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arXiv:cond-mat/0106090 · Journal of Physics Condensed Matter

The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

Axel Gelfert, Wolfgang Nolting

Published Jun 25, 2001• 69 citations
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Abstract

After a brief discussion of the Bogoliubov inequality and possible generalizations thereof, we present a complete review of results concerning the Mermin-Wagner theorem for various many-body systems, geometries and order parameters. We extend the method to cover magnetic phase transitions in the periodic Anderson model as well as certain superconducting pairing mechanisms for Hubbard films. The relevance of the Mermin-Wagner theorem to approximations in many-body physics is discussed on a conceptual level.

DOI
10.1088/0953-8984/13/27/201

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After a brief discussion of the Bogoliubov inequality and possible generalizations thereof, we present a complete review of results concerning the Mermin-Wagner theorem for various many-body systems, geometries and order parameters. We extend the method to cover magnetic phase transitions in the periodic Anderson model as well as certain superconducting pairing mechanisms for Hubbard films. The relevance of the Mermin-Wagner theorem to approximations in many-body physics is discussed on a conceptual level.
In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature T_c T c . The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green’s function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement. The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the \chi χ -sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high- T_c T c cuprate superconductors.
We make a short overview of the recent analytic and numerical studies of the classical two-dimensional XY and Heisenberg models at low temperatures. Special attention is being paid to an influence of finite system size L on the peculiarities of the low-temperature phase. In accordance with the Mermin-Wagner-Hohenberg theorem, spontaneous magnetisation does not appear in the above models at infinite L. However it emerges for the finite system sizes and leads to new features of the low-temperature behaviour.
The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at T>0T>0T>0 in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$ with a sufficiently large exponent $\ensuremath{\alpha}$. For oscillatory interactions, ferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ or $\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}2D$.
This review is based on the authors' report at the Scientific Session of the Physical Sciences Division of the Russian Academy of Sciences (RAS).The review discusses the Berezinskii±Kosterlitz±Thouless (BKT) transition in two-dimensional systems with continuous symmetry, including the transition in two-dimensional magnetic systems (X ± Y-model), the transition to the superfluid state in 4 He thin films, and the superconducting transition in thin films.Scenarios of twodimensional melting in various systems are considered, including soft matter, vortex lattices in superconducting films, and skyrmion lattices.Particular attention is paid to deviations from the standard BKT scenario for various systems depending on the parameters characterizing the system: the type of potential, which is especially important for two-dimensional melting, and the energy of the core of a topological defect.