Abstract
We study trace estimators for equilibrium thermodynamic observables that rely on the idea of typicality and derivatives thereof such as the finite-temperature Lanczos method (FTLM). As numerical examples quantum spin systems are studied. Our initial aim was to identify pathological examples or circumstances, such as strong frustration or unusual densities of states, where these methods could fail. Instead we failed with the attempt. All investigated systems allow such approximations, only at temperatures of the order of the lowest energy gap the convergence is somewhat slower in the number of random vectors over which observables are averaged.
Highlights
One way to approximate thermodynamic quantities is to rely on trace estimators
The latter fact is the essential difference to the eigenstate thermalization hypothesis (ETH) [14,15,16], which assumes that the approximation in Eq (1) holds if |r is replaced by an individual energy eigenstate
The idea of typicality in the context of trace estimators for physical quantities such as the partition function means that the overwhelming majority of all random vectors that one can draw in a high-dimensional Hilbert space consist of virtually equivalent vectors and correspond to a situation of infinite temperature, where all expansion coefficients of the density matrix with respect to an orthonormal basis would be just about the same
Summary
One way to approximate thermodynamic quantities is to rely on trace estimators. These schemes became very popular in recent years since they turn out to be rather (or astonishingly) accurate and in addition a valuable alternative in cases where quantum Monte Carlo suffers from the sign problem. Under unitary basis transforms this distribution remains Gaussian, i.e, in the energy eigenbasis The latter fact is the essential difference to the eigenstate thermalization hypothesis (ETH) [14,15,16], which assumes that the approximation in Eq (1) holds if |r is replaced by an individual energy eigenstate (with the trace operation being performed in a microcanonical energy shell). The traces to be dealt with in this article appear in equilibrium statistical physics, where they are evaluated for static operators such as exp{−βH } and O exp{−βH }, yielding the partition function and thermal expectation values of observables O, respectively.
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