# Tag Archives: Sum Rules

## Reflecting on General Ideas

In condensed matter physics, it is easy to get lost in the details of one’s day-to-day work. It is important to sometimes take the time to reflect upon what you’ve done and learned and think about what it all means. In this spirit, below is a list of some of the most important ideas related to condensed matter physics that I picked up during my time as an undergraduate and graduate student. This is of course personal, and I hope that in time I will add to the list.

1. Relationship between measurements and correlation functions
2. Relationship between equilibrium fluctuations and non-equilibrium dissipative channels (i.e. the fluctuation-dissipation theorem)
3. Principle of entropy maximization/free-energy minimization for matter in equilibrium
4. Concept of the quasi-particle and screening
5. Concept of Berry phase and the corresponding topological and geometrical consequences
6. Broken symmetry, the Landau paradigm of phase classification and the idea of an order parameter
7. Sum rules and the corresponding constraints placed on both microscopic theories and experimental spectra
8. Bose-Einstein and Cooper Pair condensation and their spectacular properties
9. Logical independence of physical theories on the theory of everything
10. Effects of long-range vs. short-range interactions on macroscopic properties of solids
11. Role of dimensionality in observing qualitatively different physical properties and phases of matter

The first two items on the list are well-explained in Forster’s Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions without the use of Green’s functions and other advanced theoretical techniques. Although not yet a condensed matter phenomenon, Bell’s theorem and non-locality rank among the most startling consequences of quantum mechanics that I learned in graduate school. I suspect that its influence will be observed in a condensed matter setting in due time.

## Some Words on Sum Rules

In condensed matter physics, sum rules are used widely by both experimentalists and theorists. One can even go as far to say that sum rules provide us with a framework within which theories must exist, i.e. theories cannot violate the constraints put forth by these sum rules. In this sense, they are of vast importance, and any theory of, for example, the dielectric function should be checked against these constraints.

Even though these sum rules are used often, their physical meaning is not always apparent because they can be written in many forms. Let me use the Thomas-Reiche-Kuhn sum rule (a.k.a the f-sum rule) to illustrate some of these points. This sum rule can be formulated as so:

$\sum_m(E_m - E_0)|\langle{m}|n(\textbf{q})|0\rangle|^2 = \frac{n\hbar^2q^2}{2m}$

where $n(\textbf{q})$ is the Fourier-transformed number density operator. In this formulation, one can see the physical principles behind the sum rule most clearly:

If one adds up the energies of the transitions made from the ground state to higher energy states (in this case by perturbing the density), this should be equal to the total energy put into the system.

The TRK sum rule can be understood quite simply, therefore, as an energy conservation law for a many-body system. This is why these sum rules are so important — they are many-body manifestations of conservation laws.

The Thomas-Reiche-Kuhn sum rule is often written in the following way as well:

$\int_0^\infty \omega S(\textbf{q},\omega) d \omega = \frac{n\hbar^2 q^2}{2m}$

where $S(\textbf{q},\omega)$ is the dynamic structure factor.

Furthermore, TRK can be formulated in terms of the inverse longitudinal dielectric function as so:

$\int_0^\infty \omega \textrm{Im}(-1 /\epsilon_L(\textbf{q},\omega))d \omega = \frac{\pi}{2}\omega_p^2$

where $\omega_p$ is the plasma frequency. Also, it can be written in a form more familiar to optical spectroscopists, who often plot the optical conductivity:

$\int_0^\infty \textrm{Re}(\sigma_L(\textbf{q},\omega))d \omega = \frac{\omega_p^2}{8}$

So while there are many sum rules (and many formulations of each sum rule as seen above for the TRK), one should always keep in mind that they derive from rather general physical principles, which are unfortunately sometimes hidden in the way they are written.