New way to resum the lattice QCD Taylor series equation of state at finite chemical potential

Taylor expansion of the thermodynamic potential in powers of the (baryo)chemical potential ${\mu }_B$ is a well-known method to bypass the sign problem of lattice QCD. Due to the difficulty in calculating the higher order Taylor coefficients, various alternative expansion schemes as well as resummation techniques have been suggested to extend the Taylor series to larger values of ${\mu }_B$. Recently, a way to resum the contribution of the first $N$ charge density correlation functions $D_1,\dots ,D_N$ to the Taylor series to all orders in ${\mu }_B$ was proposed in [Phys. Rev. Lett. 128, 022001 (2022)]. The resummation takes the form of an exponential factor. Since the correlation functions are calculated stochastically, the exponential factor contains a bias which can be significant for large $N$ and ${\mu }_B$. In this paper, we present a new method to calculate the QCD equation of state based on the well-known cumulant expansion from statistics. By truncating the expansion at a maximum order $M$, we end up with only finite products of the correlation functions which can be evaluated in an unbiased manner. Although our formalism is also applicable for ${\mu }_B\ne 0$, here we present it for the simpler case of a finite isospin chemical potential ${\mu }_I$ for which there is no sign problem. We present and compare results for the pressure and the isospin density obtained using Taylor expansion, exponential resummation and cumulant expansion, and provide evidence that the absence of bias in the latter actually improves the convergence.

Figure 1

Comparison of the results for the excess pressure $\mathrm{\Delta }P(T,{\mu }_I)$ obtained using Taylor series expansion, exponential resummation and cumulant expansion. The resummed results were obtained for $N_{\mathrm{rv}}=500$ and $N_{\mathrm{rv}}=250$ (red and yellow bands). The cumulant expansion was calculated using both biased and unbiased estimates (upright and inverted triangles). Top: comparison between 2nd and 4th order Taylor expansions, 2nd order resummation and $(N,M)=(2,4)$ cumulant expansion. Bottom: comparison between 4th and 6th order Taylor expansions, 4th order resummation and $(N,M)=(4,4)$ cumulant expansion.

Figure 2

Comparison of the results for the isospin density $\mathcal{N}(T,{\mu }_I)$ obtained using Taylor series expansion, exponential resummation and cumulant expansion. All symbols and colors are the same as in Fig. 1. Top: comparison between 2nd and 4th order Taylor expansions, 2nd order resummation and $(N,M)=(2,4)$ cumulant expansion. Bottom: comparison between 4th and 6th order Taylor expansions, 4th order resummation and $(N,M)=(4,4)$ cumulant expansion.