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
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
Multiloop pseudofermion functional renormalization for quantum spin systems: Application to the spin-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-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. 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.
Abstract For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.
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.
For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo fermion PF and pseudo Majorana PM functional renormalization group FRG and their specific ability to address higher dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions
For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.
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 J1>0 and second-neighbor J2>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.