High-temperature series expansion of the dynamic Matsubara spin correlator

This paper extends high-temperature series expansions to dynamic Matsubara spin correlators for Heisenberg models, providing precomputed exact expansion coefficients up to the 12th order for arbitrary lattices to enable the calculation of static susceptibilities and real-frequency dynamic structure factors.

The Gist

Imagine you are trying to predict the weather in a complex, chaotic city. You know the basic rules of physics (wind, temperature, pressure), but calculating the exact weather for every single street corner is impossible because there are too many variables interacting at once.

This paper introduces a new, powerful tool to solve a similar problem, but instead of weather, the authors are studying quantum spins—tiny, invisible magnets inside materials like metals or crystals.

Here is a breakdown of what they did, using simple analogies:

1. The Problem: The "High-Temperature" Puzzle

Scientists have long used a method called High-Temperature Series Expansion (HTE) to understand how these tiny magnets behave when things are hot. Think of this like trying to predict how a crowd of people behaves in a hot room. When it's very hot, everyone is moving randomly, and the interactions are simple enough to calculate step-by-step.

However, there was a major gap: This old method could only tell you about the magnets' static state (where they are pointing right now). It couldn't tell you about their dynamics (how they wiggle, vibrate, or change over time). It was like knowing where the people in the crowd are standing, but having no idea if they are dancing, running, or sleeping.

2. The Solution: "Dynamic HTE" (Dyn-HTE)

The authors, Ruben Burkard, Benedikt Schneider, and Björn Sbierski, have upgraded the old tool. They created a new version called Dyn-HTE.

• The Analogy: Imagine the old method was a photo album of a party. You could see who was standing next to whom. The new method is a video camera. It captures the movement, the rhythm, and the flow of the party.
• What it does: It calculates how these quantum magnets interact over time, specifically looking at their "wiggles" at different frequencies (how fast they vibrate).

3. The Secret Weapon: The "Kernel Trick"

Calculating how these magnets move involves solving incredibly complex math equations involving time and space. Usually, this is like trying to untangle a knot of 100 headphones while blindfolded.

The authors used a clever mathematical shortcut they call the "Kernel trick."

• The Analogy: Instead of trying to untangle the whole knot at once, they found a way to break the knot down into tiny, pre-solved pieces. They realized that for this specific type of problem, the math simplifies drastically, allowing them to solve the "time" part of the equation exactly, rather than guessing or approximating.

4. The "Lego" Approach

To handle the massive number of possible interactions, they didn't try to calculate the whole material at once. Instead, they treated the material like a giant structure built from Lego bricks.

• They broke the problem down into tiny snippets called graphs (small clusters of magnets).
• They calculated the behavior of every possible small Lego cluster (up to a very high level of complexity).
• Then, they provided a "recipe" (an algorithm) that tells you how to snap these pre-calculated Lego pieces together to describe any material, whether it's a simple line of magnets or a complex 3D lattice.

5. The Result: A Massive Library of Answers

The team didn't just write a theory; they did the heavy lifting.

• They pre-calculated the answers for about 1 million different Lego clusters.
• They stored these answers as exact fractions (rational numbers), meaning there is no rounding error or guesswork.
• They made this data available for other scientists to download and use.

6. Why This Matters (According to the Paper)

The paper highlights two main uses for this new tool:

1. Checking the Statics: They tested their method on a simple chain of magnets and a triangular pattern. The results matched perfectly with other highly accurate computer simulations, proving their new "video camera" works.
2. Unlocking Real-Time Physics: The most exciting part is that this method allows scientists to figure out the real-time behavior of these magnets without having to do a notoriously difficult and error-prone mathematical conversion (called "analytical continuation").
   • The Analogy: Usually, to see the real-time movie, you have to take a blurry photo and try to guess the motion, which often leads to mistakes. The authors' method gives you the exact script of the movie (the frequency moments) directly. You can then use standard tools to reconstruct the full movie (the dynamic structure factor) with high precision.

Summary

In short, these scientists built a universal calculator for the movement of quantum magnets at high temperatures. They broke a massive, impossible math problem into millions of small, solvable puzzles, solved them exactly, and gave the world the answers. This allows researchers to finally "watch" how these quantum systems dance, rather than just taking a snapshot of where they stand.

Technical Summary

Technical Summary: High-temperature series expansion of the dynamic Matsubara spin correlator

Problem Statement
The high-temperature series expansion (HTE) is a standard tool for computing thermodynamic quantities and equal-time spin correlations in quantum spin systems, particularly those with frustrated interactions. However, conventional HTE is limited to static, equal-time observables. To understand collective phenomena in equilibrium quantum spin systems, such as spin liquids, it is necessary to access spin dynamics. The dynamical spin-spin correlator $G_{ii'}(t)$ and its Fourier transform, the dynamical structure factor, contain rich information about excitation spectra and quasiparticle stability. While these can be calculated via methods like exact diagonalization or tensor networks, these approaches often struggle with the thermodynamic limit or high dimensions. Furthermore, obtaining real-frequency dynamical data from imaginary-time (Matsubara) data via analytical continuation is an ill-conditioned numerical problem. This paper addresses the gap by extending HTE to the dynamic Matsubara spin-spin correlator, $G_{ii'}(i\nu_m)$, in imaginary frequency.

Methodology: Dynamic HTE (Dyn-HTE)
The authors develop a method termed "Dynamic HTE" (Dyn-HTE) to compute the expansion of the Matsubara correlator in powers of the dimensionless coupling $x = J/T$ and inverse Matsubara frequency $1/\nu_m$.

1. Formalism: The expansion is derived for Heisenberg models with a single coupling constant $J$ and spin lengths $S \in \{1/2, 1\}$. The correlator is expanded as:
   $$T G_{ii'}(i\nu_m) = \sum_{n=0}^{n_{max}} (-x)^n c^{(n)}_{ii'}(i\nu_m)$$
   where the coefficients $c^{(n)}_{ii'}$ are further expanded in powers of $1/\nu_m$. The final form involves dimensionless polynomials $p^{(2r)}_{ii'}(x)$ with rational coefficients.

2. Graph-Based Evaluation: The calculation relies on a graph-based approach where the sum over lattice bonds is reorganized into a sum over topologically distinct graphs (lattice snippets) $g^{(n)}$ with $n$ edges.

   • Embedding: The contribution of each graph to a specific lattice is determined by an embedding factor $e(L, i, i', g^{(n)})$, calculated using graph-theoretical algorithms and space-group symmetries.
   • Recursive Subtractions: To isolate connected contributions (removing vacuum and disconnected diagrams), the authors employ a recursive subtraction scheme based on the "connected determinant formula" (adapted from fermionic diagrammatic Monte Carlo). This involves subtracting contributions from subgraphs $g^{(k)} \subset g^{(n)}$.

3. Kernel Trick for Time Integrals: A central technical innovation is the evaluation of the $(n+2)$-fold imaginary-time integrals required at order $n$. Instead of numerical integration, the authors utilize a "Kernel trick" (based on Ref. 22) which provides closed-form analytic expressions for these integrals.

   • Since the unperturbed Hamiltonian $H_0 = 0$ (non-interacting spins), the eigenstates are trivial product states.
   • The time-ordered integrals are expressed exactly using "kernel functions" $\tilde{K}_{n+2}$ that depend on the positions of the external operators and the Matsubara frequency index $m$.
   • This allows the expansion coefficients to be computed exactly as rational numbers, avoiding stochastic errors.

4. Algorithmic Optimization: The implementation exploits symmetries (e.g., cyclic properties of the trace, spin-flavor symmetries) to reduce the computational cost. The authors precompute the exact rational coefficients for all possible graphs up to order $n_{max} = 12$ (approximately $1.27 \times 10^6$ graphs) for spin lengths $S=1/2$ and $S=1$.

Key Contributions

• Extension of HTE: The work successfully extends the HTE framework from static, equal-time correlations to the dynamic Matsubara spin-spin correlator.
• Exact Rational Coefficients: The authors provide an algorithm that yields exact expansion coefficients in the form of rational numbers, precomputed for all graphs up to the 12th order in $J/T$.
• Public Repository: The precomputed coefficients and tools for arbitrary lattice embedding are made available in a repository, enabling the calculation of static momentum-resolved susceptibilities for arbitrary site-pairs or wavevectors.
• Kernel Trick Application: The application of the kernel trick to spin systems allows for the analytical solution of high-order imaginary-time integrals, a task previously difficult for spin operators where Wick's theorem does not apply.

Results and Validation

• Benchmarking: The method is tested on the antiferromagnetic $S=1/2$ Heisenberg chain and the triangular lattice model.
• Comparison with QMC: For the $S=1/2$ chain, the static susceptibility derived from Dyn-HTE is compared against error-controlled Quantum Monte Carlo (QMC) simulations. The results show excellent agreement, validating the expansion coefficients.
• Triangular Lattice: The static susceptibility for the frustrated triangular lattice model is reported, demonstrating the method's applicability to frustrated systems.
• Approximation Testing: The authors use the static susceptibility to benchmark a "renormalized mean-field form" parametrization for momentum dependence, showing its accuracy.

Significance and Claims
The paper positions Dyn-HTE as a robust tool for studying spin dynamics in the high-temperature regime, particularly for frustrated systems where other methods may fail or yield featureless results.

• Analytical Continuation: A primary significance claimed is the ability to bypass the ill-conditioned analytical continuation problem. By identifying the high-frequency expansion coefficients of the Matsubara correlator with the frequency moments of the real-frequency spectral function, the method allows for the reconstruction of the real-frequency dynamical structure factor using standard moment-based techniques (e.g., continued fractions). This is detailed in a companion letter.
• Efficiency: The authors note that while the dynamic expansion involves higher-dimensional integrals, the numerical cost is only modestly increased (by a factor of $n$ in the worst case) compared to conventional HTE for equal-time correlators, thanks to the analytical kernel trick.
• Scope: The method is presented as a general tool for equilibrium quantum spin systems, with current limitations restricted to single-coupling Heisenberg models and specific spin lengths, though the formalism allows for future extensions (e.g., magnetic fields, different couplings).

The authors maintain a modest tone regarding the immediate physical interpretation of the low-temperature behavior, acknowledging that HTE is primarily a high-temperature tool, but emphasizing its utility in providing exact constraints (moments) for the spectral function that are difficult to obtain otherwise.