Grassmann time-evolving matrix product operators for fermionic impurities coupled to a superconducting bath

arXiv:2604.23301 · cond-mat.str-el, cond-mat.supr-con · Submitted 2026-04-25 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Grassmann time-evolving matrix product operators for fermionic impurities coupled to a superconducting bath".

Kai: The Nambu-GTEMPO method extends Grassmann time-evolving matrix product operators to solve fermionic impurity problems in the Nambu formalism, specifically addressing superconducting baths.

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So, we're looking at this paper today, "Grassmann time-evolving matrix product operators for fermionic impurities coupled to a superconducting bath," and it’s written by Chu Guo, Wei Wu, Xiansong Xu, Ping-Xing Chen, Changming Yue, Tian Jiang and Ruofan Chen.

Mira: It’s interesting because it takes the existing GTEMPO method—which is used for impurity problems—and modifies it specifically to handle superconducting baths using a Bogoliubov transformation.

Kai: So what does that mean for us on the experimental side?

Lev: It means they are looking at how this framework might map onto actual quantum hardware setups, where you’ve got these bosonic and fermionic degrees of freedom interacting in a non-trivial way.

Mira: Exactly. They're essentially taking a known technique, which is the time-evolving matrix product operator idea, and adapting it to this new physics involving superconductors.

Kai: It’s about making the math work for real systems where you have these kinds of complex couplings between an impurity and a superconducting environment.

Lev: If this works out on paper, it suggests a pathway for tackling impurity problems that involve superconductivity, which is a major area in condensed matter physics right now.

The paper's summary: Kai: So the paper summarizes how they’ve adapted the GTEMPO method to solve the Superconducting Anderson Impurity Model, or SAIM.

Mira: They start by using that Bogoliubov transformation on the bath Hamiltonian to get a clean expression for the Feynman-Vernon influence functional. This turns it into something that looks very similar to a standard normal bath problem.

Kai: So they’re essentially simplifying the setup so they can use the established GTEMPO algorithms without having to reinvent everything from scratch for this new superconducting case.

Mira: Right, and then they discretize those Grassmann trajectories and construct a discrete hybridization function as a Grassmann Matrix Product State, or GMPS.

Kai: And finally, they get the full influence functional by multiplying the discrete K and I sigma states together to create what they call the Augmented Density Tensor, also as a GMPS.

Lev: That’s significant because it means their method has a concrete computational structure. It’s not just some abstract idea; it has defined steps for how you actually compute these quantities.

The paper's improvements: Kai: So they point out a few key improvements in this approach, and they really focus on the flexibility of the method.

Mira: They highlight that because GTEMPO only relies on the analytical expression of the influence functional, it can be straightforwardly applied on several different types of contours.

Kai: Like imaginary time, real-time Keldysh contours, or even those L-shaped Kadanoff contours. This means you’re not locked into just one way to do the calculation.

Mira: And they emphasize that the method can handle the simultaneous presence of bosonic and fermionic baths as well, which is a big deal for realistic models.

Kai: So it's about making this tool adaptable to different computational setups without needing a completely new algorithm every time you want to look at something.

Lev: From an error-correction standpoint, that flexibility is good because it means you can test the method on various physical scenarios and see where the Trotter decomposition or bond truncation errors are most likely to creep in.

Conclusion: Kai: So wrapping up the discussion on this paper, what we’re seeing is a robust framework for solving fermionic impurity problems when there’s a superconducting bath involved.

Mira: The main implication is that this method allows us to study superconducting states in both equilibrium and non-equilibrium settings within the Nambu formalism, which is pretty powerful for understanding real physics.

Kai: Specifically, they can calculate correlation functions for real-time evolution, like G↑¯↑●(t) and G↑↓●(t), which is something that’s often tricky to get right in these kinds of models.

Lev: For running this on hardware, the paper’s mention of being free of sampling noises and the sign problem when compared to continuous-time quantum Monte Carlo on the imaginary contour is a big plus for implementation.

Mira: It shows that while it might not be faster than some methods yet, its ability to avoid those specific computational hurdles makes it a promising candidate for non-equilibrium DMFT studies.

Kai: So, the Nambu-GTEMPO method for fermionic impurities coupled to a superconducting bath is definitely a flexible tool we can use when studying these complex systems.

Lev: I just want to say that having this structured approach helps immensely when you’re trying to translate theory into something that can actually run on a computer.

College of Science, National University of Defense Technology · Hunan Key Laboratory of Mechanism and technology of Quantum Information · College of Physics and Electronic Engineering, and Center for Computational Sciences, Sichuan Normal University · State Key Laboratory of Quantum Functional Materials, Department of Physics, and Guangdong Basic Research Center of Excellence for Quantum Science · College of Advanced Interdisciplinary Studies, National University of Defense Technology · Hunan Research Center of the Basic Discipline for Physical States

cond-mat.str-el, cond-mat.supr-con

Submitted: 2026-04-25

Updated: 2026-10-08

Comments: 15 pages, 8 figures

Journal ref: Phys. Rev. B 114, 225115 (2026)

DOI: 10.1103/w2b4-tm7c

Code: https://github.com/guochu/GTEMPO

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 81/100

The gist: The Nambu-GTEMPO method extends Grassmann time-evolving matrix product operators to solve fermionic impurity problems in the Nambu formalism, specifically addressing superconducting baths.

Key concepts

Nambu formalism: This is a mathematical framework used to describe fermionic systems, especially those involving superconductivity. It combines particle and hole degrees of freedom into a single mathematical structure, which simplifies the treatment of phenomena like pairing in superconducting baths.
Grassmann time-evolving matrix product operators (GTEMPO): These are advanced computational tools that use matrix product states to represent the time evolution of quantum systems. The Nambu-GTEMPO method adapts these tools to solve complex impurity problems, allowing for the study of real-time dynamics.
Superconducting Anderson Impurity Model (SAIM): This model describes a localized electron interacting with a superconducting bath. It is used to study how an impurity behaves when coupled to a material exhibiting superconductivity, which is crucial for understanding many condensed matter phenomena.

Terminology

Hubbard Model — 8× in this paper · explained in 3 episodes
Core — 7× in this paper · explained in 2 episodes
Benchmarking — 5× in this paper · explained in 3 episodes
Superconducting Anderson Impurity Model — 5× in this paper
Accuracy — 4× in this paper · explained in 1 episode
Antiferromagnetism — 3× in this paper · explained in 1 episode
Benchmark — 3× in this paper · explained in 2 episodes
Augmented Density Tensor — 2× in this paper
Dynamical Mean Field Theory (DMFT) — 2× in this paper · explained in 2 episodes
Grassmann Matrix Product State — 2× in this paper
Ab Initio — 1× in this paper · explained in 1 episode
Entanglement — 1× in this paper · explained in 4 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

The Nambu-GTEMPO method extends Grassmann time-evolving matrix product operators to solve fermionic impurity problems in the Nambu formalism, specifically addressing superconducting baths.

How it works

The core idea involves employing the Bogoliubov transformation for the superconducting bath to obtain an analytic expression of the Feynman-Vernon influence functional in a form similar to a normal bath case<ref:2604.23301#pg6>. This allows the core algorithms of GTEMPO to be straightforwardly adapted<ref:2604.23301#pg6>.
The method is applied to the Superconducting Anderson Impurity Model (SAIM), where a localized electron interacts with a superconducting bath<ref:2604.23301#pg6>.
The Feynman-Vernon influence functional (IF) for the SAIM is derived by transforming the bath Hamiltonian into a normal bath form using Bogoliubov transformations, resulting in an analytic expression that has a similar quadratic form to the standard case<ref:2604.23301#pg6>.
The GTEMPO method discretizes Grassmann trajectories and constructs the discrete hybridization function as a Grassmann Matrix Product State (GMPS)<ref:2604.23301#pg6>.
The final influence functional is obtained by multiplying the discrete K and Iσ as GMPSs together to obtain the Augmented Density Tensor (ADT) as a GMPS<ref:2604.23301#pg6>.

Key Methodological Steps

The GTEMPO method proceeds in three main steps<ref:2604.23301#pg6>:

(i) Discretizing the Grassmann trajectories and obtaining the discrete expressions of K and Iσ as in Eq.(12) and Eq.(13)<ref:2604.23301#pg6>.

(ii) Constructing the discrete K and Iσ as GMPSs, and obtaining the ADT by multiplying them together (on the fly) using Eq.(16)<ref:2604.23301#pg6>.

(iii) Calculating impurity observables based on A<ref:2604.23301#pg6>.

Validation and Benchmarking

The method's accuracy is validated against several benchmarks<ref:2604.23301#pg6>:

It is benchmarked against exact diagonalization (ED) in several exactly solvable cases, including a toy model with a single fermionic orbital and the non-interacting case<ref:2604.23301#pg6>.
For the SAIM, results are compared against continuous-time quantum Monte Carlo (CTQMC) using converged dynamical mean field theory (DMFT) iterations on the imaginary contour<ref:2604.23301#pg6>.
The method is tested for both imaginary- and real-time calculations to illustrate its flexibility<ref:2604.23301#pg6>.

Results and Significance

The Nambu-GTEMPO method proves to be an accurate and flexible solver for fermionic impurity problems with a superconducting bath<ref:2604.23301#pg6>. The results show that the method can study superconducting states in both equilibrium and non-equilibrium systems within the Nambu formalism<ref:2604.23301#pg6>. Specifically, it successfully calculates correlation functions for real-time evolution, such as G↑¯↑●(t) and G↑↓●(t)<ref:2604.23301#pg6>. The method's advantages over CTQMC on the imaginary contour include being free of sampling noises and being free of the sign problem<ref:2604.23301#pg6>. Although its efficiency is not yet comparable to CTQMC, it demonstrates potential for use in non-equilibrium DMFT<ref:2604.23301#pg6>.

Model Details

The total Hamiltonian for the QIP is defined as Hˆ = Hˆ imp + Hˆ int = Hˆ imp + Hˆ hyb + Hˆ bath<ref:2604.23301#pg6>. The SAIM Hamiltonian includes an on-site energy εd and an interaction strength U, coupled to a superconducting bath described by the BCS Hamiltonian with s-wave pairing, characterized by the energy gap ∆sc<ref:2604.23301#pg6>.

Conclusion

The Nambu-GTEMPO method provides a flexible and potentially scalable solution for fermionic impurity problems in the Nambu formalism, making it useful as a quantum impurity solver to study superconducting states in both equilibrium and non-equilibrium systems<ref:2604.23301#pg6>. The extension of GTEMPO to handle the analytic expression of the Feynman-Vernon influence functional for superconducting baths is a key contribution<ref:2604.23301#pg6>. The results confirm its potential as a powerful candidate quantum impurity solver in DMFT and non-equilibrium DMFT<ref:2604.23301#pg6>. The method can be straightforwardly extended to treat other forms of pairing in both multi-orbital and multi-site QIP<ref:2604.23301#pg6>. The implementation of the Nambu-GTEMPO method on the imaginary and Keldysh, as well as the L-shaped Kadanoff contours could be found at [91]<ref:2604.23301#pg6>.

The gist

The Nambu-GTEMPO method extends Grassmann time-evolving matrix product operators to solve fermionic impurity problems in the Nambu formalism, specifically addressing superconducting baths.

Results and Significance

Improvements for AI systems

Bold header: Nambu-GTEMPO Method Implementation for Superconducting Impurities

This method enables solving fermionic impurity problems in the Nambu formalism by deriving an analytic expression of the Feynman-Vernon influence functional using a Bogoliubov transformation, allowing for direct adaptation of core GTEMPO algorithms to superconducting baths.

Bold header: Benchmarking Against Exact and Quantum Monte Carlo Methods

The system can be validated against "exact diagonalization in several exactly solvable cases, and against the continuous-time quantum Monte Carlo method using converged dynamical mean field theory (DMFT) iterations on the imaginary contour in the non-integrable case."

Bold header: Non-Equilibrium Real-Time Evolution of SAIM

The improved system can study the non-equilibrium real-time evolution of the SAIM from an uncorrelated impurity-bath initial state, which is hard for non-equilibrium CTQMC suffering from the dynamical sign problem, by calculating correlation functions like G↑¯↑●(jδt) = Z−1imp(t) Tr [aˆ↑(jδt)aˆ†↑ρˆ0].

Bold header: Scalable Tensor Network Algorithm for Impurity Problems

The method is designed to be potentially scalable to large impurity problems by iteratively tracing out the parts of impurity that are not directly used in the interested observables, suggesting efficiency for larger systems beyond simple toy models.