Complex-time Singularity and Locality Estimates for Quantum Lattice Systems

Released on by Gabriel D. Bouch
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  • Complex-time Singularity and Locality Estimates for Quantum Lattice Systems Book Detail

  • Author : Gabriel D. Bouch
  • Release Date : 2011
  • Publisher :
  • Genre : Localization theory
  • Pages : 57
  • ISBN 13 :
  • File Size : 19,19 MB
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Complex-time Singularity and Locality Estimates for Quantum Lattice Systems by Gabriel D. Bouch PDF Summary

Book Description: In a very general class of one-dimensional quantum spin systems, the infinite volume limit of the complex-time evolution of a local observable is an entire analytic function of the time variable and obeys a locality principle. This result has recently been used to prove a number of important results in statistical mechanics. In dimensions greater than one, although it has not been expected that the infinite volume limit of the complex-time evolution of a general local observable will be entire analytic, nothing rigorous has been established concerning the breakdown of analyticity or the nature of the singularities, if they exist. In this work we begin by presenting a possible approach to proving locality bounds for the complex-time dynamics of a general class of quantum spin systems in any dimension. Then we specifically apply this approach to the one-dimensional case, and establish entire analyticity of the dynamics as a corollary. In dimensions greater than one, ideas related to the much studied Eden growth process suggest that a similar locality result will also hold. In particular, we establish an upper bound on the expected perimeter of lattice animals grown according to an Eden growth process, and note that a similar upper bound on a closely related average perimeter would lead to a locality result in the plane. Finally, and perhaps unexpectedly, we demonstrate through a specific construction that such a locality result does not hold in general and that the infinite volume limit of the complex-time dynamics can blow up a finite distance along the imaginary-time axis.

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