Abstract
Abstract Trace estimators allow us to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.
Highlights
The exact evaluation of thermodynamic quantum equilibrium observables is restricted to small systems due to the exponential growth of the Hilbert space for systems with finite-size single-site Hilbert spaces suchA prominent formulation of the method is the finite temperature Lanczos method (FTLM) [4, 21, 40,41,42] which employs a Krylov space expansion for exp{−βH}
A prominent representative is the finitetemperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights
The major argument is that this expansion does not suffer from the loss of orthogonality during recursive state generation used in Krylov space methods
Summary
The (numerically) exact evaluation of thermodynamic quantum equilibrium observables is restricted to small systems due to the exponential growth of the Hilbert space for systems with finite-size single-site Hilbert spaces such. It turns out that FTLM produces very accurate approximations when estimates are averaged over random vectors (order of ∼ 100, fewer for larger spaces); compare [10, 16, 31, 36, 43,44,45] Despite this success, the authors of [8] suggest that an alternative approximation using an expansion of the density of states in terms of Chebyshev polynomials should be more accurate [8]. The major argument is that this expansion does not suffer from the loss of orthogonality during recursive state generation used in Krylov space methods This property is certainly responsible for the high accuracy obtained in numerical unitary time evolution using a Chebyshev expansion, see e.g., [8, 46,47,48,49,50].
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