moon before the deflected light begins to converge.) Suppose we asked our neighbour if we could borrow their particle detector to check whether the light energy is still travelling in ‘whole photons’, rather than splitting into two and going through both slits and thereby getting in and out of phase with each other. We would find, indeed, that it was still travelling as a whole photon, and that this attempted explanation does not fit with observation. But we would find something else too. The mere act of putting the particle detector next to one slit causes the wavy pattern, the ‘interference’ effect, to disappear!
The Schrödinger equation describes the motion of the microscopic particles that constitute our world such as the electrons or atomic nuclei. Albeit being applicable to the smallest particle that we know of, it has observable consequences in the macroscopic world. It determines the conductivity of metals, it tells us which materials are magnetic and whether they show exotic behaviour such as super-conductivity. Unfortunately, solving the Schrödinger equation directly for any piece of material that is visible for the human eye is practically impossible. Already a grain of sand contains 1023 (that is written out 10.000.000.000.000.000.000.000) electrons and atomic nuclei. This means that only specifying the initial positions of the particles requires to safe an incredible amount of data; a procedure which is unfeasible for any human or computer. Due to the fundamental problem of applying quantum mechanics to practically relevant scenarios, a number of eective and approximate methods have been developed. In essence, they all try to reduce the dimension of the problem, i.e., the curse of the enormous amount of data required to simulate the Schrödinger equation. In this thesis, we try to analyze and expand one of those method called Density Matrix Embedding Theory (DMET). In a lot of physical system, especially when considering solid states, we can already learn a lot about its physics when describing its properties on a small fragment of the whole system. In a system with interacting particles though, we cannot simply consider just a subsystem and describe its properties without taking into account its interactions with the rest of the system. The basic idea of DMET is to divide the considered system into two parts called impurity and envi- ronment. The impurity is chosen to be so small that its wave function can be computed exactly. In the environment, only those degrees of freedom directly interacting with the impurity are considered and are included in our description. The physics on the environment itself is neglected. In the following, we will explain in detail how this can be done specically. In part I of this thesis, we will set the stage for the considered systems and present well-known and established methods to solve them. In the next part II, we will present DMET in great mathematical detail, which allows us to illustrate the advantages of DMET, but also some problems and drawbacks. We proceed by expanding DMET to the treatment of coupled electron-phonon system in part III and applying this new method to the Hubbard-Holstein model. Part of this work has been published in [53]. Finally, in part IV, we discuss some problems of DMET and, by combining DMET with functional theories, solve these problems. These insights, together with the extensive discussion of the DMET algorithm, will be published soon [59]. We illustrate this new method with an example system. This work will be published in paper [44] soon. We conclude this thesis by a summary and outlook (part V).
Abstract I would claim that the most important advances in this area (condensed matter physics) come about by the emergence of qualitatively new concepts at the intermediate or macroscopic levels-concepts which, one hopes, will be compatible with one’s information about the microscopic constituents, but which are in no sense logically dependent on it. These new concepts . . . have in common. . . that they provide a new way of classifying a seemingly intractable mass of information of selecting the important variables from the innumerable possible variables which one can identify in a macroscopic system.
Good order is the foundation of all good things. E. Burke, Reflections on the Revolution in France , 1790 Josephson effect Superconductivity: A crash course The transition from the theoretical description of hypothetical building blocks of a quantum coherent device to something which can be actually fabricated and controlled is made much easier by the existence of superconductivity . This phenomenon, roughly speaking, allows a macroscopic quantum coherent flow of electrons in a sufficiently cold piece of an appropriate material by establishing a specific long-range order among them. Due to this one can, for example, use macroscopically different states of a superconductor as quantum states of a qubit, with the obvious advantage over “microscopically quantum” systems (like actual atoms) from the point of view of control, measurement and, last but not least, fabrication of structures with the desired parameters and on the desired scale. The phenomenon of superconductivity – the history of its discovery, experimental manifestations, theoretical explanation, open questions, relevance to other branches of physics, and technological applications – requires a thorough treatment, which can be found in any number of books. For our purposes Tinkham (2004) will provide more than sufficient background. We will not wander into the field of exotic/high-temperature superconductors for two simple reasons: the quantum coherent behaviour of the kind we need to realize qubits or other quantum coherent devices has not yet been properly and/or routinely achieved in these systems; and the fabrication of such devices is not yet reliable enough or even feasible.
Abstract 1. GENERALITIES Much of solid state theory and statistical physics is concerned with the properties of macroscopic systems. These are often considered while using the “thermodynamic limit” (system’s volume, Ω, and particle number, N, tending to infinity with n = N /Ω kept constant), which is a convenient mathematical device for obtaining bulk properties. Usually, the system approaches the macroscopic limit once its size is much larger than some correlation length ζ (or, more generally, than all such relevant lengths).