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Untersuchungen von konkurrierenden Ordnungen im Quadratgitter mittels funktionaler Renormierungsgruppe

Julian Lichtenstein

Published Jan 1, 2018• 3 citations
Publisher

Abstract

In this thesis a novel computational approach for functional Renormalization Group (fRG) investigations regarding interacting fermions on two-dimensional lattices is derived. The approach is based on the exchange-parametrization fRG [C. Husemann and M. Salmhofer. Phys. Rev. B, 79:195125.] concerning the two-fermion interaction, with additional insertions of truncated partitions of unity that are inspired by the singular-mode fRG [W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, et al.. Phys. Rev. B, 85:035414.]. These insertions decouple the fermionic propagators from the exchange propagators and thereby lead to a separation of the underlying equations. It is argued that this separation is numerically advantageous and may pave the way for refined, large-scale computational fRG investigations. Furthermore, on the basis of speedup data gained from an implementation of the novel scheme, it is shown that the new equation structure facilitates efficient calculations on a large number of multi-core CPUs. Utilizing the methodical development, two applications are presented afterwards: Firstly, the scheme is applied to the t-t' Hubbard model on a square lattice. In order to test the method, the convergence of the results with respect to increasing truncation length of the partition of unity is analyzed and a comparison to findings from previous fRG studies is drawn. The test yields a fast convergence in most parts of the investigated parameter range. Moreover, it was found that the older findings are approximately reproduced using a short truncation length, while quantitative corrections to the previous results are obtained using longer truncation lengths. In addition to these method focused investigations, the high momentum-space resolution of the implementation was utilized to study ordering vectors of antiferromagnetic phases. A comparison to calculations solely based on the magnetic channel revealed that the inter-channel feedback, as contained in the fRG, generates support for commensurate antiferromagnetic orderings. The second application addresses a modified version of the previous model, in which a Coulomb-like long-range interaction term is added to the local Hubbard interaction. In this work a solution is presented on how the long-range interaction can be implemented in a momentum-space fRG scheme without encountering singularities. Afterwards, phase diagrams are given that compare the results of calculations using short- and long-ranged interactions, showing significantly lower critical scales in the long-range scenario. Finally, screening lengths are extracted from the fRG outcome and are compared to those calculated in a 'random phase approximation'. The comparison reveals quantitative differences between both values. However, the results indicate that the charge screening is not changed qualitatively by the feedback from other channels, i.e., at van Hove filling the screening length tends to zero in both cases.

DOI
10.18154/rwth-2018-225781

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In this thesis a novel computational approach for functional Renormalization Group (fRG) investigations regarding interacting fermions on two-dimensional lattices is derived. The approach is based on the exchange-parametrization fRG [C. Husemann and M. Salmhofer. Phys. Rev. B, 79:195125.] concerning the two-fermion interaction, with additional insertions of truncated partitions of unity that are inspired by the singular-mode fRG [W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, et al.. Phys. Rev. B, 85:035414.]. These insertions decouple the fermionic propagators from the exchange propagators and thereby lead to a separation of the underlying equations. It is argued that this separation is numerically advantageous and may pave the way for refined, large-scale computational fRG investigations. Furthermore, on the basis of speedup data gained from an implementation of the novel scheme, it is shown that the new equation structure facilitates efficient calculations on a large number of multi-core CPUs. Utilizing the methodical development, two applications are presented afterwards: Firstly, the scheme is applied to the t-t' Hubbard model on a square lattice. In order to test the method, the convergence of the results with respect to increasing truncation length of the partition of unity is analyzed and a comparison to findings from previous fRG studies is drawn. The test yields a fast convergence in most parts of the investigated parameter range. Moreover, it was found that the older findings are approximately reproduced using a short truncation length, while quantitative corrections to the previous results are obtained using longer truncation lengths. In addition to these method focused investigations, the high momentum-space resolution of the implementation was utilized to study ordering vectors of antiferromagnetic phases. A comparison to calculations solely based on the magnetic channel revealed that the inter-channel feedback, as contained in the fRG, generates support for commensurate antiferromagnetic orderings. The second application addresses a modified version of the previous model, in which a Coulomb-like long-range interaction term is added to the local Hubbard interaction. In this work a solution is presented on how the long-range interaction can be implemented in a momentum-space fRG scheme without encountering singularities. Afterwards, phase diagrams are given that compare the results of calculations using short- and long-ranged interactions, showing significantly lower critical scales in the long-range scenario. Finally, screening lengths are extracted from the fRG outcome and are compared to those calculated in a 'random phase approximation'. The comparison reveals quantitative differences between both values. However, the results indicate that the charge screening is not changed qualitatively by the feedback from other channels, i.e., at van Hove filling the screening length tends to zero in both cases.
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Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.
In this thesis we apply the functional renormalisation group (fRG) to the two-dimensional Hubbard model (2D-HM). We do this by solving approximate renormalisation group equations for the self energy and the two-particle interaction numerically. The purpose of these calculations is two-fold in the sense that we want to learn about the reliability of the method on the one hand, and about the physics of the 2D-HM on the other. We find that different fRG schemes agree in describing the mutual interplay and the resulting competition between different correlations, but also that the results sometimes differ quantitatively. This sharpens our perception when it comes to interpreting fRG results, and tells us that we have to separate the indicative character from the quantitative aspects of the results. The main line of arguments presented in previous fRG studies is confirmed, that in the weak-coupling region particle-hole correlations feed back into the particle-particle channel, triggering strong superconducting correlations in a non-zero parameter range. Concerning the physics of the 2D-HM show new results for the single-particle spectral functions for the case of a Fermi surface which is not perfectly nested and intersects the Umklapp surface at so-called hot spots. These results are obtained from the Wick-ordered fRG scheme, and the calculations are done directly on the real frequency axis. We find that close to a critical temperature TcT_cTc​ where correlations become strong, a dip appears in the single-particle spectral functions for momenta close to and at the hot spots. Away from the van Hove scale, this is traced back to strong correlations in the particle-hole channel and attributed to antiferromagnetic fluctuations, in agreement with numerous previous studies. For a Fermi surface at or very close to the van Hove scale, there exists a degeneracy in the sense that the strongest correlations are found around a point in coupling space which is relevant for both superconductivity and antiferromagnetism, and we can thus not distinguish between the importance of the respective fluctuations.
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model.In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for twodimensional lattice problems.The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence.The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy.We quantitatively compare our fRG results for the self-energy and momentumdependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths.The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
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).