⚛️ high-energy theory

Calculating Feynman diagrams with matrix product states

This pedagogical review outlines a method for automatically calculating and resumming Feynman diagrams in quantum nanoelectronics using matrix product states and the Tensor Cross Interpolation algorithm, specifically applied to the non-equilibrium Kondo effect in the single impurity Anderson model.

Original authors: Xavier Waintal

Published 2026-02-04

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Xavier Waintal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Taming the Quantum Chaos

Imagine you are trying to predict the behavior of a tiny, chaotic party happening inside a quantum dot (a microscopic island for electrons). You want to know exactly how many electrons are there and how much current is flowing.

In physics, the "gold standard" for solving these problems is a method involving Feynman diagrams. Think of these diagrams not as drawings, but as a massive, multi-layered recipe. To get the final answer, you have to:

Write down every possible way the electrons can interact (the ingredients).
Calculate a giant math integral (the cooking process) for every single recipe.
Add up all the results.

The problem? The number of recipes grows so fast it becomes impossible. If you want to calculate the interaction for just a few steps, you might have a few recipes. But for a slightly more complex scenario, the number of recipes explodes into the billions, then trillions, then factorials (like $100!$ ). It's like trying to count every grain of sand on a beach, but the beach keeps growing faster than you can count.

This paper describes a new "kitchen" that allows scientists to cook this meal without getting lost in the sand.

The Three Big Problems (The "Three Headed Monster")

The author identifies three specific nightmares that have stopped computers from solving this problem for decades:

Problem A: The Explosion of Options.
- The Analogy: Imagine a choose-your-own-adventure book where every time you make a choice, the number of new pages doubles. By page 10, you have more pages than there are atoms in the universe.
- The Paper's Fix: Instead of treating every single "path" (diagram) as a unique story, the authors realized that many of these paths are actually just different versions of the same underlying structure. They found a way to group millions of chaotic paths into a much smaller, manageable set of "determinants" (like organizing a messy closet into neat, labeled boxes). This reduced the workload from a factorial explosion to a much more manageable exponential growth.
Problem B: The "Sign Problem" (The Oscillating Wave).
- The Analogy: Imagine trying to measure the average height of a crowd, but half the people are standing on stilts (positive numbers) and half are hanging upside down in a pit (negative numbers). If you use a random sampling method (like picking people at random), you might pick 10 people from the pit and get a wildly wrong average. The positive and negative numbers cancel each other out so perfectly that the signal gets lost in the noise. This is the famous "Sign Problem" in physics.
- The Paper's Fix: The authors stopped using random sampling (Monte Carlo). Instead, they used a technique called Tensor Cross Interpolation (TCI).
- The Analogy: Think of the math function they need to solve as a giant, complex 3D landscape. Instead of randomly throwing darts at the map to guess the shape (which fails when the landscape has hills and valleys that cancel out), TCI is like a smart surveyor. It picks a few key "pivot" points (peaks and valleys) and uses them to reconstruct the entire map perfectly. Because it reconstructs the whole shape mathematically rather than guessing, the positive and negative parts cancel out exactly as they should, eliminating the noise.
Problem C: The Infinite Series.
- The Analogy: Imagine you are trying to predict the weather for next year. You have data for the first 20 days. If you just add up the first 20 days, you can't predict the winter. But if you try to predict too far ahead, your math breaks down.
- The Paper's Fix: The authors used a technique called Cross Extrapolation.
- The Analogy: Imagine you have a photo of a landscape, but the top-right corner is torn off (missing data). You know the pattern of the trees and clouds in the bottom-left. The authors realized that the physics of this system is "low rank"—meaning the complex pattern is actually built from a few simple, repeating layers. By analyzing the known part of the photo, they could mathematically "fill in" the missing corner with high precision, allowing them to predict the behavior for very long times and strong interactions.

The "Secret Sauce": Tensor Cross Interpolation (TCI)

The core innovation of this paper is TCI.

What it is: It's a way to compress a massive, multi-dimensional math problem into a chain of smaller, connected matrices (called Matrix Product States).
How it works: Think of a giant, multi-dimensional Rubik's cube. Usually, to solve it, you have to look at every single sticker. TCI is like realizing that the cube is actually just a few simple patterns stacked on top of each other.
The "Learning" Aspect: The paper compares TCI to machine learning. Instead of a computer blindly trying millions of random numbers, TCI is an "active learner." It asks, "If I check this specific point, will it teach me the most about the whole picture?" It picks the most informative points (pivots) to build its model.
The Result: Once the computer builds this compressed model, it can calculate the final answer (the integral) instantly and exactly, without needing any random guessing or Monte Carlo simulations.

The Real-World Test: The Quantum Dot

To prove this works, the authors applied it to a model called the SIAM (Single Impurity Anderson Model).

The Setup: A tiny quantum dot connected to two wires, with electrons flowing through it.
The Challenge: They wanted to calculate the current flowing through the dot when a voltage is applied, specifically looking for two famous quantum phenomena:
1. Coulomb Diamonds: A pattern showing that electrons block each other from entering the dot (like a bouncer at a club).
2. The Kondo Ridge: A specific feature where, at very low temperatures, the electrons suddenly start flowing perfectly smoothly due to quantum entanglement.

The Outcome:
The authors successfully calculated the current and conductance across a wide range of voltages and temperatures. Their results matched the "exact" theoretical predictions (calculated using other, much slower methods) with high precision. They were able to see the "Kondo ridge" and the "Coulomb diamonds" clearly, proving that their new "Non-diagrammatic, Non-Monte-Carlo" technique works.

The Takeaway

The author concludes that we are entering a new era of computational physics.

Open Source is Key: The code used to do this is open, allowing others to build on it.
Structure over Brute Force: The biggest breakthrough wasn't just faster computers; it was finding the hidden "structure" in the math (using TCI) that allowed them to bypass the impossible calculations.
The Future: The author suggests that problems we thought were unsolvable (like the 2D Hubbard model) might soon be solved using these types of "smart" algorithms that teach computers how to find the patterns in the chaos.

In short: They taught a computer to stop counting every grain of sand and instead learn the shape of the beach, allowing it to solve quantum puzzles that were previously impossible.

Technical Summary: Calculating Feynman Diagrams with Matrix Product States

1. Problem Formulation

The paper addresses the challenge of calculating physical observables in quantum nanoelectronic systems, specifically out-of-equilibrium interacting models like the Single Impurity Anderson Model (SIAM). The goal is to compute observables $Q(U, t)$ (such as charge or current) as a systematic perturbative expansion in the interaction strength $U$ :
$Q(U, t) = \sum_{n=0}^{\infty} Q_n(t) U^n$
Calculating the coefficients $Q_n(t)$ involves evaluating high-dimensional integrals of products of non-interacting Green's functions. The authors identify three fundamental difficulties in this approach:

Problem A (Combinatorial Explosion): The number of Feynman diagrams grows factorially ( $O(n!)$ ) with the perturbation order $n$ , making direct summation impossible beyond $n \approx 6-7$ .
Problem B (The Sign Problem and Integration): The integrands in these diagrams often oscillate rapidly, leading to the "sign problem" in Monte-Carlo methods. Furthermore, the sheer number of function evaluations required (often $\sim 10^9$ ) makes standard sampling inefficient, especially as the cost per evaluation scales exponentially with $n$ .
Problem C (Convergence and Resummation): Even if coefficients can be calculated, the perturbation series may have a finite radius of convergence (especially in the steady-state limit $t \to \infty$ ), requiring sophisticated resummation techniques to extract physical results at large interaction strengths.

2. Methodology

The authors propose a three-step framework to address these problems, moving away from traditional Diagrammatic Monte-Carlo toward a "Non-diagrammatic Non-Monte-Carlo" approach.

Step 1: Interaction Expansion (Addressing Problem A)

Instead of summing individual Feynman diagrams, the authors utilize the Keldysh formalism combined with Wick's theorem.

They express the expansion coefficients as multidimensional integrals over time variables and Keldysh indices.
Crucially, they recognize that the sum over all permutations of fermionic operators (which generates the $n!$ diagrams) is mathematically equivalent to the determinant of a matrix constructed from non-interacting Green's functions.
This reduces the complexity from $O(n!)$ to $O(2^n)$ (specifically, evaluating determinants of $(2n+1) \times (2n+1)$ matrices).
A key insight is that summing over Keldysh indices before time integration cancels out disconnected diagrams, resulting in a well-behaved, positive-definite integrand that is concentrated around the measurement time.

Step 2: Tensor Cross Interpolation (Addressing Problem B)

To evaluate the resulting high-dimensional integrals without Monte-Carlo sampling, the authors employ Tensor Cross Interpolation (TCI).

Concept: TCI is a learning algorithm that approximates a high-dimensional tensor (the integrand) as a Matrix Product State (MPS). Unlike neural networks, TCI does not use gradient descent; it uses an active learning strategy to select "pivots" (specific grid points) that maximize the determinant of submatrices (the maxvol principle).
Mechanism: The algorithm iteratively sweeps through the tensor dimensions, performing Cross Interpolation (CI) on matrices formed by slicing the tensor. CI approximates a low-rank matrix $A$ using only a small subset of its rows and columns (pivots): $A \approx A_{I,J} A_{I,I}^{-1} A_{I,J}$ .
Integration: Once the integrand is compressed into an MPS representation, the multidimensional integral can be calculated exactly and efficiently via matrix-vector multiplications, bypassing the curse of dimensionality and the sign problem entirely.
Performance: This method allows calculations up to $n \approx 20-25$ , far exceeding previous limits.

Step 3: Cross Extrapolation (Addressing Problem C)

To reconstruct the full function $Q(U, t)$ from a finite number of calculated coefficients $Q_n(t)$ , the authors use Cross Extrapolation.

Strategy: The function $Q(U, t)$ is treated as a matrix where rows correspond to interaction $U$ and columns to time $t$ . The authors observe that this matrix is approximately low-rank.
Execution: Using the same mathematical machinery as TCI (Cross Interpolation), they extrapolate the unknown regions of the matrix (large $U$ and large $t$ ) based on the known "corner" data (small $U$ or small $t$ ).
Validity: This relies on the assumption that $Q(U, t)$ can be approximated by a sum of separable functions $\sum g_k(U)h_k(t)$ . The error is controlled by varying the rank $\chi$ of the extrapolation.

3. Key Contributions

Determinant Reformulation: Demonstrated that the perturbative expansion of out-of-equilibrium observables can be reduced to the integration of determinants, eliminating the factorial growth of diagram counts.
TCI for Integration: Introduced Tensor Cross Interpolation as a superior alternative to Monte-Carlo for integrating oscillatory, high-dimensional functions arising in quantum many-body physics. It provides exact integration without stochastic noise.
Cross Extrapolation: Developed a robust method to resum perturbation series by exploiting the low-rank structure of the observable in the $(U, t)$ plane, enabling access to regimes where the series would naively diverge.
Open Source Implementation: The work is supported by the tensor4all library, providing accessible tools for these tensor network techniques.

4. Results

The methodology was applied to the out-of-equilibrium SIAM (a quantum dot coupled to two leads) to calculate charge and current.

Precision: The authors computed expansion coefficients up to $n=18$ with high precision (up to 8 digits), matching exact Bethe Ansatz results in the steady state.
Physical Observables: The method successfully reproduced the full phase diagram of the SIAM, including:
- Coulomb Diamonds: Regions of suppressed conductance due to charging energy.
- Kondo Ridge: A zero-bias anomaly in differential conductance arising from the Kondo effect.
Current-Voltage Characteristics: The authors calculated the current $I(V_b)$ $I (V_{b})$ across the Kondo ridge at strong interaction ( $U=12$ $U = 12$ ). The results captured the expected physical regimes:
- Saturation at high bias ( $V_b > U$ ).
- A plateau associated with the Kondo temperature $T_K$ at low bias ( $V_b < T_K$ ), where perfect transmission ( $I \approx V_b$ ) is expected.
- The transition between these regimes spans multiple energy scales ( $U$ , $\sqrt{\Gamma U}$ , $\Gamma$ , and $T_K$ ).

5. Significance and Claims

The paper positions this work as a significant step in "teaching a computer to calculate Feynman diagrams."

Controlled Accuracy: Unlike mean-field approximations, this technique is "controlled," meaning it provides known error bars that can be systematically improved by increasing computational resources (time and rank).
Paradigm Shift: The authors highlight a shift from manually deriving diagrammatic rules to using algorithms (TCI) that discover the underlying low-rank structure of the problem automatically.
Scalability: The combination of determinant reduction and TCI integration allows for calculations in regimes previously inaccessible to diagrammatic methods, bridging the gap between theoretical understanding and experimental observables in non-equilibrium quantum transport.
Modesty: The authors acknowledge that while the technique is powerful, it is not a universal solver for all many-body problems (e.g., the 2D Hubbard model is mentioned as a future target). They emphasize that the success relies on the specific low-rank structure of the integrands in these nanoelectronic systems.