Time discretization of functional integrals

Numerical evaluation of functional integrals usually involves a finite (Lslice) 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 ρ̂L(β) = [ρ̂1(β/L)] , if the density matrix ρ̂1(β) in the static approximation is known. We investigate the convergence of the partition function ZL(β) ≡ Tr ρ̂L(β), the internal energy and the density of states gL(E) (the inverse Laplace transform of ZL), as L → ∞. For the simple harmonic oscillator, gL(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-1/2 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, ZL 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 → ∞. PACS numbers: 05.30.-d, 31.15.Kb, 75.10.Jm Submitted to: J. Phys. A: Math. Gen. † Electronic address: [email protected] Time discretization of functional integrals 2