dorsal/arxiv
View SchemaTime discretization of functional integrals
| Authors | J. H. Samson |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0003109 |
| URL | https://arxiv.org/abs/quant-ph/0003109 |
| DOI | 10.1088/0305-4470/33/16/305 |
| Journal | J Phys A: Math Gen 33, 3111-3120 (2000) |
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 $L\to\infty$.
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"abstract": "Numerical evaluation of functional integrals usually involves a finite\n(L-slice) discretization of the imaginary-time axis. In the auxiliary-field\nmethod, the L-slice approximant to the density matrix can be evaluated as a\nfunction of inverse temperature at any finite L as\n$\\rho_L(\\beta)=[\\rho_1(\\beta/L)]^L$, if the density matrix $\\rho_1(\\beta)$ in\nthe static approximation is known. We investigate the convergence of the\npartition function $Z_L(\\beta)=Tr\\rho_L(\\beta)$, the internal energy and the\ndensity of states $g_L(E)$ (the inverse Laplace transform of $Z_L$), as\n$L\\to\\infty$. For the simple harmonic oscillator, $g_L(E)$ is a normalized\ntruncated Fourier series for the exact density of states. When the\nauxiliary-field approach is applied to spin systems, approximants to the\ndensity of states and heat capacity can be negative. Approximants to the\ndensity matrix for a spin-1/2 dimer are found in closed form for all L by\nappending a self-interaction to the divergent Gaussian integral and\nanalytically continuing to zero self-interaction. Because of this continuation,\nthe coefficient of the singlet projector in the approximate density matrix can\nbe negative. For a spin dimer, $Z_L$ is an even function of the coupling\nconstant for L\u003c3: ferromagnetic and antiferromagnetic coupling can be\ndistinguished only for $L\\ge 3$, where a Berry phase appears in the functional\nintegral. At any non-zero temperature, the exact partition function is\nrecovered as $L\\to\\infty$.",
"arxiv_id": "quant-ph/0003109",
"authors": [
"J. H. Samson"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech"
],
"doi": "10.1088/0305-4470/33/16/305",
"journal_ref": "J Phys A: Math Gen 33, 3111-3120 (2000)",
"title": "Time discretization of functional integrals",
"url": "https://arxiv.org/abs/quant-ph/0003109"
},
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