Numerical Finite-Size Calculations

Deepak Iyer, Mark Srednicki, Marcos Rigol

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.

Deepak Iyer, Mark Srednicki, Marcos Rigol

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 for translationally invariant systems in phases without long range order at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that scale as the inverse length scale of the system. 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.

A. P. Young

Djalma Medeiros, G. G. Cabrera

Highly precise numerical calculations using a variation of the Lanczos method devised by Paige were performed for s=1/2 antiferromagnetic Heisenberg finite chains up to sizes N=28. Several interesting physical quantities, including the ground-state energy, the mass gap, and the spin-wave velocity, were computed and fitted for logarithmic finite-size corrections, as suggested by conformal invariance and Bethe ansatz calculations. A crossover size for a maximum of the spin-wave velocity is predicted for N\ensuremath{\sim}50, well above the current available sizes in a typical Lanczos simulation. A list of our numerical results is given along with extrapolated values using standard algorithms.