research.md

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I study complex systems, particularly those with disorder and "frustration", where random interactions between a large number of parts give rise to emergent behavior which is difficult to understand from first principles.

Spin glasses

Mainly I have worked on spin glasses, materials containing magnetic atoms ("spins") that interact in a random way with their neighbors. Some pairs of neighbors prefer to align in the same direction, and others in opposite directions. To make it a bit more precise, for a pair of neighbors $$i$$ and $$j$$, we define a "coupling" $$J_{ij}$$. If the coupling is positive, the spins tends to align in the same direction in the absence of external influences and we say the interaction is "ferromagnetic". If negative, the pair tends to align in opposite directions and the interaction is "antiferromagnetic". The magnitude of the coupling describes the strength of the preference for alignment or anti-alignment.

In a ferromagnet, the couplings are all positive and thus at low-enough temperatures all of the spins will align, giving a net magnetization. At higher temperatures the couplings are overwhelmed by thermal fluctuations, resulting in the spins pointing in random directions and vanishing magnetization.

In a spin glass $$J_{ij}$$ is a random variable with zero mean. In contrast to a ferromagnet, where the spins are randomly-oriented at high temperature and tend to align below the Curie temperature, in a spin glass the orientation of the spins looks random at all temperatures. However, despite the lack of obvious ordering, experiments show clear evidence of a "freezing transition" at some special temperature $$T_f$$. In the "spin-glass phase" below $$T_f$$ the system tends to a random-looking but stable configuration which depends on the particular realization of the random interactions. The spin glass phase can be viewed as an ordering of the system in time, analogous to the spatial ordering seen in the ferromagnetic phase.

Spin glass (top) versus ferromagnet (bottom) [source]

The nature of the spin-glass phase remains controversial. Are there just two symmetry-related thermodynamic states, analogous to the "up" and "down" states of the ferromagnet, or is the situation more complicated for spin glasses? Analytical results for unphysical models with infinite-range interactions (called "mean-field" theories) actually predict infinitely many states, but it's not known to what extent these results generalize. More realistic models with only short-range interactions are much more difficult to study analytically, and much of what we know comes from computer simulations. So far simulations haven't shown conclusive evidence of many states in spin glasses with short-range interactions. In two recent papers, arXiv:1504.07709, arXiv:1410.5296, I've looked for evidence of many states in spin glasses with long-range interactions that decay as a power law in distance.

Connection with optimization

The competition between ferromagnetic and antiferromagnetic interactions in spin glasses creates "frustration", which means that in general there is no way of choosing an orientation for each spin that simultaneously makes every pair "happy". The combination of disorder (i.e. random interactions) and frustration leads to a complex, "rough" energy landscape with local minima on many scales separated by high energy barriers. From a physical perspective the upshot is slow dynamics and unique non-equilibrium effects. Similar behavior is seen in contexts outside of physics, notably in the study of optimization algorithms, which sometimes exhibit a "spin-glass phase" in certain regions of parameter space. In these regions the cost function looks like the energy function of a spin glass, with very many local minima of varying optimality which can make greedy algorithms such as gradient descent ineffective at finding a good solution. In particular, finding the ground state (lowest-energy state) of a spin glass in more than two dimensions is an NP-hard problem.

Publications

• The connection between statics and dynamics of spin glasses
  Matthew Wittmann and A. P. Young
  J. Stat. Mech. Theor. Exp. 2016, 013301 (2016)
  [arXiv] [DOI]

• Finite-size critical scaling in Ising spin glasses in the mean-field regime
  T. Aspelmeier, Helmut G Katzgraber, Derek Larson, M. A. Moore, Matthew Wittmann, Joonhyun Yeo
  Phys. Rev. E 93 (2014)
  [arXiv] [DOI]

• Finite-size scaling above the upper critical dimension
  Matthew Wittmann and A. P. Young
  Phys. Rev. E 90 (2014)
  [arXiv] [DOI]

• Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models
  Matthew Wittmann, B. Yucesoy, Helmut G. Katzgraber, J. Machta, A. P. Young
  Phys. Rev. B 90 (2014)
  [arXiv] [DOI]

• Spin glasses in the nonextensive regime
  Matthew Wittmann and A. P. Young
  Phys. Rev. E 85 (2012)
  [arXiv] [DOI]

Conferences

• Distinguishing graphs with a quantum annealer using susceptibility measurements
  with Itay Hen and A. P. Young
  APS March Meeting, 2014
  [abstract] [slides]

• Low-temperature behavior of the spin overlap distribution in one-dimensional long-range diluted spin glasses
  with B. Yucesoy, Helmut G. Katzgraber, J. Machta, and A. P. Young
  APS March Meeting, 2013
  [abstract]

• Scheduling: a good candidate for quantum annealing?
  with Itay Hen and A. P. Young
  Berkeley Mini Stat Mech Meeting, 2014