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arXiv:cond-mat/0703586 · Physical Review B

Contractor-renormalization approach to frustrated magnets in a magnetic field

A. Abendschein, Sylvain Capponi

Published Aug 13, 2007• 14 citations
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Abstract

We propose to use the contractor-renormalization (CORE) technique in order to derive effective models for quantum magnets in a magnetic field. CORE is a powerful nonperturbative technique that can reduce the complexity of a given microscopic model by focusing on the low-energy part. We provide a detailed analysis of frustrated spin ladders, which have been widely studied in the past: in particular, we discuss how to choose the building block and emphasize the use of their reduced density matrix. With a good choice of basis, CORE is able to reproduce the existence or not of magnetization plateaux in the whole phase diagram contrary to a usual perturbation theory. We also address the issue of plateau formation in two-dimensional bilayers and point out the analogy between nonfrustrated strongly anisotropic models and frustrated SU(2) ones.

DOI
10.1103/physrevb.76.064413

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Numerical methods for frustrated magnetism

Maria Laura Baez

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$_3$.

•Jan 1, 2018•5 citations
We propose to use the contractor-renormalization (CORE) technique in order to derive effective models for quantum magnets in a magnetic field. CORE is a powerful nonperturbative technique that can reduce the complexity of a given microscopic model by focusing on the low-energy part. We provide a detailed analysis of frustrated spin ladders, which have been widely studied in the past: in particular, we discuss how to choose the building block and emphasize the use of their reduced density matrix. With a good choice of basis, CORE is able to reproduce the existence or not of magnetization plateaux in the whole phase diagram contrary to a usual perturbation theory. We also address the issue of plateau formation in two-dimensional bilayers and point out the analogy between nonfrustrated strongly anisotropic models and frustrated SU(2) ones.
We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagom\'e lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagom\'e lattice we find that the previously proposed twofold degenerate S=1∕2S=1∕2S=1∕2 basis can account for a large number of phenomena of the spin 1∕21∕21∕2 kagom\'e system. For spin 3∕23∕23∕2, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice for the plaquette lattice or a 48-site kagom\'e lattice, which are beyond the possibilities of a standard exact diagonalization approach.
A review of the Contractor Renormalization (CORE) method, as a systematic derivation of the low energy effective hamiltonian, is given, with emphasis on its differences and advantages over traditional perturbative (weak/strong links) real space RG. For the low energy physics of the 2D Hubbard model, we derive the plaquette bosons (projected SO(5)) model which connects the microscopic model to phases and phenomenology of high‐Tc cuprates. For the S = 1/2 Pyrochlore and Kagomé antiferromagnets, the effective hamiltonians predict spin‐disordered, lattice symmetry breaking, ground states with a large density of low energy singlets as found by exact diagonalization of small clusters.
We use the nonperturbative Contractor-Renormalization method (CORE) in order to derive an effective model for triplet excitations on the Shastry-Sutherland lattice. For strong enough magnetic fields, various magnetization plateaux are observed, e.g., at 1/8, 1/4, 1/3 of the saturation, as found experimentally in a related compound. Moreover, other stable plateaux are found at 1/9, 1/6, or 2/9. We give a critical review of previous works and try to resolve some apparent inconsistencies between various theoretical approaches.
The magnetic phase diagram of a plaquette dimerized and frustrated antiferromagnetic ladder system is studied by using a combination of numerical and analytical techniques. For the strongly frustrated regime, series expansions and bond operator techniques are employed to analyze zero magnetic field, whereas low-energy effective models are used to study the complete magnetization process. The interplay between frustration and dimerization gives rise to a rich plateau structure that is captured by effective models and corroborated by numerical density matrix renormalization group simulations, in particular, the emergence of intermediate plateaus at M=14M=\frac{1}{4}M=41​ and 34\frac{3}{4}43​ of saturation in the magnetization curve is clarified.
We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied in two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut‐off using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite‐size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semi‐quantitative numerical method. For ladder type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results both in the doped and undoped case are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagomé lattice as non‐trivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagomé lattice we find that the previously proposed twofold degenerate S = 1/2 basis can account for a large number of phenomena of the spin 1/2 kagomé system and gives a good starting point to study the doped case.
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​.