Thermodynamic potentials from shifted boundary conditions: the scalar-field theory case

Abstract

In a thermal field theory, the cumulants of the momentum distribution can be extracted from the dependence of the Euclidean path integral on a shift in the fields built into the temporal boundary condition. When combined with the Ward identities associated with the invariance of the theory under the Poincaré group, thermodynamic potentials such as the entropy or the pressure can be directly inferred from the response of the system to the shift. Crucially the argument holds, up to harmless finite-size and discretization effects, even if translational and rotational invariance are broken to a discrete subgroup of finite shifts and rotations such as in a lattice box. The formulas are thus applicable at finite lattice spacing and volume provided the derivatives are replaced by their discrete counterpart, and no additive or multiplicative ultraviolet-divergent renormalizations are needed to take the continuum limit. In this paper we present a complete derivation of the relevant formulas in the scalar field theory, where several technical complications are avoided with respect to gauge theories. As a by-product we obtain a recursion relation among the cumulants of the momentum distribution, and formulae for finite-volume corrections to several well-known thermodynamic identities.

Equation of state of the SU(3) Yang–Mills theory: A precise determination from a moving frame

“…These obstacles are not rooted in the physics of the EoS, but in the method adopted for its computation. Recently there has been an intense activity to design new numerical strategies for simulating thermal field theories on the lattice and in particular to compute the EoS [8,9,10,11,12,13,14,15].…”

Energy-momentum tensor on the lattice: recent developments

“…[44,45] for a recent attempt), we will be able to use our universal formula to study physics related to EMT on a curved manifold. 9 Now, although our formula is expected to be universal, this universality holds only after the renormalization and sending the cutoff to infinity. With the lattice regularization, the universality hence holds only in the continuum limit; we have to first set a → 0 while t is kept fixed in physical unit and then take the t → 0 limit as Eq.…”

“…• Use physical normalization conditions implied by shifted boundary conditions; Z 6 can also be obtained [8,9,10,11,12,13,14,15].…”

“…Shifted boundary conditions [8,9,10,11,12,13,14,15] By employing boundary conditions with a spatial shift, one can derive physical normalization conditions which can be used to determine renormalization constants for EMT in Eq. (3.10), especially Z 6 for the off-diagonal elements.…”

QCD in a moving frame: an exploratory study

“…In this contribution we consider, for the first time, the framework of shifted boundary conditions (SBC) [10][11][12] applied to QCD. This set-up has been successfully applied to the case of SU(3) Yang-Mills theory [13][14][15][16][17], leading above all to a determination of the EoS over two orders of magnitude in the temperature with half a per-cent accuracy [14,16,17].…”