动力学平均场近似在量子场论中的应用
Dynamical Mean Field Approximation Applied to Quantum Field Theory
摘要：
我们将动力学平均场（DMFT）近似应用于实标量φ ^4量子场论。通过与晶格蒙特卡罗计算、微扰理论和标准平均场论的比较，我们在二维、三维、四维和五维上检验了近似的质量。在这些测试中考虑的量是过渡到有序相的临界耦合和相关的临界指数nu和beta。我们还绘制了四维相图。在二维和三维空间中，DMFT错误地预测了所有裸四次耦合的一级相变，这是有问题的，因为点阵φ ^[4]理论的相变的二级性质对于取连续极限至关重要。然而，通过外推远离相变的行为，可以获得临界耦合和临界指数。它们与平均场论的结果不同，并且更接近正确值。在四个维度的过渡是二阶的小四次耦合和转弱的一阶耦合增加超过三临界值。在5维和更高的，DMFT给出了定性正确的结果，预测合理的值的临界指数和更准确的临界耦合比标准的平均场论。近似最适合于四次耦合的小值。我们调查的变化，从一阶到二阶过渡的DMFT的局部极限，这是计算少得多密集。我们还讨论了与非线性自洽方程求解器的收敛性以及在存在φ ^4相互作用的情况下使用傅立叶空间蒙特卡罗更新求解有效单中心模型有关的技术问题。
英文摘要：
We apply the Dynamical Mean Field (DMFT) approximation to the real, scalar phi^4 quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents nu and beta. We also map out the phase diagram in four dimensions. In two and three dimensions, DMFT incorrectly predicts a first order phase transition for all bare quartic couplings, which is problematic, because the second order nature of the phase transition of lattice phi^4-theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behaviour away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first to second order transition in the local limit of DMFT which is computationally much less intensive. We also discuss technical issues related to the convergence of the non-linear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a phi^4-interaction.
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