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Numerical methods for frustrated magnetism

Maria Laura Baez

Published Jan 1, 2018• 5 citations
Publisher

Abstract

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, $S \geq 1/2$. 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 ($S = 1/2$) to the classical limit ($S \rightarrow \infty$), 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 $\alpha$-Li$_2$IrO$_3$, 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 $\alpha$-Li$_2$IrO$_3$.

DOI
10.17169/refubium-638

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

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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 $S=1/2$ 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.

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Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.

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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​.
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.
Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.
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 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.
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 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.