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arxiv: 2604.23301 · v1 · submitted 2026-04-25 · ❄️ cond-mat.str-el · cond-mat.supr-con

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

Chu Guo , Wei Wu , Xiansong Xu , Ping-Xing Chen , Changming Yue , Tian Jiang , Ruofan Chen This is my paper

Pith reviewed 2026-05-08 07:23 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con
keywords GTEMPOinfluence functionalBogoliubov transformationNambu formalismsuperconducting bathfermionic impuritydynamical mean-field theory
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0 comments X

The pith

The Bogoliubov transformation enables direct adaptation of Grassmann time-evolving matrix product operators to fermionic impurities in superconducting baths via the Nambu formalism.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work extends the GTEMPO method to fermionic impurity problems coupled to a superconducting bath using the Nambu formalism. The central technique applies the Bogoliubov transformation to the bath to derive an analytic Feynman-Vernon influence functional that has the same form as in the normal bath case. With this similarity, the existing GTEMPO algorithms for matrix product representation of the influence functional are adapted with minimal changes. Accuracy is verified through benchmarks with exact diagonalization for solvable models and with CT-QMC DMFT for non-integrable cases, in both imaginary and real time. The extension positions GTEMPO as a viable impurity solver for DMFT calculations involving superconducting baths and their non-equilibrium dynamics.

Core claim

By employing the Bogoliubov transformation for the superconducting bath, one obtains the analytic expression of the Feynman-Vernon influence functional in a similar form to the case of a normal bath, after which the core algorithms of GTEMPO can be straightforwardly adapted to the Nambu formalism.

What carries the argument

The Bogoliubov transformation on the superconducting bath that yields a Feynman-Vernon influence functional with structure preserved for temporal matrix product operator encoding in GTEMPO.

If this is right

  • The method matches exact diagonalization results in several solvable cases for imaginary- and real-time evolution.
  • It converges with CT-QMC in DMFT iterations for non-integrable interacting impurity models.
  • Both equilibrium and non-equilibrium calculations are supported within the same adapted framework.
  • GTEMPO becomes usable as an impurity solver in Nambu-formalism DMFT and its dynamical extensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The preserved structure suggests GTEMPO can now address pairing correlations at the impurity without redesigning the algorithm.
  • Real-time results may be used to study quench dynamics in superconducting environments.
  • Connections to other symmetry-broken baths could be explored using analogous transformations.

Load-bearing premise

The influence functional after the Bogoliubov transformation retains the exact structural properties required by the GTEMPO algorithm, without introducing any loss of accuracy or need for further approximations.

What would settle it

A mismatch between the adapted GTEMPO results and exact diagonalization in an exactly solvable model with a superconducting bath would falsify the claim that the adaptation works without accuracy loss.

Figures

Figures reproduced from arXiv: 2604.23301 by Changming Yue, Chu Guo, Ping-Xing Chen, Ruofan Chen, Tian Jiang, Wei Wu, Xiansong Xu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A schematic illustration of the partial IF algo view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a,b) The imaginary-time correlation functions view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a,b) The real-time correlation functions view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a,c) The imaginary-time correlation functions view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a,c) The imaginary-time correlation functions view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Time evolution of the probability of the four impurity view at source ↗
read the original abstract

The Grassmann time-evolving matrix product operator (GTEMPO) method, which represents the Feynman-Vernon influence functional as a temporal matrix product state, has been shown to be a flexible and potentially scalable solution for fermionic quantum impurity problems. In this work, we extend GTEMPO to solve fermionic impurity problems in the Nambu formalism, in which the impurity is coupled to a superconducting bath. A key insight is that by employing the Bogoliubov transformation for the superconducting bath, one could obtain the analytic expression of the Feynman-Vernon influence functional in a similar form to the case of a normal bath, after which the core algorithms of GTEMPO can be straightforwardly adapted. We demonstrate the accuracy of our method by benchmarking it 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. In all cases, we perform both imaginary- and real-time calculations to illustrate the flexibility of our method. These results illustrate that our method could be potentially useful as an impurity solver in DMFT as well as its non-equilibrium extension for fermionic impurity problems in the Nambu formalism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript extends the Grassmann time-evolving matrix product operator (GTEMPO) method to fermionic impurity problems coupled to superconducting baths within the Nambu formalism. By applying the Bogoliubov transformation to the quadratic bath, an analytic expression for the Feynman-Vernon influence functional is obtained that is structurally similar to the normal-metal case. This allows direct adaptation of the GTEMPO algorithms for representing the influence functional as a temporal matrix product state. The method is benchmarked against exact diagonalization for solvable models and against converged CT-QMC DMFT for interacting cases, with calculations performed on both the imaginary and real-time axes.

Significance. If the central claim holds, the work supplies a flexible and potentially scalable impurity solver for DMFT and its non-equilibrium extensions in the presence of superconductivity. The exact analytic influence functional (arising because the bath remains Gaussian after the unitary transformation) together with explicit benchmarks on both time axes against ED and CT-QMC constitute concrete strengths that could enable studies of unconventional superconductors.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, which correctly identifies the extension of GTEMPO to the Nambu formalism for superconducting baths via the Bogoliubov transformation, the resulting analytic influence functional, and the benchmarks against exact diagonalization and CT-QMC on both imaginary and real-time axes. We appreciate the recognition of the method's potential utility as an impurity solver for DMFT and non-equilibrium extensions in superconducting systems. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard external transformation and prior independent method

full rationale

The paper's central derivation applies the established Bogoliubov transformation (a standard technique from superconductivity literature, independent of this work) to the quadratic bath, yielding an exact analytic Feynman-Vernon influence functional whose 2-point correlator structure remains Gaussian and thus directly compatible with the existing GTEMPO MPO representation. This adaptation is presented as a straightforward substitution of Nambu-space correlators rather than a re-derivation or redefinition of GTEMPO itself. The prior GTEMPO framework is cited as an external algorithmic foundation, not as a self-referential load-bearing premise. Independent validation is supplied via benchmarks against exact diagonalization on solvable models and converged CT-QMC DMFT on interacting cases, confirming that the extension preserves accuracy without hidden reductions to fitted inputs or self-definitional loops. No steps in the chain reduce by construction to the paper's own outputs or self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Bogoliubov transformation yields an influence functional whose structure is identical to the normal-bath case, allowing unmodified GTEMPO algorithms; this is a domain assumption from BCS theory.

axioms (1)
  • domain assumption Bogoliubov transformation diagonalizes the superconducting bath and produces an influence functional of the same form as a normal bath
    Invoked as the key step that permits straightforward adaptation of GTEMPO core algorithms.

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Reference graph

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