In this thesis, the functional renormalization group (FRG) method is applied to quantum spin chains. FRG is a well-known method in the treatment of correlated fermion systems, but the method has been applied to spin systems only recently[1]. The method has been used, however, only as a qualitative tool to find phase boundaries for frustrated spin systems with multiple competing interactions. The main goal of this work is to critically assess the general accuracy of the method by studying toy models, for which quantitative comparison with exact results is possible. After a full derivation of the flow equations for spin systems, the numerical solution of the resulting non-linear system of differential equations is presented. It is shown that in the antiferromagnetic two-site Heisenberg model, FRG can reproduce some known properties of the singlet state but the ground state energy, for example, disagrees with the exact result by 36 %. In addition, the quasiparticle self-energy and dynamic spin-spin correlation functions are of an incorrect form. Unphysical fluctuations in the particle number are shown to be large. In the antiferromagnetic Heisenberg chain, it is shown that whereas the use of properly truncated flow equations regularizes the unphysical antiferromagnetic instability, the resulting ground state is disordered and not critical as expected. [1] J. Reuther and P. Wölfle, Phys. Rev. B 81, 144410 (2010).
In this thesis, the functional renormalization group (FRG) method is applied to quantum spin chains. FRG is a well-known method in the treatment of correlated fermion systems, but the method has been applied to spin systems only recently[1]. The method has been used, however, only as a qualitative tool to find phase boundaries for frustrated spin systems with multiple competing interactions. The main goal of this work is to critically assess the general accuracy of the method by studying toy models, for which quantitative comparison with exact results is possible. After a full derivation of the flow equations for spin systems, the numerical solution of the resulting non-linear system of differential equations is presented. It is shown that in the antiferromagnetic two-site Heisenberg model, FRG can reproduce some known properties of the singlet state but the ground state energy, for example, disagrees with the exact result by 36 %. In addition, the quasiparticle self-energy and dynamic spin-spin correlation functions are of an incorrect form. Unphysical fluctuations in the particle number are shown to be large. In the antiferromagnetic Heisenberg chain, it is shown that whereas the use of properly truncated flow equations regularizes the unphysical antiferromagnetic instability, the resulting ground state is disordered and not critical as expected. [1] J. Reuther and P. Wölfle, Phys. Rev. B 81, 144410 (2010).
We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated J1J2J3 quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin vertices, which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most 4.
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 (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)-spin systems with arbitrary N and demonstrate that the scheme becomes exact in the large-N limit. Numerically solving the generalized real-space renormalization group equations for arbitrary N, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible SU(N) models. In a case study of the square-lattice 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 N. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for SU(N) spin models that allows us to explicitly study the large-N limit and access the low-temperature spin liquid phase.
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
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.
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account. In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0. Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic second-magnon regime.