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arXiv:cond-mat/9903379

Ordered Phase in the Fermionized Heisenberg Antiferromagnet

S. Azakov, M. Dilaver, Oztas, A. M.

Published Mar 25, 1999• 0 citations
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Abstract

Thermal properties of the ordered phase of the spin 1/2 isotropic Heisenberg Antiferromagnet on a d-dimensional hypercubical lattice are studied within the fermionic representation when the constraint of single occupancy condition is taken into account by the method suggested by Popov and Fedotov. Using saddle point approximation in path integral approach we discuss not only the leading order but also the fluctuations around the saddle point at one-loop level. The influence of taking into account the single occupancy condition is discussed at all steps.

DOI
10.48550/arxiv.cond-mat/9903379

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Thermal properties of the ordered phase of the spin 1/2 isotropic Heisenberg Antiferromagnet on a d-dimensional hypercubical lattice are studied within the fermionic representation when the constraint of single occupancy condition is taken into account by the method suggested by Popov and Fedotov. Using saddle point approximation in path integral approach we discuss not only the leading order but also the fluctuations around the saddle point at one-loop level. The influence of taking into account the single occupancy condition is discussed at all steps.
The nearest-neighbor quantum antiferromagnetic (AF) Heisenberg model for spin-1∕21∕21∕2 on a two-dimensional square lattice is studied in the auxiliary-fermion representation. Expressing spin operators by canonical fermionic particles requires a constraint on the fermion charge Qi=1{Q}_{i}=1Qi​=1 on each lattice site iii, which is imposed approximately through the thermal average. The resulting interacting fermion system is first treated in mean-field theory (MFT), which yields an AF ordered ground state and spin waves in quantitative agreement with conventional spin-wave theory. At finite temperature a self-consistent approximation beyond mean field is required in order to fulfill the Mermin-Wagner theorem. We first discuss a fully self-consistent approximation, where fermions are renormalized due to fluctuations of their spin density, in close analogy to FLEX. While static properties like the correlation length, $\ensuremath{\xi}(T)\ensuremath{\propto}\mathrm{exp}(aJ∕T)$, come out correctly, the dynamical response lacks the magnon-like peaks which would reflect the appearance of short-range order at low TTT. This drawback, which is caused by overdamping, is overcome in a ``minimal self-consistent approximation'' (MSCA), which we derive from the equations of motion. The MSCA features dynamical scaling at small energy and temperature and is qualitatively correct both in the regime of order-parameter relaxation at long wavelengths $\ensuremath{\lambda}>\ensuremath{\xi}$ and in the short-range-order regime at $\ensuremath{\lambda}<\ensuremath{\xi}$. We also discuss the impact of vertex corrections and the problem of pseudo-gap formation in the single-particle density of states due to long-range fluctuations. Finally we show that the (short-range) magnetic order in MFT and MSCA helps to fulfill the constraint on the local fermion occupancy.
We present a restricted path integral approach to the two-dimensional and three-dimensional repulsive Hubbard model. In this approach the partition function is approximated by restricting the summation over all states to a (small) subclass that is chosen so as to well represent the important states. This procedure generalizes mean-field theory and can be systematically improved by including more states or fluctuations. We analyze in detail the simplest of these approximations, which corresponds to summing over states with local antiferromagnetic (AF) order. If in the states considered the AF order changes sufficiently little in space and time, the path integral becomes a finite-dimensional integral for which the saddle point evaluation is exact. This leads to generalized mean-field equations allowing for the possibility of more than one relevant saddle point. In a big parameter regime (both in temperature and filling), we find that this integral has two relevant saddle points, one corresponding to finite AF order and the other without. These degenerate saddle points describe a phase of AF ordered fermions coexisting with free, metallic fermions. We argue that this mixed phase is a simple mean-field description of a variety of possible inhomogeneous states, appropriate on length scales where these states appear homogeneous. We sketch systematic refinements of this approximation, which can give more detailed descriptions of the system.