Strange connections to strange metals

The June 2012 issue of Physics Today has, beginning on page 68 , the Quick Study “From black holes to strange metals,” by Hong Liu. It is one of many quasi-journalistic discussions I have seen of results using the AdS/CFT (anti–de Sitter/conformal field theory) correspondence from quantum gravitation theory ostensibly to solve condensed-matter physics problems such as the “strange metal” in the cuprate (high T_c) superconducting metals. As the probable source of the buzzword phrase “strange metal” to describe the phenomena observed in the cuprates and of a theory that bids well to explain those phenomena in detail, I think I have a reasonable motivation to object to the publication of those claims, even though advanced tentatively, when so much is known about this particular phase.

The strange-metal region of the cuprate phase diagram exhibits not only a linear dependence on temperature of the conductivity relaxation rate, which is generally taken by string theorists as the characteristic symptom identifying a strange metal and is the only feature they discuss. The region also exhibits several additional anomalies that in my experience are unique to this phase:

‣ The IR conductivity—the “Drude tail” of the DC conductivity—falls off with frequency with a noninteger power law, and the exponent apparently varies continuously with doping. That behavior was demonstrated by Nicole Bontemps and coworkers ¹ in 1993 and further nailed down by Dirk van der Marel and coworkers ² in 1995.

‣ The relaxation rate as measured by the Hall angle θ_H of deviation of the current from the electric field direction, using the formula θ_H = 1/ω_cτ, is quite different from that of the conductivity, and has a different, T² temperature dependence, as N. P. Ong and coauthors demonstrated ³ in 1989.

‣ Over broad regions of doping, the two kinds of relaxation rates, the one for the conductivity and the one for the Hall rotation, seem to add as inverses: Conductivity is proportional to 1/T + 1/T²—that is, it obeys an anti-Matthiessen law.

‣ Angle-integrated photoelectron spectra, tunneling spectra, and angle-resolved photoemission spectra all fit better to power-law dependences on energy than to conventional fits.

All of those symptoms were explained without the use of arbitrary free parameters in the final papers of a long sequence dating back to 2004, by me and by Philip Casey and me, ⁴ while the AdS/CFT literature holds nothing resembling a connection back to the parameters of the real solids, nor any discussion of the other anomalies. I believe that our theories are exact, in the sense of continuation, in a considerable region of the phase diagram.

Incidentally, the phase diagram of the real cuprates is only vaguely similar to the conventional diagram shown in Liu’s figure. For instance, the strange metal shows no evidence of terminating on the right in a true Fermi liquid.

It is amusing that the methods we use are closely related to results in quantum field theory, but to discoveries of three decades or more ago about “anomalies” such as the well-known chiral anomaly of Roman Jackiw and Claudio Rebbi. At about the same period, we condensed-matter theorists were concerned with what we called “x-ray edge anomalies,” but we did not realize they were related to our colleagues’ anomalies.

As a very general problem with the AdS/CFT approach in condensed-matter theory, we can point to those telltale initials “CFT”—conformal field theory. Condensed-matter problems are, in general, neither relativistic nor conformal. Near a quantum critical point, both time and space may be scaling, but even there we still have a preferred coordinate system and, usually, a lattice. There is some evidence of other linear-T phases to the left of the strange metal about which they are welcome to speculate, but again in this case the condensed-matter problem is overdetermined by experimental facts.