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Journal of Physics Conference Series

A diagrammatic theory of the antiferromagnetic frustrated 2d Heisenberg model

Johannes Reuther, P. Wölfle

Published Jan 1, 2010• 5 citations
Publisher

Abstract

We consider the ground state properties of the two-dimensional spin-1/2 J 1 - J 2 -Heisenberg model on a square lattice. Within a diagrammatic formulation in terms of auxiliary fermions we consider two different approximation schemes. First we use a phenomenological assumption on the width of the pseudofermion spectral function to calculate the magnetization and the correlation length within RPA with qualitatively correct results. Secondly we employ a Functional Renormalization Group formulation to obtain quantitative results on the ground state correlations. We discuss two different projection schemes to account for the auxiliary particle constraint.

DOI
10.1088/1742-6596/200/2/022051

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We consider the ground state properties of the two-dimensional spin-1/2 J 1 - J 2 -Heisenberg model on a square lattice. Within a diagrammatic formulation in terms of auxiliary fermions we consider two different approximation schemes. First we use a phenomenological assumption on the width of the pseudofermion spectral function to calculate the magnetization and the correlation length within RPA with qualitatively correct results. Secondly we employ a Functional Renormalization Group formulation to obtain quantitative results on the ground state correlations. We discuss two different projection schemes to account for the auxiliary particle constraint.
We study the ground state properties of the two-dimensional spin-1/2 J_1-J_2-Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary fermion formulation with exact projection. In a first approximation we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities and the spin correlation length within RPA, demonstrating the appearance of a paramagnetic phase between the Neel ordered and Collinear ordered phases, at sufficiently large pseudo fermion damping. Secondly we use a Functional Renormalization Group formulation. We find that the conventional truncation scheme omitting three-particle and higher order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find Neel order at g = J_2/J_1 0.66 ... 0.68, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase we find enhanced columnar dimer and plaquette fluctuations of equal strength.
We study the ground-state properties of the two-dimensional spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary-fermion formulation with exact projection. In a first approximation, we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities, and the spin-correlation length within random-phase approximation, demonstrating the appearance of a paramagnetic phase between the N\'eel-ordered and Collinear-ordered phases, at sufficiently large pseudofermion damping. Second we use a functional renormalization-group formulation. We find that the conventional truncation scheme omitting three-particle and higher-order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find N\'eel order at $g={J}_{2}/{J}_{1}\ensuremath{\lesssim}{g}_{\text{c}1}\ensuremath{\approx}0.4\dots{}0.45$ and Collinear order at $g\ensuremath{\gtrsim}{g}_{\text{c}2}\ensuremath{\approx}0.66\dots{}0.68$, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase, we find enhanced columnar dimer and plaquette fluctuations of equal strength.
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 diagrammatic approximations to the spin dynamics of the 2d Heisenberg antiferromagnet for all temperatures, employing an auxiliary-fermion representation. The projection onto the physical subspace is effected by introducing an imaginary-valued chemical potential as proposed by Popov and Fedotov. The method requires that the fermion number at any lattice site is strictly conserved. We compare results obtained within a self-consistent approximation using two different auxiliary-particle projection schemes, (1) exact and (2) on average. Significant differences between the two are found at higher temperatures, whereas in the limit of zero temperature (approaching the magnetically ordered ground state) identical results emerge from (1) and (2), providing the qualitatively correct dynamical scaling behavior. An interpretation of these findings is given. We also present in some detail the derivation of the approximation, which goes far beyond mean-field theory and is formulated in terms of complex-valued spectral functions of auxiliary fermions.
We analyze the thermodynamic properties of the spin-SSS two-dimensional quantum Heisenberg antiferromagnet on a square lattice with nearest and next-nearest neighbor couplings in the N\'eel phase (J2∕J1<0.4)({J}_{2}∕{J}_{1}<0.4)(J2​∕J1​<0.4) employing the quantum hierarchical reference theory, a nonperturbative implementation of the renormalization group method to quantum systems. We investigate the staggered susceptibility, the structure factors, and the correlation length at finite temperature and for different values of the frustration ratio. From the finite temperature results, we also extrapolate ground state properties, such as spin stiffness and spontaneous staggered magnetization, providing an estimate of the extent of quantum corrections. The behavior of these quantities as a function of frustration may provide some hints on the breakdown of the N\'eel phase at zero temperature for larger values of J2{J}_{2}J2​.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.