On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods
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On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods M. Fischer1, B. Kostrzewa1, J. Ostmeyer1, K. Ottnad2 , M. Ueding1, C. Urbach1,a 1 2
Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, Bonn, Germany PRISMA+ Cluster of Excellence and Institut für Kernphysik, University of Mainz, Johann-Joachim-Becher-Weg 45, Mainz, Germany
Received: 28 April 2020 / Accepted: 16 July 2020 © The Author(s) 2020 Communicated by William Detmold
Abstract We discuss the relation of a variety of different methods to determine energy levels in lattice QCD simulations: the generalised eigenvalue, the Prony, the generalised pencil of function and the Gardner methods. All three former methods can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy El in all three methods due to unresolved states decay asymptotically exponentially like exp(−(E n − El )t). Using synthetic data we show that these corrections behave as expected also in practice. We propose a novel combination of the generalised eigenvalue and the Prony method, denoted as GEVM/PGEVM, which helps to increase the energy gap E n − El . We illustrate its usage and performance using lattice QCD examples. The Gardner method on the other hand is found less applicable to realistic noisy data.
1 Introduction In lattice field theories one is often confronted with the task to extract energy levels from noisy Monte Carlo data for Euclidean correlation functions, which have the theoretical form C(t) =
∞
ck e−E k t
(1)
k=0
with real and distinct energy levels E k+1 > E k and real coefficients ck . It is well known that this task represents an ill-posed problem because the exponential functions do not form an orthogonal system of functions. Still, as long as one is only interested in the ground state E 0 and the statistical accuracy is high enough to be able to work
a e-mail:
[email protected] (corresponding author)
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at large enough values of t, the task can be accomplished by making use of the fact that lim C(t) ≈ c0 e−E 0 t ,
t→∞
(2)
with corrections exponentially suppressed with increasing t due to ground state dominance. However, in lattice quantum chromodynamics, the non-perturbative approach to quantum chromodynamics (QCD), the signal to noise ratio for C(t) deteriorates exponentially with increasing t [1]. Moreover, at large Euclidean times there can be so-called thermal pollutions (see e.g. Ref. [2]) to the correlation functions, which, if not accounted for, render the data at large t useless. And, once one is interested in excited energy levels E k , k > 0, alternatives to the ground state dominance principle need to be found. The latter problem can be tackled applying the so-called generalised eigenvalue method (GEVM) – originally proposed in Ref. [3] and further developed in Ref. [4]. It is by now well
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