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.

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11 responses to “Some Words on Sum Rules

  1. Are you sure the f-sum rule represents conservation of energy? I thought it results from particle number conservation (see Pines and Nozierres for example). The only things you need for the f-sum rule are the canonical commutation relations and a momentum independent potential.

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    • Yes, you can actually interpret it as either, depending on your point of view.

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      • I guess my point is that the requirement that the Hamiltonian is not explicitly time-dependent (energy conservation) is a distinct requirement from particle number conservation. For example, if there is a time-dependent field term that couples to the charge, the usual commutation relations still hold and, as far as I can tell, you would still get the f-sum rule to work.

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  2. I think it is more helpful to think of it in a scattering context, in the sense that the f-sum rule is basically expresses the fact that if you add up all the transitions to higher energy states caused by density perturbations (the LHS of the first equation in this post), this is exactly equal to the amount of energy you put into the system (RHS of the same equation).

    The f-sum rule is independent of the Hamiltonian of the system as long as the potential does not depend on the velocity of the particles and the system obeys time-reversal symmetry. I think it is a little misleading to think about the Hamiltonian in the way you mention above because even for the BCS Hamiltonian (where particle is not conserved), the f-sum rule is still obeyed.

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    • Good point regarding the bcs hamiltonian. I guess my ultimate confusion is now the statement that the trk sum rule represents energy conservation (or any conservation law at all! ). I say this because the assumptions needed for the sum rule are not simply energy conservation. Adding spin orbit coupling would spoil the sum rule, but clearly energy would still be conserved. I understand that the sum rule is suggestive of energy conservation, but there doesn’t seem to be one-to-one correspondence between the two.

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      • Perhaps this is a silly oversight, but I don’t immediately see why the sum rule shouldn’t be obeyed in the presence of spin-orbit coupling…?

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  3. The potential would have a component that depends on the (angular) momentum, which would get in the way because of the commutation relations.

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    • That is a very interesting point. I think I would have to work it out to believe it 100%, but it seems likely that you are correct. I would contend, however, that the f-sum rule would still be applicable in a scattering experiment in the presence of spin-orbit coupling, which makes me doubt your contention somewhat.

      Regardless, your are right that the correspondence is not one-to-one, but just a simplified physical picture to understand the origin of the basic sum rule. If we take the perspective of number-particle conservation, we may have to re-define the current and it may still work out. I will get back to you on this after I’ve worked it out. In the meantime, take a look here to see what I mean (at the bottom of this set of my notes, where I derive the f-sum rule):

      https://www.dropbox.com/s/glpajahyz95wjx4/Random%20Phase%20Approximation.pdf?dl=0

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      • I agree with your sentiment. I am sure there is an extended f-sum rule for other types of couplings as well. I think the reason other couplings usually don’t cause major sum rule deviations is probably because of how much stronger coulomb repulsion, which even includes phonons etc., is compared to spin-orbit and the like.

        Nonetheless, I just wanted to emphasize there is a bit more subtlety to this sum rule, which I think is why in the literature it usually isn’t associated with a particular conservation law.

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  4. Actually, if you take a look at D. Forster’s Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions, he explicitly claims that the sum rules are expressions of conservation laws. But the conservation laws are local conservation laws, not global ones — such as in the case of particle number conservation. It says that the change in number of electrons going into a region is equivalent to the divergence of the current. Perhaps this is where the confusion is arising.

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    • That is confusing, local conservation is a much stronger condition than global conservation. If he’s saying sum rules are associated with local conservation laws, then they necessarily imply global conservation.

      If there has to be a fundamental conservation law associated with the f-sum rule, it would probably have to be something like conservation of ‘longitudinal’ current because of the restriction on velocity-terms. But then again, I don’t know if there is much meaning to such a current.

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