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Title:  Time discretization of functional integrals 
Authors:  Samson, J.H. 
Keywords:  quantum physics statistical mechanics 
Issue Date:  2000 
Abstract:  Numerical evaluation of functional integrals usually involves a finite (Lslice) discretization of the imaginarytime axis. In the auxiliaryfield method, the Lslice 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 auxiliaryfield 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 spin1/2 dimer are found in closed form for all L by appending a selfinteraction to the divergent Gaussian integral and analytically continuing to zero selfinteraction. 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 nonzero temperature, the exact partition function is recovered as $L\to\infty$. 
Description:  This is a preprint. It is also available at: http://arxiv.org/abs/quantph/0003109. The definitive version: SAMSON, 2000. Time discretization of functional integrals. Journal of Physics A: Mathematical and General, 33, 31113120, is available at: http://www.iop.org/EJ/journal/JPhysA. 
URI:  https://dspace.lboro.ac.uk/2134/1228 
Appears in Collections:  PrePrints (Physics)

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