This past Friday, Professor Philip W. Phillips, a professor of physics and faculty scholar at the University of Illinois at Urbana-Champaign spoke at the EE Just Symposium at Dartmouth College. He discussed his recent work with then-undergraduate Kiaran Dave, Professor Charlie Kane from University of Pennsylvania, and UIUC graduate student Brandon Langley, in which he argues that the behavior of unparticles in high temperature superconducters disproves the fundamental concepts of Luttinger’s Theorem.
All particles exhibit well defined charge and fixed mass. Charge in particles is conserved while mass is not. Scale invariance is an invariance of all length scales. Since mass changes, mass is scale invariant. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
According to free field theory, a free field is a field whose equations of motion are given by linear partial differential equations. Phillips proposes a way to think about field theory: “the way to think about it is not thinking about just one field theory but taking a field theory and taking that field theory on different energy scales.” Therefore, the field theory should be viewed as changing with energy.
If mass is introduced to the field theory, invariance is introduced. Rather than letting the constant of mass change, this invariance could be solved by introducing a theory in which mass evolves. The theory would then be an integral of all possible mass, under which scale invariance is restored. It is key to note that even with mass evolving, not all particles obey this theory.
Unparticles (if they occur in nature), have all possible mass. According to Einstein’s photoelectric theory, it is possible to shine light on electrons and detect their energy in order to count the particles.
Luttinger’s Theorem considers electrons by using Green’s function (a type of function that involves an initial condition.) According to Luttinger’s theorem, there is one change in sign per particle, so particles can be counted by observing the number of times the function changes sign.
According to Phillips, the theorem is incorrect. Luttinger’s theorem is based on the fact that the function changes sign when it approaches infinity. However, the function can also change sign simply by changing sign (as linear line can simply pass through 0). Divergence is therefore unnecessary to achieve a change in sign.
A closer look at Luttinger’s theorem shows that divergences and zeros are taken on equal footage. However, divergences give you particles, while the zeroes tell you about unparticles. Luttinger’s theorem makes no distinction between particles and unparticles. Luttinger is therefore saying that particle density is equal to the number of particles plus the number of unparticles.
In his study, Phillips came up with a model with zeros in which Luttinger’s theorem fails. Luttinger’s theorem fall apart if more than just two electron spins (spin up and down) are introduced. For example, when choosing N (number of spins) to be equal to 5, the theorem produces a result of “2 equals 3,” which is known to be incorrect.. In fact, the theorem fails anytime “n” is even, and “N” is odd.
When determining particle density in high-temperature superconductors, particle density cannot equal conserved charge in the system, and there will be charges that are left over. Expectation of zeros in systems, however, can be seen in systems with very large self-energy, which is generally in well-correlated systems.
Theoretically, if Luttinger’s theorem is correct, two crossings of energy excitations should be observed when light is shined on electrons. Experimentally, double crossings are not seen–only one crossing can be observed. Therefore what can be seen are the infinites, and what are not seen are the zeros. When particle density is counted, zeros do not affect particle density. Ultimately, Phillips concludes that Luttinger’s theorem expects results that deviate from experimental data.