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Generalizations of Pseudo Majorana Functional Renormalization Group and its Application to Highly Frustrated Spin Systems

Schaden-Thillmann, Yannik

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Spin systems are known for their purely interacting nature, causing them to lack a canonical perturbative limit, which usually helps to understand the fundamental physics of a system. A method that has proven effective for the description of strongly correlated systems is Functional Renormalization Group (FRG), which can be thought of as an alternative to the path integral formalism as an approach to many-body quantum mechanics. The basic concept is to cure infrared divergences by introducing a cutoff to correlation functions. Continuously lowering the infrared cutoff and hence allowing the system to occupy lower-lying energy states is called the FRG flow and is formulated in terms of differential equations. The solutions to these equations are functions which describe effective interactions between particles and hence contain information about particle correlations and thermodynamic quantities. Formulating FRG in terms of spins turns out to be a subtle endeavor, though, as the spin algebra relations are relatively complicated compared to, for example, the canonical anti-commutation relations of fermions. In order to use the advantages of FRG for spin systems, one can map spins onto fermionic operators, which leads to pseudo-fermion FRG (PFFRG), developed by Reuther and Wölfle.[198] Alter- natively, one can express spins in terms of Majorana operators, leading to pseudo- Majorana FRG (PMFRG), which has been developed by Niggemann, Sbierski, and Reuther.[163] PMFRG is particularly powerful at finite temperatures, where useful properties of Majorana operators render the method more accurate than PFFRG. In contrast to many other methods, PFFRG and PMFRG are applicable to any spin system, even frustrated ones. In this work, PMFRG is being generalized and applied to highly frustrated spin systems. In particular, spin representations in terms of Majorana operators for spins with arbitrary large spin magnitude S are being investigated and classified thoroughly, which closes a gap in the literature about the second quantization of spin operators. Moreover, it is presented how to generalize PMFRG for Heisenberg models with full SU(2) symmetry to XXZ models exhibiting only a U(1) symmetry. This opens up a wide range of interesting applications for PMFRG. An example is the XXZ model on the pyrochlore lattice, which is known to exhibit conventional long-range order but also exotic spin liquid ground states in the so-called spin ice phase. PMFRG is applied to this model, and the results are being compared to experiments. Furthermore, the entire phase diagram of the XXZ model is being mapped out. Also, the capability of PMFRG to reproduce low-energy field theory predictions and to determine critical exponents is being tested.

DOI
10.17169/refubium-50193

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Spin systems are known for their purely interacting nature, causing them to lack a canonical perturbative limit, which usually helps to understand the fundamental physics of a system. A method that has proven effective for the description of strongly correlated systems is Functional Renormalization Group (FRG), which can be thought of as an alternative to the path integral formalism as an approach to many-body quantum mechanics. The basic concept is to cure infrared divergences by introducing a cutoff to correlation functions. Continuously lowering the infrared cutoff and hence allowing the system to occupy lower-lying energy states is called the FRG flow and is formulated in terms of differential equations. The solutions to these equations are functions which describe effective interactions between particles and hence contain information about particle correlations and thermodynamic quantities. Formulating FRG in terms of spins turns out to be a subtle endeavor, though, as the spin algebra relations are relatively complicated compared to, for example, the canonical anti-commutation relations of fermions. In order to use the advantages of FRG for spin systems, one can map spins onto fermionic operators, which leads to pseudo-fermion FRG (PFFRG), developed by Reuther and Wölfle.[198] Alter- natively, one can express spins in terms of Majorana operators, leading to pseudo- Majorana FRG (PMFRG), which has been developed by Niggemann, Sbierski, and Reuther.[163] PMFRG is particularly powerful at finite temperatures, where useful properties of Majorana operators render the method more accurate than PFFRG. In contrast to many other methods, PFFRG and PMFRG are applicable to any spin system, even frustrated ones. In this work, PMFRG is being generalized and applied to highly frustrated spin systems. In particular, spin representations in terms of Majorana operators for spins with arbitrary large spin magnitude S are being investigated and classified thoroughly, which closes a gap in the literature about the second quantization of spin operators. Moreover, it is presented how to generalize PMFRG for Heisenberg models with full SU(2) symmetry to XXZ models exhibiting only a U(1) symmetry. This opens up a wide range of interesting applications for PMFRG. An example is the XXZ model on the pyrochlore lattice, which is known to exhibit conventional long-range order but also exotic spin liquid ground states in the so-called spin ice phase. PMFRG is applied to this model, and the results are being compared to experiments. Furthermore, the entire phase diagram of the XXZ model is being mapped out. Also, the capability of PMFRG to reproduce low-energy field theory predictions and to determine critical exponents is being tested.
The pseudofermion functional renormalization group (PFFRG) method has proven to be a powerful numerical approach to treat frustrated quantum spin systems. In its usual implementation, however, the complex fermionic representation of spin operators introduces unphysical Hilbert-space sectors which render an application at finite temperatures inaccurate. In this work we formulate a general functional renormalization group approach based on Majorana fermions to overcome these difficulties. We, particularly, implement spin operators via an SO(3)\text{SO}(3)SO(3) symmetric Majorana representation which does not introduce any unphysical states and, hence, remains applicable to quantum spin models at finite temperatures. We apply this scheme, dubbed pseudo-Majorana functional renormalization group (PMFRG) method, to frustrated Heisenberg models on small spin clusters as well as square and triangular lattices. Computing the finite-temperature behavior of spin correlations and thermodynamic quantities such as free energy and heat capacity, we find good agreement with exact diagonalization and the high-temperature series expansion down to moderate temperatures. We observe a significantly enhanced accuracy of the PMFRG compared to the PFFRG at finite temperatures. More generally, we conclude that the development of functional renormalization group approaches with Majorana fermions considerably extends the scope of applicability of such methods.
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.
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 mmm-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 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.
This thesis is devoted to diagrammatic frameworks for computing the dynamic structure factor (DSF) and the static susceptibility of spin systems at finite temperature in the thermodynamic limit. We consider two complementary diagrammatic frameworks. First, we study functional renormalization group (FRG) approaches based on pseudo-fermion (pf-FRG) and pseudo-Majorana (pm-FRG) representations. Second, we develop high-order strong-coupling and high-temperature expansions (HTE) using free and restricted-graph techniques. To enable finite-temperature calculations in pf-FRG, we eliminate unphysical states by incorporating the Popov–Fedotov trick. Within pm-FRG, we introduce a temperature-flow scheme that significantly improves the efficiency and accuracy of finite-size scaling analyses of thermal phase transitions. We further extend pm-FRG to include external magnetic fields, allowing treatment of XXZ models with a field along the z direction. While the pseudo-Majorana representation maps local spin-1/2 operators to free Majorana fermions, higher spin values S>1/2 can instead be represented as strongly interacting real bosons, an idea underlying the recently proposed spin-FRG. Rather than implementing the spin-FRG itself, we analyze its perturbation series using a strong-coupling expansion. To evaluate the high-dimensional imaginary-time integrals arising in this expansion analytically with efficient bookkeeping, we introduce the kernel trick, which rewrites any Matsubara diagram in terms of the partial spectral functions of its constituent Green’s functions. The resulting fourth-order strong-coupling expansion for general spin S provides systematic beyond-mean-field corrections to critical temperatures and fields in both 1/d and large-interaction-range expansions. More importantly, it provides an analytic explanation of the long-standing quantum-to-classical correspondence observed in diagrammatic Monte Carlo simulations: the normalized static susceptibilities of quantum and classical Heisenberg models coincide after a simple temperature rescaling. Our analysis shows that this arises from the fact that the static susceptibility is well approximated by a simple functional form akin to a mean-field susceptibility, which we call the renormalized-mean-field form. To reach higher orders, we generalize the HTE to Matsubara-frequency-resolved Green’s functions (Dyn-HTE). This includes the static susceptibility (at zero Matsubara frequency) as a special case and validates the renormalized-mean-field form up to 12th order in perturbation theory. By analytically continuing the frequency dependence obtained from Dyn-HTE using a continued-fraction expansion, we compute DSFs for one-parameter Heisenberg models on arbitrary lattices, which can be directly compared with experimental inelastic neutron-scattering spectra. All Dyn-HTE calculations are implemented in a publicly available, well-documented open-source code. Finally, we clarify the relation between pseudo-Majorana and spin-based diagrammatic approaches by deriving the exact pseudo-Majorana action directly from the spin Hamiltonian, without invoking any parton representation. This derivation paves the way for extending Majorana-based methods to higher-spin systems.