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arxiv: 2501.18591 · v3 · submitted 2025-01-30 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.supr-con· hep-th

Non-Hermitian catalysis of spontaneous symmetry breaking on Euclidean and hyperbolic lattices

Christopher A. Leong , Bitan Roy This is my paper

Pith reviewed 2026-05-23 04:21 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.supr-conhep-th
keywords non-Hermitian systemsspontaneous symmetry breakingcharge-density wavespin-density waveEuclidean latticeshyperbolic latticesmean-field theorytight-binding model
0
0 comments X

The pith

Non-Hermiticity in nearest-neighbor hopping catalyzes charge-density-wave and spin-density-wave orders at weaker repulsions on Euclidean and hyperbolic lattices.

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

The paper establishes that an imbalance in hopping amplitudes between opposite directions on nearest-neighbor bonds produces a non-Hermitian tight-binding model whose spectrum remains real over a finite parameter range. This imbalance narrows the electronic bandwidth while leaving the scaling of the density of states near zero energy unchanged, whether the underlying lattice is flat Euclidean or negatively curved hyperbolic. The reduced bandwidth in turn allows both charge-density-wave order driven by nearest-neighbor Coulomb repulsion and spin-density-wave order driven by on-site Hubbard repulsion to appear at interaction strengths smaller than those required in the corresponding Hermitian models. The conclusions rest on self-consistent Hartree-channel mean-field calculations performed with biorthogonal quantum mechanics, supplemented by analysis of mass-gap scaling and finite-size behavior on open hyperbolic lattices. A general mathematical criterion for the catalysis is also supplied.

Core claim

On two-dimensional bipartite Euclidean and hyperbolic lattices, a non-Hermitian generalization of the tight-binding model arising from imbalanced hopping amplitudes in opposite directions between nearest-neighbor sites reduces the bandwidth without changing the characteristic scaling of the density of states near zero energy. This non-Hermiticity catalyzes the formation of charge-density-wave and spin-density-wave orders at weaker nearest-neighbor Coulomb and on-site Hubbard repulsions, respectively, compared to Hermitian systems. Both orders produce staggered patterns that insulate the half-filled systems. These conclusions follow from lattice-based self-consistent numerical mean-field分析 in

What carries the argument

The non-Hermitian imbalance of hopping amplitudes between opposite directions on nearest-neighbor bonds, which reduces bandwidth while keeping density-of-states scaling intact near half-filling.

If this is right

  • Both ordered states cause insulation in half-filled systems.
  • The associated mass gaps near zero-energy scale with the non-Hermitian parameter.
  • Order parameters on hyperbolic lattices with open boundary conditions exhibit specific finite-size scaling.
  • A robust general mathematical criterion exists for the non-Hermitian catalysis mechanism of ordered phases.

Where Pith is reading between the lines

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

  • The catalysis mechanism may be tested in engineered non-Hermitian platforms such as photonic lattices or cold-atom arrays where hopping imbalance can be controlled independently of interaction strength.
  • Because the density-of-states scaling is preserved, the same non-Hermitian imbalance could lower thresholds for other symmetry-breaking channels not studied here.
  • On hyperbolic lattices the combination of negative curvature and non-Hermiticity may produce boundary-localized order that differs qualitatively from Euclidean behavior.

Load-bearing premise

The Hartree-channel mean-field treatment combined with biorthogonal quantum mechanics accurately captures the non-Hermitian catalysis of spontaneous symmetry breaking without requiring corrections from fluctuations or higher-order interactions.

What would settle it

Exact diagonalization or quantum Monte Carlo simulations on small lattices that check whether charge-density-wave or spin-density-wave order parameters remain finite at interaction strengths below the Hermitian critical values predicted by mean-field.

Figures

Figures reproduced from arXiv: 2501.18591 by Bitan Roy, Christopher A. Leong.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representations of non-Hermitian (a) Eu [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Density of states ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Density of states ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Density of states ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. A schematic flow chart to arrive at the self [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scaling of the self-consistent solutions of the charge-density-wave (CDW) order ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Computation of the critical strength of the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Scaling of the self-consistent solutions of the spin-density-wave (SDW) order ( [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Computation of the critical strength of the [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. BCS-like scaling for the spin-density-wave order [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The local density of states ( [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The local density of states ( [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Self-consistent solution of the charge-density-wave order parameter ( [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Self-consistent solution of the spin-density-wave order parameter ( [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Lattice structure of the Bernal-stacked honeycomb [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Self-consistent solutions for the (a) charge-density [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
read the original abstract

Depending on the lattice geometry, the nearest-neighbor (NN) tight-binding model for free fermions gives rise to particle-hole symmetric emergent Dirac liquid, Fermi liquid, and flat bands near the half-filling or zero-energy on bipartite Euclidean and hyperbolic lattices, respectively embedded on the flat and negatively curved spaces. Such noninteracting electronic fluids are characterized by a vanishing, a finite, and a diverging density of states near half-filling, respectively. A non-Hermitian generalization of this scenario resulting from an imbalance of the hopping amplitudes in the opposite directions between any pair of NN sites continues to accommodate a real eigenvalue spectrum over an extended non-Hermitian parameter regime. Most importantly, it reduces the band width without altering the characteristic scaling of the density of states close to the zero-energy. Here, we show that on two-dimensional bipartite Euclidean and hyperbolic lattices such a non-Hermiticity catalyzes the formation of both charge-density-wave and spin-density-wave orders at weaker (in comparison to the counterparts in conventional or Hermitian systems) NN Coulomb and on-site Hubbard repulsions, respectively. These two ordered states correspond to staggered patterns of average electronic density and spin between the NN sites, respectively, and both cause insulation in half-filled systems. We arrive at these conclusions by combining biorthogonal quantum mechanics and lattice-based self-consistent numerical mean-field analysis in the Hartree channel. We discuss the scaling of the associated mass gaps near the zero-energy with the non-Hermitian parameter, and also address the finite size scaling of the order parameters specifically on hyperbolic lattices with open boundary conditions. A robust general mathematical criterion for the proposed non-Hermitian catalysis mechanism for ordered phases is showcased.

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

2 major / 1 minor

Summary. The manuscript claims that a non-Hermitian imbalance in nearest-neighbor hoppings on bipartite Euclidean and hyperbolic lattices reduces the bandwidth without changing the characteristic density-of-states scaling near zero energy, and that this catalyzes charge-density-wave and spin-density-wave orders at weaker NN Coulomb and on-site Hubbard interactions than in the Hermitian case. These conclusions are reached via biorthogonal quantum mechanics combined with self-consistent numerical mean-field analysis restricted to the Hartree channel; the work also reports the scaling of mass gaps with the non-Hermitian parameter and finite-size scaling of order parameters on hyperbolic lattices with open boundaries, together with a general mathematical criterion for the catalysis mechanism.

Significance. If the mean-field results survive beyond the approximation, the work supplies a concrete mechanism by which non-Hermiticity can lower the threshold for spontaneous symmetry breaking while preserving real spectra and DOS scaling. The explicit general mathematical criterion and the extension to hyperbolic lattices with open-boundary finite-size analysis are concrete strengths that could be useful for subsequent studies of non-Hermitian fermionic models.

major comments (2)
  1. [Numerical mean-field analysis and results sections] The catalysis claim (weaker critical U for CDW/SDW) rests entirely on self-consistent Hartree-channel mean-field numerics in the biorthogonal basis; no comparison to fluctuation-corrected treatments (RPA, Goldstone-mode analysis at T=0) or to exact methods is supplied, even though the manuscript acknowledges that the non-Hermitian term modifies the spectrum and the 2D mean-field critical point is known to be sensitive to such corrections.
  2. [Results on mass gaps and order-parameter scaling] No data tables, error estimates, or convergence diagnostics for the self-consistent solutions are provided, making it impossible to quantify the reported reduction in critical interaction strength or to assess numerical robustness of the mass-gap scaling with the non-Hermitian parameter.
minor comments (1)
  1. The precise range of the non-Hermitian hopping-imbalance parameter over which the spectrum remains real and the DOS scaling is unchanged should be stated explicitly with the relevant equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Numerical mean-field analysis and results sections] The catalysis claim (weaker critical U for CDW/SDW) rests entirely on self-consistent Hartree-channel mean-field numerics in the biorthogonal basis; no comparison to fluctuation-corrected treatments (RPA, Goldstone-mode analysis at T=0) or to exact methods is supplied, even though the manuscript acknowledges that the non-Hermitian term modifies the spectrum and the 2D mean-field critical point is known to be sensitive to such corrections.

    Authors: Our work is focused on demonstrating the non-Hermitian catalysis mechanism within the self-consistent mean-field approximation, which is a widely used method for investigating interaction-driven orders in lattice models. The mathematical criterion we derive for the catalysis is general and based on the modification of the bandwidth while preserving the DOS scaling near zero energy; this criterion does not rely on the mean-field treatment per se. We note that providing comparisons to RPA or exact methods would require substantial additional computations beyond the scope of the current study. However, we will add a discussion section addressing the potential impact of fluctuations on the observed catalysis effect. revision: partial

  2. Referee: [Results on mass gaps and order-parameter scaling] No data tables, error estimates, or convergence diagnostics for the self-consistent solutions are provided, making it impossible to quantify the reported reduction in critical interaction strength or to assess numerical robustness of the mass-gap scaling with the non-Hermitian parameter.

    Authors: We will include in the revised manuscript supplementary material with data tables listing the critical interaction strengths, mass gap values, and order parameters for various non-Hermitian parameters. Additionally, we will provide error estimates from the self-consistent iterations and details on convergence criteria and diagnostics to allow assessment of the numerical robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is numerical self-consistent mean-field

full rationale

The paper introduces an explicit non-Hermitian imbalance in NN hoppings, preserves real spectrum and DOS scaling by construction of the model, then solves for CDW/SDW order parameters via standard self-consistent Hartree-channel mean-field equations in the biorthogonal basis. No step reduces the catalysis result to a fitted parameter renamed as prediction, a self-citation loop, or a definitional identity; the weaker critical U is an output of the numerics on Euclidean and hyperbolic lattices. The mathematical criterion is presented as derived from the same framework rather than smuggled in. This is a self-contained computational claim against the Hermitian baseline.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the mean-field Hartree approximation for non-Hermitian systems and the assumption that the non-Hermitian parameter regime preserves a real spectrum; no new entities are introduced.

free parameters (1)
  • non-Hermitian hopping imbalance parameter
    Tunable imbalance between opposite-direction hoppings that defines the non-Hermitian regime.
axioms (2)
  • domain assumption Mean-field Hartree approximation suffices to capture the catalysis of CDW and SDW orders
    Invoked for the self-consistent numerical analysis on both lattice types.
  • domain assumption Biorthogonal quantum mechanics correctly extends the mean-field treatment to non-Hermitian Hamiltonians
    Used to handle the non-Hermitian generalization while keeping real eigenvalues.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exceptional flat bands in bipartite non-Hermitian lattices

    cond-mat.mes-hall 2025-08 unverdicted novelty 6.0

    Bipartite non-Hermitian lattices support exceptional flat bands that arise from sublattice degeneracy mismatch and persist beyond exceptional points with biorthogonal modes spanning both sublattices.

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