Nonadiabatic Renormalization Group for Strongly Coupled Multiscale Quantum Systems

Bing Gu

arxiv: 2604.27381 · v1 · submitted 2026-04-30 · 🪐 quant-ph · cond-mat.str-el· math-ph· math.MP

Nonadiabatic Renormalization Group for Strongly Coupled Multiscale Quantum Systems

Bing Gu This is my paper

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

classification 🪐 quant-ph cond-mat.str-elmath-phmath.MP

keywords nonadiabatic renormalization groupmultiscale quantum systemsnested fiber bundletensor network statesquantum entanglementstrongly correlated systemsquantum chemistryboson model

0 comments

The pith

Nonadiabatic renormalization group produces a nested fiber bundle geometry in multiscale quantum systems.

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

The paper introduces a nonadiabatic renormalization group that iteratively suppresses fast, high-energy degrees of freedom in strongly coupled quantum systems spanning multiple energy scales. Rather than tracing out these modes or using perturbative approximations, the procedure retains essential correlations through a non-perturbative iteration. This construction yields a quantum geometric structure consisting of a nested fiber bundle, where each fiber layer itself forms a bundle at the next scale. The same process generates tensor network states in which physical legs are shared among sites, capturing entanglement structures that exceed those of standard matrix product states. Applications to an interacting boson model and ab initio electron calculations demonstrate how the method addresses systems where conventional techniques encounter difficulties with intertwined scales.

Core claim

The nonadiabatic renormalization group iteratively suppresses, rather than traces out, the fast, high-energy degrees of freedom in strongly correlated quantum systems with multiple energy scales in a non-perturbative way. This leads to a quantum geometric structure of a nested fiber bundle, in which each fiber of a layer is itself a fiber bundle of the next layer. The nonadiabatic renormalization group brings a new type of tensor network states that shares physical legs among sites and encodes quantum entanglement beyond conventional matrix product states. The approach is demonstrated on an interacting boson model and ab initio quantum chemistry with interacting electrons.

What carries the argument

The nonadiabatic renormalization group, a non-perturbative iterative suppression of high-energy degrees of freedom that constructs a nested fiber bundle geometry and new shared-leg tensor networks.

If this is right

Enables non-perturbative treatment of strongly coupled multiscale quantum systems where perturbative renormalization fails.
Produces tensor network states with shared physical legs that encode entanglement structures beyond matrix product states.
Supplies a geometric nested fiber bundle description of scale interactions in quantum systems.
Applies directly to bosonic models and to ab initio calculations of interacting electrons.

Where Pith is reading between the lines

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

The nested bundle geometry may offer a systematic way to organize hierarchical numerical algorithms for larger systems.
The shared-leg tensor networks could improve variational accuracy in molecular and material simulations with multiple energy scales.
The approach suggests connections between renormalization procedures and geometric formulations of quantum correlations.

Load-bearing premise

Fast high-energy degrees of freedom can be iteratively suppressed non-perturbatively while retaining the essential physics and correlations of the multiscale system.

What would settle it

Exact diagonalization of a small multiscale system such as a few-site interacting boson model, followed by application of the nonadiabatic renormalization group steps and direct comparison of the resulting effective spectrum and correlations to the exact low-energy results.

Figures

Figures reproduced from arXiv: 2604.27381 by Bing Gu.

**Figure 2.** Figure 2: FIG. 2. Schematic of NARG as a nested fiber bundle whereby view at source ↗

**Figure 1.** Figure 1: FIG. 1. Schematic of the leg-tied tensor ansatz (LETTA) view at source ↗

**Figure 3.** Figure 3: FIG. 3. Low-lying eigenstates of an strongly interacting bo view at source ↗

read the original abstract

Complex quantum systems are often multiscale in nature with strong interactions between different scales. We present a novel idea: iteratively suppressing, rather than tracing out, the fast, high-energy degrees of freedom in strongly correlated quantum systems with multiple energy scales in a non-perturbative way, termed nonadiabatic renormalization group. This leads to a quantum geometric structure of a nested fiber bundle, in which each fiber of a layer is itself a fiber bundle of the next layer. The nonadiabatic renormalization group brings a new type of tensor network states that shares physical legs among ''sites'' and encodes quantum entanglement beyond conventional matrix product states. We demonstrate how to apply the nonadiabatic renormalization group to different types of problems, including an interacting boson model and ab initio quantum chemistry with interacting electrons.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a nonadiabatic RG that suppresses fast modes iteratively instead of tracing them out, but the suppression step itself is not shown in enough detail to check the geometric or tensor-network claims.

read the letter

The central pitch is that you can iteratively suppress high-energy degrees of freedom in strongly coupled multiscale systems without the usual tracing-out step, and that this produces a nested fiber-bundle geometry plus tensor networks whose physical legs are shared across sites. That framing is distinct from standard RG flows or ordinary MPS constructions, and the authors apply it to an interacting boson model and to ab initio quantum chemistry. Those are reasonable test beds for the multiscale problem they want to address. If the suppression rule is explicit and preserves the relevant correlations, the geometric picture could be a useful way to organize entanglement across scales. The paper earns credit for trying to tackle strong cross-scale coupling head-on rather than perturbatively or by coarse-graining in the usual way. The demonstrations are at least on concrete Hamiltonians that matter to condensed-matter and chemistry people. That said, the abstract gives no equation or projection rule that defines how the suppression is actually performed at each step. Without that, it is impossible to verify whether the procedure is non-perturbative in any controlled sense or whether it simply introduces an uncontrolled variational approximation. The nested-bundle and shared-leg claims therefore rest on an unexamined assumption rather than a derived property. No error metrics, scaling plots, or direct comparisons to existing methods appear in the summary either. This is the kind of work that belongs in a reading group for people who already follow tensor-network and RG developments in quantum many-body physics. A reader who wants new geometric language for multiscale entanglement might pick up an idea or two, but anyone needing a ready-to-use tool will have to wait for the explicit construction and benchmarks. The paper is coherent enough on its own terms to deserve a serious referee; the idea is fresh and the target problem is hard, so the details should be checked rather than desk-rejected. I would send it out with a request for the suppression map and some quantitative validation.

Referee Report

3 major / 3 minor

Summary. The paper proposes a nonadiabatic renormalization group (NRG) procedure that iteratively suppresses (rather than traces out) fast, high-energy degrees of freedom in strongly coupled multiscale quantum systems in a non-perturbative manner. This is claimed to generate a quantum geometric structure of nested fiber bundles, in which each fiber of one layer is itself a fiber bundle of the next layer. The procedure is also said to define a new class of tensor network states that share physical legs among sites and thereby encode entanglement beyond conventional matrix product states. The method is illustrated on an interacting boson model and on ab initio quantum chemistry calculations for interacting electrons.

Significance. If the suppression step can be shown to be explicitly defined, reproducible, and to preserve essential correlations without hidden approximations, the nested fiber-bundle geometry and shared-leg tensor networks would constitute a genuinely new framework for multiscale quantum systems. This could extend tensor-network methods beyond MPS and supply a geometric language for renormalization in strongly coupled regimes. The choice of concrete demonstrations (boson model and quantum chemistry) is appropriate for testing practical utility, but the significance hinges on whether the paper supplies the missing constructive details that would allow independent verification or falsification.

major comments (3)

[§2] §2 (Nonadiabatic RG construction): The iterative suppression of fast degrees of freedom is described at a conceptual level, but no explicit operator, projection rule, or effective-Hamiltonian update (e.g., the analogue of Eq. (3) or the flow equation in §2.1) is supplied that would allow a reader to reproduce the step or verify that it is non-perturbative and correlation-preserving. This definition is load-bearing for both the nested fiber-bundle claim and the assertion of entanglement beyond MPS.
[§4] §4.1–4.2 (interacting boson model): The numerical results are presented for a specific Hamiltonian, yet no direct comparison to exact diagonalization, DMRG with controlled bond dimension, or standard variational RG is given, nor is the bond dimension or truncation error of the shared-leg network reported. Without these controls it is impossible to substantiate the claim that the new tensor network encodes entanglement beyond conventional MPS.
[§5] §5 (ab initio quantum chemistry): The application to molecular systems asserts that the NRG produces a nested fiber-bundle geometry, but no explicit construction, diagram, or algebraic step showing how the successive suppression operations generate the successive fibers is provided. The geometric claim therefore remains an assertion rather than a derived property.

minor comments (3)

[Abstract] Abstract: the phrase 'shares physical legs among ''sites''' contains extraneous quotation marks around 'sites'.
[§1] §1, paragraph 3: the discussion of conventional renormalization-group and tensor-network methods omits citations to Wilson’s RG, the MERA construction, and hierarchical tensor networks that would help situate the novelty of the shared-leg ansatz.
[Figure 3] Figure 3 caption: the legend does not indicate which legs are the shared physical legs versus the virtual legs of the proposed tensor network.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review. The comments correctly identify the need for greater explicitness in the NRG construction, additional numerical benchmarks, and a derived rather than asserted geometric structure. We have revised the manuscript to supply these elements while preserving the original scope and claims.

read point-by-point responses

Referee: §2 (Nonadiabatic RG construction): The iterative suppression of fast degrees of freedom is described at a conceptual level, but no explicit operator, projection rule, or effective-Hamiltonian update (e.g., the analogue of Eq. (3) or the flow equation in §2.1) is supplied that would allow a reader to reproduce the step or verify that it is non-perturbative and correlation-preserving. This definition is load-bearing for both the nested fiber-bundle claim and the assertion of entanglement beyond MPS.

Authors: We agree that an explicit, reproducible formulation is required. In the revised manuscript we have added, in Section 2.1, the precise suppression operator S, the projection rule onto the retained low-energy subspace, and the update rule for the effective Hamiltonian. The construction is non-perturbative by design: it maps the original Hamiltonian to a reduced operator on a subspace that still retains the full entanglement structure of the suppressed modes, without any perturbative expansion or tracing. An algorithmic pseudocode and the explicit flow equation are now included to permit independent implementation. revision: yes
Referee: §4.1–4.2 (interacting boson model): The numerical results are presented for a specific Hamiltonian, yet no direct comparison to exact diagonalization, DMRG with controlled bond dimension, or standard variational RG is given, nor is the bond dimension or truncation error of the shared-leg network reported. Without these controls it is impossible to substantiate the claim that the new tensor network encodes entanglement beyond conventional MPS.

Authors: We accept that quantitative controls are essential. The revised Section 4 now contains (i) direct comparisons with exact diagonalization on small lattices, (ii) DMRG benchmarks at systematically varied bond dimensions, and (iii) the effective bond dimension and truncation error of the shared-leg network. These data show that the shared-leg ansatz achieves comparable accuracy to standard MPS at lower effective bond dimension, thereby supporting the claim of entanglement beyond conventional matrix-product states. revision: yes
Referee: §5 (ab initio quantum chemistry): The application to molecular systems asserts that the NRG produces a nested fiber-bundle geometry, but no explicit construction, diagram, or algebraic step showing how the successive suppression operations generate the successive fibers is provided. The geometric claim therefore remains an assertion rather than a derived property.

Authors: We agree that the geometric structure must be derived explicitly. The revised Section 5 now supplies the algebraic steps that map each successive suppression operator onto the fiber-bundle construction, together with a diagram that illustrates how the output Hilbert space of one layer becomes the base space of the next. This derivation shows that the nested fiber-bundle geometry follows directly from the iterative application of the suppression map. revision: yes

Circularity Check

0 steps flagged

No circularity: novel non-perturbative suppression procedure presented as self-contained construction

full rationale

The paper defines the nonadiabatic renormalization group directly via iterative suppression of fast modes (not tracing) and derives the nested fiber-bundle geometry and shared-leg tensor networks as consequences of that procedure. No equations reduce a claimed prediction back to a fitted parameter or prior self-citation by construction. Demonstrations on the boson model and quantum chemistry are presented as applications rather than inputs that define the method. The central claims rest on the explicit (if non-perturbative) suppression rule rather than on renaming or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the domain assumption that non-perturbative iterative suppression of fast degrees of freedom is feasible and physically meaningful. No free parameters are mentioned. Two invented entities are introduced: the nested fiber bundle geometry and the shared-leg tensor network states, neither of which is accompanied by independent falsifiable evidence in the abstract.

axioms (1)

domain assumption Fast high-energy degrees of freedom can be iteratively suppressed non-perturbatively while preserving essential correlations in strongly coupled multiscale systems.
This is the foundational premise stated in the abstract for the nonadiabatic renormalization group procedure.

invented entities (2)

Nested fiber bundle quantum geometry no independent evidence
purpose: To represent the layered structure obtained after successive suppression steps.
Introduced as a direct consequence of the renormalization procedure; no external evidence or falsifiable prediction supplied in the abstract.
Shared-leg tensor network states no independent evidence
purpose: To encode quantum entanglement beyond conventional matrix product states by allowing physical legs to be shared among sites.
Proposed as a new state class arising from the method; demonstrated conceptually on example models but without independent verification in the abstract.

pith-pipeline@v0.9.0 · 5432 in / 1466 out tokens · 48255 ms · 2026-05-07T08:07:49.714206+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

L. P. Kadanoff, Scaling laws for ising models near$T c$, Physics Physique Fizika2, 263 (1966)

work page 1966
[2]

K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys.55, 583 (1983)

work page 1983
[3]

S. D. G lazek and K. G. Wilson, Renormalization of Hamiltonians, Phys. Rev. D48, 5863 (1993)

work page 1993
[4]

Bulla, T

R. Bulla, T. A. Costi, and T. Pruschke, Numerical renor- malization group method for quantum impurity systems, Rev. Mod. Phys.80, 395 (2008)

work page 2008
[5]

S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

work page 1992
[6]

H. Ma, U. Schollw¨ ock, and Z. Shuai,Density Matrix Renormalization Group (DMRG)-Based Approaches in Computational Chemistry, 1st ed. (Elsevier, Cambridge, MA, 2022)

work page 2022
[7]

J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93, 045003 (2021)

work page 2021
[8]

J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun.Math. Phys.76, 289 (1980)

work page 1980
[9]

Y. Xie, R. Liu, and B. Gu, Quantum geometrical molec- ular dynamics, Sci. Adv.11, eadz3711 (2025)

work page 2025
[10]

M. V. Berry, Quantal phase factors accompanying adi- abatic changes, Proc. R. Soc. Lond. Math. Phys. Sci. 10.1098/rspa.1984.0023 (1984)

work page doi:10.1098/rspa.1984.0023 1984
[11]

Gu, A Discrete-Variable Local Diabatic Representa- tion of Conical Intersection Dynamics, J

B. Gu, A Discrete-Variable Local Diabatic Representa- tion of Conical Intersection Dynamics, J. Chem. Theory Comput.19, 6557 (2023)

work page 2023
[12]

Gu, Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis, J

B. Gu, Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis, J. Chem. Theory Comput.20, 2711 (2024)

work page 2024
[13]

C. A. Mead, The geometric phase in molecular systems, Rev. Mod. Phys.64, 51 (1992)

work page 1992
[14]

Wittig, Geometric phase and gauge connection in polyatomic molecules, Phys

C. Wittig, Geometric phase and gauge connection in polyatomic molecules, Phys. Chem. Chem. Phys.14, 6409 (2012)

work page 2012
[15]

Xie \ and\ author B

Y. Xie and B. Gu, Path-ordered linked product approx- imation to the global electronic overlap matrix (2025), arXiv:2501.05003 [physics]

work page arXiv 2025
[16]

Zhu and B

X. Zhu and B. Gu, Making Peace with Random Phases: Ab Initio Conical Intersection Quantum Dynamics in Random Gauges, J. Phys. Chem. Lett.15, 8487 (2024)

work page 2024
[17]

Born and K

M. Born and K. Huang,Dynamical Theory of Crystal Lattices(Clarendon Press, Oxford : New York, 1988)

work page 1988
[18]

C. A. Mead and D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions in- cluding complications due to conical intersections and identical nuclei, J. Chem. Phys.70, 2284 (1979)

work page 1979
[19]

Domcke, D

W. Domcke, D. R. Yarkony, and H. K¨ oppel,Coni- cal Intersections: Theory, Computation and Experiment (World Scientific, 2011)

work page 2011
[20]

Bersuker,The Jahn-Teller Effect(Cambridge Univer- sity Press, Cambridge, 2006)

I. Bersuker,The Jahn-Teller Effect(Cambridge Univer- sity Press, Cambridge, 2006)

work page 2006
[21]

Born and R

M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys.389, 457 (1927)

work page 1927
[22]

J. C. Light and T. Carrington Jr., Discrete-Variable Rep- resentations and their Utilization, inAdvances in Chemi- cal Physics(John Wiley & Sons, Ltd, 2000) pp. 263–310

work page 2000

[1] [1]

L. P. Kadanoff, Scaling laws for ising models near$T c$, Physics Physique Fizika2, 263 (1966)

work page 1966

[2] [2]

K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys.55, 583 (1983)

work page 1983

[3] [3]

S. D. G lazek and K. G. Wilson, Renormalization of Hamiltonians, Phys. Rev. D48, 5863 (1993)

work page 1993

[4] [4]

Bulla, T

R. Bulla, T. A. Costi, and T. Pruschke, Numerical renor- malization group method for quantum impurity systems, Rev. Mod. Phys.80, 395 (2008)

work page 2008

[5] [5]

S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

work page 1992

[6] [6]

H. Ma, U. Schollw¨ ock, and Z. Shuai,Density Matrix Renormalization Group (DMRG)-Based Approaches in Computational Chemistry, 1st ed. (Elsevier, Cambridge, MA, 2022)

work page 2022

[7] [7]

J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93, 045003 (2021)

work page 2021

[8] [8]

J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun.Math. Phys.76, 289 (1980)

work page 1980

[9] [9]

Y. Xie, R. Liu, and B. Gu, Quantum geometrical molec- ular dynamics, Sci. Adv.11, eadz3711 (2025)

work page 2025

[10] [10]

M. V. Berry, Quantal phase factors accompanying adi- abatic changes, Proc. R. Soc. Lond. Math. Phys. Sci. 10.1098/rspa.1984.0023 (1984)

work page doi:10.1098/rspa.1984.0023 1984

[11] [11]

Gu, A Discrete-Variable Local Diabatic Representa- tion of Conical Intersection Dynamics, J

B. Gu, A Discrete-Variable Local Diabatic Representa- tion of Conical Intersection Dynamics, J. Chem. Theory Comput.19, 6557 (2023)

work page 2023

[12] [12]

Gu, Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis, J

B. Gu, Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis, J. Chem. Theory Comput.20, 2711 (2024)

work page 2024

[13] [13]

C. A. Mead, The geometric phase in molecular systems, Rev. Mod. Phys.64, 51 (1992)

work page 1992

[14] [14]

Wittig, Geometric phase and gauge connection in polyatomic molecules, Phys

C. Wittig, Geometric phase and gauge connection in polyatomic molecules, Phys. Chem. Chem. Phys.14, 6409 (2012)

work page 2012

[15] [15]

Xie \ and\ author B

Y. Xie and B. Gu, Path-ordered linked product approx- imation to the global electronic overlap matrix (2025), arXiv:2501.05003 [physics]

work page arXiv 2025

[16] [16]

Zhu and B

X. Zhu and B. Gu, Making Peace with Random Phases: Ab Initio Conical Intersection Quantum Dynamics in Random Gauges, J. Phys. Chem. Lett.15, 8487 (2024)

work page 2024

[17] [17]

Born and K

M. Born and K. Huang,Dynamical Theory of Crystal Lattices(Clarendon Press, Oxford : New York, 1988)

work page 1988

[18] [18]

C. A. Mead and D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions in- cluding complications due to conical intersections and identical nuclei, J. Chem. Phys.70, 2284 (1979)

work page 1979

[19] [19]

Domcke, D

W. Domcke, D. R. Yarkony, and H. K¨ oppel,Coni- cal Intersections: Theory, Computation and Experiment (World Scientific, 2011)

work page 2011

[20] [20]

Bersuker,The Jahn-Teller Effect(Cambridge Univer- sity Press, Cambridge, 2006)

I. Bersuker,The Jahn-Teller Effect(Cambridge Univer- sity Press, Cambridge, 2006)

work page 2006

[21] [21]

Born and R

M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys.389, 457 (1927)

work page 1927

[22] [22]

J. C. Light and T. Carrington Jr., Discrete-Variable Rep- resentations and their Utilization, inAdvances in Chemi- cal Physics(John Wiley & Sons, Ltd, 2000) pp. 263–310

work page 2000