Optimization of finite-size errors in finite-temperature calculations of unordered phases

TL;DR

This paper investigates finite-size errors in finite-temperature calculations, revealing that grand canonical ensemble errors are exponentially small in translationally invariant unordered phases, unlike canonical ensemble errors which are polynomial.

Contribution

It provides a detailed analysis of finite-size effects in different ensembles and boundary conditions, highlighting the advantages of grand canonical ensemble and linked cluster expansions.

Findings

Grand canonical ensemble errors are exponentially small in system size.

Canonical ensemble and open boundary conditions exhibit polynomial finite-size errors.

Numerical linked cluster expansions minimize finite-size effects.

Abstract

It is common knowledge that the microcanonical, canonical, and grand-canonical ensembles are equivalent in thermodynamically large systems. Here, we study finite-size effects in the latter two ensembles. We show that contrary to naive expectations, finite-size errors are exponentially small in grand canonical ensemble calculations of translationally invariant systems in unordered phases at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that are only polynomially small in the system size. We further show that finite-size effects are generally smallest in numerical linked cluster expansions. Our conclusions are supported by analytical and numerical analyses of classical and quantum systems.