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.
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.
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.
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) 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.
The numerical study of high-dimensional frustrated quantum magnets remains a challenging problem. Here we present an extension of the pseudo-Majorana functional renormalization group to spin-1/2 XXZ-type Hamiltonians with field or magnetization along spin- Z direction at finite temperature. We consider a U(1) symmetry-adapted fermionic spin representation and derive the diagrammatic framework and its renormalization group flow equations. We discuss benchmark results and application to two antiferromagnetic triangular lattice materials recently studied in experiments with applied magnetic fields: First, we numerically reproduce the magnetization data measured for CeMgAl 11 O 19 confirming model parameters previously estimated from inelastic neutron spectrum in high fields. Second, we showcase the accuracy of our method by studying the thermal phase transition into the spin solid up-up-down phase of Na 2 BaCo ( PO 4 ) 2 in good agreement with experiment.
The high-temperature series expansion for quantum spin models is a well-established tool to compute thermodynamic quantities and equal-time spin correlations, in particular for frustrated interactions. We extend the scope of this expansion to the dynamic Matsubara spin-spin correlator and develop an algorithm that yields exact expansion coefficients in the form of rational numbers. We focus on Heisenberg models with a single coupling constant J and spin lengths S=1/2,1. The expansion coefficients up to 12th order in J/T are precomputed on all possible ∼106 graphs embeddable in arbitrary lattices and are provided in a repository. This enables calculation of static momentum-resolved susceptibilities for arbitrary site-pairs or wavevectors. We test our results for the antiferromagnetic S=1/2 chain and triangular lattice model. An important application that we discuss in a companion letter is the calculation of real-frequency dynamic structure factors. This is achieved by identifying the high-frequency expansion coefficients of the Matsubara correlator with frequency moments of the spectral function.
We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by ∼5% and we further confirm the established values for the critical exponent ν and the anomalous dimension η. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T→0. While extensions towards higher loop orders ℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.