We show that the high-temperature expansion of the free energy and arbitrary imaginary-time-ordered connected correlation functions of quantum spin systems can be recursively obtained from the exact renormalization group flow equation for the generating functional of connected spin correlation functions derived by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)]. Our recursive algorithm can be explicitly written in closed form including all combinatorial factors. We use our method to estimate critical temperatures of Heisenberg magnets from low-order truncations of the inverse spin susceptibility in the static limit. We also calculate the connected correlation function involving three different spin components (chiral nonlinear susceptibility) of quantum Heisenberg magnets up to second order in the exchange couplings.
We show that the high-temperature expansion of the free energy and arbitrary imaginary-time-ordered connected correlation functions of quantum spin systems can be recursively obtained from the exact renormalization group flow equation for the generating functional of connected spin correlation functions derived by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)]. Our recursive algorithm can be explicitly written in closed form including all combinatorial factors. We use our method to estimate critical temperatures of Heisenberg magnets from low-order truncations of the inverse spin susceptibility in the static limit. We also calculate the connected correlation function involving three different spin components (chiral nonlinear susceptibility) of quantum Heisenberg magnets up to second order in the exchange couplings.
We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbit{k},\ensuremath{\omega})$ of quantum Heisenberg magnets at infinite temperature. For small wave vectors $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ we find that $G(\mathbit{k},\ensuremath{\omega})$ assumes in dimensions d>2 the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient D is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about 30 larger than the measured high-temperature value of D in the Heisenberg ferromagnet ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. In reduced dimensions $d\ensuremath{\le}2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\ensuremath{\omega})$ which diverges logarithmically in d=2, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ in d=1. Our result in one dimension implies scaling with dynamical exponent z=3/2, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ for all wave vectors. We show how the short-wavelength behavior of $S(\mathbit{k},\ensuremath{\omega})$ at high temperatures reflects the relative sign and strength of competing exchange interactions.
We investigate dimerized quantum spin systems using the spin functional renormalization group approach proposed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] which directly focuses on the physical spin correlation functions and avoids the representation of the spins in terms of fermionic or bosonic auxiliary operators. Starting from decoupled dimers as initial condition for the renormalization group flow equations, we obtain the spectrum of the triplet excitations as well as the magnetization in the quantum paramagnetic, ferromagnetic, and thermally disordered phases at all temperatures. Moreover, we compute the full phase diagram of a weakly coupled dimerized spin system in three dimensions, including the correct mean-field critical exponents at the two quantum critical points.
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.
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.
We show that the diagrammatic approach to quantum spin systems developed in a seminal work by Vaks, Larkin, and Pikin [Sov. Phys. JETP 26, 188 (1968)] can be embedded in the framework of the functional renormalization group. The crucial insight is that the generating functional of the time-ordered connected spin correlation functions of an arbitrary quantum spin system satisfies an exact renormalization group flow equation which resembles the corresponding flow equation of interacting bosons. The SU(2) spin algebra is implemented via a nontrivial initial condition for the renormalization group flow. Our method is rather general and offers a different nonperturbative approach to quantum spin systems.
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.