FiniteSize Scaling of TypicalityBased Estimates
Abstract
According to the concept of typicality, an ensemble average can be accurately approximated by an expectation value with respect to a single pure state drawn at random from a highdimensional Hilbert space. This randomvector approximation, or trace estimator, provides a powerful approach to, e.g. thermodynamic quantities for systems with large Hilbertspace sizes, which usually cannot be treated exactly, analytically or numerically. Here, we discuss the finitesize scaling of the accuracy of such trace estimators from two perspectives. First, we study the full probability distribution of randomvector expectation values and, second, the full temperature dependence of the standard deviation. With the help of numerical examples, we find pronounced Gaussian probability distributions and the expected decrease of the standard deviation with system size, at least above certain systemspecific temperatures. Below and in particular for temperatures smaller than the excitation gap, simple rules are not available.
 Publication:

Zeitschrift Naturforschung Teil A
 Pub Date:
 May 2020
 DOI:
 10.1515/zna20200031
 arXiv:
 arXiv:2002.00411
 Bibcode:
 2020ZNatA..75..465S
 Keywords:

 Spin Systems;
 Thermodynamic Observables;
 Trace Estimators;
 Typicality;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 9 pages, 11 figures