SciPost Phys. 10, 007 (2021) ·
published 13 January 2021
We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.
SciPost Phys. 8, 032 (2020) ·
published 2 March 2020
Based on the results published recently [SciPost Phys. 7, 026 (2019)], the
influence of surfaces and boundary fields are calculated for the ferromagnetic
anisotropic square lattice Ising model on finite lattices as well as in the
finite-size scaling limit. Starting with the open cylinder, we independently
apply boundary fields on both sides which can be either homogeneous or
staggered, representing different combinations of boundary conditions. We
confirm several predictions from scaling theory, conformal field theory and
renormalisation group theory: we explicitly show that anisotropic couplings
enter the scaling functions through a generalised aspect ratio, and demonstrate
that open and staggered boundary conditions are asymptotically equal in the
scaling regime. Furthermore, we examine the emergence of the surface tension
due to one antiperiodic boundary in the system in the presence of symmetry
breaking boundary fields, again for finite systems as well as in the scaling
limit. Finally, we extend our results to the antiferromagnetic Ising model.
SciPost Phys. 7, 026 (2019) ·
published 2 September 2019
We present detailed calculations for the partition function and the free
energy of the finite two-dimensional square lattice Ising model with periodic
and antiperiodic boundary conditions, variable aspect ratio, and anisotropic
couplings, as well as for the corresponding universal free energy finite-size
scaling functions. Therefore, we review the dimer mapping, as well as the
interplay between its topology and the different types of boundary conditions.
As a central result, we show how both the finite system as well as the scaling
form decay into contributions for the bulk, a characteristic finite-size part,
and - if present - the surface tension, which emerges due to at least one
antiperiodic boundary in the system. For the scaling limit we extend the proper
finite-size scaling theory to the anisotropic case and show how this anisotropy
can be absorbed into suitable scaling variables.