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arXiv:2011.01268

Multiloop pseudofermion functional renormalization for quantum spin systems: Application to the spin-$\frac{1}{2}$ kagome Heisenberg model

Julian Thoenniss, Marc K. Ritter, Fabian B. Kugler, Jan von Delft, Matthias Punk

Published Nov 2, 2020• 17 citations
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Abstract

We present a multiloop pseudofermion functional renormalization group (pffRG) approach to quantum spin systems. As a test case, we study the spin-$\tfrac{1}{2}$ 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 $\ell \geq 5$. The multiloop flow remains stable as the infrared cutoff $Λ$ is reduced relative to the microscopic exchange interaction $J$, allowing us to reach values of $Λ/J$ 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 $Λ/J$. 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.

DOI
10.48550/arxiv.2011.01268

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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.
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
Renormalization group methods are well-established tools for the (numerical) investigation of the low-energy properties of correlated quantum many-body systems, allowing us to capture their scale-dependent nature. The functional renormalization group (FRG) allows us to continuously evolve a microscopic model action to an effective low-energy action as a function of decreasing energy scales via an exact functional flow equation, which is then approximated by some truncation scheme to facilitate computation. Here, we report on our implementation of multiloop FRG, an extended truncation scheme recently developed for electronic FRG calculations, within the pseudofermion functional renormalization group (pf-FRG) framework for interacting quantum spin systems. We discuss in detail the conceptual intricacies of the flow equations generated by the multiloop truncation, as well as essential refinements to the integration scheme for the resulting integrodifferential equations. To benchmark our approach, we analyze antiferromagnetic Heisenberg models on the pyrochlore, simple cubic, and face-centered cubic lattice, discussing the convergence of physical observables for higher-loop calculations and comparing with existing results where available. Combined, these methodological refinements systematically improve the pf-FRG approach to one of the numerical tools of choice when exploring frustrated quantum magnetism in higher spatial dimensions.
The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to SU(N)\mathrm{SU}(N)SU(N)-spin systems with arbitrary NNN and demonstrate that the scheme becomes exact in the large-NNN limit. Numerically solving the generalized real-space renormalization group equations for arbitrary NNN, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible SU(N)\mathrm{SU}(N)SU(N) models. In a case study of the square-lattice SU(N)\mathrm{SU}(N)SU(N) Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of NNN. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for SU(N)\mathrm{SU}(N)SU(N) spin models that allows us to explicitly study the large-NNN limit and access the low-temperature spin liquid phase.
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.
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 $S\ensuremath{\rightarrow}\ensuremath{\infty}$. 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 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ 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 ${J}_{2}/{J}_{1}\text{\ensuremath{-}}S$ 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 $S\ensuremath{\rightarrow}\ensuremath{\infty}$, 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.
We enhance the pseudo-fermion functional-renormalisation-group (PFFRG) method for the investigation of highly frustrated quantum spin systems in two significant ways. On one hand, we generalise the method towards off-diagonal Dzyaloshinsky-Moriya (DM) interactions and single-ion anisotropies. On the other hand, we develop a self-consistent Fock-like mean-field technique which is based on vertex functions from a PFFRG analysis and calculates effective free spinon models for the considered systems at low energies. In Chap. 1, we introduce the fundamental mathematical concepts on which our studies rely. The so-called flow equations for the PFFRG are then derived in Chap. 3. The first part of our original work is presented in Chap. 4 where we, for the first time, extend the PFFRG scheme towards off-diagonal spin interactions and therefore confirm the quite general applicability of this method. There are two important results from our investigations of the spin-1/2 Heisenberg-DM model on the kagome lattice in this chapter. One is that a finite DM interaction enlarges magnetically ordered phases and thereby shrinks the paramagnetic phases which possibly house spin liquid states. The other important result is that the mineral herbertsmithite which inspired this project lies in close proximity to a quantum critical point. We obtain a phase diagram for the considered Hamiltonian and qualitatively good results in comparison to experimental data for herbertsmithite [68,69]. Our collaboration with a research group from Cologne is summarised in Chap. 5. Here, we investigate two different XXZ models with tetragonally split-up second-neighbour Heisenberg interactions and single-ion anisotropies on the diamond lattice in order to study the spin-1 A-site spinel NiRh2O4. We are able to confirm the existence of a paramagnetic phase for one of the considered models. However, such a paramagnetic behaviour cannot be verified for the Hamiltonians suggested in Refs. [29] and [30]. The second main part of this thesis is contained in Chap. 6. The developed PFFRG plus mean-field method enables us to characterise the effective low-energy spinon band structure of the spin-1/2 J1-J2 (nearest-neighbour) Heisenberg model on the square (kagome) lattice. This is achieved by comparing self-consistently determined overall amplitudes for different mean-field ansätze which are taken from corresponding PSG classifications in Refs. [182] and [111]. Our studies imply that the square lattice model realises a SU(2) spin liquid with Pi-fluxes through elementary plaquettes and that the spin-1/2 kagome antiferromagnet is most likely described by a Z2 spin liquid called Z2[0,Pi]Alpha. We argue that our analysed models have current experimental relevance and that the developed techniques show plenty of possibilities for future applications. A detailed summary and discussion of all methods and results is provided in Chap. 7.