A paper, published at Science Express, looks at the non-obvious connections that exist between string theory in a hyperboloid spacetime and high temperature cuprate superconductors.
The work, carried out by a trio of physicists from Leiden University in the Netherlands, looks at the possibility of using the mathematics of string theory to describe quantum phase transitions in fermionic liquids. In particle physics, the 16 basic building blocks of the universe are either bosons, the force-carrying particles that obey Bose-Einstein statistics, or fermions, the constituents of matter that are described by Fermi-Dirac statistics. While quantum physics can easily deal with systems of bosons, there is no general mathematical theory that describes fermions at non-zero density.
Methods known to work well for bosonic systems break down when applied to fermions due to what is known as the “fermion sign problem.” Computationally, describing a fluid system of fermions (such as the electron “sea” present in metals) bogs down because the problem scales with exponential complexity, making all but the most trivial systems intractable.
In order to understand these systems, educated guesswork or simplifications are often applied. Unfortunately, the simplifications typically cannot describe the symmetry breaking that occurs near a quantum critical state (a region believed to be of key importance in the understanding of high temperature superconductors).
In order to explore this critical region of quantum phase space, the authors attempt to borrow a technique from the string theorist’s playbook. While string theory is considered by some to be nothing more than elegant mathematics, the mathematics themselves have proven useful.
In 1998 a paper came out that proposed a duality between quantum field theory (conformal field theory) and string theory in an Anti-deSitter space. The so called AdS/CFT correspondence allows problems that are intractable in one domain to potentially be solved in the other. Since a correspondence is hypothesized to exist between the two, variables and concepts have meanings (albeit different ones) on each side of the correspondence.
Fermion density behavior happens to be an example of a quantum field theory problem, so the trio of researchers attempted to use the AdS/CFT correspondence to gain insight into this problem by attacking it from the spacetime and string theory side of the universe. The researchers were able to derive a set of functions that describe a spectrum that can be inferred experimentally from angular resolved photoemission spectroscopy and scanning tunneling microscopy.
By describing the fermions in three dimensions (two spatial, one temporal) in the quantum side of the correspondence, the paper used string theory to calculate some of the properties that arise when varying the strength of the fermionic field and the chemical potential of the overall system. The authors were able to identify three key elements in their computed spectrum: the peak approaches a delta function at the Fermi momentum, there is a linear dispersion relationship between energy and velocity, and, at low temperatures, the width of the peak grows with temperature. Reasonable matches to these predictions were observed experimentally, leading the authors to conclude that they’ve managed to describe the characteristics of fermi liquids emerging from a quantum critical state.
While the main thrust of the paper is a mathematical description of a quantum phenomenon, the work also demonstrates how seemingly distant fields can be intimately related in the world of science. The paper applies math that describes a string world and weaves it into the world of electron “seas” in metals. As an interesting note, the authors point out that the peaks in the spectral functions computed here (which give support to their stated conclusion) correspond to various modes of black holes when the functions are used to describe the string world.