2000/03/22 by J. H. Samson
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Coupling constant #Density matrix #Inverse #Mathematical analysis #Mathematical physics #Mathematics #Partition function (quantum field theory) #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Quantum, superfluid, helium dynamics #cond-mat.stat-mech #quant-ph
paper · pdf · doi:10.1088/0305-4470/33/16/305
published as J Phys A: Math Gen 33, 3111-3120 (2000) · 11 pages, to appear J Phys A: Math Gen. Uses iopart.cls
arxiv created 2000/03/22 · openalex publication_date 2000/04/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .