Finite-size scaling of typicality-based 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 high-dimensional Hilbert space. This random-vector approximation, or trace estimator, provides a powerful approach to, e.g., thermodynamic quantities for systems with large Hilbert-space sizes, which usually cannot be treated exactly, analytically or numerically. Here, we discuss the finite-size scaling of the accuracy of such trace estimators from two perspectives. First, we study the full probability distribution of random-vector 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 system-specific temperatures. Below and in particular for temperatures smaller than the excitation gap, simple rules are not available.

arXiv comment9 pages, 11 figures

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Cited by in corpus (5)

• The pyrochlore S=1/2 Heisenberg antiferromagnet at finite temperature
• Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles
• Density dynamics in the mass-imbalanced Hubbard chain
• Accuracy of the typicality approach using Chebyshev polynomials
• A spectrum adaptive kernel polynomial method