Time discretization of functional integrals

Abstract

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 ${\rho }_L(\beta )=[{\rho }_1(\beta /L){]}^L$, if the density matrix ${\rho }_1(\beta )$ in the static approximation is known. We investigate the convergence of the partition function $Z_L(\beta )=Tr{\rho }_L(\beta )$, the internal energy and the density of states $g_L(E)$ (the inverse Laplace transform of $Z_L$), as $L\to \infty$. 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-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, $Z_L$ is an even function of the coupling constant for L<3: ferromagnetic and antiferromagnetic coupling can be distinguished only for $L\ge 3$, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as $L\to \infty$.

11 pages, to appear J Phys A: Math Gen. Uses iopart.cls

References in corpus (1)

• Total and Parity-Projected Level Densities of Iron-Region Nuclei in the Auxiliary Fields Monte Carlo Shell Model

Cited by in corpus (3)

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• Rényi entropy of quantum anharmonic chain at non-zero temperature
• Coherent-state path integral calculation of the Wigner function