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

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.

Figure 1

Coefficients (absolute value) of the $\beta$ expansion of the free energy vs. the expansion order $l$. We show the coefficients in Eq. (7) for the 1D Ising model, and a fit to the function $a{b}^{-l}/l$ with $a=2.0000$ and $b=1.5708$ (the difference between the exact result and $2{(\pi /2)}^{-l}/l$ vanishes exponentially fast with $l$), and from Eq. (15) for the 2D Ising model. The coefficients in the latter case do not fall off exponentially fast. Because of this, the $\beta$ expansion only converges for $\beta <{\beta }_c$. It is only in this regime that finite-size errors in grand-canonical ensemble calculations of translationally invariant systems are exponentially small in system size. Inset: Shows the rational part of the coefficients in Eq. (15). They decrease at first but increase after $O\left({\eta }^{24}\right)$. See text for further discussion.

Figure 2

Comparison of results of canonical and grand-canonical ensembles with open and periodic boundary conditions, and NLCEs, for noninteracting fermions hopping on a one-dimensional lattice [corresponds to Eq. (41) with $t=1$, $V=V^{\prime }=0$]. The energy difference $\delta E_l$ (see text) is plotted for two different temperatures in panels (a) and (b). Panel (c) shows the occupation of the $k=0$ mode, or, equivalently, the correlation $L^{-1}{\sum }_{ij}\langle c_i^{†}{c}_j\rangle$. $\delta n_{k=0}$ is zero for the grand-canonical ensemble with PBCs; see text.

Figure 3

(a) $\delta E_l$ and (b) $\delta K_l$ for interacting fermions [Eq. (41)] with $t=1$ (unit of energy), $V=1.0,V^{\prime }=0$, and $T=1.0$.

Figure 4

(a) $\delta E_l$ and (b) $\delta K_l$ for interacting fermions [Eq. (41)] with $t=1$ (unit of energy), $V=1.0,V^{\prime }=1.0$, and $T=1.0$.

Figure 5

$\delta E_l$ in NLCE calculations of the Heisenberg model in four different two-dimensional lattices, as indicated. The approach to the thermodynamic limit result is again exponentially fast as one increases the order of the expansion. The results presented here were taken from Refs. [5, 11, 15, 16]. $J=1$ is taken to be the unit of energy.