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arXiv:2505.14571 · Physical Review Letters

Dynamic Correlations of Frustrated Quantum Spins from High-Temperature Expansion

Ruben Burkard, Benedikt Schneider, Björn Sbierski

Published Jan 7, 2026• 3 citations
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Abstract

For quantum spin systems in equilibrium, the dynamic structure factor (DSF) is among the most feature-packed experimental observables. However, from a theory perspective it is often hard to simulate in an unbiased and accurate way, especially for frustrated and high-dimensional models at intermediate temperature. To address this challenge, we compute the DSF from a dynamic extension of the high-temperature expansion to frequency moments. We focus on nearest-neighbor Heisenberg models with spin lengths S=1/2 and 1. We provide comprehensive benchmarks and consider a variety of frustrated two- and three-dimensional antiferromagnets as applications. In particular we shed new light on the anomalous intermediate temperature regime of the S=1/2 triangular lattice model and reproduce the DSF measured recently for the S=1 pyrochlore material NaCaNi_{2}F_{7}. An open-source numerical implementation for arbitrary lattice geometries is also provided.

DOI
10.1103/jtjk-x2lw

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For quantum spin systems in equilibrium, the dynamic structure factor (DSF) is among the most feature-packed experimental observables. However, from a theory perspective it is often hard to simulate in an unbiased and accurate way, especially for frustrated and high-dimensional models at intermediate temperature. To address this challenge, we compute the DSF from a dynamic extension of the high-temperature expansion to frequency moments. We focus on nearest-neighbor Heisenberg models with spin lengths S=1/2 and 1. We provide comprehensive benchmarks and consider a variety of frustrated two- and three-dimensional antiferromagnets as applications. In particular we shed new light on the anomalous intermediate temperature regime of the S=1/2 triangular lattice model and reproduce the DSF measured recently for the S=1 pyrochlore material NaCaNi_{2}F_{7}. An open-source numerical implementation for arbitrary lattice geometries is also provided.
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\sim 10^6∼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 study the low temperature dynamics of the classical Heisenberg antiferromagnet with nearest neighbor interaction on the frustrated pyrochlore lattice. We present extensive results for the wave vector and frequency dependence of the dynamical structure factor, obtained from simulations of the precessional dynamics. We also construct a solvable stochastic model for dynamics with conserved magnetization, which accurately reproduces most features of the precessional results. Spin correlations relax at a rate independent of the wave vector and proportional to the temperature.
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 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%\sim5\%∼5% and we further confirm the established values for the critical exponent ν\nuν and the anomalous dimension η\etaη. 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→0T\rightarrow0T→0. While extensions towards higher loop orders ℓ\ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ\ellℓ 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.
We analyze the thermodynamic properties of the spin-SSS two-dimensional quantum Heisenberg antiferromagnet on a square lattice with nearest and next-nearest neighbor couplings in the N\'eel phase (J2∕J1<0.4)({J}_{2}∕{J}_{1}<0.4)(J2​∕J1​<0.4) employing the quantum hierarchical reference theory, a nonperturbative implementation of the renormalization group method to quantum systems. We investigate the staggered susceptibility, the structure factors, and the correlation length at finite temperature and for different values of the frustration ratio. From the finite temperature results, we also extrapolate ground state properties, such as spin stiffness and spontaneous staggered magnetization, providing an estimate of the extent of quantum corrections. The behavior of these quantities as a function of frustration may provide some hints on the breakdown of the N\'eel phase at zero temperature for larger values of J2{J}_{2}J2​.
Motivated by recent experiments on low-dimensional frustrated quantum magnets with competing nearest-neighbor exchange coupling J 1 and next nearest-neighbor exchange coupling J 2 we investigate the magnetic susceptibility of two-dimensional J 1 - J 2 Heisenberg models with arbitrary spin quantum number s . We use exact diagonalization and high- temperature expansion up to order 10 to analyze the influence of the frustration strength J 2 / J 1 and the spin quantum number s on the position and the height of the maximum of the susceptibility. The derived theoretical data can be used to get information on the ratio J 2 / J 1 by comparing with susceptibility measurements on corresponding magnetic compounds.