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arXiv:1612.05074 · Physical review. B./Physical review. B

Numerical treatment of spin systems with unrestricted spin length S : A functional renormalization group study

Maria Laura Baez, Johannes Reuther

Published Jul 28, 2017• 66 citations
arXivHTMLPDF

Abstract

We develop a generalized pseudofermion functional renormalization group (PFFRG) approach that can be applied to arbitrary Heisenberg models with spins ranging from the quantum case $S=1/2$ to the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$. Within this framework, spins of magnitude $S$ are realized by implementing $M=2S$ copies of spin-1/2 degrees of freedom on each lattice site. We confirm that even without explicitly projecting onto the highest spin sector of the Hilbert space, ground states tend to select the largest possible local spin magnitude. This justifies the average treatment of the pseudofermion constraint in previous spin-1/2 PFFRG studies. We apply this method to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ honeycomb Heisenberg model with nearest-neighbor ${J}_{1}>0$ and second-neighbor ${J}_{2}>0$ interactions. Mapping out the phase diagram in the ${J}_{2}/{J}_{1}\text{\ensuremath{-}}S$ plane, we find that upon increasing $S$, quantum fluctuations are rapidly decreasing. In particular, already at $S=1$ we find no indication for a magnetically disordered phase. In the limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$, the known phase diagram of the classical system is exactly reproduced. More generally, we prove that for $S\ensuremath{\rightarrow}\ensuremath{\infty}$ the PFFRG approach is identical to the Luttinger-Tisza method.

DOI
10.1103/physrevb.96.045144

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We develop a generalized pseudofermion functional renormalization group (PFFRG) approach that can be applied to arbitrary Heisenberg models with spins ranging from the quantum case S=1/2S=1/2S=1/2 to the classical limit S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞. Within this framework, spins of magnitude SSS are realized by implementing M=2SM=2SM=2S copies of spin-1/2 degrees of freedom on each lattice site. We confirm that even without explicitly projecting onto the highest spin sector of the Hilbert space, ground states tend to select the largest possible local spin magnitude. This justifies the average treatment of the pseudofermion constraint in previous spin-1/2 PFFRG studies. We apply this method to the antiferromagnetic J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ honeycomb Heisenberg model with nearest-neighbor J1>0{J}_{1}>0J1​>0 and second-neighbor J2>0{J}_{2}>0J2​>0 interactions. Mapping out the phase diagram in the J2/J1\ensuremath-S{J}_{2}/{J}_{1}\text{\ensuremath{-}}SJ2​/J1​\ensuremath-S plane, we find that upon increasing SSS, quantum fluctuations are rapidly decreasing. In particular, already at S=1S=1S=1 we find no indication for a magnetically disordered phase. In the limit S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞, the known phase diagram of the classical system is exactly reproduced. More generally, we prove that for S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞ the PFFRG approach is identical to the Luttinger-Tisza method.
In this thesis, we will concentrate on the numerical study of classical and quantum frustrated magnets. If we focus on the full set of problems which are contained in the class of quantum spin systems, we see that the techniques which can deal with them are plenty, but that the presence of frustration introduces several methodological hindrances. Whether it is the presence of a sign problem, or the extensive resources needed to perform the computation, many of these methods are efficient only on a reduced subset of frustrated systems. We will employ the pseudofermion-Functional Renormalization Group (pfFRG) method as our method of choice to tackle two dimensional frustrated quantum magnets, all of them modeled via a Heisenberg Hamiltonian. We will extend the pfFRG formalism to include spin systems where the spin length is unrestricted, S≥1/2S \geq 1/2S≥1/2. With this extension we are able to study in detail what the effect of quantum fluctuations are on the system. From the extreme quantum limit (S=1/2S = 1/2S=1/2) to the classical limit (S→∞S \rightarrow \inftyS→∞), the large-S extension of pfFRG allows us to manipulate the strength of the quantum fluctuations and study in detail how the quantum to classical transition happens. We apply this method to the study of incommensurate phases in the Heisenberg honeycomb model. We map the phase diagram for different values of the spin length and analyze how this length, and subsequently quantum fluctuations, affect the phases found in the classical limit. Furthermore, we prove that in the classical limit, pfFRG reduces to the Luttinger-Tisza formalism. When quantum fluctuations can be neglected, we model our frustrated magnet as a classical spin system. While this limit can easily be obtained from the large-S extension of pfFRG, this methodology is constrained to the study of two point correlators at zero temperature. To collect information regarding the behavior of our classical magnet beyond the two point correlators we study classical spin systems via Monte Carlo simulations. In this case we show how the method can be applied to continuous spin systems with strong anisotropic interaction. Furthermore, we employ this technique to numerically study α\alphaα-Li2_22​IrO3_33​, a material which exhibits an incommensurate ground state but for which a minimal model has not been determined. We study and compare the possible minimal models that have been proposed. We reduce the number of minimal models by showing that many of those that have been proposed do not reproduce the full set of experimental results. Furthermore, we predict the magnetic behavior (for those models which reproduce the experimental results) in the presence of an external magnetic field. With this study we obtain magnetization processes and propose experiments which can confirm if one of the studied models is the correct minimal model for α\alphaα-Li2_22​IrO3_33​.
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.
We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated J1J2J3{J}_{1}{J}_{2}{J}_{3}J1​J2​J3​ quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin vertices, which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most 44%4.
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.
Abstract The pseudofermion functional renormalization group (pffRG) is a computational method for determining zero-temperature phase diagrams of frustrated quantum magnets. In a recent methodological advance, the commonly employed Katanin truncation of the flow equations was extended to include multiloop corrections, thereby capturing additional contributions from the three-particle vertex (Thoenniss et al. https://arxiv.org/abs/2011.01268 ; Kiese et al. https://arxiv.org/abs/2011.01269 ). This development has also stimulated significant progress in the numerical implementation of pffRG, allowing one to track the evolution of pseudofermion vertices under the renormalization group flow with unprecedented accuracy. However, cutting-edge solvers differ in their integration algorithms, heuristics to discretize Matsubara frequency grids, and more. To lend confidence in the numerical robustness of state-of-the-art multiloop pffRG codes, we present and compare results produced with two independently developed and algorithmically distinct solvers for Heisenberg models on three-dimensional lattice geometries. Using the cubic lattice Heisenberg (anti)ferromagnet with nearest and next-nearest neighbor interactions as a generic benchmark model, we find the two codes to quantitatively agree, often up to several orders of magnitude in digital precision, both on the level of spin-spin correlation functions and renormalized fermionic vertices for varying loop orders. These benchmark calculations further substantiate the usage of multiloop pffRG solvers to tackle unconventional forms of quantum magnetism. Graphic abstract
We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon Ans\"atze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, we find that in the regime of maximal frustration a SU(2) $\ensuremath{\pi}$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid with a small circular spinon Fermi surface and approximate Dirac cones at low but finite energies.