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arXiv:cond-mat/0503076

Quantum Phase Transitions of Frustrated Heisenberg Antiferromagnets: a Comprehensive Renormalization Approach

Frank Krüger, Scheidl, Stefan

Published Mar 3, 2005• 0 citations
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Abstract

We develop a novel renormalization approach for frustrated quantum antiferromagnets. It is designed to consistently treat spin-wave interactions all over the magnetic Brillouin zone, including high-energy modes in outer regions as well as low energy modes in the center. Focusing on the paradigmatic $J_1$-$J_2$ model, we find a unifying description of the second-order transition between the Néel phase and the paramagnetic phase and the first-order transition between the Néel phase and the columnar phase. Our approach provides explicit results for the renormalized values of the spin stiffness and spin-wave velocity characterizing the low-energy magnons in the Néel phase.

DOI
10.48550/arxiv.cond-mat/0503076

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Quantum phase transition at the saturation field is studied for a class of frustrated quantum antiferromagnets. The considered models include (i) the ${J}_{1}\mathrm{\text{\ensuremath{-}}}{J}_{2}$ frustrated square-lattice antiferromagnet with ${J}_{2}=\frac{1}{2}{J}_{1}$ and (ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic lattice. In the fully saturated phase the magnon spectra for the two models have lines of degenerate minima. Transition into a partially magnetized state is treated via a mapping to a dilute gas of hard-core bosons and by complementary spin-wave calculations. Momentum dependence of the exact four-point boson vertex removes the degeneracy of the single-particle excitation spectra and selects the ordering wave vectors at $(\ensuremath{\pi},\ensuremath{\pi})$ and $(\ensuremath{\pi},0,0)$ for the two models. We predict a unique form for the magnetization curve $\ensuremath{\Delta}M=S\ensuremath{-}M\ensuremath{\simeq}{\ensuremath{\mu}}^{(d\ensuremath{-}1)/2}(\mathrm{log}\ensuremath{\mu}{)}^{(d\ensuremath{-}1)}$, where $\ensuremath{\mu}$ is a distance from the quantum critical point.

arXiv:cond-mat/0312076•Physical Review Letters•Jul 2, 2004•28 citations
We develop a novel renormalization approach for frustrated quantum antiferromagnets. It is designed to consistently treat spin-wave interactions all over the magnetic Brillouin zone, including high-energy modes in outer regions as well as low energy modes in the center. Focusing on the paradigmatic J1J_1J1​-J2J_2J2​ model, we find a unifying description of the second-order transition between the Néel phase and the paramagnetic phase and the first-order transition between the Néel phase and the columnar phase. Our approach provides explicit results for the renormalized values of the spin stiffness and spin-wave velocity characterizing the low-energy magnons in the Néel phase.
We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the ${J}_{1}{\ensuremath{-}J}_{2}$ model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the N\'eel state to the spin liquid state at ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38,$ and the first-order transition from the spin liquid state to the collinear state at ${(J}_{2}{/J}_{1}{)}_{c4}\ensuremath{\approx}0.60.$ We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at ${(J}_{2}{/J}_{1}{)}_{c3}=0.50\ifmmode\pm\else\textpm\fi{}0.02,$ and the transition from the simple N\'eel state to the N\'eel state with spin columnar dimerization at ${(J}_{2}{/J}_{1}{)}_{c1}=0.34\ifmmode\pm\else\textpm\fi{}0.04.$ We also present an independent calculation of ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38$ using another approach.
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.
We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with LLL === 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1{J}_{2}/{J}_{1}J2​/J1​ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high J2/J1{J}_{2}/{J}_{1}J2​/J1​, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.
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.
Quantum phase transition at the saturation field is studied for a class of frustrated quantum antiferromagnets. The considered models include (i) the ${J}_{1}\mathrm{\text{\ensuremath{-}}}{J}_{2}$ frustrated square-lattice antiferromagnet with J2=12J1{J}_{2}=\frac{1}{2}{J}_{1}J2​=21​J1​ and (ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic lattice. In the fully saturated phase the magnon spectra for the two models have lines of degenerate minima. Transition into a partially magnetized state is treated via a mapping to a dilute gas of hard-core bosons and by complementary spin-wave calculations. Momentum dependence of the exact four-point boson vertex removes the degeneracy of the single-particle excitation spectra and selects the ordering wave vectors at $(\ensuremath{\pi},\ensuremath{\pi})$ and $(\ensuremath{\pi},0,0)$ for the two models. We predict a unique form for the magnetization curve $\ensuremath{\Delta}M=S\ensuremath{-}M\ensuremath{\simeq}{\ensuremath{\mu}}^{(d\ensuremath{-}1)/2}(\mathrm{log}\ensuremath{\mu}{)}^{(d\ensuremath{-}1)}$, where $\ensuremath{\mu}$ is a distance from the quantum critical point.