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arXiv:1712.02535 · Physical review. B./Physical review. B

Effects of two-loop contributions in the pseudofermion functional renormalization group method for quantum spin systems

Marlon Rück, Johannes Reuther

Published Apr 9, 2018• 29 citations
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Abstract

We implement an extension of the pseudofermion functional renormalization group method for quantum spin systems that takes into account two-loop diagrammatic contributions. An efficient numerical treatment of the additional terms is achieved within a nested graph construction which recombines different one-loop interaction channels. In order to be fully self-consistent with respect to self-energy corrections, we also include certain three-loop terms of Katanin type. We first apply this formalism to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and benchmark our results against the previous one-loop plus Katanin approach. Even though the renormalization group (RG) equations undergo significant modifications when including the two-loop terms, the magnetic phase diagram, comprising N\'eel ordered and collinear ordered phases separated by a magnetically disordered regime, remains remarkably unchanged. Only the boundary position between the disordered and the collinear phases is found to be moderately affected by two-loop terms. On the other hand, critical RG scales, which we associate with critical temperatures ${T}_{\text{c}}$, are reduced by a factor of $\ensuremath{\sim}2$ indicating that the two-loop diagrams play a significant role in enforcing the Mermin-Wagner theorem. Improved estimates for critical temperatures are also obtained for the Heisenberg ferromagnet on the three-dimensional simple cubic lattice where errors in ${T}_{\text{c}}$ are reduced by $\ensuremath{\sim}34%$. These findings have important implications for the quantum phase diagrams calculated within the previous one-loop plus Katanin approach which turn out to be already well converged.

DOI
10.1103/physrevb.97.144404

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We implement an extension of the pseudofermion functional renormalization group method for quantum spin systems that takes into account two-loop diagrammatic contributions. An efficient numerical treatment of the additional terms is achieved within a nested graph construction which recombines different one-loop interaction channels. In order to be fully self-consistent with respect to self-energy corrections, we also include certain three-loop terms of Katanin type. We first apply this formalism to the antiferromagnetic J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model on the square lattice and benchmark our results against the previous one-loop plus Katanin approach. Even though the renormalization group (RG) equations undergo significant modifications when including the two-loop terms, the magnetic phase diagram, comprising N'eel ordered and collinear ordered phases separated by a magnetically disordered regime, remains remarkably unchanged. Only the boundary position between the disordered and the collinear phases is found to be moderately affected by two-loop terms. On the other hand, critical RG scales, which we associate with critical temperatures Tc{T}_{\text{c}}Tc​, are reduced by a factor of \ensuremath∼2\ensuremath{\sim}2\ensuremath∼2 indicating that the two-loop diagrams play a significant role in enforcing the Mermin-Wagner theorem. Improved estimates for critical temperatures are also obtained for the Heisenberg ferromagnet on the three-dimensional simple cubic lattice where errors in Tc{T}_{\text{c}}Tc​ are reduced by \ensuremath∼34\ensuremath{\sim}34%\ensuremath∼34. These findings have important implications for the quantum phase diagrams calculated within the previous one-loop plus Katanin approach which turn out to be already well converged.
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 employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by ∼5%\sim5\%∼5% and we further confirm the established values for the critical exponent ν\nuν and the anomalous dimension η\etaη. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T→0T\rightarrow0T→0. While extensions towards higher loop orders ℓ\ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ\ellℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.
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.
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Multiloop pseudofermion functional renormalization for quantum spin systems: Application to the spin-12\frac{1}{2}21​ kagome Heisenberg model
We present a multiloop pseudofermion functional renormalization group (pffRG) approach to quantum spin systems. As a test case, we study the spin-12\tfrac{1}{2}21​ Heisenberg model on the kagome lattice, a prime example of a geometrically frustrated magnet believed to host a quantum spin liquid. Our main physical result is that, at pure nearest-neighbor coupling, the system shows indications for an algebraic spin liquid through slower-than-exponential decay with distance for the static spin susceptibility, while the pseudofermion self-energy develops intriguing low-energy features. Methodologically, the pseudofermion representation of spin models inherently yields a strongly interacting system, and the quantitative reliability of a truncated fRG flow is \textit{a priori} unclear. Our main technical result is the demonstration of convergence in loop number within multiloop pffRG. Through correspondence with the self-consistent parquet equations, this provides further evidence for the internal consistency of the approach. The loop order required for convergence of the pseudofermion vertices is rather large, but the spin susceptibility is more benign, appearing almost fully converged for loop orders ℓ≥5\ell \geq 5ℓ≥5. The multiloop flow remains stable as the infrared cutoff ΛΛΛ is reduced relative to the microscopic exchange interaction JJJ, allowing us to reach values of Λ/JΛ/JΛ/J on the subpercent level in the spin-liquid phase. By contrast, solving the parquet equations via direct fixed-point iteration becomes increasingly difficult for low Λ/JΛ/JΛ/J. We also scrutinize the pseudofermion constraint of single occupation per site, which is only fulfilled on average in pffRG, by explicitly computing fermion-number fluctuations. Although the latter are not entirely suppressed, we find that they do not affect the qualitative conclusions drawn from the spin susceptibility.